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(«I it.) :1 slut... : A: ...: “INESIS Li, LIBRARY 20W Michioan State Unwersuy This is to certify that the dissertation entitled SIMULTANEOUS WALL-PRESSURE AND VELOCITY MEASUREMENTS IN THE FLOW FIELD DOWNSTREAM OF AN AXISYMMETRIC BACKWARD-FACING STEP presented by LAURA MICHELE HUDY has been accepted towards fulfillment of the requirements for the PhD. degree in MECHANICAL ENGINEERING Maw Major Professor’s SigfinatVu're ”(/(‘f/loog Date MSU is an Affirmative ActiorVEquaI Opportunity Institution Anew/nun 91918 uefiiuom AHVHBI'I c.-o-o-A-l-.---l-'-C-0-0-0-0-0-0-I-A-n-.. — - PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 2/05 p:lClRC/DateDue.indd-p.1 SIMULTANEOUS WALL-PRESSURE AND VELOCITY MEASUREMENTS IN THE FLOW FIELD DOWNSTREAM OF AN AXISYMMETRIC BACKWARD-FACING STEP By Laura Michele Hudy A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 2005 ABSTRACT SIMULTANEOUS WALL-PRESSURE AND VELOCITY MEASUREMENTS IN THE FLOW FIELD DOWNSTREAM OF AN AXISYMMETRIC BACKWARD-FACING STEP By Laura Michele Hudy Surface-pressure and planar, Particle Image Velocimetry measurements were obtained in the separating/reattaching flow region downstream of an axisymmetric, backward-facing step. Data were acquired for a two-dimensional (2D) separating boundary layer at four different Reynolds numbers based on step height (Reh), spanning 5900-18500. The experimental set-up consisted of an array of 32 flush-mounted microphones beneath the separation bubble. These microphones were used to detect the wall-pressure signature that was acquired simultaneously with the PIV measurements of the velocity field above the surface. The surface-pressure and PIV measurements were used to characterize the wall- pressure distribution and the flow field in terms of Reynolds number dependencies. Statistics from these measurements were consistent with findings in the literature. In addition, the evolution of coherent structures in the flow field was investigated using proper orthogonal decomposition (POD) and multi-point, linear, stochastic estimation (LSE). POD was used to determine the dominant convective modes in the pressure signature. Two convective modes were found to be responsible for 60% of the energy of the pressure signature. Multi-point LSE was used to estimate the dominant flow structures above the wall from convective-mode, wall-pressure signatures over a series of time steps. It was found that a large-scale, coherent structure develops in place at approximately half the reattachment distance. Once this structure reaches a height equivalent to the step, it sheds and accelerates downstream. This growth in place and then shedding resembles a wake mode, which describes the flow field downstream of bluff bodies. The wake mode has been observed in numerical simulation studies of long cavities and backward-facing steps, where two dimensionality is controllable. The present study shows for the first time evidence for the existence of a wake mode in an experimental study of a backward-facing step. This is believed to be related to the good two dimensionality (i.e., axisymmetry) of the test geometry and the ability to track the temporal-evolution of structural features through LSE. Finally, it was observed that, unlike behind bluff bodies, the wake mode in the back-step flow is intermittent, switching on/off in an apparently random manner. Copyfightby LAURA MICHELE HUDY 2005 DEDICATION To my parents, for their unwavering support, dedication, and love ACKNOWLEDGEMENTS I would like to acknowledge support for this research project from the National Science Foundation (Contract #CTSOI 16907) and supplemental support from the Office of Naval Research. I would also like to acknowledge the support of William (Tony) M. Humphreys and Scott Bartram at NASA Langley Research Center in Hampton, Virginia. Both Tony and Scott provided their expertise and gave countless hours towards the successful completion of this project. I am forever grateful for their contribution. On a personal note, there are several individuals that I must acknowledge for the role they played in my 11 years at Michigan State University. First, I must take a moment to reflect on undergrad. Thank you to my freshman roommate, Eve, who patiently listened to me (for hours) regarding whether I should change my major from Journalism to Mechanical Engineering and to Jil, for introducing me to the Society of Women Engineers. I must also thank my friend, Kristine, for answering all the “91 1- phone-calls” regarding classes. Thank you to my friend, Judy, for acting as a sounding board all these years as I worked to meet the challenges of both undergrad and grad school. Finally, I must mention my friends for making the time outside of school fun including Kate, Brad, Mark, Matt, Brady, Scott, Floria, Correna, Heather, and KJ. From soccer games to tailgating, Florida to Aachen, themed parties to Thanksgiving dinners, Albert to Elizabeth Street, the fun never stopped! Thanks for the memories! As for grad school, I need to start by thanking Dr. Naguib, who has been a wonderful teacher, mentor, and friend. I have enjoyed working with Dr. Naguib over the past six years and I could not imagine going through this degree with any other advisor. vi Dr. Naguib is the reason I stayed at MSU for my Ph.D. and I could not be happier with my decision. You have changed my life, Doc, and I am forever grateful to you. Thank you to several professors at MSU including Dr. Radcliffe, Dr. Pierre, Dr. Shih, Dr. Koochesfahani, Dr. Zhuang, and Dr. Foss for their input regarding my research and their support during my graduate career. I would also like to extend my appreciation to Jill for all our wonderful chats and to Roy and Mike for their help in the machine shop. A thank you must also be given to my friends at both EB and RCE. Thank you to Scott for his guidance regarding classes, research, and especially presenting at conferences. Also, thank you to the individuals of the Flow Physics and Controls Lab including Yongxiang, Mohamed, Chad, Tony, and Barry for helping me take data and for your company throughout the years. A special thanks goes out to Al and Chee who I have known Since undergrad and who have offered so much peer support over the years. Outside of the academic world, I have spent countless hours on the soccer field with Melinda, Parks, H, Cara, Sarah, Shannon, and Nat. I wilI miss playing the game with “the ladies of soccer.” Also, here’s to the Art’s soccer team, which has had many faces over the years and many more shoopers. Oh and I cannot forget the 2002 MSU Floor Hockey Champion Intramural team Jedi Mind Trick —- wisdom beats age any day! A special thanks to my roommates in grad school: Shannon, Nikki, and Sarah. Woohoo! You girls rock! We have had a lot of fun over the years. . .may it continue even though we are miles apart. Big Squeeze and Sweet Cheeks - here’s to the Hog Wild Roadhouse and parking tickets. And let us not forget game nights (Gestures, anyone?) at Wardcliff, Wardcliff West and Wardcliff West-North. I also must mention the MSU vii Spartan basketball games and loyal fans: Brad, Kate, Drew, Nikki, Chee, and Kristen. I had a blast yelling “Go Spartans” with y’all!!! I must to extend a very special thank you to three wonderful people in my life. Brad, all I can say is THANK YOU for lunch. Kate, we have been friends all my years at MSU. You are the “bestest” of best friends! I cannot thank you enough for all you have done for me and for all the fun times we have had together. And Barry, your support this past year has been amazing. Thank you for always being there for me. Next to last, I need to recognize my extended family, especially my Grandma. Grandma always knows how to make you smile no matter the type of day. In addition, thank you to the family members of the “Spartan Tailgating Crew” who started tailgating and attending MSU football games back in the Fall of 1994. Eleven years later the same group is still cheering at Spartan football games. Some of my greatest memories come from standing with them on cold fall nights under the lights at Spartan Stadium. I would like to say a special thanks to my Uncle Tom for keeping it fun during college and for all his study advice. Also, thank you to my cousin, Kristen, for allowing me to make dinners for her over the years at MSU. And to all my extended family on both sides, thank you! Finally, I must thank my immediate family. First, thank you to my brother Dave, for always bringing me back down to earth and for serving in the United States Navy. I am so proud of him. Last, but not least, I must thank my mom and my dad. My dad encouraged me to go into engineering and I thank him for his vision. My parents have always been my number one fan and they have given so much to me over the years. For their love and devotion, I dedicate this work to them. Thank you, Mom and Dad! viii TABLE OF CONTENTS LIST OF TABLES -- xii LIST OF FIGURES - - - - - xiii LIST OF SYMBOLS xx 1 Introduction 1 1.1 Background ......................................................................................................... 1 1.2 Organization of document ................................................................................... 6 1 .3 References ........................................................................................................... 7 2 Experimental Apparatus, Instrumentation, and Measurement Techniques ........ 8 2.1 Wind Tunnel Facility .......................................................................................... 8 2.2 The Model ........................................................................................................... 9 2.3 Instrumentation and Measurement Techniques ................................................ 14 2.3.1 Static Pressure Taps ...................................................................................... 15 2.3.2 Microphones ................................................................................................. 1 7 2.3.3 Particle Image Velocimetry .......................................................................... 18 2.3 .3. 1 Experimental Set-up .............................................................................. 21 2.4 Experimental Procedure .................................................................................... 26 2.4.1 Static Pressure System .................................................................................. 26 2.4.2 Microphone Measurement System ................................................................ 28 2.4.3 Particle Image Velocimetry System .............................................................. 29 2.4.3.1 Synchronizing the PIV images and the microphones ........................... 30 2.4.3.2 Post-processing the PIV images ............................................................ 34 2.5 Test matrix ........................................................................................................ 34 2.6 References ......................................................................................................... 36 3 General Characterization of the Flow Field 37 3.1 Reattachrnent Length ........................................................................................ 38 3.2 Boundary Layer Velocity Profiles .................................................................... 47 3.3 Mean-pressure distribution ............................................................................... 53 3.4 References ......................................................................................................... 56 4 Wall-pressure Measurements - 58 4. 1 Introduction ....................................................................................................... 5 8 4.2 Results and Discussion ...................................................................................... 64 4.2.1 Fluctuating-pressure distributions ................................................................. 64 4.2.2 Auto-correlations .......................................................................................... 68 ix 4.2.3 Frequency spectra ......................................................................................... 76 4.2.4 Cross-correlation with respect to the pressure signal at reattachment .......... 83 4.2.5 Frequency-wavenumber spectra ................................................................... 87 4.3 Conclusions ..................................................................................................... 100 4.4 References ....................................................................................................... 102 5 Velocity Measurements using PIV-- 103 5.1 Introduction ..................................................................................................... 103 5.2 Results and Discussion .................................................................................... 106 5.2.1 Mean-Velocity Field ................................................................................... 106 5.2.2 F luctuating-velocity field: uRMs and vRMS ................................................... 111 5.2.3 Reynolds Shear Stress ................................................................................. 118 5.2.4 Vorticity and shear-layer thickness ............................................................. 122 5.3 Conclusions ..................................................................................................... 128 5.4 References ....................................................................................................... l 30 6 Pressure and Velocity Measurements 131 6. 1 Introduction ..................................................................................................... 1 3 1 6.2 Results and Discussion .................................................................................... 135 6.2.1 Average Wall-pressure Signature versus Mean Velocity. Field .................. 135 6.2.2 Single-point Linear and Quadratic Stochastic Estimation .......................... 145 6.2.3 Instantaneous, Pressure-Signature Characteristics ...................................... 162 6.2.3.1 Instantaneous, pressure signature ........................................................ 162 6.2.3.2 Proper Orthogonal Decomposition ..................................................... 167 6.2.4 Multi-point Linear Stochastic Estimation ................................................... 173 6.2.4.1 Velocity field associated with the pressure signature of POD modes 173 6.2.4.2 The evolution of large-scale structures in the flow field .................... 180 6.2.4.3 Instantaneous versus estimated flow field .......................................... 192 6.3 Conclusions ..................................................................................................... l 97 6.4 References ....................................................................................................... 200 7 Conclusions with Discussion and Future Work -_ - - 203 7.1 Conclusions with Discussion .......................................................................... 203 7.2 Future Work .................................................................................................... 210 7.3 References ....................................................................................................... 212 8 Appendix 213 8.1 Wind Tunnel Facility ...................................................................................... 213 8.2 Axisymmetric BF S Design ............................................................................. 218 8.2.1 Skeleton shaft .............................................................................................. 219 8.2.2 Nose ............................................................................................................ 220 8.2.3 Upstream Stationary Section ....................................................................... 221 8.2.4 Rotating Section .......................................................................................... 223 8.2.5 Step Section and Step Ring ......................................................................... 224 8.2.6 Instrument Plates (I-plates) ......................................................................... 230 8.2.7 Motor Section .............................................................................................. 230 8.2.8 Drive Shaft and Motor ................................................................................ 231 8.2.9 Tail .............................................................................................................. 233 8.2. 1 O Wire-Support System .................................................................................. 234 8.2.11 Support System ........................................................................................... 236 8.2.12 Model Assembly ......................................................................................... 240 8.3 Azimuthal Uniformity ..................................................................................... 241 8.4 Additional Microphone Information ............................................................... 245 8.4.1 Microphone Specifications ......................................................................... 245 8.4.2 Circuitry ...................................................................................................... 245 8.4.3 Calibration ................................................................................................... 247 8.4.4 Preliminary tests .......................................................................................... 251 8.4.4.1 Cross talk check .................................................................................. 251 8.4.4.2 Laser Noise ......................................................................................... 252 8.4.4.3 Mirror Interference .............................................................................. 254 8.5 Additional PIV Information ............................................................................ 258 8.5.1 Dot Card ...................................................................................................... 258 8.5.2 Grid card ..................................................................................................... 260 8.6 Stochastic Estimation ...................................................................................... 262 8.7 Shear Strain ..................................................................................................... 272 8.8 Appendix References ...................................................................................... 274 xi LIST OF TABLES Table 2.1 Laser pulse separation times for all four Reynolds numbers tested .................. 23 Table 2.2 Experimental parameters for the four cases investigated ................................ 35 Table 4.1 Frequency values of spectrum peaks and Re for five different studies ........... 78 xii LIST OF FIGURES Figure 1.1 Example of separated flow in a backward-facing step ..................................... 2 Figure 2.1 Axisymmetric, backward-facing-step configuration ........................................ 9 Figure 2.2 Axisymmetric, backward-facing-step model — “Images in this dissertation are presented in color” .................................... I ....................................................................... 11 Figure 2.3 Side view of the axisymmetric, backward-facing step ................................... 11 Figure 2.4 Labeling system for the axisymmetric, backward-facing step viewed from the approaching flow direction ............................................................................................... 13 Figure 2.5 A schematic of the top view of the Microphone and static-pressure-tap instrmnent-plate - located on the top side of the model .................................................... 15 Figure 2.6 View of the I-plate Showing provisions for embedding the microphones and static pressure into the I-plate ........................................................................................... 16 Figure 2.7 Schematic of instrument plate containing only static pressure taps ............... 16 Figure 2.8 Image of the fog generator used in the PIV set-up, placed upstream of the inlet section to the wind tunnel ......................................................................................... 19 Figure 2.9 Example of a typical PIV system [Humphreys] ............................................. 20 Figure 2.10 Camera set-up used in experiment ................................................................ 22 Figure 2.11 PIV set-up in SBRT ....................................................................................... 25 Figure 2.12 PIV optics set-up on top of the wind tunnel ................................................. 25 Figure 2.13 Streamwise distribution of the mean-pressure coefficient, depicting the quality of alignment of the model ..................................................................................... 27 Figure 2.14 Timing diagrams showing how the microphones data and PIV images were synchronized ..................................................................................................................... 3 1 Figure 2.15 Timing diagrams for the PIV image acquisition system — Copied from Wemet ............................................................................................................................... 32 Figure 3.1 Streamwise distribution of the forward flow probability for all four Reynolds numbers ............................................................................................................................. 39 Figure 3.2 Normalized reattachment distance versus normalized momentum thickness 41 Figure 3.3 Streamwise distribution of the mean, streamwise velocity component at y/h = 0.05 for all four 2D Reynolds number .............................................................................. 43 xiii Figure 3.4 Secondary separation region from the streamwise distribution of the mean, streamwise velocity component at y/h = 0.05 for all four 2D Re ..................................... 43 Figure 3.5 Reynolds number dependence of x,, calculated using F PP and U = 0 and compared to Li and Naguib data ....................................................................................... 46 Figure 3.6 Comparison of the measured boundary layer profile with the Spalding equation ............................................................................................................................. 48 Figure 3.7 Boundary-layer mean velocity profiles from present experiment and Naguib’s data .................................................................................................................................... 50 Figure 3.8 Boundary-layer turbulent-streamwise-velocity profiles from present experiment and Naguib data ............................................................................................. 51 Figure 3.9 Comparison of fiiction coefficient value from the present experiment with the Ludweig and Tillmann relation ......................................................................................... 51 Figure 3.10 Mean-pressure distribution for all Reynolds numbers, 2D case only, with a line marking the corresponding x, value for each case ..................................................... 54 Figure 3.11 Mean-pressure coefficient distribution for the four Re cases ....................... 55 Figure 4.1 RMS pressure distribution for all Reynolds numbers with a line marking the corresponding xr value for each case ................................................................................ 67 Figure 4.2 RMS pressure distributions for all Re as a function of x/xr ............................ 67 Figure 4.3 Flooded-color, contour maps of the wall-pressure auto-correlation at Reh = a) 8081 and b) 18588 ............................................................................................................. 73 Figure 4.4 Auto-correlation function for six streamwise positions upstream of xr — red line is for Reh = 8081 and green line is for Reh = 18588 .................................................. 74 Figure 4.5 Auto-correlation function for six streamwise positions downstream of xr - red line is for Re}I = 8081 and green line is for Re.1 = 18588 .................................................. 75 Figure 4.6 Frequency spectra for: a) Reh = 8081 and b) Re}I = 18588. Eleven spectra are plotted relative to different origins, at the following streamwise positions starting from the top: x/h = 1) 0.822, 2) 1.60, 3) 2.38, 4) 3.16, 5) 3.94, 6) 4.72, 7) 5.50, 8) 7.07, 9) 8.63, 10) 10. 2, 11) 11.7, and 12) 12.5 ....................................................................................... 81 Figure 4.7 Spectra from a) Reh = 8081 and b) Reh = 18588 cases plotted at three streamwise positions ......................................................................................................... 82 Figure 4.8 Flooded-color, contour maps of the cross-correlation and implied convection velocities at ReyI = a) 8081 and b) 18588 .......................................................................... 84 xiv Figure 4.9 Frequency-wavenumber spectrum obtained from all 32 microphones at Re = a) 8081 and b) 18588 ........................................................................................................ 91 Figure 4.10 Normalized convection velocity as a function of normalized frequency for Reh = 8081 and 18588 ....................................................................................................... 92 Figure 4.11 Flooded-color, cross-correlation contour plots for a) Rel1 = 8081 using microphone #2 (x/h = 0.82) as the reference microphone and b) Reh = 18588 and reference microphone #3 (x/h = 1.21) ............................................................................... 93 Figure 4.12 Frequency-wavenumber spectra for Re;1 = 8081 calculated using data from the microphones located: a) upstream of xr and b) downstream of xr ............................... 95 Figure 4.13 Uc versus normalized frequency for Rel1 = 8081, comparing results from full spectrum with spectrum results from downstream of xr ................................................... 97 Figure 4.14 Frequency-wavenumber spectrum for x/xr > 1.7 at Reh = 8081 for fli/Uoo > 0.085 .................................................................................................................................. 99 Figure 4.15 Uc versus normalized frequency obtained from microphones located at x/xr > 1.7 and frequencies fh/U.o > 0.086 for Reh = 8081 ............................. . .............................. 99 Figure 5.1 Mean-velocity vector fields and associated streamlines for Re;1 = a) 8081 and b) 18588 .......................................................................................................................... 107 Figure 5.2 Mean-streamwise-velocity profiles at selected streamwise locations: (O)Reh = 8081, and (CI) Reh = 18588 ......................................................................................... 108 Figure 5.3 Flooded-color, contour maps of uRMg/Uoo at Re.1 = a) 8081 and b) 18588 114 Figure 5.4 F looded-color, contour maps of vRMS/U.o at Reh = a) 8081 and b) 18588 115 Figure 5.5 Streamwise-velocity RMS profiles: (O)Reh = 8081 and (Cl) Re}, = 18588. 116 Figure 5.6 Normal-velocity RMS profiles: (O)Reh = 8081 (El) Reh = 18588 .............. 117 Figure 5.7 F looded-color, contour maps of — u—'v-'/ U f, at Re;1 = a) 8081 and b) 18588. 120 Figure 5.8 Reynolds-shear—stress profiles: (O)Reh = 8081 and (El) Re}I = 18588, ....... 121 Figure 5.9 Demonstration of the eight-point grid used to calculate the circulation for estimating the vorticity at point (i, j). Copied from Raffel et al. .................................... 123 Figure 5.10 Flooded-color, contour plots of vorticity, 032* = wzh/Uw, for Reh = a) 8081 and b) 18588 .................................................................................................................... 125 Figure 5.11 Vorticity, 032* = (nzh/Uoo, profiles as selected streamwise locations : (O)Reh = 8081 and (El) Reh = 18588 .............................................................................................. 127 XV Figure 5.12 Shear-layer edges from Figure 5.11: Reh = 8081 (red line) and Rel| = 18588 (green line) ...................................................................................................................... 127 Figure 6.1 Auto-correlation from wall-pressure signature (top plot) and uRMs/Uw of the velocity field (bottom plot) for Reh = 8081 .................................................................... 141 Figure 6.2 Auto-correlation from wall-pressure signature (top plot) and local mean velocity at the center of the shear layer versus streamwise position (bottom plot) for Reh = 8081 ............................................................................................................................. 142 Figure 6.3 Cross-correlation from wall-pressure signature (top plot) and local mean velocity versus streamwise position (bottom plot) for Reh = 8081 ................................. 143 Figure 6.4 Local, mean velocities from velocity field and average convection velocities calculated from wall-pressure, cross-correlation peaks for Reh = 8081 ......................... 144 Figure 6.5 Flooded-color, contour plots of Ru'pv/(Vszaf) for Reh = 8081 ..................... 154 Figure 6.6 Flooded-color, contour plots of RV-pr/(Vszof) for Reh = 8081 ..................... 155 Figure 6.7 Single-point stochastic estimation results for a negative pressure event (p' = - SpRMs) at x/h = 3.51 for Re;I = 8081. Linear results: a) vorticity and b) shear strain rate. Quadratic results: c) vorticity and d) shear strain rate. ................................................... 156 Figure 6.8 Single-point stochastic estimation results for a positive pressure event (p' = +5pRMg) at x/h = 3.51 for Reh = 8081. Linear results: a) vorticity and b) shear strain rate. Quadratic results: c) vorticity and d) shear strain rate. ................................................... 157 Figure 6.9 Single-point stochastic estimation results for a negative pressure event (p' = - SpRMs) at x/h = 4.68 for Re}1 = 8081. Linear results: a) vorticity and b) shear strain rate. Quadratic results: c) vorticity and d) shear strain rate. ................................................... 158 Figure 6.10 Single-point stochastic estimation results for a positive pressure event (p' = +5pRM3) at x/h = 4.68 for Re.1 = 8081. Linear results: a) vorticity and b) shear strain rate. Quadratic results: 0) vorticity and d) shear strain rate. ................................................... 159 Figure 6.11 Single-point stochastic estimation results for a negative pressure event (p' = - Spams) at x/h = 5.46 for Reh = 8081. Linear results: a) vorticity and b) Shear strain rate. Quadratic results: c) vorticity and d) shear strain rate. ................................................... 160 Figure 6.12 Single-point stochastic estimation results for a positive pressure event (p' = +5pRMs) at x/h = 5.46 for Reh = 8081. Linear results: a) vorticity and b) shear strain rate. Quadratic results: c) vorticity and d) Shear strain rate. ................................................... 161 Figure 6.13 Spatial, pressure signature for Re}I = 8081 case at three instants of time: a) 12.00 s, b) 12.01 s, and c) 12.02 s ................................................................................... 165 xvi Figure 6.14 Spatial, pressure signature for Reh = 8081 case at three instants of time: a) 8.80 s, b) 8.81 s, and c) 8.84 s ......................................................................................... 166 Figure 6.15 First four mode shapes defined by POD for the pressure distribution at Reh = 8081 ................................................................................................................................. 170 Figure 6.16 The cumulative distribution of energy over all 32 modes for the pressure distribution at Reh = 8081 ............................................................................................... 171 Figure 6.17 Vorticity field associated with mode-one, pressure Signature .................... 175 Figure 6.18 Shear strain field associated with mode-one, pressure signature ............... 175 Figure 6.19 Vorticity field associated with mode-two, pressure signature .................... 176 Figure 6.20 Shear-strain field associated with mode-two, pressure signature ............... 176 Figure 6.21 Vorticity field associated with mode-three, pressure signature .................. 178 Figure 6.22 Shear strain field associated with mode-three, pressure signature ............. 178 Figure 6.23 Vorticity field associated with mode-four, pressure signature ................... 179 Figure 6.24 Shear strain field associated with mode-four, pressure signature .............. 180 Figure 6.25 Multi-point LSE of the velocity field associated with mode A pressure signature .......................................................................................................................... 185 Figure 6.26 Multi-point LSE of the velocity field associated with mode B pressure signature .......................................................................................................................... 186 Figure 6.27 Multi-point LSE of the velocity field associated with mode AB pressure signature .......................................................................................................................... 187 Figure 6.28 Frames showing evolution of flow structure, At = 0.4 ms .......................... 190 Figure 6.29 Location of the flow structure, x/h, versus time used in tracking the evolution of a flow structure developing between x/h = 2-3 .......................................... 191 Figure 6.30 Tracking the location of a flow structure as it forms between x/h = 2-3 and then accelerates downstream ........................................................................................... 191 Figure 6.31 Vorticity field at instant of acquisition of PIV image #1299: a) instantaneous, vorticity field and pressure signature, b) multi-point LSE using instantaneous, pressure signature, and c) multi-point LSE using mode A, pressure signature .......................................................................................................................... 195 xvii Figure 6.32 Vorticity field at instant of acquisition of PIV image #67: a) instantaneous, vorticity field and pressure signature, b) multi-point LSE using instantaneous, pressure signature, and c) multi-point LSE using mode B, pressure signature ............................. 196 Figure 7.1 Idealized sketch of the flow structures developing in “Shear-layer mode” downstream of a backward-facing step ........................................................................... 205 Figure 7.2 Idealized sketch of the flow structures developing in “wake mode” downstream of a backward-facing step ........................................................................... 205 Figure 7.3 Instantaneous, pressure signature and associated, vorticity field ................. 208 Figure 8.1 Subsonic Basic Research (Wind) Tunnel at NASA Langley ....................... 213 Figure 8.2 Axisymmetric, backward-facing step inside the test section of SBRT ........ 215 Figure 8.3 False floor in test section of SBRT ............................................................... 216 Figure 8.4 Static pressure distribution along false floor in SBRT test section ............... 217 Figure 8.5 Side view of the axisymmetric, backward-facing step ................................. 218 Figure 8.6 Schematic of nose section ............................................................................. 221 Figure 8.7 Schematic of the stationary section .............................................................. 222 Figure 8.8 Rotator schematic ......................................................................................... 224 Figure 8.9 Schematic of step section ............................................................................. 226 Figure 8.10 Picture of step ring on model ...................................................................... 227 Figure 8.11 Motor section schematic ............................................................................. 231 Figure 8.12 Schematic of drive shaft ............................................................................. 232 Figure 8.13 Sketch of tail section ................................................................................... 233 Figure 8.14 View of one corner of the wire-support system ......................................... 235 Figure 8.15 View of the wire-support system in use in SBRT ...................................... 235 Figure 8.16 Model support stand ................................................................................... 237 Figure 8.17 Picture of the stand used to support the model ........................................... 238 Figure 8.18 Model-stand set-up in SBRT at NASA Langely ........................................ 239 Figure 8.19 Set-up for azimuthal, uniformity test .......................................................... 242 xviii Figure 8.20 Dial-gauge-test set-up ................................................................................. 243 Figure 8.21 Results from the dial-gauge test ................................................................. 243 Figure 8.22 Azimuthal symmetry of the flow “at” separation for Re.1 = 8081 .............. 244 Figure 8.23 Typical F G-3629—P16 Frequency-Response, Curve - Emkay data sheet .. 245 Figure 8.24 Wiring for the I-plate .................................................................................. 246 Figure 8.25 Microphone, calibration assembly .............................................................. 249 Figure 8.26 Sample of calibration results for the FG-3629 microphone ....................... 250 Figure 8.27 Microphone sensitivities from bench calibration ....................................... 251 Figure 8.28 Surface-pressure power spectra at microphones: a) 2, b) 11, and c) 26 ..... 254 Figure 8.29 Mirror mounted on tail of model ................................................................ 255 Figure 8.30 Static pressure data with and without mirror in tunnel for the Reh = 8081 255 Figure 8.31 Auto spectra of microphone data with and without mirror: a) microphone closest to the step and b) microphone nearest to reattachment ....................................... 257 Figure 8.32 Example of dot card image from camera #1 ............................................... 259 Figure 8.33 Flow regions captured by the two cameras used ........................................ 259 Figure 8.34 Grid cards from cameras 1 and 2 - after dewarping, prior to merging ....... 261 Figure 8.35 Example of merged, grid card from PIV set-up ......................................... 261 Figure 8.36 Path of integration used to calculate the shear strain at point (i, j). Copied from Raffel et al. ............................................................................................................. 273 xix Au.quad Av,quad 3.1 LIST OF SYMBOLS Definition linear stochastic estimation coefficient term for streamwise velocity component linear stochastic estimation coefficient term for normal velocity component linear stochastic estimation coefficient term for streamwise velocity component (in quadratic estimation) linear stochastic estimation coefficient term for normal velocity component (in quadratic estimation) amplitude coefficient for POD mode functions quadratic stochastic estimation coefficient term for streamwise velocity component quadratic stochastic estimation coefficient term for normal velocity component intercept of the least squares line skin-friction coefficient = 2(u,/U.,.,)2 mean pressure coefficient RMS fluctuating pressure coefficient intercept of the mean-velocity distribution in the log region, when plotted on semi-log scale average displacement vector over a PIV interrogation cell expectation or average of the square of the error error between the estimated and measured fluctuating velocity components frequency [Hz] total height of the fence XX Syr_nbol H12 h iwidth k kt Owidth Ps Re Reh R89 2611M boundary layer shape factor = 670 step height image width Von Karman constant time-delay index in correlation function streamwise wavenumber camera magnification for PIV images slope total number of points measured in the time record time shifi, or delay object width mean surface-pressure fluctuating surface-pressure root mean square value of p’ reference mean pressure surface mean pressure Reynolds number Reynolds number based on step height Reynolds number based on momentum thickness thickness of the boundary layer at separation normalized auto-correlation function normalized cross-correlation between streamwise velocity and surface- pressure xxi 0* C‘. C‘. C‘. C + 8 0 fi Definition normalized cross-correlation between normal velocity and surface- pressure discrete-time auto-correlation function time delay between light pulses from the two different lasers time mean streamwise velocity component mean convection velocity freestream velocity non-dimensional mean streamwise velocity component = U/ut streamwise velocity component fluctuating streamwise velocity component root mean square value of u' wall-friction velocity = (tw/p)”2 Reynolds shear stress mean normal velocity component power source voltage for the microphones velocity vector field normal velocity component fluctuating normal velocity component root mean square value of v' streamwise coordinate reattachment distance wall-normal coordinate xxii ¢I (I’P'P' Tw (Oz in éxy E312 éyy Definition non-dimensional wall-normal coordinate = yuT/v spanwise coordinate boundary layer thickness displacement thickness of boundary layer circulation azimuthal angle mode functions for ith POD mode power spectrum of p' wavelength kinematic viscosity momentum thickness Of boundary layer fluid density time shift average (mean) wall-shear stress out-of—plane vorticity component vorticity vector strain rate tensor field in-plane shear strain component out-of-plane linear strain component in-plane linear strain component in streamwise direction in-plane linear strain component in normal direction xxiii 1 Introduction 1.1 Background One important class of fluid flow that is encountered frequently in engineering applications is separated flows. These applications include flow over airplane wings, in dump combustors, and through turbines and compressors, to mention a few. Extensive research has been done in the area of separated flows in order to understand the dominant flow structures, or flow motions, in the separated shear layer and the general characteristics of the flow field. Some of the research has been focused on understanding the flow generation mechanisms of wall-pressure fluctuations, which is important in engineering applications involving flow-induced noise/vibrations and flow-structure interaction. The ability to understand these and other flow physics concerning separated flows can lead to the development of simplified models for prediction of wall-pressure fluctuations and the implementation of active or passive flow control techniques for optimizing the flow state above the surface and/or minimizing the adverse flow effects on the wall. An example of a canonical separated-flow configuration is the backward—facing step (BFS) geometry shown in Figure 1.1 along with the average flow pattern above the surface. This flow is frequently adopted for the purpose of research studies as an idealization of real, separated-flow problems. Generally speaking, flow separation occurs where the flow detaches from the boundary of a solid surface, which is at the step edge in the case of the BFS. When the boundary layer separates, it forms a shear layer that may reattach along the same surface depending on the length and contour of the geometry. Beneath the shear layer is a primary and a secondary, recirculation region. Beyond the mean reattachment point, the reattached boundary layer begins to “relax” as it redevelops into an equilibrium turbulent boundary layer for a sufficiently long surface. The generally accepted view of the BFS flow field in the literature is that large- scale, turbulent structures develop in the highly-unstable, separated shear layer and grow in size and strength as they convect downstream towards reattachment. As will become clear in Chapters 6 and 7, the findings in this study indicate, in agreement with a handful existing studies, an alternate view. Specifically, it will be Shown that the large-scale turbulent structures found in the separation zone grow in size and strength while stationary at about half of the reattachment length before shedding from the shear layer and accelerating downstream. U... Secondary recirculation ——> region 6 ‘ ' Recirculation ._, region . —> ______ Separation W WWI/1 ' ‘ ' ‘ streamline %’ //// /// //// ///// //// 7 / /////A A V xr Fl Figure 1.1 Example of separated flow in a backward-facing step Most of the separating/reattaching flow research to date has been on “2D flows” in planar canonical geometries such as a backward-facing step, a splitter plate, or a splitter-plate-with-fence. Classic studies of these geometries include, but are not exclusive to, Eaton and Johnston1 (BF S), Cherry et al.2 (splitter plate), Castro and Haque3 (Splitter-plate-with-fence), and many others. In these studies, mean flow two dimensionality was assumed due to the large aspect ratio (width of the model divided by the step height). Brederode and Bradshaw4 showed that three-dimensional effects along the centerline in a planar BF S may be neglected by ensuring a model aspect ratio greater than ten. However, Ruderich and F emholz5 provide striking evidence of strong secondary flow patterns within the recirculation region as well as near reattachment using Oil-flow pictures in a Splitter-plate/fence configuration, which has similar flow characteristics to a BFS. This was true for aspect ratios as high as 22, even with the utilization of end plates. More recently, in Hudy6, a large aspect ratio of 36 was used for a splitter-plate-with-fence model. It was found that the spanwise distribution of the surface pressure varied between 10 — 30% from the centerline values, suggesting that the flow field was inherently three dimensional. Thus, although three dimensionality effects may be reduced along the centerline of planar separated flow models, the flow exhibits large variation in the spanwise direction, which could substantially influence the flow features observed along the centerline. The current research project investigates the separating/reattaching flow over an axisymmetric, backward-facing step (BFS). The general, mean-flow characteristics surrounding the axisymmetric BF S are similar to those seen in a planar BFS, except in the axisymmetric BFS the mean flow characteristics are the same in every azimuthal plane. This invariance in the azimuthal direction eliminates the three-dimensional effects that are typically seen in planar BFS and improves the quality of the two dimensionality along the measurement plane. Figure 1.1 shows the ideal two-dimensional, mean-flow field downstream of a planar BFS, which is the same as seen in a single, azimuthal plane downstream of the axisymmetric BF S. An infinite number of these azimuthal planes make up the axisymmetric geometry; and, thus, the mean flow characteristics are the same all the way around the cylindrical model. Based on this explanation and for clarification purposes, the axisymmetric flow developing around the step will frequently be referred to as two dimensional (2D) in this document since the average flow field in the plane perpendicular to the wall and parallel to the freestream velocity is invariant in one of the three spatial dimensions, i.e., the azimuthal direction. One study that did investigate the flow field in an axisymmetric backward-facing step is Li and Naguib7. In 2005, the authors employed the same axisymmetric BFS model used in the present study. Li and Naguib7 measured the wall-Shear stress at different Reynolds numbers using a high-frequency oscillating hot-wire sensor. The sensor was embedded in the wall of the model at different streamwise positions spanning 0.3 to 10 step heights downstream of the step edge and measured both skin friction magnitude and direction. Data were acquired within the viscous sublayer at a height of 97 um above the wall, which corresponds to y+ = 3.1 (y+ = yut/v and u, = (cw/mm, where y is the distance normal to the wall, v is the kinematic viscosity of the fluid, ‘Ew is the shear stress at the wall and p is the fluid density). Nonetheless, the study was limited to using skin friction measurements and to comparing results with existing planar BFS studies. The present study adds to the work completed by Li and Naguib7 by investigating both the wall-pressure signature and the flow field above the wall in the axisymmetric geometry. This is accomplished by using simultaneous wall-pressure-array and Particle Image Velocimetery (PIV) measurements. It is the overall goal of this research project to bring further understanding to the flow-physics community in the area of 2D (i.e., axisymmetric) flow separation. Although the backward-facing-step flow field has been investigated extensively over the past few decades, only one other study (Lee and Sungg) has utilized an array of microphones to investigate the spatio-temporal statistics. Nonetheless, this study only examined the flow in a planar BFS. In addition, the few PIV studies of the BFS flow have all been conducted in a planar flow geometry. Finally, to the best of the author’s knowledge, no studies have investigated the relation between the velocity field and the wall-pressure in an axisymmetric backward-facing step in order to identify the flow structures/mechanisms generating the surface-pressure signature. Thus, considering these motivating factors, the objectives for the present study are the following: 1. To investigate the wall-pressure signature and the velocity field in an axisymmetric separating boundary layer flow field in order to understand the flow and wall-pressure physics using spatio-temporally-resolved measurements of the wall-pressure and spatially-resolved flow-field data 2. To explore and identify Reynolds-number dependencies in the flow 3. To identify the flow structures that generate the surface-pressure signature using wall-pressure-based, single-point linear- and quadratic-stochastic estimation, and multi-point, linear, stochastic estimation 4. To employ stochastic estimation to facilitate an investigation into the time evolution of the flow structures that are linked to the wall-pressure-generation- process within the shear layer and downstream of reattachment 1.2 Organization of document The present document is divided into seven chapters including the Introduction. The axisymmetric, backward-facing step model is described in Chapter two in terms of the general design and overall construction. Chapter two also highlights the experimental set-up used to acquire surface pressure measurements and Particle Image Velocimetry images simultaneously in the axisymmetric, backward-facing-step model. The general flow field characteristics are discussed in detail in Chapter three, including the reattachment distance, the boundary-layer state at separation, and the mean-pressure distribution along the surface of the model. Chapter four is dedicated to investigating Reynolds-number effects on the wall-pressure measurements. Auto-correlation; cross- correlation; and frequency-wavenumber contour plots are utilized in exploring the spatio- temporal characteristics of the wall-pressure. Similar analyses are presented in Chapter five for the velocity field measurements. The streamwise and normal fluctuating velocity components for different Reynolds-number cases are discussed along with the Reynolds shear stress in each flow field. Mean-vorticity plots are also compared between Reynolds -number cases. Chapter six links all the information together regarding the wall-pressure and above-surface, velocity measurements. In this chapter, the flow structures that generate the wall-pressure signature are investigated using stochastic estimation and proper orthogonal decomposition. A final discussion is given in Chapter seven. The Appendix provides supplementary information regarding the construction of the axisymmetric, backward-facing-step model, measurement techniques implemented, and Signal processing techniques utilized. Finally, each chapter contains its own literature review and list of references. 1.3 References I Eaton, J. K. and Johnston, J .P., “A Review of Research on Subsonic Turbulent Flow Reattachrnent,” AIAA Journal, Vol. 19, No. 9, 1981, pp. 1093-1099. 2 Cherry, N. J., Hillier, R., Latour, M.E.M.P., “Unsteady Measurements in a Separated and Reattaching Flow,” Journal of Fluid Mechanics, Vol. 144, 1984, pp. 13- 46. 3 Castro, LP. and Haque, A., “The Structure of a Turbulent Shear Layer Bounding a Separation Region,” Journal of Fluid Mechanics, Vol. 179, 1987, pp. 439-468. 4 Brederode, V. dc and Bradshaw, P., “Three-dimensional flow in nominally two- dimensional separation bubbles. 1. Flow behind a rearward-facing step,” Imperial College Aero Report 72-19, Imperial College of Science and Technology, London, 1972. 5 Ruderich, R. and Femholz, H. H., “An Experimental Investigation of a Turbulent Shear Flow with Separation, Reverse Flow, and Reattachrnent,” Journal of Fluid Mechanics, Vol. 163, 1986, pp. 283-322. 6 Hudy, L.M., “Simultaneous wall-pressure array and PIV measurements in a separating/reattaching flow region,” Masters thesis, Michigan State University, 2001. 7 Li, Y. and Naguib, A.M., “High-Frequency Oscillating-Hot-Wire Sensor for near- Wall Diagnostics in Separated Flows,” AIAA Journal, Vol. 43, No. 3, 2005, pp. 520. 8 Lee, I. and Sung, H.J., “Characteristics of wall pressure fluctuations in separated and reattaching flows over a backward-facing step,” Experiments in Fluids, Vol. 30, 2001, pp. 262-272. 2 Experimental Apparatus, Instrumentation, and Measurement Techniques 2.1 Wind Tunnel Facility The experiment was completed in the Subsonic Basic Research (Wind) Tunnel (SBRT) at NASA Langley Research Center in Hampton, Virginia. The open-circuit, low-Speed, wind tunnel has a 6:1 contraction ratio upstream of a 0.57 m-wide by 0.82 m- high by 1.85 m-long test section. An adjustable, false floor was placed in the test section and was set at a slight angle so that the pressure gradient in the test section was zero over most of the test-section’s length (see Appendix 8.1). The nominal height of the test section with the false floor installed was 0.62 m. The blockage ratio with the model inside the test section was 3.6% and the turbulence intensity within the test section was less than one percent. More details describing the wind tunnel facility are located in Appendix 8.]. 2.2 The Model The model used for generating the separating/reattaching flow was an axisymmetric backward-facing step as shown in Figure 2.1, where x represents the streamwise coordinate, y is the wall-normal coordinate, I!) is the azimuthal angle, h represents the step height, and U... represents the freestream velocity. The model was a circular cylinder with its axis parallel to the x direction, thus making the test geometry axisymmetric. ——-> .___, —> try ‘I‘ """"""""""" ’ \ \ h it ‘_, «a Figure 2.1 Axisymmetric, backward-facing-step configuration The design of the axisymmetric BF S is shown in Figure 2.2 and Figure 2.3 with intricate construction detail outlined in Appendix 8.2. The model was designed and manufactured in cooperation with JBJ Products and Machinery located in Williamston, Michigan. The model measures 2.37 m in length, 0.124 m in diameter upstream of the step, and 0.10 m in diameter downstream of the step. The height of the step is 12.2 mm, and details regarding the selection of the height of the step can be found in Appendix 8.2. A spherical nose at the leading edge transitions the flow over the model relatively smoothly; although, there is a step discontinuity in the radius of curvature between the nose and the stationary section immediately downstream that may cause a flow disturbance. Because it was desired to establish a turbulent boundary layer at separation, this flow disturbance only aided in the quicker development of the boundary layer to a turbulent state; and, therefore, it was considered beneficial. Downstream of the nose is the 2D section module, measuring 0.35 m in length. This region allows the flow to settle into a 2D boundary layer after transitioning over the nose. In addition, at the upstream end of the 2D section, sandpaper was used to hasten the boundary-layer transition. At the aft end of the model, a conical tail prevents an abrupt transition of the flow surrounding the model into the flow region farther downstream. Moderating the step-like feature at the end of the model helps to prevent the development of disturbances that could propagate upstream, i.e. acoustically, and possibly change the flow conditions in the separating/reattaching flow zone of interest. 10 'I‘ran'crsc 'l‘alrlc MIMI-I Sh mull Support Figure 2.2 Axisymmetric, backward-facing-step model — “Images in this dissertation are presented in color” Flow S u - te (1% (h = 1233 mm) Motor/Support 2D Section ROWOT Module Tail (350.5 mm) (749.3 mm) / (406.4 mm) (254 m) T H W 9 I \kl I . * JL V 1 f \ Nose 0.124 m 05,0 (70 mm dia.) dia- ‘a‘ Model Sensors Support Figure 2.3 Side view of the axisymmetric, backward-facing step The model was designed to allow the establishment of boundary layers with/without cross-flow velocity component. To accommodate the provisions for establishing the cross flow, the model was hollow with a 25.4 mm-diameter steel shaft that runs the entire length of the model at its center for increased rigidity. Inside the model, a drive shalt, connected to a 1/15 hp motor, rotated the 0.75 m long rotator section II upstream of the step to introduce the cross-flow component, establishing what is known in the literature as three-dimensional (3D) boundary layer at separation. Without rotation, the boundary layer is 2D. Since both 2D and 3D boundary layers can develop along the surface, a comparison between the two cases within the same geometry can be made. The results for the 3D boundary layer case are not the focus of this thesis but, for the interested reader, they may be found in Hudy et al.1 Approximately 0.37 m or 30 step heights (6.7 x,, where xr is the reattachment length) downstream of the step is a steel motor/support module, which housed the pulley for the electric motor and provided the stability necessary to connect the model to the support stand as shown in Figure 2.2 and Figure 2.3. In addition, four stretches of piano wire threaded through the nose of the model and secured near the four corners of the test section were used to support the front end of the cantilever-beam-like model. Figure 2.4 Shows a front view of the model as “seen” by the freestream flow. In this figure, the model is divided into four quadrants and labeled north, south, east, and west with the corresponding azimuthal angle given. These labels will be used throughout the text to explain the location around the model of the area being discussed. In Figure 2.4, the instrument plates (i.e., plugs employed for the insertion of wall sensors) can be seen. The main instrument plate that houses both microphones and static pressure taps is located on the north side of the model. The instrument plate on the east side contained static pressure taps. For the study of Liz, this plate also contained two oscillating hotwire sensors that were used for acquiring wall-shear-stress measurements. Finally, two more instrument plates containing only static pressure taps were placed on the south and west 12 sides for the purpose of alignment of the model. All wiring and static-pressure tubes were stored inside the model as discussed in further detail in sections 2.3. North side is main North (0°) measurement region for microphones and PIV West (270°) (90°) 1 . Shear stress nstrument plates wrth measurements only static pressure completed on taps were placed on the east side of south and west SldeS South (180°) model Figure 2.4 Labeling system for the axisymmetric, backward-facing step viewed from the approaching flow direction It is important to note that the results found in this study are limited to a specific geometric ratio of the step height to the radius (r) of the model. Changing the ratio can affect the flow field downstream of separation. For instance, if the radius is large compared to the step height (r >> h), then the affect due to the curvature of the model is reduced. In such a case, the geometry would be similar to a planar BFS and this may cause the reattachment length to be longer. The r/h for the present model is 5.2. Thus, the results presented are specific to the axisymmetric geometry used in this experiment. 13 2.3 Instrumentation and Measurement Techniques Several measurement techniques were used to acquire an extensive pressure and velocity dataset. A single hotwire was used to examine the alignment of the model inside the test section in terms of azimuthal uniformity. Appendix 8.3 has the procedure and the results of this test. The axisymmetric, backward-facing step described in Section 2.2 was also instrumented, downstream of the step, with 56 static pressure taps and 32 Emkay microphones. The microphones and taps were embedded in a 222.25 mm (18 h) long by 20.6 mm (1.7 h) wide anodized instrument plate (I-plate) shown in Figure 2.5. The I- plate was located on the top side of the model and was secured to the model by two lA-20 screws located at either end of the I-plate. The upstream portion of the I-plate was instrument-free for placement of a 50.3 mm-long “step ring”, which was used to form the separation edge. The center of the most upstream microphone was located 5 .3 mm (0.43 h) downstream of the step-ring edge. The wiring and tubing for the instrumentation was stored inside the model within a 12.7 mm-wide annular space (see Appendix 8.2 for model construction details). The taps were used to characterize the mean flow surrounding the model and to align the model in the wind tunnel. The microphones were used to measure the unsteady pressure along the surface. 14 Through hole for securing I-plate to model Flgw /Microphones ........................................................................... r 2% 3% 3: 3 3 3 Through hole for . Static pressure taps securing I-plate to model Figure 2.5 A schematic of the top view of the Microphone and static-pressure-tap instrument-plate - located on the top side of the model 2.3.1 Static Pressure Taps On the top side of the model, 32 static pressure taps were located beside the microphones at a spacing of 3.2 mm center to center. The static pressure taps were secured in counter-bored holes, as shown in Figure 2.6, using a non-conductive adhesive. A through hole equal to the 1 mm (0.08 h) inside diameter of the tap was drilled through to the top surface of the I-plate, shown in Figure 2.6. This ensured a smooth top surface and avoided the curvature mismatch between the taps and the top surface. The Urethane tubing for the static pressure taps was threaded through the 12.7 mm clearance inside the model and through the shroud to the Scanivalve, mechanical, pressure scanner. The Scanivalve system used is described in Hudy3. The remaining 24 static pressure taps were placed in three arrays of eight taps located on the two sides and bottom of the model as seen in Figure 2.7. These pressure tap holes were spaced 9.5 mm (0.8 h) apart center to center with the first tap starting at 19.05 mm (1.6 h) downstream of the step. The I-plates were secured to the model using two lA-20 screws at each end. The tubing for the taps was routed through the inside of the model and the shroud out to the Scanivalve. 15 Through hole 0.75 mm deep Two through holes and equal to sensing diameter Through hole 3. 5 mm for the '/4-20 screws of microphone (0.77 mm)/ deep and 1 mm in used to secure I-plate I diameter Microphone counter-bored |_\1 Static pressure tap hole (2.5 mm diameter, 5.6 “P counter-bored hole mm deep) (1.5 mm diameter, 3 3.18 mm deep) Figure 2.6 View of the I-plate showing provisions for embedding the microphones and static pressure into the I-plate Through holes for securing I-plate to model nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn ._-—-—-—-_-_-—-_- .............................................................................. Figure 2.7 Schematic of instrument plate containing only static pressure taps The output port of the Scanivalve was connected to one of two pressure transducers from the Setra 239 series. One transducer measured differential pressure in the range of 0-12.7 mm H20, outputting a corresponding 0-5 V signal. The other transducer ranged from 0-127 mm H20 over the same voltage range. The output port of the pressure scanner was connected to the negative input of one of the transducers depending on the expected pressure range of the measurements. In turn, the positive pressure port of the transducer was connected to a reference pressure tap located in the side wall of the wind-tunnel contraction section at its exit. 16 2.3.2 Microphones The microphones used in the experiment were Emkay (FG-3629-Pl6) Electret Condenser Microphones. The diameter of the microphones is nominally 2.6 mm (0.2 h). There were 32 microphones embedded in the I-plate with a center-to-center spacing of 4.8 mm (0.4 h). The microphones were secured in counter-bored holes, as shown in Figure 2.6, using a non-conductive adhesive. A through hole equal to the 0.77 mm (0.06 h) microphone sensing diameter was drilled through to the top surface of the I-plate. This ensured a smooth, top surface and avoided the curvature mismatch between the microphones and the top surface. Each microphone had a nominally reported sensitivity of -53i3 dB (relative to 10 V/Pa) over the frequency range of 100-10,000 Hz (Figure 8.23 in Appendix 8.4). Since the frequency range of interest extended below 100 Hz, and to account for variations in the response of individual units, the microphones were bench calibrated between 20 and 15,000 Hz using a B&K 4226 multifunctional calibrator. Details regarding the calibration are provided in Appendix 8.4. The microphones were powered utilizing a custom-made circuit box. The box was also used to break out the output of the microphone array into 32 individual coaxial cables for connection to the A/D board, as described in Appendix 8.4. The outputs of the microphones were connected to four National Instruments A/D boards (SCXI 1141), placed in a SCXI 1001 chassis. Each board had an input signal range of iSV, eight differential analog-input channels, and a variable channel gain that was set to ten for this experiment. The highest sampling rate for which the board is capable is 1.25 MHz for one channel. In all, 35 channels were acquired: 32 microphone l7 output channels, one channel for B&K reference microphone located in the ceiling, and two channels dedicated for the PIV synchronization with the microphones (see Section 2.4.3.1 for more detail). Because the channels were multiplexed, the maximum sampling rate was 35,714 Hz per channel with an inter-channel time delay of 0.8ps. This delay results in a systematic error, which is small compared to the flow convective time scale, 476us, over a distance equal to the microphone spacing at the freestream velocity. Each channel of microphones could also be connected to an ONO SOKKI FFT Analyzer (CF940) using a matrix switch in order to spot check the data during the experiment. Data from the microphones were collected using PC software written in Visual Basic by William M. Humphreys. 2.3.3 Particle Image Velocimetry Two-component (2C) Particle Image Velocimetry (PIV) measures two-velocity- field components at a number of points in a plane (Humphreys et al.4). There are four basic steps to PIV. First, the flow is uniformly seeded. In this test, pharmaceutical grade mineral oil was dispersed using a commercially available smoke generator. This smoke generator consisted of a shearing atomizer feeding a vaporization/condensation section and was capable of producing a cloud of smoke that could be varied in density by adjusting the atomizer supply pressure. The smoke generator was placed at the inlet section of the wind tunnel, thus allowing the incoming flow to carry the seed into the test section‘ as shown in Figure 2.8. The mean aerodynamic size of a particle was approximately 0.2-0.4 pm. This same set-up was used in a unified instrumentation test by Fleming et al.5 The group found the small particle size scattered enough light to be 18 imaged by CCD cameras. In addition, each image contained a sufficient amount of seeding to accurately cross-correlate particles within the interrogation region, resulting in a low rate of bad velocity vectors. The particles were also evenly dispersed throughout the image and the particle diameter was consistent across the image. Figure 2.8 Image of the fog generator used in the PIV set-up, placed upstream of the inlet section to the wind tunnel The second step is to illuminate the particles in the flow region being studied. The illumination is from two Nd:YAG laser systems. Beams from the systems are combined through optics to form a laser sheet that is used to illuminate the flow area of interest. A record of this illuminated area is then captured using CCD cameras as the third step. The idea is that one of the laser beams is pulsed and an image is captured; then at a known time delay the other laser beam is pulsed and a second image is taken. With proper selection of the time delay, the seed particles are displaced while mostly remaining within the illuminated region. The PIV records acquired are stored and analyzed to create a velocity vector field of the flow. The analysis requires locating the particles in one image and “matching” them with particles in the time-delayed image. This is accomplished by dividing the image into small “interrogation” windows (e.g., 16 by 16 pixels) and calculating the two- dimensional cross correlation between the first and second images for each of these sub- regions. The pixel offset at which the highest correlation is found gives the average particle displacement at the center of the interrogation cell. By dividing this displacement by the time delay between the two images, the velocity is calculated. Figure 2.9 shows an example of a PIV system set-up. ‘ Nd:YAG % Laser System _ , i w'"; GOO-m], 532-nm ”i“ :23, . Bea Beam Combining / Light Sheet Optics ' ‘ . » . Light Sheet Imaging System A__AA Wm, M _ 7 _ ,‘Li ,, MM __1 Figure 2.9 Example of a typical PIV system [Humphreys‘] 20 2.3.3.1 Experimental Set-up The PIV system used in this study was composed of two lasers, two cameras, and optical components. The two lasers were NDzYAG from the Continuum Powerlite series. Each laser was capable of producing up to 600 mJ per pulse of green light (wavelength of 532 nm) at a nominal repetition frequency of 10 Hz. The nominal pulse width for both lasers was 10 ns, and they were pulsed in synchronization with a pre-set, time separation between the lasers. The laser-lit particles were captured by two MegePlus ES-1.0 high resolution CCD cameras by Redlake. Each camera had a detector size of 1008 pixel wide by 1018 pixel high with a pixel size of 9x9 pm2 and with a 55% fill, meaning no gap between pixels. The two cameras, shown in Figure 2.10, were used to obtain simultaneous PIV measurements over two planes that overlapped partially along the streamwise direction (about l/l6th of the image width). This enabled coverage of all of the streamwise extent of the separating/reattaching flow region and part of the reattached boundary layer region (corresponding to a total length of 8.4h (102.5 mm) downstream of the step). 21 7! ,7 , " "I _ * I Figure 2.10 Camera set-up used in experiment The Redlake cameras utilize the “frame-straddling” mode of image acquisition. In this technique, the first frame integration period is 250us and the second frame integrates for a full 33.3 ms. During the integration time of the second frame, the information from the first frame is a readout of the sub-register. This method of operation allows the use of one camera for acquiring two consecutive images synchronized with the firing of each laser7. The time delay between the lasers was estimated based on the pixel size, the magnification factor, and the “PIV equation”. The CCD size is 1018 pixels high by 1008 pixels wide and the pixel size is 9 pm on the side. Therefore, the total width of the CCD element is 9.07 mm, also called the image view. The field of view was set to be 57.2 mm. From the field of view and the image view, the magnification can be calculated as follows: 22 Magnification = M’— = M = 0.159 (2.1) owm 57.2mm where, iwidth represents the image width (height) and owidth equals the object width (height). The object in this case is the field of View along the model. The equation for extracting the velocity from displacement information is given by ‘ - d V =— mAT (2.2) where 17 represents the speed, 3 represents the displacement, m is the magnification, and AT is the laser pulse separation (i.e., the time delay between the light pulses from the two different lasers. A 32 by 32 interrogation box was used for the cross correlation, and the maximum displacement was estimated as approximately 40% of the interrogation window width, or 13 pixels (117 um). Using equation (2.2) and solving for AT, the laser pulse separation for each test speed was calculated and is presented in Table 2.1. Table 2.1 Laser pulse separation times for all four Reynolds numbers tested Test Speed (m/s) Laser pulse separation (us) 7.4 102 10 77 18 42 23 33 The optics used to change the polarization of the lasers, form the light sheet, and direct the beam into the test section consisted of a variety of components. Turning mirrors, from CVI Laser Optics (Y2-2037-45-UNP), were used to redirect the path of the 23 two lasers. In this case, they were used to direct the laser beams to the top of the wind tunnel. The light sheet was then formed using optics along the top of the wind tunnel and then redirected inside the test section and upstream toward the step using a turning mirror as shown in Figure 2.11. These mirrors were 50.8 mm in diameter with 9.5 mm thickness, allowing a wavelength of 532 nm to reflect with a minimum un-polarized reflectance of 99%. This means that 99% of the laser light received by the mirror is reflected, resulting in 1% loss of energy, without affecting the polarization of the beam. From the laser, one of the beams passed through a M2 waveplate from Newport to rotate its polarization, which was necessary in order to combine the beams using the thin, film- plate polarizer. The combined beams had perpendicular polarization with respect to each other. The CVI Laser Optics thin, film-plate polarizer (TFP-532-PW-2025-C) had a transmission efficiency of 95% at a wavelength of 532 nm. The polarized beams were then redirected to the optics on top of the wind tunnel using turning mirrors. The optics set-up is pictured in Figure 2.12. Two spherical lenses by Newport were used to focus the laser sheet on the model and limit its width. The first was a -100 mm singlet spherical lens, followed by a +150 mm bi-convex spherical lens. The distance between the two lenses was varied to change the focal point of the beam on the model. The polarized beam then passed through a BK7 cylindrical plano-concave round lens (CLCC- 50.8-38.1-C-532) from CVI Laser Optics. This lens had a nominal focal length of 75 mm and was used to lengthen the laser sheet. Finally, a turning mirror was attached to the tail end of the model and used to turn the laser sheet upstream, towards the measurement region downstream of the step. 24 ()ptics on top (Il‘tunncl \ Turning mirror I i l I .1. Turning Mirror Turning Mirror r 150 mm and-IOU mm Spherical lenses LSICP ’ . \ ( )lin/tliical ‘ I'IN' I .. ...C‘ lens (75 mm) "l/ ' \ \(mtics I'Ihlc 7‘ . “I~-"I—

<——— 30.33 e Figure 2.14 Timing diagrams showing how the microphones data and PIV images were synchronized 31 —>' it— Frame Offset I Nl-6602 Trig _II . Camera ‘ ' " Trigger —i na— 20 Its Camera Strobe I —p Feedback to Nl-6602 —D§ F— TPD Transfer 1 Flashlamp 1 Q-Switch l Flashlamp 2 Q-Switch 2 Frame #1 I‘— 255 Ira—III Frame #2 I —p 33 ms —9 4— AT Laser Output I L Figure 2.15 Timing diagrams for the PIV im7age acquisition system — Copied from Wernet The true acquisition time for the PIV images was 30.3 seconds due to both software and hardware issues. The first pulse in the external trigger signal sent by the Dell PC laptop was dropped by the NI-6602 board. In addition, the PIVACQ program dropped the first two images. Therefore, to acquire 300 images, three extra pulses needed to be added to the external trigger signal. The external trigger contained 30.3 seconds worth of pulses as seen in Figure 2.14. The external trigger signal that was sent to the NI-6602 and the camera strobe that fired the lasers were the two extra channels acquired simultaneously with the microphone signals. Acquiring the external trigger gave the number of pulses sent by the Dell PC lapt0p. The camera strobe pulse train gave the pulses associated with image acquisition, 32 and it was used to match each image with the wall-pressure pattern acquired by the microphones at the same time as the image. In the post-processing phase, the first two pulses from the camera strobe signal were ignored. The third individual pressure reading after the rising edge of the third pulse was the point in the time series that matched with the first PIV image, followed by the remaining 299 images and pulses. The third pressure reading from the rising edge was acquired 250us after the camera strobe fired the lasers and opened the camera shutter and marked the point at which the image was transferred to the sub-register as Shown in Figure 2.15. At this point, the first laser had definitely fired and the second laser was about to fire within microseconds. In order to test the synchronization between the PIV and the microphones, a strobe light was used. The set-up for the test required a Single strobe lamp that could be externally triggered. All the lights in the room containing the wind tunnel were turned off so that the only light source would be from the strobe light. A simple target was placed on the model and the strobe was set-up so it illuminated the target when triggered. A pulse train was then created in LabView that externally triggered the strobe during randomly selected frames. This trigger pulse was one millisecond in duration. The strobe only fired during the second camera frame since the integration time was longer thim the duration of the strobe. For this particular test, the strobe was triggered in frames 1, 2, 8, 111, 223, and 291. Subsequent inspection of the 300 acquired images, proved that in fact an illuminated target was seen only in the second of the “PIV” image pair 1, 2, 8. 111, 223, and 291. In addition, by acquiring the strobe light trigger pulse, it was Verified that the strobe pulses overlapped with the PIV synchronization TTL pulses 33 number 1, 2, 8, 111, 223 and 291 (ignoring the first three pulses for the reasons described above). These completed the verification process of the synchronization procedure. 2.4.3.2 Post-processing the PIV images The post-processing of the PIV images required a few steps including merging of the images, as explained in Appendix 8.5, from the two separate cameras into one image representing the full extent of the measurement domain. Once merged, the PIV images were processed using Mark Wemet’s PIVPROC processing code8 and bad vectors were removed based on Chauvenet’s criterionq. The images were processed using a multi-pass correlation technique with 50% overlap for enhanced spatial resolution. The multi-pass approach utilized an initial 64 x 64 pixel integration box followed by a 32 x 32 pixel integration box for the second pass. This resulted in a vector Spacing of 0.92 mm. Overall, the vector field map covered 101.6 mm (8.4 h) in the streamwise direction and 45.6 mm (3.8 h) in the normal direction. 2.5 Test matrix The test matrix for the experiments done in SBRT was extensive. Four test cases, Corresponding to freestream velocities (U...) of 7.4, 10, 18, and 23 m/s, were tested, thus resulting in a Reynolds-number, test range based on a step height of 5900-19000. This test range was selected to allow comparison with the wall-shear-stress study by Li2 (Reh = 4300-13000), which utilized the same axisymmetric, BFS geometry. Both microphone and PIV data were acquired at all speeds reported in the present study. A list of the cases is shown in Table 2.2 along with the corresponding Reynolds numbers based on step 34 u 9). ,cfir-v "x...I tab ...-\I\I\.. height (Reh) and momentum thickness of the separating boundary layer (Ree), reattachment distances divided by step height (xr/h), and boundary layer momentum thicknesses at separation divided by step height (0/h). Note that xr/h was calculated using “Forward Flow Probability”, or FFP. Details of FFP will be discussed in Section 3.1. Due to the vast scope of this research project, the amount of information reported has been limited. Only two of the four 2D flow conditions will be analyzed in the following pages. A complete wall-pressure and velocity field analysis will be presented. In addition, data from the Reh = 8081 case only will be analyzed using stochastic estimation in effort to reveal the flow structures that generate the wall-pressure signature. Table 2.2 Experimental parameters for the four cases investigated U... Re.I xrlh from FFP Ree Olh 7.4 m/s 5980 4.29 889 0.149 10 m/s 8081 4.48 1237 0.153 18 m/s 14547 4.79 1751 0.122 23 m/s 18588 4.94 1979 0.107 35 2.6 References 1 Hudy, L. M., Naguib, A., Humphreys, W. M., and Bartram, S., “Particle image velocimetry measurements of a two/three-dimensional separating/reattaching boundary layer downstream of an axisymmetric backward-facing step,” 43rd AIAA Aerospace Sciences Meeting and Exhibit, AIAA-2005-0114, Reno, Neveda, January 10-13, 2005. 2 Li, Y., “Investigation of the wall-shear-stress signature in a backward-facing-step flow using oscillating hot-wire sensors,” Ph. D. dissertation, Michigan State University, 2004. 3 Hudy, L.M., “Simultaneous wall-pressure array and PIV measurements in a separating/reattaching flow region,” Masters thesis, Michigan State University, 2001. 4 Humphreys, W. M. and Bartram, S. M., “Measurement of separated flow structures using a multiple-camera DPIV system,” Presented at the 19’” International Congress on Instrumentation in Aerospace Simulation Facilities, 2001 . 5 Fleming, G. A., “Unified Instrumentation: Examing the Simultaneous Application of Advanced Measurement Techniques for Increased Wind Tunnel Testing Capability,” 22nd AIAA Aerodynamic Measurement Technology and Ground Testing Conference, St. Louis, Missouri, AIAA 2002-3244, June 24-26, 2002. 6 Humphreys, W. M., “Introduction to Particle Image Velocimetry,” Presented at the Wind Tunnel University workshop, NASA Langley Research Center, 2000. 7 Wemet, M. P., Paticle Image Velocimetry Acquisition Manual PIVACQ, NASA Glenn Research Center, Version 2.2, 2002. 8 Wemet, M. P., Particle Image Velocimetry Processing Manual PIVPROC, NASA Glenn Research Center, Version 6.03, 2002. 9 Dally, J. W., Riley, W. F., and McConnell, K. G., Instrumentation for Engineering Measurements, 2Ind ed., John Wiley & Sons, Inc., New York, 1984, pp. 538. 36 ‘1‘! . \ ‘2 5‘ 413.. '0 ‘1 ‘F .j I; 2““ II t to": .7. ,4 , I k ‘n 3 General Characterization of the Flow Field Prior to presenting the results from the experiments it is necessary to examine the characteristics of the general, flow field surrounding the axisymmetric, backward-facing step. First, the reattachment length is determined using the PIV data and compared to wall-shear-stress data acquired in the same model geometry. The average reattachment length is an important parameter in the flow field in that it gives the length of the separation bubble. The second form of useful information presented regarding the flow field is the state of the boundary layer at separation. Sandpaper was used at the upstream end of the model in order to trip the boundary layer with the aim of establishing a turbulent boundary layer at separation. A single hotwire was used approximately 1 mm upstream of the step edge to acquire the boundary-layer velocity profile. This profile is then compared to classic, fully-developed, boundary-layer profiles from literature in order to characterize the state of the boundary layer at separation. Finally, the mean-pressure distribution downstream of the step is presented and the Reynolds-number effect on this distribution is discussed. Note that hereafter, upper case letters represent time-averaged quantities (e.g., U indicates the mean, streamwise velocity) and lower case letters with a prime indicate fluctuating quantities (e.g., u' is the turbulent, streamwise velocity). 37 3.1 Reattachrnent Length Two methods were used to determine the reattachment distance for all four cases listed in Table 2.2. The results from these methods were compared with results from Li and Naguibl who were able to determine the reattachment distance within the same axisymmetric backward-facing step geometry using an oscillating hotwire that measured the shear stress along the surface. The first method used to determine the reattachment distance was forward-flow probability (F FP), which was also employed in past studies by Spazzini et al.2, Eaton3, Westphal and Johnston4, and Tihon et al.5 F FP is the fraction of the time that the flow is in the forward direction. In each of the 1500 vector fields, the direction of the streamwise velocity measured at the lowest position of y/h = 0.05 (y = 0.6 mm) is recorded and the FFP is determined as the number of images for which the streamwise velocity is positive divided by the total number of images. The results are presented as percentages along the ordinate as shown in Figure 3.1, with the normalized streamwise distance along the abscissa. Figure 3.1 presents the F F P for all four Reynolds. numbers. A 0% FF P means the flow at that particular x/h position is moving toward the step in all realizations of the 1500-sample dataset; whereas, 100% FFP indicates that the flow is moving in the downstream direction in all realizations for the same sample set. The reattachment distance is located at the streamwise position where the FFP = 50%. As a result of the Slight scatter in the data, a linear fit to the data near x,, as Shown in Figure 3.1, was employed to estimate the streamwise position at which the FFP equaled 50%. 38 Figure 3.1 Streamwise distribution of the forward flow probability for all four Reynolds numbers In Figure 3.1, there are two streamwise positions where the FFP = 50%. Depending on the Reynolds number, the first streamwise position is between x/h = l — 1.5 and the second position is roughly between x/h = 4 — 5. The first position gives the location of the secondary separation point close to the step, which separates the secondary and primary re-circulating regions. The second FFP = 50% position is the point of reattachment of the Shear layer. The four cases plotted in Figure 3.1 Show that increasing the Reynolds number results in a longer reattachment length and a shorter secondary separation point. Similar results were found for the same test model by Li and Naguib' from wall-shear measurements using an oscillating hotwire at y/h = 0.008. Spazzini et al.2 also report the same trend for the Reh range 3500 through 16000. 39 1.. “IL oh i.) Li and Naguibl attribute this Reynolds number trend to the variation in the boundary-layer thickness. This reasoning is supported by the earlier observations of Eaton and Johnston6 and Adams and Johnston7, both of whom found that xr increased as the boundary layer thickness decreased. Since in this study the Reynolds number increases by increasing the freestream velocity, the boundary-layer thickness should decrease as Re is increased. Referring to Table 2.2 and Figure 3.2, this is seen to be in fact the case for U,o of 10 m/s (Reh = 8081) and higher where the value of 0 is found to decrease with Reh. However, the lowest Reh case (U,o = 7.4 m/s) deviates from this trend where the momentum thickness is seen to be smaller than that at ReI| = 8081. This is attributed to possible "under stimulation" of the boundary layer at the lowest freestream speed, suggesting that the boundary layer may not be sufficiently developed to a turbulent state as in the other cases. Such a low Reynolds number effect on turbulent boundary layers established in the laboratory is typical (e. g., see Naguibg). It is important to note that it would be ideal to increase Reh without changing the ratio of the boundary-layer thickness to step height (i.e., keeping 0/h invariant). This would make the shear layer development solely dependent on the change in the freestream velocity and step height, while decoupling the connection between freestream velocity and momentum-thickness variation. 40 ‘52—.— I d Reh=5980 4.9% I = l3 ' l o R h 8081 l 4 8% l p Reh - 14547 [.1 . l l D Reh =18588 'l l: I D I- ...V __ ._ .... _ 4 I 4'7F l 4.6} I z 4 5' l \b . I 4.42 f | ’ l 4.31 O I l : I 4.2'I I, l I 4.1!- I I I 4% _ __ . l _ l_ _. ._ .. l 2, .. _.. .L A 'I __l _ __l 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 O/h Figure 3.2 Normalized reattachment distance versus normalized momentum thickness The second method used to estimate the mean reattachment distance required using the mean streamwise velocity component. The premise of the method is similar to that based on measurements of the mean, wall-shear stress. More specifically, the reattachment point may be determined accurately as the point of zero streamwise wall shear stress; i.e., the point at which the stress switches from a negative to positive value. Since this study had no access to wall-shear information, the analysis was conducted on the mean, streamwise velocity at the lowest y position (y = 0.6 mm). The results should be the same as that based on the wall-shear information if the y position is within the viscous sub-layer (i.e., within the region where variation of u with y is linear). If this is not true, general knowledge of the streamline pattern in this flow suggests that the results Will be biased towards the upstream direction, or shorter x,. This bias will increase with 41 the value of y at which the analysis is done. Based on this discussion, the direction of the mean, streamwise component of the velocity (U) located at the height closest to the wall was analyzed. Figure 3.3 and Figure 3.4 Show the distribution of U for the four 2D Reh cases. The U profile is qualitatively similar to the FF P profile shown in Figure 3.1. The U changes direction, or crosses zero (U = 0), two times. The smaller x location at which the profile crosses zero indicates the location of the secondary separation point. As seen with the FF P data, the distance between the secondary separation point and the step decreases with increasing Reynolds number as shown in Figure 3.4. When U drops below zero, this indicates the reverse flow region, or the re-circulating flow zone as shown in Figure 3.3. The larger x position at which U crosses zero is the location of the mean reattachment point. The mean reattachment distance increases with increasing Reynolds number as observed using the F FP method. 42 Figure 3.3 Streamwise distribution of the mean, streamwise velocity component at y/h = 0.05 for all four 2D Reynolds number I7 7 7 7 717 7 7 7 I7 7 7 7 7 7 7 7 7 7T 7‘ 0.12 I .7R€h=39807II l I D Reh=8081 I ‘ 0.08‘ I D Reh=14547 {I l ”I Reh=18588 I I 0.06;- - - L,, -»I 0.0.; I :8 ‘ :1 [5 I \ use . I3 , D 0.02I SHEERDDHBD I i 7 *339 dfi- i 0177 7 7 7 ‘3pkcué' 77 7 7777 I oozI E? U . 3‘ 004I 33‘:I . I ' I *DD I l *pc'i I L, ,7 7 . .l ,3, . .i 7 I 7 ,7 I AU M2. him—l o 0.5 1 1.5 2 2.5 x/h Figure 3.4 Secondary separation region from the streamwise distribution of the mean, streamwise velocity component at y/h = 0.05 for all four 2D Re 43 Figure 3.5 shows the reattachment distance as function of Reynolds number using both the FF P and U = 0 methods. For comparison, the results of Li and Naguibl are also included in the figure. All four profiles show the same trend: as the Reynolds number increases, xr increases. Comparing the two methods, the FF P estimates a longer reattachment distance than the U = 0 method, which was also seen in the Li and Naguibl study. This difference may be inherent to the methods used. The F F P method determines the streamwise position that delineates where 50% of the records show the direction of the flow is in the downstream direction. This will only coincide with the location of U = 0 if the probability density function (pdf) of u is not skewed. In addition, it is good to compare the xr values obtained here to those from other studies in the literature, which generally find xr to be equal to roughly 5-6h for a planar backward-facing step. Spazzini et al.2 reported x,/h = 5.39 in their planar BFS study at Reh = 5100, Tihon et al.5 found xr/h = 5.1 at ReyI = 4800, and Jovic and Driver9 measured xr/h = 6.0 for Reh = 6000. Eaton and Johnston6 reported in their review of planar backward-facing step studies that the xr values measured ranged from 4.9 to 8.2 step heights. The shorter xr values reported in this study differ from most BFS studies in the literature. However, these studies are either investigations using planar, backward— facing-step geometries or sudden-expansion pipes. Li and NaguibI measured the mean reattachment length in the same axisymmetric, backward-facing-step geometry employed here using an oscillating hotwire that measures the wall shear stress and found xr/h = 4.6 at Reh= 8700. They determined reattachment as the point where the wall shear stress equaled zero. Their reattachment length was also shorter than those reported in the 44 .b. i ..I . I], at u. ‘1‘ literature. Li and Naguibl attributed the shorter reattachment length to the difference in geometry, stating that even though a sudden expansion is axisymmetric, the geometry has a concave curvature and the boundary layer characteristics at separation may be different. Comparing the current study to that of Li and Naguib], the xr values measured by Li and Naguibl were at most 6% longer than those measured in the present study (Figure 3.5). This is expected because Li and Naguibl were able to measure three times closer to the wall (approximately 100nm above the wall). If data had been obtained closer to the wall in the present study, then the xr value would be longer, in qualitative agreement with the Li and Naguibl data. This is because of the bias towards the upstream in the values of xr measured at y locations that are close to the wall but above the viscous sub-layer, as discussed above. Nevertheless, it seems that the F F P method provides xr values closer to the results of Li and Naguib]. Therefore, xr values calculated using the FFP method were used throughout the present study as the best estimate of the mean reattachment length. The xr values calculated by Li and Naguibl were not used in this study because the Re range used in their study stretched only from 4000-13000. In the present study, the highest Re case is 18588. 45 Figure 3.5 Reynolds number dependence of x,, calculated using FFP and U = 0 and compared to Li and Naguibl data 46 3.2 Boundary Layer Velocity Profiles Boundary layer velocity profiles were acquired using hotwire anemometry in order to characterize the state of the boundary layer at separation. As mentioned in the experimental set-up, strips of sandpaper were placed near the nose of the model in order to trip the boundary layer and accelerate its development to a turbulent, boundary layer at separation. The state of the boundary layer was determined by comparing the mean streamwise velocity profile from the current study with that from Naguibs, who acquired hotwire measurements within a turbulent, boundary layer. In order to make the comparison, the measured boundary-layer profiles were normalized and displayed in the classic U+ = U/ut versus y+ = quv profile on a semi-log plot (ut = (tw/p)”2 where, 1?“, is the shear stress at the wall and p is the fluid density). Since u, was not measured in the experiment, it was estimated by comparing the measured profile with the law of the wall empirical equation provided by Spaldingl0 y = U+ + e”“{e"U‘ — 1 — kU“ +1/2(kU+)2 —1/6(kU+)3} (3.1) The ut value used to normalize was iterated until the mean, squared error between the first five values of the measured boundary layer in the y‘r versus U+ profile and the Spalding equation was minimized (within 2%). This percentage was determined by dividing the mean, squared error by the maximum y+ value in the first five points. The Spalding fit and measured boundary layer profile with the calculated ut value are shown in Figure 3.6. This technique was first introduced by Kendall and Koochesfahani”. Although Clauser-fit is a common way of estimating the shear stress at the wall, it requires being able to locate the beginning and end of the log region. By using the Spalding equation, the data is fit to the boundary layer profile in wall units and stretches 47 from the wall to the end of the log region as shown in Figure 3.6. This removes some of the uncertainties associated with determining the log region. Kendall and Koochesfahanill were able to use the technique to estimate the wall shear stress with an accuracy of within two percent. 9007777777;7::_i 7 — —I — —- — .— -— — —--_ — — —7I_ ___. flaws—LI I5 §7palding I 800 I 3391562 189 I I 700 7 I I 600~ I 500 , I +> I 400 — I I 300 ~ I I 200 ” I I 100 ~ I ..w I 0I777 .L_.___ -L __ _ I _ _ l ___ _ __ .__ l__ __ M F _I 14 1 5 16 1 7 1 8 1 9 20 21 22 U4- Figure 3.6 Comparison of the measured boundary layer profile with the Spalding equation Once the ut was determined to be approximately 0.46 m/s, the boundary layer from the present experiment could be compared with previously measured turbulent boundary layer profiles to verify if the present boundary layer was turbulent at separation. Figure 3.7 shows this comparison between the present measurements, R69 = 1237, and Naguib’s8 boundary layer measurements at R69 = 1577. In Figure 3.7, mean-velocity Profiles are scaled using wall variable and are plotted on a semi-log axis. In addition to Naguib’s8 data, the standard log-law in log-linear coordinates is presented, U+ = 48 2.44*ln(y+)+5.5 as given in Hinze”. It can be observed that the present data collapse with Naguib’s data and the log law over a range of 50 400. Traditionally, when comparing equilibrium boundary layers at different Reynolds numbers using wall scaling, the uRMs values in this region do not collapse but increase in value with increasing Reynolds number. To further examine the reasonableness of the estimated ut magnitude, Figure 3.9 shows a comparison between the friction coefficient calculated from the relationship Cf = 2(ur/Um)2 and the Ludweig and Tillmann (Hinze'z) empirical relation for Cf versus Reg (Reg = UwB/v, where 0 is the momentum thickness). The Ludweig and Tillmann relation, which is based on direct wall shear stress measurements, is given by Cf = 0.246 X 1070-678”l2 RCEO'268 (32) 49 where H12 = 57/0 is the shape factor and 6* is the displacement thickness. The relationship between parameter H12 and Reg is defined by H12 = 1.773 Ref-"276 (3.3) Comparing the present Cf value with the Ludweig and Tillmann relation, it can be seen that the present value falls just outside of the 5% range from the curve. This deviation is believed to be caused by “underdevelopment” of the outer-layer turbulence as discussed in the following paragraphs. 24 ’1 r—r- 17——' A77 #1— v u ", ,,,,...V 7—1- r—V‘YT‘TT — <‘v— ,._,,A..fi., +"r~} O 7 Naguib data Re97=17437 I __U Present data, Ree- - 1237 I I | I [8 fiwLWfi ' I I I i I 20; (1+ = 2.44y+ + 5.5 ..s N ;_.__.4L,_z_ur L. u m -L. 4. Ar; L 4 .1 0° 101 1o2 103 10‘ + Y Figure 3. 7 Boundary-layer mean velocity8 profiles from present experiment and Naguib’s8 data 6'». L. ..,1 lx_I..‘.l,. W. n". .5...I ,1 A7... L_,_i l . l;_A_.Ll 50 3L.-. ,.,.,a,fi.,,,,-z...,..., . Loafinwr _ .,.fif.-.q I o Naguib data, Ree=1437 II I E] Present data, Ree=1237 II 00 I,,,,,,., .,,,,,II I I 'I I I I I I 'I I I *I I I 0L0, ,. . -..I.L;1, ,_ . .....IIé ,. A.#._..3 I. .7. PHQI‘ 10 10 10 10 10 Figure 3.8 Boundary-layer turbulent-streamwise-velocity profiles from present experiment and Naguib8 data 5I"' I ,,,,i. . - 77——77LudMeg & fillmahn --- i 5% I I 53.5 .. 2I,,,.,I . . i, , ..,,, , . 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Re 0 Figure 3.9 Comparison of friction coefficient value from the present experiment with the Ludweig and Tillmann relation 51 IIIc :.. I ‘ uCICIO'I‘CQ. I ..I .. . ISL-INTI reach equ It's-mac .Li\5 \. 0'. Kilt {III ' I 9.. |.H‘T‘.’1. \ui Halli II I ‘II'J “I? uL “h ECI \f} The finding that the boundary layer state at separation has developed into a fully- developed, or equilibrium, state in the inner portion but not in the outer region is a reflection of the need for longer boundary layer development length for the outer layer to reach equilibrium. This is consistent with the fact that the time scales of the outer-layer structures is substantially higher than that of the inner-layer turbulence. In fact, the ratio of the outer to inner time scales is the same as the ratio of the outer to inner length scales, or 8+ (for this study, this ratio is approximately 1000). However, it is important to note that the recent study of Morris and Fossl3 has demonstrated that for shear layers resulting from the separation of a turbulent boundary layer, the dynamics of the flow is dictated by the inner layer with the outer layer playing a passive role. Therefore, whether the state of the turbulence in the outer portion of the flow is that of an equilibrium boundary layer or not should be of no particular significance to the current study. 52 77"“ K >&.J\\ 3M0 3 L13: ' II‘ .2. . .1“, «I, I.“ I .*‘ \‘ k.\ I: . II lit v ”a ~. M'I ' “Ikn H gr» 5.] P- ..,\ rm. hair 3.3 Mean-pressure distribution The mean-pressure, streamwise distribution, acquired at four Reynolds numbers (Reh = 5980 — 18588) downstream of the axisymmetric, backward-facing step using static pressure taps, are shown in Figure 3.10. The mean-pressure coefficient, Cp= (pS — p,)/(‘/2pU.o2), is plotted along the ordinate as a fimction of the streamwise coordinate, which is plotted on the abscissa as x/h (note that ps is the pressure measured along the surface of the model and pr is the reference pressure measured with a static pressure tap located at the exit of the wind-tunnel contraction). The mean-pressure distributions show a classical, backward-facing—step, pressure profile with mean reattachment located between xr = 4-5h (xr found from PIV analysis in Section 3.1). Immediately downstream of the step, the pressure distribution decreases until about 0.5xr where the pressure begins to recover, reaching a peak slightly downstream of x,. The mean-pressure distribution then drops slightly as the shear layer reattaches to form a boundary layer. Comparing the four Reh cases, a trend is easily observed in Figure 3.10. Immediately downstream of the step, the pressure drops between the step and 0.5x,. In this region, it can be seen that the Cp values increase with increasing Reynolds number. Furthermore, the mean-pressure distribution for the lower Reh values recovers over a shorter distance, consistent with the shorter mean reattachment length found earlier. Beyond reattachment, the Cp values are lower for lower Reh values. This trend is explained below. 53 -0.2 0° I , I I -0.3 - I I7-*— Reh- - 5980 I I I .9. Reh=8081 I I I - I I .9. Reh-14547 I I I I + Reh“ = 18588 I I -0.4I- _ _ _I_ ._ __ __I I1 __; L. T ,-_ :' :: _ .I 0 2 4 6 8 10 xlh Figure 3.10 Mean-pressure distribution for all Reynolds numbers, 2D case only, with a line marking the corresponding xr value for each case Previous planar, backward-facing-step studies, such as Westhal”, Roshko and Lau”, and Adams et al.“, have shown that static pressure data collapse to a universal curve when normalized using the following equation and plotted versus x/xr ‘ C p7Cpmin Cp =—”——1 C (3.4) pmm where C p... is the minimum CI) in the mean-pressure distribution. Figure 3.11 shows the present static pressure data for all four Re normalized as C9. and plotted along the ordinate. Streamwise distance with respect to reattachment is represented on the abscissa. The C; distribution for all four Re numbers collapsed through reattachment. 54 Adams et al. '6 found in their planar, backward-facing-step study that static pressure profiles for several Reynolds numbers, spanning 8500 to 41000, collapsed through reattachment. The boundary-layer thickness at separation for all cases was approximately 5/h = 1.1. The pressure-recovery curves for these thick boundary layers were independent of Reynolds number. Westphall4 and Roshko and Laul5 found the same conclusion for thin boundary layers. Beyond reattachment, the pressure profiles differ from each other with a larger Cp‘ associated with a lower Re. The Cp’ of Adams et al.16 exhibited a trend similar to that seen in Figure 3.11 and suggested that downstream pressure recovery is influenced by blockage within the wind tunnel. 0.14I 0.12 I- o.1 I a O. 0 one 0.06 — 0.04 » 0.02 ~ OI Figure 3.11 Mean-pressure coefficient distribution for the four Re cases 55 3.4 References l Li, Y. and Naguib, A.M., “High-Frequency Oscillating-Hotwire Sensor for near-Wall Diagnostics in Separated Flows,” AIAA Journal, Vol. 43, No. 3, 2005, pp. 520. 2 Spazzini, P.G., Iuso, G., Onorato, M., Zurlo, N., and Di Cicca, G.M., “Unsteady behavior of back-facing step flow,” Experiments in Fluids, Vol. 30, 2001, pp. 551-561. 3 Eaton, J. K., “Turbulent flow reattachment: an experimental study of the flow and structure behind a backward—facing step,” Ph.D. Dissertation, Stanford University, 1980. 4 Westphal, RV. and Johnston, J .P., “Effect of initial conditions on turbulent reattament downstream of a backward-facing step,” AIAA Journal, Vol. 22, 1984, pp. 1727-1732. 5 Tihon, J ., Legrand, J ., and Legentilhomme, P., “Near-wall investigation of backward-facing step flows,” Experiments in Fluids, Vol. 31, 2001, pp. 484-493. 6 Eaton, J. K. and Johnston, J .P., “A Review of Research on subsonic Turbulent Flow Reattachrnent,” AIAA Journal, Vol. 19, No. 9, 1981, pp. 1093-1099. 7 Adams, E.W. and Johnston, J.P., “Effects of the separating shear layer on the reattachment flow structure, Part 2: Reattachrnent length and wall shear stress,” Experiments in Fluids, Vol. 6, 1988, pp. 493-499. 8 Naguib, A.M., Wark, CE, and Juckenhofel, 0., “Stochastic estimation and flow sources associated with surface pressure events in a turbulent boundary layer,” Physics of Fluids, Vol. 13, No. 9, 2001, pp. 2611-2626. 9 Jovic, S. and Driver, D., “Reynolds number effect on the skin friction in separated flow behind a backward-facing step,” Experiments in Fluids, Vol. 18, 1995, pp. 464-467. '0 Spalding, D. B., “A Single Formula for the Law of the Wall,” Journal of Applied Mechanics, Vol. 28, 1961, pp. 455-457. 1' Kendall, A. and Koochesfahani, M. M., “A new method of estimating wall shear stress is turbulent boundary layers,” 57’” Annual Meeting of the American Physical Society Division of Fluid Dynamics, Seattle, Washington, November 21-23, 2004. '2 Hinze, J. 0., Turbulence: An Introduction to Its Mechanism and Theory, McGraw- Hill Book Company, Inc., New York, 1959, pp. 486. 13 Morris S. C., and Foss, J. F., “Turbulent boundary layer to single-stream shear layer: the transition region,” Journal of Fluid Mechanics, Vol. 494, 2003, pp. 187-221. 56 '4 Westphal, R.V., “Experimental study of flow reattachment in a single-sided sudden expansion,” Ph.D. thesis, Stanford University, 1983. '5 Roshko, A. and Lau, J. C., “ Some observations of transition and reattachment of a free shear layer in incompressible flow,” Proceddings of 1965 Heat Transfer and Fluid Mechanics Institute, Stanford University Press, pp. 157-167, 1965 '6 Adams, E. W., Johnston, J. P., and Eaton, J. K., “ Experiments on the structure of turbulent reattaching flow,” Report MD-43, Department of Mechanical Engineering, Stanford University. 57 4 Wall-pressure Measurements 4.1 Introduction The array of microphones embedded in the surface downstream of the axisymmetric, backward-facing step was utilized to investigate the space-time character of the wall-pressure field beneath the shear layer and the reattached, boundary layer. Although separated flows have been studied extensively over the years, especially in the classic geometries such as the backward-facing step, splitter plate, and splitter-plate-with— fence, most studies have been planar geometries. In addition, only a few of these studies utilized an array of microphones embedded in the surface. Several studies over the years have investigated the flow field downstream of a planar backward-facing step. Eaton and Johnston], in 1981, reviewed the research done on the classic backward-facing step geometry in effort to summarize the data available at the time and to suggest areas worth investigating. Most of the studies up until that time had been conducted using hotwire, pulsed-wire, and laser anemometry. Since then, there have been a good number of studies in this classic geometry that have used more current techniques such as surface pressure sensors and particle image velocimetry (PIV). The ones containing mostly surface-pressure data are summarized below. The studies that utilized PIV will be highlighted in the next chapter (Section 5.1), which provides analysis of the velocity field. Farabee and Casarellaz, in 1986, studied fluctuating wall pressures in a forward- facing and backward-facing step using a flush-mounted B&K l/8in. condenser microphone to measure the surface pressure fluctuations. They characterized the backward-facing step as having an “attached shear layer” that separates from the step at 58 the edge and reattaches on the wall a short distance later. From the root-mean-squared (RMS) turbulent pressure values, they observed the surface-pressure fluctuations to rise rapidly with increasing streamwise distance as the shear layer structures grew in strength and moved downstream. These fluctuations reached a maximum as the flow impinged on the surface of the plate at reattachment. The measurements of Farabee and Casarclla2 were compared to those from the surface beneath an equilibrium flat-plate, boundary- layer flow. The measured RMS values at reattachment were 5 times larger than those measured in the equilibrium flow farther downstream. Using frequency domain analysis, Farabee and Casarella2 described the characteristics of the wall-pressure field as variable with streamwise distance along the wall. Close to separation, the spectra showed the highest level of energy at very low frequencies; whereas, farther downstream the spectrum containing the largest energy was found at reattachment. This was a manifestation of the increase in the energy of the organized, turbulent structures as they convected downstream. A corresponding shift was seen in the spectrum as the dominance of the low-frequency disturbances gave way to the dominance of the high-frequency structures downstream. F arabee and Casarella2 also stated that the energized structures diffused and decayed beyond reattachment; however, these structures were still identifiable 72 step heights downstream of the step. In 1987, Driver et al.3 identified two types of fluctuating motion in their backward-facing-step study of the time-dependent character of the separated, shear layer at a Reynolds number of 37,000, based on the step height. The first type was the flapping of the shear layer that has been identified by many researchers, and the second was a quasi-periodic vortical type motion, which, they stated, needed further investigation. 59 Driver et al.3 noticed abnormal contraction and elongation of the separation bubble as a result of the shortening and lengthening of the reattachment length. This was labeled as the flapping motion of the shear layer with amplitude estimated to be 20% of the shear layer width, or vorticity thickness. They also found the direction of the flow to change intermittently over the range of i1 step height from the reattachment point. In addition, Driver et al. identified a low frequency disturbance in the flow from surface-pressure and velocity measurements. They stated these low-frequency disturbances were random and contributed very little energy to the overall pressure fluctuations, but were believed to be associated with the shear layer flapping. Heenan and Morrison4 investigated wall-pressure fluctuations behind a rearward- facing step and passive control of these fluctuations using a permeable surface (at Reynolds number equal to 1.9x105 based on the step height) in 1998. A particularly interesting feature of Heenan and Morrison’s study is that they were able to remove the low-frequency surface-pressure signature associated with shear-layer “flapping” by using the permeable surface placed downstream of the step. They attributed this success to inhibiting the re-circulating flow responsible for the upstream convection of disturbances formed at reattachment. Although the characteristics of the measured, wall pressure in the non-permeable case were consistent with that summarized earlier in this section, one difference existed. Specifically, Heenan and Morrison4 found an upstream convection velocity close to separation using phase-angle analysis. They identified negative phase angles (with respect to a microphone signal measured immediately behind the step) at low frequencies and at locations from separation up to 0.4xr in the impermeable case. 60 In 2001, Lee and Sung5 used a 32-microphone array downstream of a planar backward-facing step to measure wall-pressure fluctuations in the streamwise and spanwise directions. Spatio-temporal statistics were extracted from this comprehensive data set for a Reynolds number of 33,000, based on the step height. Lee and Sung5 observed the RMS pressure fluctuations rose sharply starting around 0.5x, and peaked in the vicinity of reattachment, decaying beyond that point. In addition, pressure spectra revealed low-frequency dominance close to separation, presumably caused by the flapping of the shear layer. Farther downstream, the spectra were dominated by high- frequency components. In terms of the convection velocity, Lee and Sung5 calculated a downstream convection velocity of 0.6U,o at high frequencies, and they did not find evidence of an upstream convection velocity. Although from their phase plot used to determine the convection velocity, there were many singularities (phase discontinuities) at low frequencies. This was not the case at the higher frequencies. Also from 2001, Spazzini et al.6 concentrated their investigation on the low- frequency motion identified in their backward-facing step. In their experiment, using Forward-Flow Probability (see Section 3.1 for definition) from skin fi‘iction measurements, they studied two important areas beneath the shear layer: (1) the mean, reattachment point and (2) the secondary, separation point. Similar to wall-pressure fluctuations, the maximum of the streamwise skin—friction RMS distribution was found slightly upstream of the reattachment point. Employing Fourier analysis, 75% of the total energy in the wall-shear spectra was found to be contained at low frequencies lower than fxrlU,o = 0.4 at locations in the vicinity of the step up to 0.3 of the reattachment distance. 61 Farther downstream, x/xr = 0.7, this same frequency range contained only 30% of the total energy. At this position, the spectra were dominated by higher frequencies. Using a wavelet transform for more detailed analysis of the low frequency motion, Spazzini et al.6 suggested that at a comparable frequency to the flapping frequency there was intense activity in the secondary, re-circulation bubble. This hypothesis was examined further using PIV measurements in a water-tunnel experiment. Based on visual observation, it was proposed that the cyclic behavior of the secondary, re-circulation bubble had a frequency corresponding to the low frequency revealed by the wavelet analysis. In 2001 and 2003, Hudy7 and Hudy et al.8, respectively, compiled a database of wall-pressure-array measurements for studying the space-time characteristics of the surface-pressure field within a separating/reattaching flow region. The experimental geometry was a splitter-plate-with-fence that was instrumented with an array of flush- mounted microphones. Data were acquired for a Reynolds number of 7900, based on the height of the fence above the splitter plate. Two distinct regions, defined based on their location relative to the position of the mean reattachment point (x,) of the shear layer, were identified by the authors. The first region was located upstream from the fence to O.25x,. In this region, the surface-pressure signature was dominated by large-time-scale disturbances and an upstream convection velocity of 0.21Uoo. Beyond 0.25x,, turbulent structures with smaller time scales and a downstream convection velocity of 0.57Uoo generated most of the pressure fluctuations. In 2004, Daoud9 investigated the reattached boundary layer downstream of the separation region in the splitter-plate-with-fence geometry used by Hudy7 and Hudy et 62 al.8 Daoud9 used 16 microphones embedded in the surface, starting at x/h = 37.5 and extending to x/h = 75. The reattachment point, based on step height, was measured to be xr = 22.5h and the two Reynolds-number cases investigated were Reh = 7600 and 15700. Using a frequency-wavenumber analysis, Daoud9 found only high frequency structures within the reattached boundary layer. In addition, Daoud9 stated that these structures had a convection velocity that was independent of frequency, meaning the structures were non-dispersive. That is, structures of different scales travel at the same speed. Daoud9 determined this speed to be 0.81Uoo. The objectives for this section are to investigate only the surface-pressure signature downstream of the axisymmetric backward-facing step. Specifically, this chapter investigates the wall-pressure signature in a 2D separating boundary-layer flow field in order to understand the wall-pressure physics surrounding each case using spatio- temporally-resolved measurements of the wall-pressure. In addition, Reynolds-number effects in the 2D case are explored. 63 4.2 Results and Discussion The wall-pressure data presented below were sampled simultaneously with PIV data; thus, the mirror mounted on the tail of the model was installed when these pressure measurements were made. The effect of the mirror was checked as explained in Appendix 8.4. The sampling frequency of the pressure was 12 kHz for 35 seconds. This resulted in a total of 420,000 points acquired for each flow condition. 4.2.1 Fluctuating-pressure distributions The microphone array embedded in the wall of the model downstream of the step was used to measure the spatial distribution of the pressure fluctuations produced by the passage of various flow structures over the reattachment surface. These fluctuations represent the instantaneous deviation from the mean pressure measured by the static- pressure taps. The root-mean-square (RMS) of the pressure fluctuations (p'RMS) determines a measure of the strength of the pressure deviations from the mean. Additionally, the square of the RMS pressure fluctuations presents the ‘energy’ of the unsteady pressure time series, thus the larger the RMS, the stronger the pressure fluctuation. The present RMS measurements (plotted as C, = p'RMS/Vszmz) are shown in Figure 4.1 for all four Reh cases. The Q, is plotted along the ordinate as a function of the streamwise distance from the step, x/h, on the abscissa. A three-point average was used to smooth the RMS results shown in Figure 4.1 to reduce some data scatter associated 64 III. 1?) Lu . ,, .0). '9} A\AU P3: -Hn with uncertainties in microphone sensitivity calibration, which is explained in more detail in Appendix 8.4. The shape of the RMS distributions is consistent with that documented in the literature for a backward-facing step. At the point of separation, the shear layer is relatively far away from the wall-pressure sensors. Thus, at the first microphone location (x/h = 0.43), a low RMS pressure fluctuation is detected followed by a rapid increase of RMS pressure values with increasing downstream distance from the step up to the point of reattachment, approximately. Within this region of the flow from the step to reattachment, it is believed that the surface-pressure fluctuations are predominately associated with shear-layer vortical structures. As these structures convect downstream, growing in size and strength and moving closer to the wall, they» produce an increasingly strong, wall-pressure signature. This signature reaches a maximum level in the vicinity of where the flow impinges, or reattaches, on the wall as described by Farabee and Casarellaz. Beyond reattachment, the RMS values slowly decrease as the energized structures from the shear layer decay and diffuse downstream in the reattached boundary layer. Similar effects were recorded by Farabee and Casarella2 in their planar backward- facing step study. The peak in the present data occurs in the vicinity of reattachment. Heenan and Morrison4 found their peak RMS value at approximately one step height upstream of reattachment, which was determined by using chalk/paraffin flow visualizations as well as sublayer-fence and split-film probes. The magnitude of the RMS peak spans va = 0.025003 based on Reh, which is comparable to the peak RMS found by Farabee and Casarella2 and Driver et al.3 However, these values are about half the value recorded by 65 I: u (I) Heenan and Morrison4 in their planar BFS study. One other point worth noting is that it can be observed in Figure 4.1 that the RMS peak value decreases and shifts towards a larger streamwise position as the Reynolds number increases. This shift of the location of the peak to larger x values is consistent with the increase in reattachment length, and hence the development length, of the shear layer with Reynolds number. On the other hand, the cause of the decrease in the peak-pressure fluctuation is not known presently. Spazzini et al. found a similar trend in their fluctuating wall-shear-stress measurements over a Re range of 3500-16000, based on step height. The researchers plotted the normalized wall-shear-stress fluctuations as a fimction of x/xr and observed that the peak skin-fiiction fluctuation value occurred upstream of x,. No explanation was given regarding the decrease in skin-friction fluctuation with increasing Re. Figure 4.2 shows va, on the ordinate, as a function of x/x,, given on the abscissa, for all four Re investigated. The fluctuating-pressure distribution shows a collapse of the profile over the range stretching from separation to x/x,= 0.5. After this streamwise position, the profiles split and the trend becomes evident as seen by Spazzini et al. The trend continues downstream of xr until about x/xr = 1.75 when the three highest Re investigated collapse again. The RMS for the lowest Re remains higher than the rest of the flow cases for the extent of the streamwise distance measured. 66 I II I I . R§I=ssso II 0.03II ’ ' I 7 7 '[ .IIJ'I .I., r 7 I O Reh = 8081 Id I I‘ I .i I I, Reh=14547 II °L> . " I Re =18588 “ 0.025I .°”£*I*I~§§§§§ I. 377177 7 7 II I >*I I II I I I I O I I I 0.02I7HA-rtwa I 77777777 I .. I °2 : I I in; , I 00015. - .6? -.. «I, $53333 I I I I I j 36 I I 2* I I I I I 0.01M I- 4 A- 777 I I I I 33% I II I I 000st ,, I I I I I II I ‘ I OI 7 77 7 7 77 ,I 7 7 7 I, 7 ,7 I7 7 7IL7 7 7I o 2 4 6 8 1o 12 14 xlh Figure 4.1 RMS pressure distribution for all Reynolds numbers with a line marking the corresponding xr value for each case 0'035I’ 7,77 7 7 1* 7 7 I7 7 7»: ; 1’41? I ‘ : ‘ I Reh=5980 II °-°3I ....... III-...,“ .I o Reh=8081 II I I ,,,,, (b o ' I Re =18588 II 0 ' °-°2I” ft 0-. , I h. I ? "u. I o s I H. 0.015I~ 7c}, sow 'o'I . . up», I I «a I I 0%ng . ~ ; iiiii I I ,‘ I I 0.005Is - ..... I I I I I I77 4. 7 7 7; 7 :7 ‘ ,7; 77 o 0.5 1 1.5 2 2.5 3 x/xr Figure 4.2 RMS pressure distributions for all Re as a function of x/x, 67 4.2.2 Auto-correlations The auto-correlation is useful in identifying the dominant time scales at a particular position in a flow field from the time series measured by a single sensor, such as a microphone. In the case of the current study, the auto-correlation could be calculated at the location of each of the microphones in the array. In addition, the spatial distribution of the auto-correlation starting just downstream of the step edge and extending approximately 12 step heights downstream can be displayed using a flooded- color, contour plot. Figure 4.3 shows the flooded-color, contour plots of the auto- correlation for two Reynolds numbers: Reh = 8081 and 18588. Along the abscissa, the streamwise distance is normalized by the step height and along the ordinate, the time shift is normalized by the free stream velocity and the step height. The color bar indicates the magnitude of the auto-correlation, which is normalized by the RMS of the microphone signal as follows: r.. R . .= W (4.1) pp . 2 pRMS where, rpIpv is the discrete-time auto-correlation function for the signal from each of the microphones and p'RMs is the RMS value for that particular signal. The discrete-time auto-correlation is defined as l N -l r,,I,,I(n,x/h)=ng'(kpX/h)p'(k, —n,x/h) (4,2, where k. represents the time index, n represents a time shift, or delay, x/h is the streamwise position, and N is the total number of points measured in the time record p'(n). 68 The auto-correlation for the Re,1 = 8081 case is shown in Figure 4.3a, and four distinct regions can be seen by following the zero-contour line. The first region extends from x/h = 0, immediately downstream of the step edge, to x/h = 2. In this region, the auto-correlation function extent is wide, indicating that the pressure signature is dominated by long, time scales or low-frequency, flow disturbances. Other backward- facing-step studies, such as Eaton and Johnston', Heenan and Morrison“, Driver et 01.3, Farabee and Casarellaz, and Lee and Sungs, have observed similar time scales near the step. Some of these studies have attributed this low-frequency dominance to the flapping of the shear layer as explained in Heenan and Morrison4 as well as in Eaton and Johnston]. F lapping refers to the unsteadiness of the shear layer trajectory, which results in variations of the reattachment point location. Equivalently, a shortening and lengthening of the separation bubble results from the flapping of the shear layer. The second distinct region lies between x/h = 2 - 4. At about x/h = 2, the auto- correlation extent narrows, spanning TUw/h = -4 to 4. The narrowing of the auto- correlation is consistent with what was observed by Hudy et al.8 in their splitter-plate- with-fence study. In that particular study, Hudy et al.8 found the time-scales to narrow over the x/xr = 025-05 region, which translates to x/h = 1.12h-2.24h in the present study. The time-scales for which significant rpIpv values exist in the present study, narrow abruptly within a shorter region than in the splitter-plate-with—fence study. As explained by Hudy et 01.8, the narrowing of the auto-correlation extent is linked to the dominant, flow structures within the shear layer. It is within this region that the flow structures are growing in strength as well as moving closer to the wall-pressure array. Thus, these 69 higher-frequency, flow structures begin to dominate the pressure signature as discussed in other backward-facing-step studies such as Heenan and Morrison“. The auto-correlation narrows once again, starting approximately around x/h = 4, to form the third distinct region. The region between x/h = 4 to 8 roughly spans from the start to the end location of a change in the time scales of the flow structures dominating Rp-p'. Within this region, the time-scales narrow as the flow structures within the shear layer impinge on the wall at xr and the shear layer reattaches to form a boundary layer. The fourth region starts around x/h = 8 and continues downstream until the end of the microphone array. The auto-correlation extent within this region is the shortest compared to the other three regions. Thus, this region is dominated by the highest-frequency, flow structures. It is hypothesized that the narrowing of Rp-pv over the x/h = 4 to 8 range is caused by the dominant, flow structures either convecting at a faster speed and/or changing shape in terms of getting smaller in size. In Section 4.2.4, the cross-correlation is presented, which shows an increase in the convection speed from upstream of reattachment to downstream of reattachment. Beyond x/h = 8, the contour lines remain approximately parallel to the constant 1 lines showing very little change in RpIpv with additional increase in x. The spatial distribution of the auto-correlation for Re,I = 18588 in Figure 4.3b is similar to that seen in Figure 4.3a; although, there are slight differences. Four distinct regions can still be seen, but each region is not as well defined as in Figure 4.3a. First, the region with low-frequency dominance extends from immediately behind the step to about x/h = 2.5. Beyond x/h = 2.5, the auto-correlation starts to narrow over a long streamwise distance, stretching from x/h = 2.5 to 8. Although the overall trend is the 70 same in terms of the region closest to the step being dominated by low frequency and the regions farther downstream being dominated by higher-frequency, the imprint of the pressure signature paints a slightly different picture in terms of where the dominant time scales transition and how abrupt the transition is over the streamwise distance. In order to investigate the differences in the auto-correlations between the two Reynolds numbers in more detail, the individual auto-correlations at 12 different streamwise positions were plotted for both Reynolds numbers in the same graph as shown in Figure 4.4 and Figure 4.5. Figure 4.4 gives six streamwise positions upstream of x,, which include the following x/h = 0.822, 1.6, 2.38, 3.16, 3.96, and 4.72. The abscissa represents the time shifi normalized by freestream velocity and the step height. The ordinate shows the magnitude of the auto-correlation function as given by Equation (4.1). The same axes are used in Figure 4.5 for six streamwise positions located downstream of reattachment. The positions are x/h = 5.5, 7.07, 8.63, i0.2, 11.7, and 12.5. The first position explored in Figure 4.4, x/h = 0.822, shows that the magnitude of the auto-correlation for both cases collapses between vapI = 0.7 — 1.0. In this region, the time scale is relatively short. At vapv < 0.7, the auto-correlation function for the Reh = 18588 case becomes wider than the My for the Reh = 8081 case. In addition, the smaller Reynolds number crosses the RpIp- = 0 axis at tUoo/h = i 10; whereas, in the Reh = 18588 case the zero axis is crossed at tUoo/h = i 20, approximately. This shows that, near the step, the higher Re case experiences a longer-time-scale, pressure signature than the lower Re case, and/or that the fluctuations associated with long time scales are relatively more energetic at the higher Reynolds number. The same is true for the other streamwise 71 positions shown in Figure 4.4 and Figure 4.5, where it seems as though the Rp'pv’s for the two cases collapse for the shorter time scales. Meanwhile, the Re;I = 18588 case depicts a stronger influence of the longer time scales compared to the other case. The difference between the two cases at large time delays is evident upstream and downstream of the reattachment position. The two Rp'p"s converge in magnitude as xr is approached and then slowly diverge beyond reattachment. Near xr (x,/h = 4.72) the two cases show nearly prefect collapse. 72 2 4 6 8 10 12 xlh Figure 4.3 Flooded-color, contour maps of the wall-pressure auto-correlation at Rel. = a) 8081 and b) 18588 73 x/h=0.822 x/h=1.6 1:le h Figure 4.4 Auto-correlation function for six streamwise positions upstream of x, — red line is for Re.I = 8081 and green line is for Re.| = 18588 74 0 0 xlh=11.7 x/h=12.5 1U lh 1:U lh a) 00 Figure 4.5 Auto-correlation function for six streamwise positions downstream of x, - red line is for Re.I = 8081 and green line is for Re.I = 18588 75 4.2.3 Frequency spectra Figure 4.6 shows the evolution of the wall-pressure frequency spectra with downstream distance. This figure contains the power spectrum plots on a logarithmic scale for both axes. The spectra are normalized by (1/2pU002)2 and plotted along the ordinate relative to an arbitrary reference value. The abscissa represents the frequency normalized by the fence height and freestream velocity. The use of an arbitrary reference value for the ordinate provides a means by which many spectrum plots can be displayed on the same plot without clutter. Twelve spectra from two flow cases (Reh = 8081 and 18588) are shown in Figure 4.6. The streamwise locations of the 12 selected microphones are the same as used in the individual auto-correlation plots in Figure 4.4 and Figure 4.5. The spectra in Figure 4.6 show the distribution of pressure-fluctuation energy with frequency and how this distribution varies with streamwise position. The spectrum at the first streamwise position (x/h = 0.82) shows a peak at the low-frequency end of the spectrum. The peak for Reh = 8081 case is found approximately at fh/U.,D = 0.014 10.002 and for Reh = 18588 the peak is located around fh/U.,0 = 0.011 i0.0008 (the error band is based on the spectrum resolution). In comparison with planar backward-facing-step studies, Lee and Sung5 found a similar peak close to separation at a frequency value of fh/U.o = .015. This is comparable to values given by Heenan and Morrison4 at fli/U.o = .017 and Spazzini et (11.6 at fli/Uao = .012. The latter authors used skin-friction spectra to determine the peak in comparison to the other studies that used wall-pressure spectra. Many studies, including the three mentioned here, have associated the low-frequency peak with the flapping of the shear layer. Farabee and Casarella2 suggested that the 76 energy distribution in the spectra indicates that the wall-pressure fluctuations close to separation were caused by the unsteadiness of the low-speed, re-circulating flow, rather than the highly turbulent structures in the shear layer. This is consistent with the fact that these structures are only beginning to develop in this region and are most likely weak compared to the strength of the low frequency disturbance produced by the shear layer flapping. Farther downstream, the first noticeable shift in the peak to higher frequencies occurs at streamwise location x/h = 2.38 or x/xr = 0.53 for Reh = 8081 and x/h = 2.77 or x/xr = 0.56 for Reh = 18588. At these locations, the peak in the frequency spectra is fh/U.o = 0.039 _+. 0.002 for the Reh = 8081 case and fli/U.o = 0.036 i 0.0008 for the Reh = 18588 case. This is approximately the same location where there is a fairly abrupt narrowing of the auto-correlation in Figure 4.3. Lee and Sung5 stated that the power spectrum in their planar BF S study reached a maximum at fli/U.,o = 0.068. This peak was found in all the spectra stretching from x = 2-10h with xr = 7.4h. Both Spazzini et al.6 and Heenan and Morrison4 found their maximum at fh/U00 = 0.15 and fli/U.,o = 0.17, respectively. Spazzini et al. identifies the shifi in the peak at x/xr = 0.5, where the FFP value reduces to zero. Heenan and Mom‘son find their high-frequency peak near x/xr = 0.75, which they say correlates with the maximum pressure recovery in the separation bubble based on Cp measurements. Finally, Driver et al.3 recorded a peak frequency value close to fli/U0° = 0.098 at both x/h = 5.5 and x/h = 8. The reattachment length in their study was xr = 6.1. This higher-frequency peak has been attributed to the structures within the shear layer, as discussed previously. 77 One point worth mentioning is the range of the high versus low frequency peak values reported in this study and in the literature. The values listed for the high frequency peak near reattachment vary over a range of 0.036 to 0.17, depending on the study. All of these studies, except the present study, were on planar backward-facing steps. Yet even in the planar BFS the range is relatively large compared to the differences seen in the low-frequency peak values. Both peak values are shown in Table 4.1 along with the authors and Reynolds numbers for each study. The average low- frequency value reported is fl1/Uco = 0.0138 1: 0.0024; whereas, fit/Ugo = 0.093 i 0.057 is the average for the high frequency peak including the present study. Without the present study, the average is fh/U,o = 0.12 i 0.05 for the high-frequency peak. This could indicate that the low frequency peak is Reynolds number independent since all these studies were done at different Re and in different overall test conditions (boundary layer state, test-section blockage, etc.). The high-frequency peak seems to be more sensitive to the Reynolds number and/or changing test conditions. Table 4.1 Frequency values of spectrum peaks and Re for five different studies Low High Study Re.I frequency frequency peak peak Present 8081 0.014 0.039 Present 18588 0.011 0.036 Lee and Sung5 33000 0.015 0.068 Heena.“ arid 190000 0.017 0.17 Momson Spazzini et 111.6 16000 0.012 0.15 Driver et al.3 37000 0.098 78 The comparison studies reported here only discuss two peak frequencies: 1) near separation and 2) upstream of x,. However, in this study, it was noticed that the high- frequency peak referred to above shifts to an even higher frequency, downstream of x,, as seen in the narrowing of the auto-correlation in Figure 4.3. Figure 4.7 highlights the spectra from three streamwise positions for both Reynolds numbers in order to illustrate the frequency shifi more clearly. It can be seen in the figure that the low-frequency peak at x/h = 0.82 shifts to a higher frequency at x/h = 2.38 for Re.I = 8081 and x/h = 2.77 for Reh = 18588 which then shifts to an even higher frequency at x/h = 12.14. At that most downstream position, the peak for Reh = 8081 is fli/Uw = 0.106 i 0.002 and for Reh = 18588 is fli/U,o = 0.085 t 0.0008. Farabee and Casarellaz, who obtained measurements in a backward-facing step downstream of x,, also saw a slight shift in the peak frequency to fli/Uco = 0.14 starting around x/h = 10 and stretching to at least x/h = 16. Reattachment in their flow was xr = 6h. The observations regarding the frequency spectra are consistent with the findings discussed previously in regards to the auto-correlation function shown in Figure 4.3. In particular, the transition in the time scale of the wall-pressure- generating motion from large to small with streamwise distance downstream of the step up to xr is associated with the transition in the peak of the spectrum from a low to a high frequency. Beyond x,, another narrowing of the time scales was observed in the auto- correlation function that mirrors the second shift in the spectrum peak frequency to an even higher frequency downstream of x,. Another point worth mentioning deals with the low-frequency end of the spectra (fh/Uco < 0.02) and the energy associated with this range at different Reynolds numbers. In the Reh = 8081 case, the low-frequency peak occurs immediately downstream of the 79 step and is attributed to the flapping of the shear layer as previously explained. Although the same is true for the Reh = 18588 case, there is a noticeable difference farther downstream where the energy level at the low-frequency end of the spectra relative to that at higher frequencies does not decay as much with increasing x. Moreover, downstream of x,, this relative importance of the low-frequency, pressure fluctuations increases again. This is consistent with the auto-correlation results in Figure 4.3, where the auto-correlation function downstream of xr is in fact wider at the higher Reynolds number than the lower one, reflecting the stronger relative dominance of the low- frequency, pressure fluctuations at the higher-Reynolds number. The origin of this increase in the low-frequency energy content at the higher-Reynolds number is not known at this time. In addition, the increase of the low-frequency content at the x/h = 4.72 position in Figure 4.6a is observed but cannot be explained at this time. 80 1a) l - : ..... _. 7 1 :11 l _ 1 1T 1 1” 1 1 1- 1-1 1111 1 11 1 1 -.1 -4 m . _ I r1 11M1 -1 11 111111111 11 1 1 11 .1 hawwww-.flm1.m mummwmw .1 7H. H unnumflw 1 1 11111 _ 111111111111111111111111111111111 1 1 1 a 11111 1111111111111 .11 I11 1111111111111111 L 7 . z A w11u_1111111meuw11ur1nm-H111HHH..1.H1.1_.11111,11.1W TH 1H 1H1M1H11H 111.H,1.1,H111,1.HHHH-1H 1111.11mm r1111! 01111 1| [.1 1|.l1f1l 1l1l1l1l i1l1l1i11 1L 0 0 :w 4 0 0 1 .11 0 1 U / fh 0 1 w _ w J L i . J . fh/U..o Figure 4.6 Frequency spectra for: a) Rel. = 8081 and b) Rel. 18588. Eleven spectra ions t use posi treamw' mg s t the follow rigins, a are plotted relative to different 0 starting from the top xlh = 1) 0.822, 2) 1.60, 3) 2.38, 4) 3.16, 5) 3.94, 6) 4.72, 7) 5.50, 8) 7.07, 9) 8.63, 10) 10. 2, 11) 11.7, and 12) 12.5 81 111211;] 141,, _10 71 1' ' ”‘1 1 1 .1" ‘ 1o __ 11111 11111 11. 11 41 .1111,:‘ 1_1,1>.11 1. 711, . 1 .11. -3 -2 -1 0 1 10 10 fh/Um 10 10 10 Figure 4.7 Spectra from a) Re). = 8081 and b) Re.I = 18588 cases plotted at three streamwise positions 82 4.2.4 Cross-correlation with respect to the pressure signal at reattachment The cross-correlation is obtained from two discrete-time signals to determine the similarity between them as a function of different time shifts between the signals. If the two signals are identical (i.e., p'l = p'z), then the calculation yields the auto-correlation function, obtained earlier in Section 4.2.2. The cross-correlation values were normalized using the RMS value of p'l and p'z. rP'I Piz = 1 1 4.3 (P l,RMS )(P 211145) ( ) P't 10.2 where rpqpq is the cross-correlation (see Equation (4.3)) and p'1,RMS and #23115 are the RMS pressure fluctuations for the two different discrete-time signals. Figure 4.8 yields the cross-correlation between the microphone nearest to reattachment and all 32 microphones in the array for two different flow cases: Reh = 8081 and 18588. The streamwise coordinate is normalized by the step height and is presented along the abscissa. The time shift (I) is normalized by the step height and the freestream velocity and is shown along the ordinate. The color bar represents the magnitude of the normalized cross-correlation, which was averaged over 102 records. 83 O-‘NWAU'I 2 4 x/h Figure 4.8 Flooded-color, contour maps of the cross-correlation and implied convection velocities at Re.l = a) 8081 and b) 18588 84 In each case shown in Figure 4.8, there is a main, positive-peak lobe inclined at an angle and two negative-peak lobes on either side of the main lobe. At each x location, the main peak is centered around time shift values corresponding to the largest positive correlation between the wall-pressure signal at this x location and that at reattachment. The negative peaks give the time delay to the highest negative correlation. By finding the slope of the peak locus of the main (or negative) lobe, an average downstream convection velocity can be calculated for the flow structure dominating the generation of surface- pressure fluctuations regardless of their time-scales. In Figure 4.8, the coordinates of the positive-lobe peaks were extracted and a linear fit was applied to determine the slope of the peak-locus line. The convection velocity is calculated from the slope of the fit using m __ S 1 U0 " U (4.4) (I) where 1nS is the slope of the fit line seen in Figure 4.8, U1,o is the freestream velocity, and Uc is the convection velocity. The largest mean-squared error surrounding the linear fit lines was determined to be 10% of the smallest time step (onolh = 0.069) within the fit range for both cases. For both Reynolds numbers, the main-lobe, peak locus consists of two parts that are inclined at two distinct angles depending on whether x is upstream or downstream of reattachment. Thus, two convection velocities were calculated per flow case. These convection velocities are displayed in the flooded contour plots in Figure 4.8. There is approximately 30% difference between the convection velocity calculated upstream of reattachment and that obtained downstream of reattachment. This indicates that the dominant, wall-pressure-generating structures within the shear layer are convecting at a 85 slower speed before they impinge and interact with the wall. Once the shear layer reattaches, these flow structures advect at a faster velocity, which partially explains the reduction in the dominant time scale downstream of reattachment observed in the auto- correlation contour plots in Figure 4.3. Compared to previous planar backward-facing step studies such as Heenan and Morrison4 and Lee and Sungs, which reported convection velocities spanning the range 0.5Uco to 0.6 U10, the Uc values calculated here are rather low. This difference in convection velocity may be caused by the difference in geometry between a planar configuration and an axisymmetric geometry. Li and Naguib10 investigated the wall-shear-stress signature in the same axisymmetric backward-facing step geometry and reported “local” convection velocities upstream of reattachment, as low as 0.35Uoo. In addition, Li and Naguib10 found the “local” convection velocity to increase beyond reattachment up to about 0.5Uw. The “local” Uc values were calculated from the slope of the peak locus of the two-point cross-correlation of the fluctuating wall-shear stress. 86 4.2.5 Frequency-wavenumber spectra The frequency-wavenumber (f-kx) spectrum is obtained from the two- dimensional, Fourier transformation of the auto-correlation of the space-time wall- pressure signal (p'(x,t)). As a result, this spectrum decomposes the energy content of the wall-pressure field into contributions at different frequency and wavenumber combinations. All 32 microphones, sampled at 12 kHz for 35 seconds, were used in the f-kx spectra. A total of 417,792 (102 x 2‘2) wall-pressure samples were used out of the 420,000 acquired at each microphone. Thus, a 417,792 x 32 (time x space) two- dimensional array made up the entire data set, which then was divided into sub-arrays of 2048 rows x 32 columns. This allowed averaging of the spectrum over 204 records and produced a random uncertainty error of (204415100 = 7% in the spectral estimation. The frequency-wavenumber spectrum of each record was calculated by taking the Fast Fourier Transform (F FT) of the 2048 x 32 array relative to the dimension representing time and multiplying by the complex conjugate. This changes the representation of the data from the time to the frequency domain. A second F FT is calculated along the space dimension (32) in order to change the spatial domain representation to one based on wavenumber. The space dimension was padded with zeros so that the F FT was computed on an array of 2048 x 128. This helps to obtain the resulting spectrum at smaller wavenumber increments via interpolation. The resolution of the dimensionless frequency and wavenumber is 0.007 and 0.08, respectively, for the Reh = 8081 case and 0.003 and 0.08 for the Reh = 18588 case. Note that the resolution value for the wavenumber is that based on the true rather than the interpolated (zero-padded) spatial information. 87 Figure 4.9 shows two flooded-color, contour plots of the frequency-wavenumber spectra for each of the Re cases. The color bar indicates the magnitude of the spectrum divided by the maximum peak magnitude, making the scale zero to one. On the abscissa, the normalized wavenumber is shown and the ordinate gives normalized frequency. Depicted in each plot is a concentration of energy forming a ridge of peak spectrum values inclined at an angle and located in the right half plane, or kx>0. The peak locus of this ridge is practically linear and does not pass through the origin of the plot. The mean- squared error around the linear fit line was determined to be 0.1% of the lowest frequency (fh/Uw = 0.021) in the fit range for both Re cases. Since the convection speed at a given f and k" point is the slope of a straight line drawn from the origin of the f—kx spectrum to the point, the ridge represents downstream (positive) traveling disturbances. The offset between the origin and the intercept of the peak locus with the kx axis results in frequency (wavenumber) dependent convection velocities for different points along the ridge. These convection velocities were calculated by determining the equation of a least-squares line fit to the ridge (seen in Figure 4.9) and dividing through the frequency to obtain the convection velocity as a function of frequency. Equation (4.5) shows the details behind the calculation. 91 :1: NOW. k. m.(f(h)/U..)+b (4'5) °° ridge where, Uc represents the convection velocity, fli/Uco is the normalized frequency, mS is the slope of the least-squares line that fits the ridge, and b is the intercept of the least squares line. Figure 4.10 shows a plot of the results for both Re.1 cases. The dependence of the convection velocities on frequency suggests that the structures in the flow are dispersive, 88 meaning that flow structures of different scales travel at different speeds in the downstream direction. However, as will be shown in Chapter 6 this dependence is actually related to the acceleration of large-scale structures that have formed in place at approximately l/2xr before shedding downstream. At negative wavenumbers, two localized peaks (shown by arrows in Figure 4.9) in the f-kx spectra are visible in the lower frequency range at M.,. z 0.021 i0.004 and 0.012 i 0.002 for the Reh = 8081 and 18588 respectively. These peaks are found at a negative wavenumber of kx(h) z -0.10 for both cases, indicating an upstream convection velocity of Uc z -0.21U.o for the Reh = 8081 and -0.12U.,0 for Re)1 = 18588. These upstream convection velocities are rather rough estimates. More specifically, because of the resolution of the f-kx spectrum, the locations of the peaks could vary giving a velocity range anywhere from -0.12Um to -0.4lU.,o for the Reh = 8081 case and -0.08U.o to - 0.23Uoo for the Re)| = 18588 case. Thus, it is difficult to determine the actual upstream convection velocities from the f-kx spectra with high degree of confidence for these low- wavenumber/low-frequency peaks. Hudy et a1.8 identified an upstream convection velocity of 0.2 1 U1,o in their splitter- plate-with-fence study in both the frequency-wavenumber spectrum and a cross- correlation plot. The Reynolds number based on fence height in the study was 7885. The peak in their study was located around f‘2H/U... z 0.02 and kx*2H z -0.11 in the f—kx spectrum, where 2H equals the total height of the fence. Hudy et al.8 acquired measurements starting at 0.6 “step” heights downstream of the fence, which translates to x/xr = 0.02. In the present measurements, the first microphone was located at 0.4 step heights, or x/x, = 0.09 and 0.08 for Reh = 8081 and 18588, respectively. Hudy et al.8 89 stated that the strongest manifestation of the upstream propagating disturbances was located in the region x/xr < 0.25. This means that in the present study only two microphones are contained within the 0.25xr region for the Reh = 8081 case and three microphones for the Reh = 18588 case. Figure 4.11 shows the flooded-color, cross-correlation contour plots for both Re numbers using microphone #2 (x/h = 0.82) as the reference microphone for the lower Re case and #3 (x/h = 1.21) for the higher Re case. The positive-inclined lobes in the contour plots (highlighted with broken lines) indicate an upstream convection velocity near separation. The average upstream Uc is difficult to calculate from these plots due to limited resolution, which means there are not enough peak values upstream of the reference microphones to accurately fit a linear trend line to determine the slope for calculating Uc as done in Section 4.2.4. Thus, in this study, an upstream convection velocity has been identified, but not quantified. Due to the microphone spacing, the resolution in the region of the upstream convection is low, which makes it difficult to calculate upstream Uc with confidence. 90 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Figure 4.9 Frequency-wavenumber spectrum obtained from all 32 microphones at Re = a) 8081 and b) 18588 91 0.91 ‘_ Reh=18588l ‘ 11 _1 _11_,_1__ i l 1l___"Re 980817421 0 V l 11 11 1,1111 111111111.___L__1 l | l 0L1_11_-1L_111111_111_1111 1 0 0. 05 0.1 0.15 0.2 0.2 0. 0. 35 fh/Uw Figure 4.10 Normalized convection velocity as a function of normalized frequency for Re.I = 8081 and 18588 Lee and Sung7 also computed a frequency-wavenumber spectrum similar to the one in Figure 4.9. Their results show a ridge associated with a downstream convection velocity as well as a stationary mode along the kx = 0 axis, but they did not observe an upstream convection velocity as previously mentioned. It is believed that Lee and Sung7 could not find a negative convection speed because their most upstream measurement location extended down to only x/xr z 0.27. Similarly, the measurements of Farabee and Casarellaz, who also reported no upstream convection velocity, were limited to x/xr > 0.25. 92 xlh 2 4 6 8 10 12 xlh Figure 4.11 Flooded-color, cross-correlation contour plots for a) Re.I = 8081 using microphone #2 (xlh = 0.82) as the reference microphone and b) Rel. = 18588 and reference microphone #3 (xlh = 1.21) In order to analyze the frequency-wavenumber spectra further and in more detail, the Reh = 8081 spectrum was reprocessed using only the microphones upstream of xr and then using only the microphones downstream of x,. The results were similar for the higher Re case. Figure 4.12 shows the resulting frequency-wavenumber spectrum with the ordinate and the abscissa in the figure the same as described for Figure 4.8. The standard error for the linear fits to the spectrum ridge peaks was less than 0.2% of the smallest frequency (fit/U00 z 0.028) within the fit range for both cases. In Figure 4.12a, by removing the influence of the wall-pressure signature downstream of reattachment, the magnitude of the spectrum peak in the negative kx half plane becomes stronger. Thus, within this region there is a relatively strong upstream convection velocity at low frequencies. In Figure 4.12b, which 'shows information based only on the microphones downstream of x,, the peak on the negative wavenumber side is weak. This suggests that a weak manifestation of the upstream traveling disturbances is found downstream of x,. This may be a reflection of the fact that, on instantaneous basis, reverse flow velocities may be found as far downstream as approximately 5.5h (or 1.2 x,), as depicted from the FPP plots in Figure 3.1. 94 -O.6 -O.4 -0.2 0 0.2 0.4 0.6 Figln'e 4.12 Frequency-wavenumber spectra for Rel. = 8081 calculated using data from the microphones located: a) upstream of x, and b) downstream of x, 95 It is also evident that the point at which the straight line determined from the peaks along the ridge crosses the abscissa in Figure 4.12b shifts to lower wavenumbers when only investigating the microphones downstream of x,. In Figure 4.12a, the slope line crosses almost at kxh = 0.2 whereas, in Figure 4.12b, the intersection point is closer to kxh = 0. This shows that within the separated shear layer region the convection velocity is highly dependent on frequency. Downstream of x,, however, the convection velocity is less dependent on frequency as the ridge line intersects the abscissa closer to zero. This relationship is shown in Figure 4.13, which displays the convection velocity as a function of frequency for the filll range of microphones and for the microphones downstream of x,. It can be seen that the line associated with the microphones downstream of xr has a slightly flatter profile compared to the line for the full range of microphones. This shows that downstream of xr the convection velocity is less dependent on frequency. 96 09:“ 1 ...... Reh=8081; x/h>xr 111111 1 ___1 ._ .1 0 0 0.05 0.1 O. 15 0.2 0.25 0.3 0.35 Figure 4.13 Uc versus normalized frequency for Re). = 8081, comparing results from full spectrum with spectrum results from downstream of xr Daoud9 investigated the reattached boundary layer downstream of a separating/reattaching flow region in a fence-with-splitter—plate configuration and found a constant convection velocity equal to 81% of the freestream velocity using the f-kx spectrum. The convection velocity was independent of frequency in the reattached, boundary-layer region, which is not the case in the present study. However, Daoud’s9 microphone measurements started at x/xr > 1.7 downstream of the fence; whereas, the results shown in Figure 4.12 and Figure 4.13 are based on measurements at and downstream of x/xr = 1.05. This is near the beginning of the reattached, boundary-layer region. In order to compare the convection velocity of the structures within the reattached boundary layer as calculated by Daoud9 with those from the present study, only the microphones downstream of x/xr = 1.7 were used to determine the f-kx spectrum 97 for the Reh = 8081 case as shown in Figure 4.14. It is found that if only the spectrum- ridge peaks located at the frequency of the global spectrum peak (fh/U.,o ~ 0.1) and higher frequencies are used to determine the line fit, then in fact the line does pass through the plot’s origin (see dashed line in Figure 4.14) within the resolution of the origin. For frequencies below fli/Uco ~ 0.1, the convection speed remains dependent on frequency as evident from the deviation of the ridge peaks from the dashed line in this frequency range. In an effort to compare the convection velocity found by Daoud9 in the reattached boundary layer with the present study, the peaks along the ridge below fli/U00 < 0.1 are ignored. Figure 4.14 shows the slope of the linear fit for the peaks greater than the frequency value stated along the positive ridge in each f-kx spectrum. The linear fit lines cross at kxh = 01:001. The slope of the line indicates that the high-frequency structures within the reattached boundary layer have an average convection velocity of 0.59Uoo independent of frequency as shown in Figure 4.15. Farabee and Casarella2 found the convection speed within the reattached boundary to be roughly 0.65-0.70Uw. The difference between the two studies may be due to the geometry difference. Farabee and Casarella2 used a planar BFS with assumed two dimensionality along the centerline. 98 -O.6 -0.4 0.2 0 0.2 0.4 0.6 k h X Figure 4.14 F. 1 _, u... ' spectrum for x/xr > 1.7 at Re.I = 8081 for fin/U... > 0.085 1 +_V—fi—_Y#i +7 77—1—‘# 0.9 1 — 0.8 1 — 0.7 r 0.6 — __ 4 8 D \005 k j 3 0.4 0.3 » 1 0.2 - 0.1 L J 0 ~ ~ 11_. —1_. ‘47 11J——11 1_1¥ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 fhlUw Figure 4.15 Uc versus normalized frequency obtained from microphones located at x/xr > 1.7 and frequencies fh/Um > 0.086 for Re.I = 8081 99 4.3 Conclusions A comprehensive, wall-pressure-signature dataset has been acquired beneath the flow field downstream of an axisymmetric, backward-facing step. The dataset was acquired at four different Reynolds numbers, allowing investigation of the Reynolds- number effect. Both the mean- and fluctuating-pressure distributions exhibited Reynolds-number dependence. The mean-pressure recovery occurred over a longer, streamwise-distance with increasing Reynolds number. Similarly, the RMS pressure fluctuations reached their peak at an x value that increases with Reynolds number. Both of these effects were consistent with the lengthening of the separation bubble with Re... Another Reynolds- number effect that remains unexplained at this stage is the reduction of the RMS pressure, relative to the dynamic head of the flow, as the Reynolds number increases. The wall-pressure, auto-correlation showed a distinct region in the flow, stretching from separation to half of the reattachment distance that is dominated by low- frequency disturbances. These disturbances are characteristic of backward-facing-step flows and are associated in the literature with the flapping of the shear layer. A second noticeable region was found in the auto-correlation starting from l/2xr and stretching up to about x,. This region of the flow is dominated by higher-frequency structures, which are linked to the passage of flow structures within the shear layer. Beyond x,, up to approximately x/h = 8, the dominant time scale extracted from the auto-correlation results narrows again as the shear layer reattaches on the surface and becomes a boundary layer. The fourth distinct region delineated in the auto-correlation contour plots starts beyond x/h = 8. It is in this region that the time scales are the shortest over the entire streamwise 100 extent measured. These time scales do not change over the remainder of the region measured and indicate that this region may be dominated by flow structures that may be smaller in size and may be convecting at a faster speed than the flow structures in the shear layer upstream of reattachment. An increase in the convection speed between upstream and downstream of reattachment was seen in the wall-pressure cross-correlation results relative to a reference signal measured near reattachment. The speed increased from approximately 0.35Um upstream of xr to about 0.5Uoo downstream of x,, seemingly independent of Reynolds number. The convection velocity value upstream of reattachment is lower in magnitude than reported in planar backward-facing step studies. Overall, a weak effect was seen in terms of Reynolds number in both the auto-correlation and cross-correlation analysis. The frequency-wavenumber spectra showed that for both Reynolds number cases that the downstream convection velocity was dependent on frequency. This dependence of the convection velocities on frequency suggests that the structures in the flow are dispersive. However, as will be seen in Chapter 6, this dependence reflects the acceleration of the large-scale structures once they form. In addition, an upstream convection velocity was identified in the f—kx spectra; although, Uc was difficult to quantify due to the limited wavenumber resolution of the spectra. 101 4.4 References ' Eaton, J. K. and Johnston, J .P., “A Review of Research on Subsonic Turbulent Flow Reattachrnent,” AIAA Journal, Vol. 19, No. 9, 1981, pp. 1093-1099. 2 Farabee, T.M., and Casarella, M.J., “Measurements of fluctuating wall pressure for separated/reattached boundary layer flows,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 108, 1986, 301-307. 3 Driver, D.M., Seegmiller, H.L., and Marvin, J.G., “Time-dependent behavior of a reattaching shear layer,” AIAA Journal, Vol. 25, No. 7, 1986, pp. 914-919. 4 Heenan, AF. and Morrison, J .F., “Passive control of pressure fluctuations generated by separated flow,” AIAA Journal, Vol. 36, No. 6, 1998, pp. 1014-1022. 5 Lee, I. and Sung, H.J., “Characteristics of wall pressure fluctuations in separated and reattaching flows over a backward-facing step,” Experiments in Fluids, Vol. 30, 2001, pp. 262-272. 6 Spazzini, P.G., Iuso, G., Onorato, M., Zurlo, N., and Di Cicca, G.M., “Unsteady behavior of back-facing step flow,” Experiments in Fluids, Vol. 30, 2001, pp. 551-561. 7 Hudy, L.M., “Simultaneous wall-pressure array and PIV measurements in a separating/reattaching flow region,” Masters thesis, Michigan State University, 2001. 8 Hudy, L.M., Naguib, A.M., and Humphreys Jr., W.M., “Wall-pressure-array measurements beneath a separating/reattaching flow region,” Physics of Fluids, Vol. 15, No. 3, 2003, pp. 706-717. 9 Daoud, M.1., “Stochastic Estimation of the flow structure downstream of a separating/reattaching flow region using wall-pressure array measurements,” Ph.D. Dissertation, Michigan State University, 2004. '0 Li, Y. and Naguib, A.M., “High-Frequency Oscillating-Hot-Wire Sensor for near- Wall Diagnostics in Separated Flows,” AIAA Journal, Vol. 43, No. 3, 2005, pp. 520. 102 5 Velocity Measurements using PIV 5.1 Introduction Although the backward-facing-step flow field has been investigated extensively over the past few decades, the majority of these studies have utilized time, rather than space, resolved measurements. Only during the last decade have a few PIV studies (summarized below) of the BF S flow become available. All of these have been conducted in a planar flow geometry, which suffers from the inability to establish a truly two-dimensional mean flow because of end effects as discussed in the Introduction. The objective of this study is to explore the statistics of the velocity field above the surface using PIV measurements in a geometry that allows for axisymmetric separating boundary layer conditions. This information will give better understanding of the flow structures that develop in the shear layer beyond separation as well as provide a database that is suitable for benchmarking of computational codes based on periodic, boundary conditions in the spanwise direction. Spazzini et al.1 acquired digital PIV measurements in a planar BFS at Reynolds numbers based on step height spanning 3500-16000. These measurements were completed in a water tunnel and provided quantitative information about the flow field as well as qualitative visualizations. In addition, time-resolved skin-friction measurements were obtained in a companion experiment using a wall-mounted double hot-wire probe. Data from both velocity and wall-shear-stress measurements revealed a strong dependence between xr and Reh with the length of reattachment growing with increasing Reynolds number. In addition, Spazzini et al.1 found that the location of the secondary 103 separation point was weakly dependent on Reynolds number, moving closer to the step with rising Reh. In 2002, Kostas et a1.2 acquired PIV measurements in a planar backward-facing- step geometry at Reh = 4660. From their measurements, the mean-velocity field in the plane parallel to the streamwise and wall-normal coordinates revealed both primary and secondary recirculation regions. The mean-reattachment point recorded was 4.8h i 0.2h, which is the location that the streamline originating at the step edge impinges onto the wall. Kostas et al.2 stated xr could not be accurately obtained because the PIV data did not extend to the wall. In addition, Kostas et al.2 found xr to be shorter than reported in literature, and they attributed this to the high level of turbulence near the wall in the upstream boundary layer. Reynolds—stress components 8172,75,??? normalized by U...2 were also presented. Kostas et al.2 found appreciable, Reynolds stress values starting around two step heights downstream of separation at a vertical height of y = h. They also reported that the largest levels of Reynolds stress were located roughly one-step height upstream of reattachment at y = 0.7h. Beyond reattachment, Kostas et al.2 observed a decline in the Reynolds stress levels. They also investigated the out-of-plane (spanwise) vorticity, observing that negative vorticity was contained within the shear-layer region with a peak, vorticity value of a); = (ozh/U.o = -3.4. At separation, the vorticity is concentrated in a thin layer that grows in width with increasing streamwise distance until it takes on a more bulbous shape between x/h = 2-5 as a result of the high-degree of mixing in the region. Downstream of reattachment, the authors observed a decrease in vorticity as the reattached boundary layer began to relax back to equilibrium. 104 Scarano and Riethmuller3 published a study in 1999 describing an improved algorithm for interrogating PIV images. In the paper, they presented PIV data in the form of mean velocity and Reynolds stress quantities from a planar backward-facing-step study at Reh = 5000. The mean velocity profiles with streamlines show both a primary and secondary re-circulation region and a reattachment distance of xr = 5.9h. Reynolds stress contour maps show that the maximum values observed for uRMS, View, and u'—v' were located along the centerline of the shear layer, which was defined as the locus of inflection points for the mean streamwise velocity plots. Scarano and Riethmuller3 found the maximum uRMS value of 17% of the freestream velocity at approximately one-step height upstream of x,. As for vRMS and u’v‘ , the maximum values occurred at x/h = 5 and had a magnitude of 12% and -1.2%, respectively, when normalized using U... In the remainder of this chapter, the velocity field downstream of the axisymmetric, backward-facing step is investigated using PIV measurements. The primary focus is the examination of the Reynolds number effect on the velocity-field statistics. 105 l 5.2 Results and Discussion Details of the PIV measurements analyzed here were described in the test matrix in Section 2.4. Results presented include the mean-velocity field, the streamwise and normal fluctuating-velocity fields, the Reynolds shear stresses, and the mean-spanwise- vorticity field. Reynolds-number effects are explored between Reh = 8081 and 18588. 5.2.1 Mean-Velocity Field The mean-velocity vector fields and streamlines above the surface downstream of the axisymmetric backward-facing step are shown in Figure 5.1a—b for two of the four Reynolds numbers investigated: Reh = 8081 and 18588. The streamwise coordinate, x/h, is along the abscissa and the coordinate normal to the wall, y/h, is represented along the ordinate. The vector maps show a classical backward-facing step mean-velocity field. The boundary layer separates at the edge of the step forming a shear layer that reattaches at a distance downstream of the step. The streamlines curve towards the wall with increasing x until the shear layer reattaches at x,. Beneath the shear layer there is a clockwise recirculation zone located near the wall that stretches from x/h z 1.1, depending on Reynolds number, to x,. The reattachment point was captured for all tested Reynolds numbers and, as seen earlier in the F FP and U = 0 plots in Figures 3.1 and 3.3, the reattachment length increases with increasing Reynolds number. Near the step, within the x/h = 0-1.1 region, the streamlines indicate a secondary recirculation zone, which is a characteristic feature in BF S flows. Scarano and Riethemuller3 along with Kostas et al.2 were also able to capture the secondary recirculation zone in their recent 106 PIV studies of a planar BFS. As observed in the FFP analysis as well as recorded in studies by Spazzini et al.l and Li and Naguib“, there is a reduction in the size of the secondary re-circulation region with increasing Reynolds number. Beyond reattachment, the shear layer forms a reattached boundary layer as shown in Figure 5.1 for both Reynolds numbers. ___—a ~k“» Figure 5.1 Mean-velocity vector fields and associated streamlines for Reh = a) 8081 and b) 18588 To take a closer look at the mean-velocity-field characteristics, mean-velocity profiles extracted from the planar PIV data at six different x/h positions are shown in Figure 5.2 for Reh = 8081 and 18588. The six positions are located at x/h = 0.85, 2.08 3.54, 4.23, 6.07, and 8.06. These same six positions will be used throughout the 107 remainder of the chapter. Along the abscissa, the normalized velocity is presented as U/U..o and along the ordinate the distance normal to the wall is given as y/h for all six x/h positions. In general, the top three plots are located downstream of separation and upstream of the reattachment point. These three plots show the reverse flow below the step height, the shear layer region, and the relatively unchanged boundary layer-like profile above a y/h value that is slightly larger than 1. For the fourth plot, x/h = 4.22 is near reattachment and at this point slightly more reversed flow is seen at Reh = 18588 than Reh = 8081 since for the former the flow reattaches 0.48h farther downstream. The last two velocity-profile plots are located downstream of reattachment. It is in this region that the flow reattaches and begins to relax towards an equilibrium boundary layer state. xlh =0.85 xlh =2.08 xlh =3.54 3 8"” ""‘ ' “" ’ ' ’5 "‘1. 31' 11 1 1. 1 * ‘1 31*“ '” ‘" — ‘ 1'“? 1 1 1 11 1 1 81 I 1 5?: ‘ 123:1 l :53 l 21 3?; 1 21 {:51 21 :1; 1 11’: 1 :1; . 1 t I ‘11. ‘ 911v tj , -C :1.” .5 It: 1 2 £3 1 \ : 4.3-:9 l \ 3:: ‘ 1 1': l > < 11:1: 1 >‘ l 47:: l >‘ 3."? 1 1 -11 WI 11 1_ .11 1 1I :1“ 1 1 15* J F l 1 1283* F 1 I {If}? 1:: 1 1,1 1,198 l ‘3‘ 1 :1‘ 1 1 11114“ 0L 1;. 1 1 1 1 1 0531.. 1 1 1 J 083.-.-.1 ' __ _1 4 0 0 5 1 0 0.5 1 O O 5 1 x/h =4.23 xlh =6.07 x/h =8 06 31' T" T - T ' T 7 3"‘V' 7 'fi ' 7f, “"1 3' 11.. A _ 111*] 1 8 1 I 1 I 1 1 571‘ 1 1 E}. l l g; 1 2 1 ’53:” 'l 2 z 533 I 2 L 5'? l ' 5" , l .1511. ' 1 Law I .7: 1 _...': :13 1 £ - ‘3; E l 3'." 1 E i If}: > 1 {:13 l > 1 £5 . > i 5’; l 1 .1 1:33" '- 1 l ., (gt-I 1 1 1 1;. l 1 1 ' 1 1 1.8 . 1‘91" 1.1.1.1? 11:15" 1:11 1 1 3' 1 oil?- 1 1 1 1 ._l 011 1:3 .. ,1 11' 011,111 ELI-K11 1_l 0 0 5 1 0 0.5 1 0 0 5 1 U/U U/U U/U Figure 5.2 Mean-streamwise—velocity profiles at selected streamwise locations: (O)Re.I = 8081, and (Cl) Re.I = 18588 108 To elaborate further, starting with the x/h = 0.85 plot located downstream of the step, the two profiles show the reverse flow beneath the step height, y/h = 1. Around y/h = 1, the velocity changes rapidly over a very short y distance, corresponding to the shear- layer region. This is consistent with the idea that near the step, the shear layer is thin; that is, a large velocity gradient exists. At y/h > 1, the velocity profile takes on a profile similar to that of the boundary layer upstream (not shown here) away from the near-wall region. This gives indication that the boundary layer once it separates remains unaffected except in the region close to the wall. In their study of a single-stream shear layer, Morris and Foss5 introduced the idea that the initial shear layer instability originates only from within the near-wall region of the boundary layer, forming a “sub-shear layer” immediately downstream of separation. The rest of the boundary layer remains intact until farther downstream when the sub-shear layer grows sufficiently to affect the full width of the flow domain. Morris and F0555 stated that the mean profiles are identical for y/G > 2 for a streamwise distance between 0 < x/O < 29 (where 9 is the separating boundary layer momentum thickness). The first four plots in Figure 5.2 have x/G equal to 5.5, 13.6, 23.1, and 27.6 for Re1l = 8081. The current study provides evidence that the ideas of Morris and Foss seem to also hold for shear layers separating over a back step. Further discussion of this point will be given in Section 5.2.2 of the Results and Discussion. Farther downstream, the steep velocity gradient in the shear-layer region relaxes and the shear layer spreads out. This can be seen in the next two plots, x/h = 2.08 and 3.54, where the velocity gradient around y/h = 1 is not as steep. In addition, the vertical height of the reverse-flow region becomes smaller as the shear layer curves toward the 109 wall. At x/h = 4.23, the reverse-flow region is almost non-existent as the shear layer nears the reattachment point. Beyond reattachment, the two profiles are similar as seen in Figure 5.2 at x/h = 6.07 and 8.06. 110 5.2.2 Fluctuating—velocity field: aims and vRMS Figure 5.3-Figure 5.4 show the longitudinal (uRMS) and vertical (vRMs) turbulent fluctuations for Re}1 = 8081 and 185 88. Each figure is a flooded-color, contour map with the streamwise distance downstream of the step, x/h, along the abscissa and the distance normal to the wall, y/h, along the ordinate. These axes will be used on all flooded-color, contour plots in this chapter. The color bar on the bottom of the contour plot indicates the magnitude of the particular turbulent RMS velocity normalized by Ugo. For all plots in Figure 5.3-Figure 5.4, the peak root-mean-square values are located within the separating-shear layer, with the highest peak along the center of the shear layer near the separation edge. The data presented in Figures 5.3 and 5.4 compare well, qualitatively, with Scarano and Riethmuller3 in their PIV study of a planar, backward-facing step. In addition, the general behavior observed is consistent with the findings of Castro and Haque6 as well as Ruderich and F ernholz7 in their investigations of a fence-with-splitter- plate flow. Both studies employed the maximum loci in the turbulent intensity plots to determine the center of the separating-shear layer. Quantitatively, the RMS magnitudes in the current study are higher than in Scarano and Riethmuller’s study. In particular, the uRMs and vRMS values are almost double the values found by those authors. It is interesting to note that, in contrast to a free-shear layer, which spreads gradually and smoothly with increasing x, the reattaching, shear layer initially diverges smoothly until approximately x/h ~ 2 when a "sudden" divergence occurs and the shear layer grows to a thickness approximately equal to the step height. This is also the streamwise location at which the width of the wall-pressure auto-correlation reduces somewhat abruptly (see Figure 4.3). The RMS results show that in fact this transition in the wall-pressure field 111 characteristics is associated with the shear-layer structures approaching the wall. This was hypothesized through inference using the wall—pressure data alone. The velocity field data confirm the hypothesis. In Figure 5.3a—b, the uRMs, contour maps outline the shear layer and the “boundary layer” (i.e., the portion of the flow that seems to remain similar to the boundary layer at separation) distinctly for both Reynolds-number cases. Within the separating, sub-shear layer (the thin region of high shear near y/h = 1, immediately downstream of the step), the uRMs values change dramatically over a short x distance as a result of the high shear in the region. Above this sub-shear layer, starting around y/h = 1.25, the uRMS values decrease slowly over a large, streamwise distance in the downstream direction. That is, within this zone very little change is observed in the turbulence activity from that in the outer part of the separating, boundary layer, and hence the flow-structure characteristics are likely to be similar to those in the boundary layer. At the edge of the boundary layer, 6 = 1.73h, uRMs /Uco values are less than 0.02 as the boundary layer merges with the free stream. In Figure 5.3a-b, distinct uRMs bands can be seen at the different heights above the shear layer. These uRMS values decrease with increasing radial distance as expected. In addition, the uRMS bands show a slight curvature toward the wall upstream of reattachment. Beyond reattachment, the bands become parallel to the wall as the shear layer transitions to a reattached, boundary layer. In addition, the turbulence within the shear layer loses energy once the shear layer reattaches as can be seen by the reduction in uRMS magnitude beyond reattachment. A point of interest is the two peaks in the uRMs contour maps that can be seen within the shear layer at x/h ~ 0.1-1 and 4.2. The peak closest to separation increases in 112 magnitude with increasing Reynolds number and has a higher magnitude than the second peak. This local maximum value close to separation has also been seen by Morris and F oss5 in their single-stream, shear-layer study. Morris and F oss5 found a local maximum value located roughly at x/0 ~ 2 and y/0 ~ 0.15. Such a local maximum value has been observed in the literature only in laminar separating boundary layers. Hence, it appears that the present data support the findings of Morris and Fosss, whose boundary layer was also turbulent at separation, suggesting that the sub-shear-layer region is dominated by viscous effects from the near-wall at separation. Nonetheless, caution should be exercised since the region immediately downstream of the separation point is a high- shear region and the accuracy of the PIV results there may be questionable. Therefore, to check that the peak near separation is not an artifact of the PIV processing, the data near separation need to be re-processed using iterative methods, such as particle image distortion (Huand and Fidlers) that have been developed to remedy high-shear effects on PIV measurements. The second uRMs peak occurs slightly upstream of reattachment, x/h ~ 4.2. Scarano and Riethmuller3 find their maximum uRMS value about one-step height upstream of reattachment. The same trend has also been recorded in surface-pressure studies in planar BFS. For example, Heenan and Morrison9 observed in their study that the maximum pRMS value is located approximately one step height upstream of reattachment. In the present study, the pRMs value occurs close to reattachment as seen in Figures 4.1 and 4.2 and, due to data scatter near the peak, it is possible that the peak pRMs occurs slightly upstream of reattachment. 113 0 0.05 0.1 0.15 0.2 0.25 0.3 Figure 5.3 Flooded-color, contour maps of “RMS/U1.» at Rel. = a) 8081 and b) 18588 The comparison between the two Reh cases shows no noticeable difference in the uRMS distribution. This can be seen more clearly in the uRMs profiles plotted in Figure 5.5 for the same six streamwise positions displayed in Figure 5.2. In the first plot at x/h = 0.85, there are slight differences between the two profiles in both the separation region and near the free stream. At the remaining five streamwise positions, Figure 5.5 shows the profiles at the two different Reynolds numbers collapsing over both the separation region and the shear layer, with only slight differences near the freestream where the Reh = 8081 case shows more streamwise fluctuations. 114 o 0.05 0.1 i ‘- 1,0115 02' 0.25 0.3 Figure 5.4 Flooded-color, contour maps of vRMS/Um at Reh = a) 8081 and b) 18588 Flooded-color, contour maps of VRMS for the two different Reynolds numbers are shown in Figure 5.4a-b. The contour maps have similar characteristics compared to the longitudinal turbulent intensity maps as evidenced by the higher intensity values being contained within the shear layer. In addition, vRMs bands can be seen located above the shear layer with slight curvature toward the wall; although, these bands are not as clearly defined as the uRMs bands. Nonetheless, the vRMs values decrease with increasing height within the region of remaining boundary-layer turbulence. The effect of increasing Reynolds number is evident in the vertical, turbulent- intensity, contour maps. One of the most noticeable characteristics of increasing 115 Reynolds number is that the magnitude of the peak value, located at x/h ~ 0.1-l as seen in the uRMs contour maps, increases. The vRMS profiles for both Reh cases are shown in Figure 5.6 at the selected streamwise locations. The only difference between the two cases is that the magnitude of the peak near separation is greater for the Reh = 18588 case than the Reh = 8081 case as can be seen in the x/h = 0.85 plot. Beyond this streamwise position, the profiles collapse for the remaining x/h locations investigated. x/h =0.85 x/h =2.08 x/h =3 54 3 r k - i W J ' W i ‘ 3 1' # k _ - 7 3 1' — a f 7 _ —‘ 7 I '. z E ' E..i 1 1;; 1 1 1 1 ’ . I ‘ 53:. 1 21 1 21 1 218.1 1 .r: 1 211' , .c 1 if: I .c 1 3f?- 3. 12.8, I 3. 1 1 3. 1 8 1 1 1 ”’TTTI‘; ‘ .;1 j 1 ‘ 1'1”; 1 '1 M53431 1 ‘ 5;; 11 ‘ 7 1 I a??? 1 1 ' “ii-:2, 1 17:1 1 .-t. «2'1: . :5: i I 7.1.3“ 1 l , ; :5: 1 1 51:1 1 113.; 1 1 3,128?“ 0. ; , . . 1 0. C, _ _ 0 _331 1 .1 1 0 0.1‘ 0.2 0 .1 0.2 0 0.1 0.2 x/h =4.23 x/h =6.07 x/h =8.06 31 1 ,,_ -- 3: ”~77 -- 1. 3 -— 1 — .1 < 1 1.1 ‘ ‘ ' l r 8; £2. 1 ~. I ‘ .17... 3 -:::.1 c: 1 1 {‘33. 1 1 13".". l ‘ 11“.". S 2;}; 2 LL: 1 _c 1.»: i \ 1 7,. \ ' \ 31‘: > 1 1:51 > 1 1 > 1 4,31 1 1 '31.: _1 1 1 11 ‘78. 1 1 I 1 , 1 1 1 If; 1 :1: ‘.'; 1 i . i 1 l 1 .837." . 1. _ 2:4.“ . . 1.1 01 L13 .- . _ .1 01 IE 11. 1 1.1 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 URMs/Uoo uRMS/Uao uRMS/Uoo Figure 5.5 Streamwise-velocity RMS profiles: (())Re.I = 8081 and (D)Re1I = 18588 116 xlh =3.54 x/h =2.08 xlh =0. 85 n«u 1.. .6 . ~17:mean .. 1.1.1 1.11'1111 1 0 a}; HBWNawmmv . 51.1.1 a; _ a 2. , nnv . anvil w 83.81...me 8 4| ..1.....1 ...rwp. 0 ....L ”11“.; ..f. L 7. .13.”.V . A .2 4 . . _«wZMJ 1» . "va 88a1u1w1w....1_ if .«>«1w1..1m.:r+.t .1vt.‘L | . 1- 1 .11 11111111.11..11111 11 1 any 1 .3 1 ,1. . ,1 $8- 7 1.53.: .. » any x/h =8.06 x/h =6.07 x/h =4.23 VRMS/Uoo vRMS/[Joe Figure 5.6 Normal-velocity RMS profiles: (O)Re.. = 8081 (El) Rel. = 18588 VRMS/Uoo 117 5.2.3 Reynolds Shear Stress Figure 5.7 and Figure 5.8 provide the Reynolds shear stress, —i¢'—\;'/U: , information for the axisymmetric backward-facing step. Figure 5.7a-b contains the flooded-color, contour maps for the Reynolds-shear-stress for two Reynolds numbers. The axes for the contour plots are the same as seen previously with other contour plots. The color bar below the maps gives the magnitude of the normalized Reynolds stress. Figure 5.8 provides individual —i4—'\7'/Uf, profiles at selected streamwise locations. In these profile plots, —i¢—'_v'/ U f, is plotted along the abscissa and the ordinate shows the normal coordinate normalized by the step height. Starting with the color, contour maps in Figure 5.7, it is interesting that the magnitude of the non-dimensional, Reynolds-shear-stress decreases as the Reynolds number is increased. This may be explained by looking at the scaling used in the contour maps. The Reynolds stress values in the contour maps are all normalized by U002. However, Morris and Foss5 suggested that the Reynolds stresses in a single-stream shear layer should scale with the separating boundary-layer friction velocity (i.e., wall-shear stress), at least close to separation. Thus, since the skin-friction coefficient of the boundary layer (which is proportional to the square of the friction velocity divided by U002) decreases with increasing Reynolds number, then the ratio of friction velocity to free stream velocity decreases with Reh as well. This means that the free stream velocity changes faster than the friction velocity with increasing Reh. Thus, if — iz’_v'/ U 3, (ur is the friction velocity) does remain invariant with Reynolds number as hypothesized by Morris and Fosss, then — W/ U 1: should decrease with increasing Reynolds number. 118 The individual — u'—v'/ U 5: profiles in Figure 5.8, for the same six different streamwise positions employed earlier to display uRMs and VRMS profiles, provide a clearer support for the above observations. First, the peak for the Re.1 = 8081 case near separation is almost double the magnitude of the peak for the Re = 18588 case. This is consistent with, but does not prove, the idea that the turbulence within the shear layer, especially near separation, scales on ur and not Ugo. Throughout the shear layer, upstream of reattachment, as shown in the first four plots in Figure 5.8, the lower Reynolds number case has a larger, Reynolds-stress magnitude than the higher Re case. The gap between the two decreases, however, with increasing x/h distance. Beyond reattachment, the profiles collapse, which suggests that in this region —;'_\7'/U: is the proper non- dimensional quantity. 119 Figure 5.7 Flooded-color, contour maps of — u'—v'/ U f, at Reh = a) 8081 and b) 18588 120 x/h =0.85 _, . ,,_ . 3 r 1 3 1“ _ r f— T 1 I. g - _ T— _ _ _ _, i ‘1 2%.; 1 1 i 1 :1; I ‘7: 2:. 2.11 1 212.1. 1 2 1 ”...-..I‘ 1, .1 1; .c ‘23:? 1 .1: 1 {'1 1 .c I \ ...1 1 \ .3. 1 \ -...11 >. {'3} > ,1: 1 >~ I; 1 am?" 1 1 . f1 -2: 1 1177 ._:«:1 1 11 p 1 11 * 11.7,.-. ~ 3:1"- "‘ 1 1 .‘ 1 $7.53! ,. 1 1:“? 1 1 ”1V 1 1 z: 17 1 .72.? 1 1 7'1 1 1 1T 1 L ., '1' 1 1 1 ~" 1 l 0". 1 f 1 l o 1 ,,, i i . , ,z. _ . , ..i O ._ ... 7 7 ; 7 . .z .# #1 It. _, , -1 _ ___ _. i __4 0 0.005 0.01 0 0.005 0.01 0 0.005 0.01 x/h =4.23 xlh =6.07 xlh =8 06 3 ' ' '_ “~"‘ _. — '_'_— ” ”11 31 _ ' ~_ __ "' b _ _1 31_- # _'~_-— — _ __ ’1 1.13.; 1 531—1 1 1:1 1 :1? 1 I 1 '87-. 1 1 23.: 1 2,?w I 211:1 1 2;L 1 1‘.“ 1 :1". :1: 1 .c 1.3 .c - Z 1 .c 1'27; 1 \ ., 11 1 \ r . \ > 2.1 1 > 1 c. 1 > 1.1,}? 1 1 111'11’114 1 1 111.75%: 1 1 1 1113181 1 1 4’ 1 1 ' . ‘ 1 .131"; 1 I 5;.‘3 1 1 1 3;?“ 1 1 1.3.11} 1 1 34.1.1 1 1 .._,,11_.."?1 1 1 171.?" 1 01:11; ._. ,, _ __ 1 11 z - z, 1 11151-9I _: ”v“... ,__.,1 0 0.005 0.01 0 0.005 0.01 0.005 0.01 — {RV/U3, Figure 5.8 Reynolds-shear-stress profiles: (O)Re1. = 8081 and (El) Re.l = 18588, - W/U: 121 — {Ff/U: 5.2.4 Vorticity and shear-layer thickness Vorticity is used to describe the local rotational characteristics of fluid particles. The vorticity is calculated from the velocity field using @zva an where, (2') is the vorticity vector, V is the “del” differential operator, and I7 is the velocity vector field. For this study, velocity measurements were acquired in the x-y coordinate plane; and, therefore, only the vorticity component in the z-direction, i.e., out of plane, can be calculated. This component is given by grill” 2 ax @’ am where, (02 is the out-of-plane vorticity component. From equation (5.2), it is clear that large vorticity is associated with regions containing large velocity gradients; i.e., regions of high shear. Therefore, the vorticity distribution would be valuable in highlighting the flow within the separated shear layer. The vorticity downstream of the axisymmetric backward-facing step was calculated using the PIV data. Although the velocity gradients required to obtain the vorticity can be calculated using a finite difference scheme, a circulation-based method for calculating vorticity was used here based on the recommendation of Raffel et al.”. This relies on the fact that vorticity is related to circulation through the Stokes theorem. More specifically, the average vorticity within a particular area can be estimated by calculating the circulation along a contour surrounding the area and dividing the result by 122 the enclosed area. Here, the vorticity estimate is calculated based on the circulation estimate around eight neighboring points as shown in Figure 5.9. F (602 )1, j = m (5.3) where AX and AY represent the grid spacing in the streamwise and normal directions, respectively, and 1 1.1.9]. =§AX(U 1—l,j—l + 2(jiJ—l + Ui+l,j-l) +—;'AY(V:'+1,j—l+ 2V '+ V'+l,j+l) i+l,j l 1 - EAX(U. + 2U. 1,j+l +Ui—l,j+l) z+l,j+l (5.4) _ éAY(V + 2V—Lj + V;_],j_|) i-l,j+l I H O+—O<—Oi+1 . 0 ii i j-l Figure 5.9 Demonstration of the eight-point grid used to calculate the circulation for estimating the vorticity at point (i, j). Copied from Raffel et al.10 123 Figure 5.10 shows mean-vorticity, flooded-color, contour plots for the two Reynolds number cases being compared: Reh = 8081 and 18588. The mean vorticity was calculated from the mean-velocity field shown in Figure 5.1. The axes are the same as given in Figure 5.3. The magnitude of the vorticity, normalized as (02* = mzh/Uoo, is given by the color bar. Finally, the solid and dashed black lines mark the point where the vorticity is 8% of the global peak vorticity. These lines were determined by finding, for each streamwise location, the vertical position at which the vorticity equaled 8% of the peak vorticity in the flow (these positions are displayed using open circles in Figure 5.10). A 7th order polynomial fit of these locations was then used to obtain the two solid black lines and a 3'd-order polynomial fit was used for the dashed white line. The order of the polynomial was selected to provide a good fit quality. The fits are found to be good for the most part except near reattachment. The solid, black line along the top of the shear layer traces the outer edge of the shear layer from the step to all the way downstream. The bottom edge of the shear layer can only be traced with confidence up to reattachment. Beyond x,, the dashed line indicates where the lower edge of the shear layer may be located. It is difficult to determine the lower edge of the shear layer beyond reattachment since once the flow reattaches a new boundary layer begins to develop beneath the reattached flow. The point that delineates the two regions is difficult to determine. Looking at the mean-vorticity distribution for the two cases, the concentration of substantial negative vorticity within the flow field is confined to the shear-layer region. Within the shear layer, there is a local vorticity peak seen slightly downstream of the separating edge. At this point the shear layer is the thinnest and the vorticity is the 124 highest. The vorticity distribution grows in width in the downstream direction, and at the same time the intensity level of the vorticity decreases. The spread rate of the vorticity distribution in the wall-normal direction is particularly pronounced around x/h = 1 to 2, which is consistent with the lateral spread of the distribution of the RMS velocities and the Reynolds stress (Figure 5.3, Figure 5.4, Figure 5.7). Kostas et al.2, who investigated the flow field downstream of a backward-facing step using PIV for Rey1 = 4660, referred to the shape as bulbous and attributed the shape to the large degree of mixing in the region. Downstream of reattachment, the level of vorticity continues to decrease as the reattached shear layer begins to diffuse and a new boundary layer begins to develop. 0 Figure 5.10 Flooded-color, contour plots of vorticity, (01* = wzh/Um, for Re.I = a) 8081 and b) 18588 125 A comparison can be made between the results for the two Reynolds numbers using both the color, contour plots in Figure 5.10 and the individual vorticity profiles shown in Figure 5.11. Figure 5.11 shows vorticity profiles for both Reynolds number at the same six streamwise positions used for the RMS and Reynolds stress analysis. The normalized vorticity is shown along the abscissa, and along the ordinate, the normal direction divided by the step height is given. The peak vorticity, located near separation (x/h = 0.008), for the Reh = 8081 case was 002* = -4.7 and for the Reh = 18588, (02* = - 5.2. Kostas et al.2 found their peak vorticity value near the step to be (02* = -3.4. Although the peak levels differ slightly, overall the vorticity magnitude of the two cases is of the same order. There is very little difference between the shear-layer edge lines shown in Figure 5.10 for both cases as highlighted in more detail in Figure 5.12. This shows that the shear-layer spread rate remains unaffected by the Reynolds number (at least over the Re range covered in this study). 126 x/h =2.08 xlh =0.85 —_ . r I . u i I IllIII l. H . tr...> lie. .1 y 9. ,rrfi.mfim? HEW“ ......m 0 .... t .4... I l? . $.83... Innsflw t Al.- .. ,lllllllilli . 3....... . .. 2.4.1... .3.... .. C.» r. L.Mwmntrmumw... WM... 1.2.»...1 0 . .9 .v ...‘i . .ifiubu rt. 5; ......Zwmwumaeaaa 0 _ . . m _. . 2 8. . . ._ h , m _ x , .4 I m . _ _ _ 31.:12liiflll10fiv c; .114.....n¢ T4m)..§ ._ H .4: r... whim. 0 .5 .....- Eb... W.MWKEU .. r a”... 7 . , __ m. ...... _. . _ m _ . x .. ..4 . _ ., _ x/h =4.23 mi... n...h...m.mm.§.. -..... 0 ‘ tylvwmiri. 1L...WH‘\ . ....mwfi. _ . ._ 2 _ . A . . . 4 4 . . . . 6 3 2 1 0 (02* mzh/Uw, profiles as selected streamw (02* 002* Figure 5.11 Vorticity, 032* ions : locat [SC 18588 (O)Re.. = 8081 and (El) Reh xlh ) and Reh = ine Reg. = 8081 (red I Figure 5.12 Shear-layer edges from Figure 5.10 18588 (green line) 127 5.3 Conclusions An extensive, planar PIV dataset of the flow field downstream of an axisymmetric backward-facing step was acquired for four different Reynolds numbers in the 2D separating boundary layer case. Reynolds-number effects have been investigated using a portion of the 2D data (specifically for Reh = 8081 and 18588). The mean-velocity, streamline plots and profiles of the shear layer and the recirculation zone above the wall revealed that both the reattachment length and the location of the secondary, separation point are Reynolds-number dependent. As Reh increased, the reattachment distance grew while the secondary separation point moved closer to the step. As mentioned in Chapter three, the Reynolds-number effect on the reattachment length may be attributed to variation in the boundary layer momentum thickness as suggested by Eaton and Johnston” and Adams and Johnston”. The streamwise and wall-normal, turbulent, RMS-velocity, contour maps also indicated Reynolds number dependence near separation, where a peak in the turbulent intensity values was observed at x/h z 0.1 - l. The existence of the peak may be explained by the findings of Morris and Foss5 in a single-stream shear layer, who suggested that the sub-shear layer region is dominated by viscous effects from the near- wall at separation. This local maximum value near separation increased in magnitude with rising Reynolds number. Nonetheless, specialized PIV routines need to be used to verify that the observed peak is not an artifact of high-shear effects on PIV measurements. A second peak was also identified in uRMS and vRMS contour maps. This peak was located slightly upstream of reattachment and increased slightly in magnitude with rising Reh in uRMS maps but not in the VRMS maps. 128 The effects of changing Reynolds number were also seen in the Reynolds stress contour maps. The magnitude of the Reynolds stress decreased with increasing Reh, indicating that near separation — -u'—v' possibly scales on uI as suggested by Morris and Fosss. This observation requires further testing by directly measuring the wall-shear stress at separation. The mean-vorticity distribution was used to identify the upper and lower edges of the separating/reattaching shear layer. It was found that the extent of the shear layer, and hence its spread rate, remained unaffected by Reynolds number. 129 5.4 References I Spazzini, P.G., Iuso, G., Onorato, M., Zurlo, N., and Di Cicca, G.M., “Unsteady behavior of back-facing step flow,” Experiments in Fluids, Vol. 30, 2001, pp. 551-561 . 2 Kostas, J ., Soria, J ., and Chong, M.S., “Particle Image Velocimetry Measurements of a Backward-facing Step,” Experiments in Fluids, Vol. 33, 2002, pp. 838-853. 3 Scarano, F. and Riethmuller, M. L., “Iterative Multigrid Approach in PIV Image Processing with Discrete Window Offset,” Experiments in Fluids, Vol. 26, 1999, pp. 513- 523. 4 Li, Y. and Naguib, A.M., “High-Frequency Oscillating-Hot-Wire Sensor for near- Wall Diagnostics in Separated Flows,” AIAA Journal, Vol. 43, No. 3, 2005, pp. 520. 5 Morris S. C., and Foss, J. F., “Turbulent boundary layer to single—stream shear layer: the transition region,” Journal of Fluid Mechanics, Vol. 494, 2003, pp. 187-221. 6 Castro, LP. and Haque, A., “The Structure of a Turbulent Shear Layer Bounding a Separation Region,” Journal of Fluid Mechanics, Vol. 179, 1987, pp. 439-468. 7 Ruderich, R. and Femholz, H. H., “An Experimental Investigation of a Turbulent Shear Flow with Separation, Reverse Flow, and Reattachrnent,” Journal of Fluid Mechanics, Vol. 163, 1986, pp. 283-322. 8 Huang, HT. and F iedler, H. E., “A DPIV Study of a Starting Flow Downstream of a Backward-facing Step,” Experiments in Fluids, Vol. 23, 1997, pp. 395-404. 9 Heenan, AF. and Morrison, J. F., “Passive Control of Pressure Fluctuations Generated by Separated Flow,” AIAA Journal, Vol. 36, 1998, pp. 1014-1022. '0 Raffel, M., Willert, C., and Kompenhans, J., Particle Image Velocimetry: A Practical Guide, 3rd ed., Springer-Velag, Berlin, 1998, pp. 160-163. ” Eaton, J. K. and Johnston, J .P., “A Review of Research on Subsonic Turbulent Flow Reattachrnent,” AIAA Journal, Vol. 19, No. 9, 1981, pp. 1093-1099. ‘2 Adams, E.W. and Johnston, J.P., “Effects of the separating shear layer on the reattachment flow structure, Part 2: Reattachrnent length and wall shear stress,” Experiments in Fluids, Vol. 6, 1988, pp. 493-499. 130 6 Pressure and Velocity Measurements 6.1 Introduction Few authors have investigated the flow mechanisms that generate a specific wall- pressure signature. Most of these investigations have been done in a turbulent boundary layer (e.g. Thomas and Bull', Snarski and Lueptowz, and Johansson et al.3). Conditional sampling techniques for the pressure signature and the velocity field were used to identify these mechanisms. One of the few studies that looked at the velocity field and surface pressure measurements in a separating/reattaching flow geometry was Lee and Sung4. In 2002, they investigated the spatio-temporal characteristics of the wall-pressure fluctuations of the separating/reattaching region downstream of a planar, backward-facing step in a wind tunnel with Re.1 = 33000 based on step height. The authors used a one-dimensional array of 29 electret condenser microphones spaced over nine step heights in the streamwise direction, starting two step heights downstream of the separating edge. Pressure measurements were simultaneously acquired with velocity measurements using two split- film probes traversed over a grid of 37 x 21 that extended over 2.0 S x/h S 11.0 and 0.01 S y/h S 1.6. The simultaneous pressure-velocity measurements were used to analyze the interrelation between the flow field above the surface and the pressure signature on the surface. Lee and Sung4 cross-correlated both the u and v components of the velocity field with the pressure signature acquired at zero time delay. Results from the cross- correlation between the pressure and u (Ru'pv) showed a main positive peak inclined at a negative angle with two similar negative peaks on either side. With the Rv-pt, a negative 131 correlation peak was located directly above the pressure position with a positive peak on either side. The highest correlation was found near the streamwise location of the wall- pressure microphone. In addition, these simultaneous measurements were used to reconstruct the flow field by conditionally averaging the velocity measurements based on specific wall-pressure conditions that were obtained from a new wavenumber filtering technique. The reconstructed flow field showed a well-organized, spanwise, vortical structure with 0.6Uco convection speed. They also examined the unsteadiness of the reattachment length and were able to analyze the flapping motion of the shear layer. The authors linked the enlargement and shrinkage of the recirculating zone to the low-pass filtered component of the pressure fluctuations. Another approach for obtaining conditional averages is stochastic estimation (SE). Most of the existing SE studies have utilized velocity measurements at one or more points in the flow field to estimate the flow velocity elsewhere (i.e. Adrians, Tung and Adrian6, Guezennec7, Cole and Glausers, and Stokes and Glauserg). In 2001, Naguib et al.'0 was the first study to use SE to estimate the flow field from the wall-pressure signature in order to determine the flow sources associated with surface pressure events in a turbulent boundary layer. Simultaneous microphone and hot-wire measurements were acquired over a Reynolds number range of 1437 S R69 S 5670, based on momentum thickness. The velocity measurements were obtained in the buffer and logarithmic regions of the flow field. The study estimated the velocity above the surface using both linear and quadratic stochastic estimation (LSE and QSE, respectively). Results showed that it was necessary to include the quadratic term for estimating the flow field as LSE 132 did not capture the conditional average accurately, but the quadratic estimate showed good representation. In 2002, Murray and Ukeileyll used stochastic estimation models for resolving the temporal evolution of the velocity field above the surface from surface-pressure information in a 2D, open-cavity flow. The authors generated their surface pressure and velocity data using numerical simulation in order to test the ability of SE to estimate the flow structures in a separating-flow geometry. The flow Mach number was 1.5 and the cavity had a length to depth ratio of six. F luctuating velocity, turbulent kinetic energy, and spanwise vorticity were used to compare the estimated and simulated results. Both single-point and multi-point, linear and quadratic SE were used to predict the flow field from the wall-pressure information. Murray and Ukeileyll reported that the estimation strongly depends on the location of the pressure measurement. They found that adding more pressure measurement locations in the LSE only improves the estimation in regions close to where the pressure measurements were acquired. Murray and Ukeileyll also found that the linear estimate was able to predict the majority of the flow field, but that adding the quadratic term was necessary to accurately represent the turbulent energy of the flow. In terms of vorticity, the linear term estimated most of the vorticity. However, the addition of the quadratic term was necessary to capture the finer details. Finally, the authors were able to time-resolve the flow evolution using SE and noted that the quadratic estimate predicted the dominant features in the flow very well when compared to the instantaneous simulated data. Later in 2005, the same authors12 acquired simultaneous PIV and pressure measurements in a Mach 0.2 cavity flow with two different length to 133 depth ratios of 5.16 and 1.49. Using wall-pressure-based quadratic stochastic estimation, they were successful in using this technique to characterize the time-dependent flow behavior within the cavity from the experimental measurements. To the best of the author’s knowledge, no studies have investigated the relation between the velocity field and the wall-pressure in an axisymmetric backward-facing step in order to identify the flow mechanisms generating the surface-pressure signature. Thus, the objectives for this chapter are to identify the flow structures that generate the surface- pressure signature using wall-pressure-based single-point linear and quadratic stochastic estimation and multi-point linear stochastic estimation. Based on the instantaneous spatial pressure distribution, proper orthogonal decomposition is also used to highlight the dominant mode shapes in the pressure signature. These dominant shapes are then used along with stochastic estimation to explore the time evolution of the large-scale flow structures that are linked to the wall-pressure-generation-process within the shear layer and downstream of reattachment. For brevity, the coupling of the flow field with the surface pressure using stochastic estimation will be presented for the Reh = 808lflow case only. 134 6.2 Results and Discussion 6.2.1 Average Wall-pressure Signature versus Mean Velocity Field The purpose of this section is to compare the wall-pressure signature with the velocity field above the surface. Thus far, the pressure signature has been presented in Chapter 4 and the velocity field has been analyzed in Chapter 5. This section combines the two and looks at possible links between the behavior of the wall-pressure statistics and those of the above-surface flow field. Figure 6.1 gives the auto-correlation from the wall-pressure signature in the top plot and the uRMS/U.o of the velocity field in the bottom plot for the Re;l = 8081. The auto-correlation, contour plot and axes are the same as those given in Figure 4.3 and are described in detail in Section 4.2.2. The uRMg/U.0 contour plot and axes are presented in Figure 5.3 and described in Section 5.2.2. Both plots are reprinted here for ease of discussion. In addition, the two plots cover the same streamwise extent and are lined up with respect to each other. By aligning the two plots, a comparison between the wall- pressure and the velocity field streamwise development can be made. In addition, inferences made about the velocity field using the pressure signature in past literature by Heenan and Morrison”, Driver et al. '4, and Hudy et al. '5, to name a few, can be verified. It can be seen in Figure 6.1 that the abrupt transitions seen in the auto-correlation, contour plot are also evident in the uRMS/Uw, contour plot at the same streamwise locations. Near the step and immediately downstream of separation (0 < x/h < 2), as described in Section 4.2.2, the region is dominated by a low-frequency disturbance that has been associated with the flapping of the shear layer in literature. It has been assumed that this region is dominated by low frequency disturbances because the shear layer over 135 this streamwise extent is narrow and far away from the microphone array. Therefore, the disturbances that form in the shear layer due to the Kelvin-Helmholtz instability do not affect the pressure signature since they are weak and far away from the embedded wall- array. Examining the uRMs/Uoo, contour plot over the same streamwise extent, it can be seen that the shear layer within the 0 < x/h < 2 region is relatively thin and that most of the intense turbulent activity occurs within the shear layer. In addition, the shear layer within this region is far away from the wall-pressure array. This comparison does not affirm that the low-frequency is associated with the flapping of the shear layer but does give reason as to why the pressure signature is not dominated by high-frequency fluctuations from structures within the shear layer. The second region in Figure 6.1, 2 < x/h < x,, is dominated by higher frequencies relative to that seen in the region upstream of x = 2h in the auto-correlation. The abrupt transition seen in the auto-correlation around x = 2h mirrors the transition seen in the uRMs/Uw contour plot. At this point in the mean-velocity field, the shear layer is seen to spread vertically over a very short streamwise extent. The vertical spread of the shear layer is roughly equal to the height of the step. It has been inferred from pressure measurements that the reason the auto-correlation is dominated by shorter time scales in this region is due to the proximity of the shear layer with respect to the wall-pressure array. This is confirmed by the observation that the peak, ums/U.o values seen in this region are in fact closer to the wall-pressure array compared to the location of the peak, uRMs/Uw values in the shear layer near separation. Thus, the structures within the shear layer dominate the pressure signature in the 2 < x/h < xr region as stated in literature. 136 Beyond x,, the time scales in the auto-correlation gradually become even shorter, indicating that this region is dominated by even higher frequency structures compared to the 2 < x/h < 4 region. As suggested in Section 4.2.2, it is hypothesized that the narrowing of Rp'p' over the x/h = 4 to 8 range is caused by the dominant flow structures either convecting at a faster speed and/or changing shape in terms of getting smaller in size. What is evident from the uRMS/Uoo, contour plot is that the vertical extent of the shear layer narrows as the shear layer pulls away from the wall on the lower end and the contours curve toward the wall on the upper end of the shear layer. The y/h position of the center of the shear layer, taken at the postion of the peak uRMS/U.o value at each streamwise location and marked by “x” in Figure 6.1, increases slightly in the region downstream of reattachment. To understand more about this region, it is necessary to look at Figure 6.2. Figure 6.2 gives the same auto-correlation, contour plot as shown in Figure 6.1. In addition, a U/U.o versus x/h plot is also given. The velocity values plotted in this figure are the local U values at the center of the shear layer, shown in the bottom plot of Figure 6.1. In free, shear-layer studies, such as Ho and lHuerrel6 and Brown and Roshko”, it is recognized that the average velocity at the center of the layer is approximately equal to the convection velocity of flow structures within the shear layer. Thus, as an estimate, it may be assumed that the velocity at the center of the shear layer in this BF S flow is an approximation of Uc of the flow structures within the shear layer. Focusing only on the U/IJ.,o values downstream of x,, it can be seen that the local, mean velocity at the center of the shear layer increases with increasing streamwise distance. This supports the hypothesis that downstream of reattachment the dominant structures in 137 the flow are convecting faster with increasing x (i.e., accelerating) and therefore a shift in dominance towards smaller times scales (or higher frequencies) of the corresponding wall-pressure signature is observed. Upstream of reattachment, the local mean velocity at the center of the shear layer is around U z 0.45Uoo, in the 0 < x/h < 2 region, except for the two streamwise locations that show a dip in the velocity roughly around x = 0.3h and 0.5h (Figure 6.2, bottom). This dip most likely is due to an error in the PIV measurements since the locations of these measurements are near the step and PIV has difficulty with near-wall accuracy. At the downstream end of this region, the first transition in the convection velocity is observed when the shear layer balloons out toward the wall. Morris and Foss18 found Uc to increase steadily through x/0 = 120 in their single-stream shear layer study. Here, the local convection velocity shows a dramatic decrease in the velocity at x/h z 2 (x/0 = 13), as the shear layer spreads out and the dominant flow structures slow down. The local mean velocity at the center of the shear-layer transitions again and starts to increase around x/h = 3. In investigating the uRMS/Uoo, contour plot in Figure 6.1 again, it appears that the shear layer reaches its full vertical extent at the streamwise location of x/h = 3. It is difficult to verify this in the mean-vorticity-based shear-layer edges plot shown in Figure 5.10. Although it is evident that the shear layer reaches the wall beyond x/h = 2. Once the shear layer reaches its full extent, the flow structures start to increase in speed and continue to do so beyond reattachment. Another interesting comparison is the local, mean-velocity plot with the wall- pressure cross-correlation contour plot from Figure 4.8 for the Reh = 8081. This comparison is shown in Figure 6.3. In Chapter 4, the cross-correlation, contour plot was 138 used to identify an average, convection velocity upstream and downstream of reattachment. It was found that the average convection velocity upstream of reattachment was lower than Uc downstream of x,. This same trend is found in the local, mean-velocity plot in Figure 6.3. By averaging the local, mean velocities over the same region from which the convection velocities were calculated, the results showed an increase downstream of x,. More specifically, from the cross-correlation, Uc was found to be 0.34U.o upstream of xr and 0.46Uoo downstream of x,. From averaging the local mean velocities, Uc = 0.26U.o and 0.41U.0 for upstream and downstream of x,, respectively. While the trend of Uc increasing downstream of xr is the same between the two methods, the actual calculated convection velocity values differ. This difference could be due to how the two Uc values were calculated from the cross-correlation contour plot, as explained in the following paragraph. Figure 6.4 investigates the convection velocity calculated from the positive peaks in the cross-correlation contour plot and compares the results with the local mean velocity values located at the center of the shear layer. The center of the shear layer in Figure 6.4 has been defined based on: 1) the peak of the uRMg/U.o at each x/h location, and 2) the peak of the mean vorticity at each x/h location. It can be seen that the local, mean velocities at the center of the shear layer based on peak, mean vorticity are greater in value than the velocities at the center of the shear layer based on the location of the peak uRMs/Um values. On the other hand, the convection velocity of the dominant wall- pressure-generating flow structures were calculated in two ways using the peaks of the positive lobe from the cross-correlation shown in Figure 6.3. The first method involved fitting a linear trend to the positive peaks as described in Section 4.2.4. This gives two 139 convection velocities: 1) upstream of x,, Uc = 0.34Uoo, and 2) downstream of x,, Uc = 0.46Uoo. The other method explored fitting a parabola to the positive peaks of the cross- correlation. The results from the parabola fit are given in Figure 6.4. It can be seen that while the Uc values from the linear fit do a decent job estimating the convection velocity within the particular streamwise extent, the Uc versus x/h line from the parabola describes the velocity variation along the shear layer center line more accurately within the region. In addition, it seems as though the parabolic fit describes Uc within the reattached boundary layer better than in the shear-layer region. Collectively, the above analysis establishes on a firm basis what has been inferred in the past from wall-pressure or limited wall-pressure/velocity measurements concerning the relationship between the shear layer flow structures and the wall-pressure signature. 140 _. ,IIIIOIIIIOI x/h Figure 6.1 Auto-correlation from wall-pressure signature (top plot) and uRMSIUw of the velocity field (bottom plot) for Reh = 8081 141 O-IINOOAU'I tUw/h c'nchtb-K 0.5- b - 0.45L . 0.4» ‘ i ' ~ U/Um 0.35— ' ‘ — 0.3 F ‘2 . - 0.257 “ ' — 0.2 . , _ 015,#'_l*.g»_l—l x J_ 1 | 41— 0 1 2 3 4 5 6 7 8 Figure 6.2 Auto-correlation from wall-pressure signature (top plot) and local mean velocity at the center of the shear layer versus streamwise position (bottom plot) for Reh = 8081 142 O-‘NOOAU'I lllfi tU/h u..=o.34 <——> u..=o.26 U°°=0.46 u.,=o.41i 0.55 r 0.5 ~ 0.45 - D 0.35 - 0.3 - 0.25 L ‘ 0.15 _.___._ # ~—J—— . 0 1 2 3 4 Figure 6.3 Cross-correlation from wall-pressure signature (top plot) and local mean velocity versus streamwise position (bottom plot) for Re.I = 8081 143 7 ‘ l —a— Peak um I l 0.9|t 1+ Peak vorticity H b Polynomial fit 08 l * Linearfitxr :1 Figure 6.4 Local, mean velocities from velocity field and average convection velocities calculated from wall-pressure, cross-correlation peaks for Re.I = 8081 144 6.2.2 Single-point Linear and Quadratic Stochastic Estimation Linear stochastic estimation was used to estimate the u' and v' components of the velocity based on a known wall-pressure signature according to l7'(xo + may. + Ayat) = A.(Ax,Ay;xosy.,)p'(xoaymt) (6.1) 3'06. + may. + Ayat) = AJAX.Ay;xo,y.,)P'(xoay0J) (6.2) where 5' and it" are the estimated streamwise and normal fluctuating velocities, respectively, x0 is the streamwise position of the known variable (wall-pressure condition), Ax is the streamwise distance between the known and estimated variables, y is the normal coordinate (also the normal distance between the known and estimated variables), t is the instant in time the pressure event occurs, p' is the measured fluctuating pressure on the wall, and Au and Av are the linear estimation coefficient terms for 17' and 37' respectively. For more detailed information regarding single-point linear stochastic estimation, refer to Appendix 8.6. Single-point, linear, stochastic estimation requires information from the cross- Correlation between the fluctuating pressure signature at one streamwise position and the fluctuating velocity field in order to determine the linear-estimation coefficient. Figure 6.5 shows the flooded-color, contour plots of the cross-correlation between the fluctuating pressure at a particular streamwise position and the mean-removed, streamwise-velocity component (i.e., Ruvpo) at every node in the flow field. Figure 6.6 display similar plots for the mean-removed, normal-velocity component (i.e., RV-p-). The cross-correlations are shown for three different streamwise positions of wall-pressure measurements, x/h = 3.51, 4.68, and 5.46 at Reh = 8081. The first position is located upstream of reattachment, and the last position is located downstream of reattachment as 145 indicated by the arrows in the figures. The middle position, x/h = 4.68, is located near reattachment, approximately x/h = 0.20 downstream of x,. These positions were selected in order to highlight the main correlation features at different points in the flow field including within the separated shear layer, around the mean reattachment point, and in the reattached boundary layer. In addition, it is known that in these regions the wall- pressure fluctuations are generally high as shown in the RMS pressure distributions in Figures 4.1 and 4.2. In Figure 6.5 and Figure 6.6, the streamwise coordinate is shown on the abscissa as x/h and the normal direction is plotted along the ordinate as y/h. The color bar, which gives the magnitude of the cross-correlation normalized by (‘/2pr2)Uw, has been custom- made so that the transition from red to blue indicates the narrow region around zero correlation. The normalization used for the cross-correlation allowed for identification of Correlation regions that are strong globally (i.e., over the entire flow domain). The traditional means of normalization using the local uRMS and plum proved difficult to use to compare between the strength of the correlation at different locations in the flow because of the variation in RMS values from one point to another. However, RMS-based normalization is useful in judging the significance of the calculated, correlation values. Thus, it is mentioned here that when normalized by local RMS values, the Ru'pv and vap' correlations yielded values between -O.4 to 0.4. In Figure 6.5 and Figure 6.6, only correlation values below/above the uncertainty level (correlation value of i0.00008) are shown in order to highlight the significant correlations in the flow field (note again the apparently small values in the figure are due to normalization by the freestream velocity). 146 Finally, an arrow with the label p' indicates the streamwise position for the microphone used in each of the correlations. Starting with the first position, x/h = 3.51, the correlation between the flow field and the pressure is strong as seen in Figure 6.5. A positive, peak correlation can be seen, inclined at an angle, at the x/h = 3.51 microphone location. This positive peak stretches from x/h = 2 - 4.5 and from y/h = 0 — 1.0, indicating good, positive correlation between the streamwise velocity within this region and the pressure at the x/h = 3.51. A negative correlation region can also be seen in Figure 6.5 downstream of x/h = 3.51. It is found inclined at an angle mirroring the positive peak and stretches from x/h = 2.5 — 8, over the y/h = 0 — 2.0 range. The same observations can be made for the other two streamwise positions, x/h = 4.68 and 5.46, shown in Figure 6.5. Note, though, that in all cases the highest correlation value is found very close to the location of observation of the wall pressure. Additionally, the negative correlation region is confined closer to the wall for Wall-pressure observations near and downstream of reattachment in comparison to the most upstream location of pressure observation. Finally, a small region of negative Correlation upstream of the positive correlation region is found near the wall for x/h = 3 .5 1 . Figure 6.6 shows the vap'/('/2pU.o3) contour plots for Reh = 8081 at the same three streamwise positions. Both positive and negative peaks can be seen in the correlation plots; although, in opposition to the Ru-p' results, the main, negative peak is now located directly above the microphone used for the correlation, and the main, positive peak is located farther downstream. 147 One point worth noting is that the peak correlation values, both positive and negative, at x/h = 4.68 are noticeably stronger than those at the other streamwise positions. This location is about 0.2h downstream of xr when the structures within the reattached boundary layer are very energetic and are close to the wall. The correlation is expected to be high in this region. Using the cross-correlation results between pressure and velocity in Figure 6.5- Figure 6.6, the linear, estimation coefficients at each node in the flow field could be calculated (see Equation (8.7) in Appendix 8.6). Once the coefficients were determined, the estimation of the flow field could be performed based on a single-point pressure event. Thus, the flow field was estimated based on the condition that the pressure at a particular streamwise position was five times the RMS. This was computed for both a negative and positive pressure event for all three streamwise locations shown in the cross-correlation, contour plots. Specifically, SE was used to estimate the mean-removed velocities associated With the pressure events. Subsequently, the mean-velocity field was added to the eStimated field to provide an estimation of the full flow field rather than the turbulent field alone. The vorticity and strain rate (Em) fields were then calculated to allow the Observer to better see the coherent motions in the flow field. These two measures were used because they are calculated from velocity gradients and therefore, unlike velocity- V ector field observations, they are independent of the choice of the translation velocity of the flame of reference. The difficulty in viewing the flow field from a convective frame of reference is in determining the appropriate convection velocity of the flow structures associated with the pressure signature used in the estimate. Subtracting the incorrect 148 convection velocity from the estimated field would give false information regarding the true nature of the flow features. Moreover, as seen in section 6.2.1, the convection velocity of the flow structures responsible for the generation of the wall pressure depends on the streamwise location. Hence, a particular choice of the convection velocity may be suitable for proper observation of the estimated velocity field at one point in the flow but not at others. The description of vorticity and the equations used in its calculation were given in Section 5.2.4. On the other hand, strain rate is a measure of the rate of geometrical distortion of a fluid element. Rate of strain is a symmetric, second-order tensor, and hence is fully described using six different components (three corresponding to shear and three to elongation). With the availability of planar, velocity-field data in the current study, it was possible to compute only one shear and all three elongation, strain components. The computed shear strain rate component is the in-plane one (gxy). Because this component was found to be substantially larger than the elongation Components, only the former is presented here. The equations used to calculate the in- plane shear strain are given in Appendix 8.7, with additional information available from Raffel et al. '9 Figure 6.7 — Figure 6.12 show the results of linear, stochastic estimation using Vorticity and in-plane shear strain rate (hereafter referred to as just “strain” for brevity). In addition, results from the quadratic stochastic estimation (QSE), which is explained in more detail below, are also shown. The vorticity and strain from both the LSE and QSE for a negative-pressure event at the streamwise position x/h = 3.51 are shown in Figure 6.7 and for a positive pressure event in Figure 6.8. The same results are shown in Figure 149 6.9 and Figure 6.10 for the streamwise position x/h = 4.68 and in Figure 6.11 and Figure 6.12 for x/h = 5.46. The abscissas and the ordinates for the vorticity and strain fields are the same as described in the cross-correlation results in Figure 6.5 and Figure 6.6. The color bar indicates the magnitude of the vorticity (to; = (ozh/Um) and shear strain (éxf = éxyh/Uoo). The magnitudes were plotted on a scale of -4 to 4. Vorticity and shear strain levels between -0.4 and 0.4 were set at zero in order to minimize noise in the contour plots and enhance the visibility of the dominant features in the flow field. The single-point, quadratic, stochastic estimation was used as a means of verifying the accuracy of the single-point LSE. QSE was used as a check to see what additional information the quadratic term provides in the flow field and, more importantly, how well does LSE estimate the flow field. The equations used to estimate the u' and v' components of the velocity based on the pressure signature using quadratic stochastic estimation are the following: 17 ' (x. + Ax, y. + Ay, t) = A..q....l(Ax. Ay; x0 , y. )p'(x. , y. , t) + B..(Ax, Ay; 19.32..)19'2 (trays!) (6.3) 37' (x. + Ax, yo + Ay, t) = A.,...uw, Ay; 36.. , y. )p'(xo , yet)- + 3.14% 4y; 36.. , Y. )p'2 (X. , yo , t) (6.4) where ‘7' and ‘7' are the estimated streamwise and normal fluctuating velocities in the flow field respectively, x0 is the streamwise position of the known variable, Ax is the streamwise distance between the known and unknown variables, y0 is the normal position of the known variable, Ay is the normal distance between the known and unknown 150 variables, t is the instant in time the pressure event occurs, p' is the magnitude of the wall- pressure event, Amquad and Av,quad are the linear estimation coefficient terms, and Bu and B" are the quadratic estimation coefficient terms. More details regarding QSE, including the equations for the coefficients, are given in Appendix 8.6. At x/h = 3.51, shown in Figure 6.7 and Figure 6.8, the top two plots give the vorticity and strain fields from LSE and the bottom two plots show QSE results. In Figure 6.7a, a region of localized negative-vorticity concentration is shown directly above the microphone used in the estimation. The concentration of vorticity can be seen more clearly in the QSE results in Figure 6.7c, suggesting the presence of a vortical structure immediately on top of the point of observation of the negative pressure event. Inspection of the corresponding strain maps (both linear and quadratic) indicates no particular features in the immediate vicinity of the negative-pressure event. Instead, a hint of a localized strain peak downstream of the event location is found in the QSE results (Figure 6.7d). This feature is smeared in Figure 6.7b where the LSE results are Shown. The above suggests that the generation of negative-pressure events on the wall is linked to the passage of large-scale (order h) vortical features. A comparison between the LSE and QSE results shows that the global flow features are the same as found using LSE, but the features are enhanced and magnified by the addition of the quadratic term. Nonetheless, the basic structure of the estimated flow field is the same by both techniques. LSE is able to capture the large-scale structures with some smearing, while QSE defines the details of the large-scale structures more clearly. 151 In Figure 6.8, the same results are seen for a positive pressure event. The LSE vorticity field in Figure 6.8a shows that the estimation using this positive pressure signature at x/h = 3.51 produces localized concentration of vorticity upstream and downstream of this streamwise position. Again, this is better defined in the QSE results in part c of the same figure. Whereas, in Figure 6.8 b & d, a region of high strain is found directly above x/h = 3.51. This localized strain-rate peak being found in-between two vertical structures suggest its association with the high-strain region of a saddle point between two vortices. The above findings are consistent with the study of Naguib and Koochesfahani20 who found that for the flow field of an axisymmetric vortex ring impinging on a flat wall, strong positive-pressure-generation was associated with the large strain-rate resulting from the interaction of the primary and secondary vortices. Additionally, they also identified the core of the vortex structures to be associated with strong generation of negative pressure. These findings were related to the earlier study of Bradshaw and Koh21 in which it was shown that, in general, positive pressure generation can be solely attributed to fluid strain while negative pressure production arose due to vorticity. Similar results can be seen at the remaining two positions, x/h = 4.68 and 5.46, given in Figure 6.9 — Figure 6.12. One interesting observation made in Figure 6.11c and Figure 6.12c is the existence of a region of positive vorticity downstream of reattachment. In Figure 6.11c, this region extends roughly in the range x/h = 4.5 — 5.2 and is found very close to the wall just upstream of the vortex structure immediately above the negative- pressure event. In Figure 6.12c, the positive vorticity region is found downstream of a localized negative-vorticity structure upstream of the event location, but farther away 152 from the wall (y ~ h). It is hypothesized here that the region of positive vorticity near the wall is associated with the generation of an opposite-sign secondary vortex near and downstream of reattachment via the interaction of the large-scale negative-vorticity structures with the wall. Subsequently, the secondary vortex gets “pulled away” from the wall by the primary structure. Thus, the originally near-the-wall positive-vorticity feature orbits the primary structure reaching y ~ h. The described scenario, which is derived from the observations of Gendrich et al.22 of the interaction of an axi-symmetric vortex ring with a flat wall, was also observed in, at least, one instance of a movie of time- resolved multi-point LSE of the flow field (the multi-point estimation and associated flow field is addressed in section 6.2.4). 153 \/II = 3.51 \/h = 4.68 \/II = 5.46 Figure 6.5 Flooded-color, contour plots of Ruvp~/(‘/szw3) for Re;I = 8081 154 \/h = 4.68 \/h = 5.46 Figure 6.6 Flooded-color, contour plots of vapt/(Vszm’) for Reh = 8081 155 a) 012 3_4 5 6 7 8 3 ”1T*kfifilfi'i‘lifiiiri’fiiAfiFi'TFFFfT’: Figure 6.7 Single-point stochastic estimation results for a negative pressure event (P' = -5pRMs) at x/h = 3.51 for Re.I = 8081. Linear results: a) vorticity and b) shear strain rate. Quadratic results: c) vorticity and d) shear strain rate. 156 Figure 6.8 Single-point stochastic estimation results for a positive pressure event (p' = +5pRMS) at x/h = 3.51 for Re.I = 8081. Linear results: a) vorticity and b) shear strain rate. Quadratic results: c) vorticity and d) shear strain rate. 157 y/h 8 3F_T‘—A fii If f AYAF A? Af Afi_—T—_-FT—F4 c) Figure 6.9 Single-point stochastic estimation results for a negative pressure event (P = ‘5pnMs) at x/h = 4.68 for Re.I = 8081. Linear results: a) vorticity and b) shear strain rate. Quadratic results: c) vorticity and d) shear strain rate. 158 3"- T7 F777r_—l"_' _VflF'FiivFvT—F_l_l—_F ——I—T——l l a) Figure 6.10 Single-point stochastic estimation results for a positive pressure event (P = +5pRMS) at xlh = 4.68 for Re.I = 8081. Linear results: a) vorticity and b) shear strain rate. Quadratic results: c) vorticity and d) shear strain rate. 159 L' f’ A -7_—-T' —-T_-fl———I—a 8) 3 r—‘T—‘7 ‘— F A’AY -4 -3 -2 —1 0 1 2 3 4 Figure 6.11 Single-point stochastic estimation results for a negative pressure event (p' = -5pnms) at x/h = 5.46 for Re.I = 8081. Linear results: a) vorticity and b) shear strain rate. Quadratic results: c) vorticity and d) shear strain rate. 160 Figure 6.12 Single-point stochastic estimation results for a positive pressure event (p' = +5pams) at xlh = 5.46 for Re.I = 8081. Linear results: a) vorticity and b) shear strain rate. Quadratic results: c) vorticity and d) shear strain rate. 161 6.2.3 Instantaneous, Pressure-Signature Characteristics 6.2.3.1 Instantaneous, pressure signature Although the above single-point analysis gives information regarding the flow structures associated with the generation of strong, wall-pressure events, the results are biased to one particular microphone at a single, streamwise location. This assumes that a flow structure only affects a single sensor and therefore, only the pressure signature from that sensor is needed to reconstruct the associated flow field. In general, however, flow structures will imprint a pressure signature on multiple microphones. Additionally, it can be seen from the cross-correlation plots in Figure 6.1 and Figure 6.2 that a correlation between the velocity and a single wall-pressure sensor only extends over a limited part of the velocity field. Therefore, in order to estimate the flow structures over the entire flow field, it is necessary to have several sensors at various streamwise locations. The array of 32 microphones embedded in the wall of the axisymmetric backward-facing step model provided this level of detail by measuring the pressure signature at multiple points in space. The pressure signature from the microphone array was analyzed statistically in Chapter 4 and an average spatio-temporal picture of the pressure field was drawn based on the auto-correlation, the cross-correlation, and the frequency-wavenumber contour plots. The statistics regarding the velocity field have also been presented in Chapter 5. The question that remains, however, is what are the instantaneous characteristics of the flow structures that are associated with various instantaneous spatial pressure imprints? In other words, what flow field characteristics produce the observed pressure signature and how do these flow features develop? 162 Answering these questions requires investigating the spatial pressure signature and corresponding velocity field over time. Starting with the instantaneous, spatial- pressure distribution, Figure 6.13 - Figure 6.14 show examples of the pressure signature over all 32 microphones at different instants in time. The abscissa gives the streamwise distance and the ordinate gives the unsteady pressure in Pascals. Each signature was band-pass filtered between 20-160 Hz (M., = 0.024-0.19). According to past literature as explained in Section 4.2.3 and seen in the frequency spectra in Figure 4.6, the most energetic part of the pressure fluctuations occurs within this bandwidth. This frequency range is mostly beyond the flapping frequency and is where the frequencies associated with the passage of the shear-layer structures occur. Thus, the band-pass filtering allows one to focus the analysis on the flow features associated withthe shear layer; i.e., those associated with the downstream-traveling pressure disturbances corresponding to the main spectrum ridge in the right half plane of the wall-pressure frequency-wavenumber (see Figure 4.9a) Looking at all six spatial-pressure signatures presented in Figure 6.13 and Figure 6.14, it can be seen that the general, signature shape changes along with magnitude. At certain instances in time, the pressure signature seems random in distribution and relatively low in magnitude as seen in Figure 6.13a, Figure 6.13c, and Figure 6140. These signatures show low-pressure levels, roughly between -1 to +1 Pascals, over most of the spatial extent. In addition, spatially, the pressure distribution seems unorganized compared to, for example, the signatures in Figure 6.13b and Figure 6.14a-b, where a certain level of organization, almost a modulated “sinusoidal-like” distribution, can be 163 seen. The corresponding pressure magnitude spans a larger range (-4 to 4 Pascals) than seen with the random signatures. Similar observations were made by Cherry et al.23 in their investigation of the flow around a thick, splitter plate at Re = 32,000 based on model thickness. Two Setra 237 low-pressure transducers were mounted inside their model and were used to measure the fluctuating pressures on the wall. Smoke visualization was used to monitor the flow field simultaneously as the pressure fluctuations were acquired. From their dataset, Cherry et al.23 synchronized the pressure traces with the flow visualization and were able to describe the pressure signature in time as having various shedding phases. These 1.23 One such observation was phases were defined by observations made by Cherry eta the shedding of pseudo, periodic trains of vortical structures fiom the reattachment zone. The smoke visualization showed that these structures had a characteristic spacing of approximately 60-80% of the reattachment zone. Large-scale vorticity was also observed to have an irregular shedding pattern and there were quiescent periods where no large- scale shedding occurred. These quiescent periods in the flow field were found to be synchronized with periods in the pressure trace where the pressure fluctuations were random and low in magnitude. During a shedding phase, the pressure signature was found to be energetic, showing higher peaks and lower valleys (in magnitude) than in the quiescent phase. Based on their observations, Cherry et al.23 stated that the pressure Signature of a single microphone in time showed substantial variation. The same is true for the pressure traces measured in the current study. It is very evident from these six pressure signatures that the spatial shape of the pressure distribution varies in time between a quiescent, random signature and a more organized 164 energetic phase. Within the latter a modulated, sinusoidal-like shape seems to recur (Figure 6.13b and Figure 6.14a-b). This offers reason to believe that there is an underlying pattern to the energetic, pressure distributions that appears ever so often in the flow field. In order to investigate such an idea, a mathematical technique that extracts the various mode shapes (spatial shapes) of the most energetic pressure signature was utilized to conduct further analysis. The results of applying the technique, which is known as Proper Orthogonal Decomposition (POD), are given in the following section. Pa Pa Pa Figure 6.13 Spatial, pressure signature for Re.I = 8081 case at three instants of time: a) 12.00 s, b) 12.01 s, and c) 12.02 s 165 Figure 6.14 Spatial, pressure signature for Re.| = 8081 case at three instants of time a) 8.80 s, b) 8.81 s, and c) 8.84 s 166 6.2.3.2 Proper Orthogonal Decomposition Proper Orthogonal Decomposition (POD) is an unbiased technique for extracting organized signatures (or mode shapes), which was first introduced in 1967 by Lumley24 for use in problems of turbulence. More specifically, the technique identifies the more energetic structures in a flow field. The idea behind POD is that it assumes that a random field can be represented by a set of deterministic, spatial functions superposed with coefficients (or amplitudes) that are random functions of time. Fourier decomposition assumes the same ideology, except that the functions are predetermined to be sinusoidal and are made to fit by solving for coefficients that would make the functions define the random field. The uniqueness of POD stems for its ability to extract both the functions and the coefficients used to define the random field based on Statistical properties of the field. In essence, for the purposes of the current study, POD allows the random field to dictate the modal shapes that make up the spatial pressure field. In addition, POD “ranks” each mode according to its contribution to the wall-pressure signature with the first mode providing the most energetic contribution, followed by the second one and so on. Hence, POD was used in this study to find the dominant spatial pressure distributions underlying the measured pressure field and to rank these distributions according to their average contribution to the pressure field. For more information regarding the background of POD and the implementation of the technique, refer to Berkooz et al.25 The results for the POD analysis are shown in Figure 6.15, which gives the first four modal shapes plotted as a function of streamwise distance (x/h). A total of 32 modes were calculated using POD since there were 32 wall-pressure sensors. In order to be able to discuss these shapes in their full context, it is necessary to look at the contribution of 167 each mode to the pressure signature. Figure 6.16 shows the percentage of the fluctuating- pressure-field energy recovered from the POD representation, plotted versus the number of POD modes used to represent the pressure signature. The inclusion of all 32 modes in the representation recovers 100% of the energy in the signal. What can be seen from the percent energy distribution in Figure 6.16 is that mode one and mode two, combined, make up almost 40% of the energy content in the signal. Modes three and four make up an additional 20% of the energy content, with each mode representing approximately 10%. This is significant in that 60% of the energy in the pressure signatures can be described using only the first four modes from the POD analysis. The next two modes, five and six, add an additional 10%. Thus, 70% of the pressure signature can be recovered using six out of the 32 modes. This helps to simplify the analysis of the pressure signature and allow for focus on the most dominant spatial pressure distributions. Modes one and two in Figure 6.15 represent the largest energy content and are noticeably similar in spatial distribution. The difference between the two modes is that mode two is shifted to the right with respect to mode one. For instance, notice that when mode one peaks, mode two crosses zero approximately. The same is true when mode two peaks, mode one crosses zero as shown at x/h = 4. The similarity between the two modes with only a streamwise shift difference indicates that the two together represent a single mode that is convecting‘. The same observation can be made for modes three and ‘ It can be shown that POD analysis of a convecting sinusoidal wave gives only two modes: sine and cosine (i.e., sine with 90 degrees phase shift) functions. The combination of the sine and cosine functions with different weights (i.e., amplitudes) can then produce the proper instantaneous phase of the wave. 168 four in Figure 6.15. This is consistent with the observation that the pairs of modes have similar energy percentage in Figure 6.16. Thus, it can be stated that a convecting mode with a spatial shape given by some weighted average of modes one and two in Figure 6.15 produces 40% of the fluctuating-pressure energy. Moreover, another convecting mode, given by the combination of modes three and four in Figure 6.17, is responsible for the generation of an additional 20% of the unsteady-pressure energy. Based on this analysis, the first four modes will be the focus of the velocity field analysis undertaken in the next section. In addition, since mode one and two represent parts of a convective mode, the two modes are added together with appropriate weighting factors (as will be seen shortly) to form the convecting mode called mode A. The same can be done for modes three and four. The addition of these two modes is called mode B for discussion purposes. 169 '1; :9: mouse] l | l I l | ”..-—___... , I -1 ._ -.- ___ __VL” .7 ._ 7 ._._ ._._ _-___L____._ _ .4 _. w ._.-s _.lw _.____.__L-_. - J n 2 4 6 8 10 12 Figure 6.15 First four mode shapes defined by POD for the pressure distribution at Re.| = 8081 170 100 8 i l 6. .1 l ‘ l 4‘ .‘ . l 2 II; "t 5 7 911131517192123252 PODMode# % Energy 0 O O O C Figure 6.16 The cumulative distribution of energy over all 32 modes for the pressure distribution at Re.I = 8081 It is important to point out that the relative importance of the different modes as shown in Figure 6.16 represents an average over time. For example, although mode one is the most dominant on average there are time instants at which this mode may not be observed at all. Thus, each mode has a time-dependent coefficient, or amplitude an(t), that is employed in superposition of all modes to reconstruct the instantaneous pressure signature. The reconstruction is given by the following equation: P(Xat) = a.(t)¢1(t)+ a2(t)¢2(t)+ 33(t)¢3(t)+- . -+ 832(t)¢32(t) (65) where p(x,t) is the instantaneous spatial wall-pressure signature and at — a3; are the amplitude coefficients for each of the mode functions, which are defined as (in - (1:32. The advantage of equation (6.5) is that by limiting the summation to one or a few modes, the Space-time evolution of the mode(s)’s pressure signature can be created. For example, if 171 only the first two terms on the right hand side of equation (6.5) are used, one would obtain the spatio-temporal-pressure signature of mode A. This signature can then be combined with LSE to investigate the development of the large-scale structures associated with this mode. However, before looking at the large-scale-structure development, it is instructive to examine the vorticity and strain fields associated with the individual mode shapes, focusing only on the first four modes. Because a particular mode is defined based on the pressure values at all 32 microphones, these vorticity and strain fields were estimated using multi-point LSE employing the mode’s pressure signature (as seen in Figure 6.15). 172 6.2.4 Multi-point Linear Stochastic Estimation 6.2.4.1 Velocity field associated with the pressure signature of POD modes Multi-point, linear, stochastic estimation is an extension of the single-point LSE equations as shown in Section 6.2.2 5': Am,p'(xo,t) + Aulp'(x,,t) + Auzp'(x2,t) + ...+ Amp'(xn,t) (6.6) V'= A...p'(x.,t) + A.,.p'(x.,t) + Anp'(xz,t) + + A.,.p'(x.,t) (6.7) where, Auo-Aun and Avg-Ayn are the linear-estimation coefficients for each pressure point used in the LSE. For more detailed information, including how the coefficients are calculated, refer to Appendix 8.6. Note that the multi-point, linear, stochastic estimation is used to investigate the flow field associated with the spatial, pressure signature instead of single-point, LSE due to its increased accuracy in estimating the instantaneous flow field as will be shown in Section 6.2.4.3. Figure 6.17 - Figure 6.24 display the vorticity and shear strain field contour plots for each of the first four modes. The abscissa and the ordinate give the streamwise and normal direction, respectively, normalized by the step height. The color bar shows the magnitude of the vorticity and shear strain, ranging from -4 to +4. The contours plotted are between -4 and -0.5 for vorticity and between +0.5 to +4 for shear strain. This is in an effort to clean up the plots by removing some of the background noise associated with the accuracy of evaluating the derivatives and to highlight regions dominated by high levels of vorticity and shear strain. Beneath each contour plot is the mode-pressure signature used to estimate the vorticity and shear strain fields. The mode is given in Pa and is plotted as a function of streamwise distance. 173 Figure 6.17 shows the vorticity field associated with the first mode from the POD analysis and Figure 6.18 gives the in-plane shear strain field. It is evident in the vorticity field that a negative peak in the mode’s pressure signature (at x/h = 2 and x/h = 4.6) is associated with localized concentration of vorticity as found in the single-point LSE and QSE analysis. At the same streamwise positions in the shear strain field, these negative pressure peaks line up with a region of low shear strain. The reverse is true with the positive pressure peak at roughly x/h = 3.5 in Figure 6.17 and Figure 6.18. The positive pressure peak is associated with an area of low vorticity in Figure 6.17 and a region of peak positive shear strain in Figure 6.18. Farther downstream, a second, positive peak is seen in the mode-pressure signature within the reattached, boundary-layer region. However, both the vorticity and shear strain fields do not show a flow feature associated with this positive peak. This could be due to the level of estimation used. Multi-point QSE may be able to define this region with more clarity as mentioned in the single-point LSE versus QSE discussion. Collectively, the observations in Figure 6.17 and Figure 6.18 suggest that mode one is related to two vortical structures that are located at x/h = 2 and 4.6 with a high-strain, or saddle point, in between. 174 x/h Figure 6.17 Vorticity field associated with mode-one, pressure signature ‘ --- -.---- _ _. 3-.-- ... . e1 ~ - - _-- — - “I 4 , . 2L 1 .2 l 1 l l 3 7 1 . J: \ >~ Pa x / h Figure 6.18 Shear strain field associated with mode-one, pressure signature Figure 6.19 and Figure 6.20 show the same results for mode two as seen in the previous two figures for mode one. The results seem quite similar to those of mode one with the exception of a downstream shifi. In particular, the vorticity field shown in Figure 6.19 is no more than the vorticity field shown in Figure 6.17 shified downstream such that the localized focusing of vorticity is now found at x/h = 2.5 and 6. A similar shift is found for the shear strain field in Figure 6.18 and Figure 6.20. In summary, mode two appears to reflect the same flow structure as mode one after the translation of the latter farther downstream. Thus, a linear combination of both modes can be used to 175 represent the advection of the mode, as reasoned earlier based on the pressure signature alone. Figure 6.20 Shear-strain field associated with mode-two, pressure signature Figure 6.21 and Figure 6.22 display the vorticity and shear-strain field, respectively, for mode three. This mode shape has lower negative and positive peak magnitudes then those seen in modes one and two. Additionally, mode three’s pressure 176 signature shows two positive peaks and three negative peaks over the streamwise extent investigated. Mode one had two positive pressure peaks and one negative; whereas, mode two had the opposite of mode one. Based on the single-point, SE findings, the number of negative-pressure peaks may be used to indicate the number of vortical-flow structures associated with the particular mode shape. In Figure 6.21, is can be seen that the three negative-pressure peaks are associated with three vortical structures; whereas, in Figure 6.22, the positive peaks line up with the spaces in between the structures. An interesting observation may be made at this point. Recognizing the association of negative-pressure peaks with vortical structures, it appears that it may be possible to create a potential flow model of a train of convecting vortical structures that would produce the same mode-pressure signatures seen in Figure 6.21 and Figure 6.22. More specifically, it is known from potential flow theory that as a vortex core passes over a wall it leaves a negative pressure imprint. A series of such vortices would then produce a corresponding train of negative pressure peaks with positive pressure between the vortices. A similar model for representing the wall-pressure field downstream of reattachment was proposed by Daoud26. 177 — iri—irk" *;+ fi—IA-v— ~.—*——r7fi Jr | 1 .l: \ >~ ’71—'7'—* wa—‘Lr— 3 T lv ‘17 ,7 4—Ar x / h Figure 6.22 Shear strain field associated with mode-three, pressure signature The vorticity and shear strain fields associated with the mode-four, pressure signature are shown in Figure 6.23 and Figure 6.24 respectively. The fields are the same as seen in mode three but shified in the streamwise direction due to the convective nature of the structures that the modes represent. Finally, it is interesting to note that the vorticity field is able to capture a structural feature in the reattached boundary layer when there is a negative pressure peak at such a 178 streamwise location (see Figure 6.19 at x/h = 6 and 6.22 at x/h = 8). However, the shear strain field does not display any features in the reattached, boundary-layer region. These observations are also consistent with the single-point-estimation results (see Figure 6.10 and Figure 6.11b) where it is seen that the localization of the shear strain is more evident in the shear layer than in the reattached, boundary layer. However, the single-point QSE results within the same region (Figure 6.10 and Figure 6.11d) are successful in capturing the localization of the strain field. Hence, it appears that whereas LSE does a good job in representing localization in vorticity, it is not as good in capturing localization of strain. The reason for this is currently unknown and requires further investigation. Figure 6.23 Vorticity field associated with mode-four, pressure signature 179 x / h Figure 6.24 Shear strain field associated with mode-four, pressure signature 6.2.4.2 The evolution of large-scale structures in the flow field Figure 6.25 shows the evolution of the vorticity field using multi-point LSE based on the pressure signature of mode A. Only the vorticity field is shown since it gives the same information as the shear strain field and seems to better reveal structures within the reattached, boundary layer. The axes in Figure 6.25 are the same as given in Figure 6.23, including the color bar. The pressure signature used in the estimations starts 8.81 seconds (image 67) into the acquisition sequence and ends after an elapsed time of 125 ms. Numbers associated with arrows are used in the figure to identify coherent structures, which are defined as regions of vorticity concentration. In all four plots in Figure 6.25, it can be seen that immediately downstream of separation there exists a thin region of vorticity that extends to approximately x = 1.5-2h. This region is dominated by the mean vorticity field seen in Figure 5.10a. Multi-point LSE reveals the mean-removed quantities in regions where the correlation between the pressure signature and the flow field is high. Due to the relatively large separation between the shear layer in this region and the wall-pressure sensors, the correlation in this 180 It?“ region is low; and, therefore, no particular flow featured is captured. This can also be seen by the low magnitude of the pressure signature used in the estimation in this region shown in all four contour plots in Figure 6.25. Beyond x/h = 2, the correlation between the flow field and pressure signature is good (Figure 6.5 and Figure 6.6) and the magnitude of the pressure signature is higher; therefore, the vorticity field estimation begins to reveal some of the underlying, turbulent structures. As discussed in Section 6.2.3.1, the pressure field seems to alternate between periods of random, quiescent, pressure signatures and organized active ones. In the quiescent phase, the magnitude of the pressure signature is relatively low and the estimated, vorticity field looks similar to the mean-vorticity field shown in Figure 5.10a for reasons just explained above. Figure 6.25a shows the end of a relatively quiet phase compared to the other vorticity fields shown in Figure 6.25. Nonetheless, some strengthening in the shear layer has occurred and, as can be seen in Figure 6.25a, a large concentration of vorticity (1) seems to start to develop. During an active phase, the magnitude of the pressure signature is strong and the coherent structures in the vorticity field are well defined. As shown in Figure 6.25b, the evolution of the shear layer shows a spatial growth of vorticity that forms a coherent structure (2) between x/h = 2-3 while the large concentration of vorticity (1) sheds from the shear layer and travels downstream. This sequence of events occurs approximately 3.75 ms after Figure 6.25a. In time-dependent estimation (not shown here), the large- scale structure (2) is then seen to move downstream to approximately x/h = 3 where it shows weakening. Twelve milliseconds after Figure 6.25b, the coherent structure (2) has moved to x/h = 3-4 where it remains stationary while growing in size, shape, and strength 181 (Figure 6.25c). At this point, coherent structure (1) has moved even farther downstream and seems to have decayed. This shows a limitation of the POD technique in that the spatial modes 1 and 2 are equal to zero beyond x/h = 8 (Figure 6.15). Therefore, using mode A pressure signature in multi-point LSE would not be useful in estimating mean- removed quantities at streamwise positions greater than x/h = 8. That is, the coherent structure (1) in Figure 6.25c most likely has not completely decayed as shown, but, in terms of POD, is likely to now be defined by a higher mode shape. In Figure 6.25d, which was taken 6 ms after Figure 6.25c, coherent structure (1) is no longer in the streamwise extent shown and large-scale structure (2) is seen to be shedding from the shear layer. At the same time instant, coherent structure (3) starts to form upstream between x/h = 2-3. The concentration of vorticity (2) eventually moves downstream past the mean, reattachment point and into the reattached, boundary layer. A similar pattern can be described for the time evolution of the large-scale structures based on mode B pressure signature. The time-sequenced, estimated-vorticity fields for mode B are shown in Figure 6.26. These vorticity fields were estimated starting 17 ms before image 67 was acquired and ran 1.7 ms after the image acquisition. The region between separation and x/h = 1.5-2 is the same as described for mode A. An initial roll-up of a large-scale structure (3) occurs between x = 2-3h as seen in Figure 6.26a. Weaker coherent structures (1) and (2) are also evident in the vorticity field. Coherent structure (2) has just passed through the average reattachment point. The evolution of coherent structure (3) shows that the structure moves downstream once it reaches approximately the size of one step height. It then shows weakening between x/h = 3-4 for a moment before growing stronger as shown in Figure 6.26b (8.3 ms after 182 Figure 6.26a). Meanwhile, the concentration of vorticity (2) near the wall in Figure 6.26a has convected downstream and picked up strength in Figure 6.27b. This same structure (2) is then seen 5.4 ms later to have convected downstream as in Figure 6.27c along with the large-scale structure (3). Both of these concentrations of vorticity formed well- defined structures and shed from the shear layer. In addition, another large-scale coherent structure (4) is starting to form between x/h = 2-3 in Figure 6.26c. In Figure 6.26d, 3.3 ms later, both coherent structures (3) and (4) lose strength as seen in the vorticity field and in the mode B pressure signature. It is possible that the fast decay of the strength of the structures may be due to the structures switching to a higher mode shape; but more likely, it is related to the phenomenon of switching between active and inactive, flow periods. The physical mechanisms leading to this switching remains a mystery at the present time. Modes A and B, pressure signatures are combined in Figure 6.27, and the associated vorticity fields are shown for the same time sequence as used with the mode B pressure signature. The modes A and B combined are responsible for 60% of the pressure-fluctuation energy on average. Estimating the vorticity field from both modes A and B can provide additional information since the strength of the coefficients for modes 1-4 can vary with time. For example, at a particular instant in time, one of the modes may be more dominant or both could be equally important. In Figure 6.27, the same sequence of events described for the individual modes A and B can be seen when both modes are combined. Two coherent structures (1) and (2) are evident in Figure 6.27a. Both convect farther downstream 2.1 ms later in Figure 6.27b, while a concentration of vorticity (3) starts to form at streamwise location x/h = 2- 183 3. The roll-up of the large-scale vortex (3) continues 2.9 ms later in Figure 6.27c and coherent structure (2) is seen to have moved farther downstream. It is interesting to note here that the streamwise extent between the two concentrations of vorticity (2) and (3) is equivalent to approximately 0.67x,. Cherry et al.23 found the characteristic spacing between structures shedding from the reattachment zone to be between 60% - 80% of the separation bubble length. In Figure 6.27d, 6.25 ms later, coherent structure (3) increases in strength between x/h = 3-4 and at the same time the large-scale vortex (2) is decaying in the reattached boundary layer. One observation that can be made in the estimation using mode AB pressure signature is the period of weakening noticed in the individual mode pressure signatures does not seem to exist. This suggests that the weakening of the structures is associated with the disappearance of energy from one mode and its appearance in the other. Since the POD modes are generally mathematical, it is likely that this is a mathematical artifact. That is, modes A and B may simply be a mathematical decomposition of the same physical structure. 184 3.“! * -. -. * f "r. 7.7, ’f’i (U Ci A i i 0121‘ - . . 1. \-—/ ..2 .l ‘ 1 7. - T 7 1 71 . 1 ...‘ 0 1 2 3 4 5 6 7 8 x I h Figure 6.25 Multi—point LSE of the velocity field associated with mode A pressure signature 185 as Gil“ x/h Figure 6.26 Multi-point LSE of the velocity field associated with mode B pressure signature 186 x/h Figure 6.27 Multi-point LSE of the velocity field associated with mode AB pressure signature 187 As mentioned previously, it is best to show the evolution of the flow structures using a movie or multiple frames. Figure 6.25, Figure 6.26, and Figure 6.27 show the overall picture seen regarding the evolution of the flow structures. However, the flow- field frames given in these figures are separated by a rather coarse time step. To provide a clearer picture of the evolution of the flow features, a series of 36 plots of the vorticity field separated by a time offset of 0.4 ms between plots are shown in Figure 6.28. The axes and the color bar (not shown) are the same as described in Figure 6.27. In Figure 6.28, the flow structure of interest is marked by an “X” by visually locating the highest vorticity value within the flow structure. Note that without the fitting of a curve to the vorticity distribution in order to locate the peak vorticity, a minimum uncertainty equal to the grid spacing of the PIV data (0.92 mm or 7.5% of the Step height) is assumed in identifying the structure location. Additionally, the visual process of identification is likely to increase this uncertainty by a factor of two or three. Therefore, it is estimated here that the identified structure location is known with 15% - 20% uncertainty. Overall, detailed formation and subsequent evolution of the marked vortex structure is depicted in Figure 6.28. To see this more clearly, the streamwise coordinate of the vortex is plotted as a function of time in Figure 6.29. The figure shows the flow structure of interest remaining stationary, within the uncertainty of the data, between x/h = 2-3 for the duration of the first four to five vorticity fields (2 ms) before it starts to accelerate downstream. Finally, Figure 6.30 shows the streamwise and normal position of the flow structure of interest as it is tracked downstream in the multiple frames shown in Figure 6.28. 188 ZITM - +7 - 7 7'7 --_- a e-airex-j 2r +++’-+ filvq-?.F gm zI—+ A- ++l++ + ++ ----.__-r--_,I 2w --.; +7 ++ ++~+—+ —+ t =78?) ¥n§I ’ a t=0.4msI‘ ' t=150mst I 7 '7” {£67063}; m... «l ....~ 9 l l I o -‘ ‘ 2rfi~r—+v—IIa—'a— IF : t=2.00msI:t—640I1}SI t=6.80 msI l‘ ' z I ' I V t=2.80msI21 t—720ms‘ O — -0 277 F Fl‘f # Fri 3:20 ms? 2; t= 7.60 ms I 51 o 1‘ .1 *mfi; ""‘V l o " — -o ' ’ ..----t- ..-F- .-- - J-.- .7 - -, . \1 o 1 1";- - t ...-n 2."F”T*"" t—400msI2IY— ’fi‘ (moons f1 6 1‘ ‘ O _ -0 .— 189 zI—W #7,-.- ..-- a- g; [.7 g. 2>fi * *7 fiffiFJ—gfi’ . roams] I >. I 0 . ._ .Io , I“ ------ . Figure 6.28 Frames showing evolution of flow structure, At = 0.4 ms 190 O L_-L___...;_A_ Emu._sg;._s._l_-_ -;, 'rllr-s._ A Figure 6.29 Location of the flow structure, xlh, versus time used in tracking the evolution of a flow structure developing between xlh=2-3 2I'T iiiiii is + + + - --r- rpm! *r‘—'¢JI I I . I > W o - 1 g ! 0 1 2 3 4 5 6 7 8 x/h Figure 6.30 Tracking the location of a flow structure as it forms between xlh = 2-3 and then accelerates downstream 191 6.2.4.3 Instantaneous versus estimated flow field As mentioned previously and explained in detail in Appendix 8.6, under certain conditions, the multi-point LSE can satisfactorily estimate the instantaneous, velocity field. Multi-point LSE imposes a strict spatial condition for estimating the average flow field by using a particular, spatial—pressure-distribution condition. The estimate is stricter with this condition compared to a single-point, pressure condition and is less likely to deviate from the instantaneous flow field. This would be particularly true for flow fields where the pressure generation is dominated by coherent motions. Thus, multi-point LSE allows for a more “realistic” estimation of the instantaneous, flow field. Cole and Glauser8 addressed the accuracy of multi-point, LSE compared to single-point, LSE in their jet mixing layer study. In their study, they used velocity measurements to estimate the entire flow field using LSE. From their results, they found that single-point, velocity estimates did not adequately estimate the instantaneous, flow field and that at least two reference points were needed to represent the flow field well. The present study utilizes surface-pressure to estimate the flow field. The degree to which the estimate represents the instantaneous flow field will be assessed in this section. The instantaneous pressure signatures shown in Figure 6.13b and Figure 6.14b were simultaneously acquired with PIV images 1299 and 67, respectively. The instantaneous vorticity fields associated with the pressure signatures are shown in Figure 6.31a and Figure 6.32a. The instantaneous vorticity fields have been smoothed using a 3x3 filter in order to remove the effect of small-scale random turbulence and some of the noise resulting from calculating derivatives. The abscissa, the ordinate, and the color bar are the same as described for Figure 6.23. In Figure 6.31b and Figure 6.32b, the vorticity fields estimated using multi-point LSE based on the instantaneous pressure signatures 192 occum'ng simultaneously with the sampling of images 1299 and 67, respectively, are displayed. Finally, the vorticity fields estimated using multi-point LSE based on one of the dominant mode shapes is shown in Figure 6.31c and Figure 6.32c. The mode shape used for image 1299 is mode A and for image 67 is mode B, as will be explained below. The instantaneous vorticity field in Figure 6.31a shows small-scale concentrations of vorticity spread across the streamwise and normal extent measured by the PIV. The vorticity field shown in Figure 6.31b gives an estimate of both small- and large-scale structures. Since the technique used was LSE, the large-scale structures are better defined. The organization of the flow field is difficult to see in Figure 6.31a, yet the instantaneous, pressure signature suggests some organization and Figure 6.31b shows large-scale structures. This organization is similar in shape to the spatial pressure distribution shown for modes one and two. Since these modes form the convective mode A, the pressure signature for mode A at the time image 1299 was acquired is shown in Figure 6310 along with the estimated flow field from multi-point LSE. Comparing the three vorticity fields in Figure 6.31 shows that while mode A does not capture all the small-scale coherent structures, it does give a good representation of the large-scale organization of the flow field, which was seen in Figure 6.31b. One item to note is that the magnitude of the mode A, pressure signature is smaller compared to the magnitude of the instantaneous pressure signature. Wee et al.27 used POD to analyze BFS flow and found that the reconstruction accuracy of the flow field is dependent on the dominant dynamics at a location and the number of modes used in the reconstruction. The more significant the higher frequencies, the larger the number of modes needed to accurately reconstruct the flow field. This seems to be the case in Figure 6.31. 193 Figure 6.32a shows that at the instant of time that image 67 was acquired the flow field was dominated by large-scale concentrations of vortical structures. Three distinct structures can be identified near x/h = 1.5, 4, and 6.5. This distribution of the coherent structures in the flow field matches the instantaneous, pressure signature, which shows three minimum peaks at the same streamwise positions. The multi-point, linear, stochastic estimation of the vorticity field in Figure 6.32b shows a similar large-scale vortical distribution as seen in the instantaneous, vorticity field. Looking at the spatial, pressure signature, it can be seen that the distribution is similar to that seen in mode B. Thus, at this moment in time mode B is more dominant in the pressure signature. The pressure signature associated with the large-scale structures is captured in mode B from POD and estimated using multi-point LSE. The reconstruction of the flow field from mode B in Figure 6.32c gives a good estimate of the instantaneous flow field in Figure 6.32a. This discussion is not to say that mode A or B are physical representations of the instantaneous flow field, but it is encouraging to see how well each represent the flow field at certain time instances. At times both modes are needed to represent the flow field and still at other times modes beyond A and B are needed. 194 Figure 6.31 Vorticity field at instant of acquisition of PIV image #1299: a) instantaneous, vorticity field and pressure signature, b) multi-point LSE using instantaneous, pressure signature, and c) multi-point LSE using mode A, pressure signature 195 Figure 6.32 Vorticity field at instant of acquisition of PIV image #67: a) instantaneous, vorticity field and pressure signature, b) multi-point LSE using instantaneous, pressure signature, and c) multi-point LSE using mode B, pressure signature 196 6.3 Conclusions The goal of this chapter was to investigate the flow features that generate the wall-pressure signature motivated by the physical link between these features and flow- induced vibration and noise in practical applications. Comparing statistics from the simultaneously acquired wall-pressure and above-surface-velocity measurements showed that the abrupt transitions seen in the wall-pressure characteristics are also evident in the velocity field. The links between these wall-pressure and velocity field characteristics associated with the transition regions have been inferred in previous literature from wall- pressure information alone or in conjunction with limited velocity information. Here, these links have been verified as well as expanded based upon the comprehensive wall- pressure and velocity measurements that have been measured. The simultaneously acquired database was also used to estimate the coherent structures in the flow field based on the wall-pressure signature. Linear stochastic estimation was used to highlight the flow structures associated with various types of the wall-pressure signature. Single-point LSE revealed that a negative pressure event was associated with the passage of vortices; whereas, a positive pressure event correlated with a saddle point between two vortices. Single-point QSE was used to verify the accuracy of the single-point LSE. Although QSE does add additional flow-field information, particularly in relation to clearly identifying localized regions of large strain rate, LSE was able to estimate the global features of the large-scale, vortical structures in the flow field. One interesting finding from the single-point QSE analysis was the possible existence of a secondary vortex that is formed due to the interaction of a passing, large- 197 scale vortex with the wall at or downstream of the point of reattachment. This secondary vortex was seen when a negative pressure event was applied approximately one step- height downstream of the mean, reattachment location. In addition, when a positive pressure event was applied at the same location, the secondary vortex was found farther away from the wall in the reattached boundary layer region. Proper Orthogonal Decomposition (POD) was used to determine the spatial modes that dominated the generation of the fluctuating-pressure signature. Spatial- pressure distribution patterns were noticed in the instantaneous, pressure signature, suggesting periods of active and organized, flow features followed by periods of quiescent and random activity. POD helped extract the modes of organization in efforts to analyze each more closely. It was found that the first four POD modes were responsible for 60% of the energy in the instantaneous, pressure signature. In addition, POD also revealed that modes one and two formed convective mode A and modes three and four formed convective mode B. Multi-point, linear, stochastic estimation was then used to estimate the flow field based on the spatial-pressure distribution for each mode shape. Valleys in the mode- pressure signature showed concentrations of vorticity and absence of shear strain in the respective fields at nominally the same streamwise locations; whereas, mode-pressure signature peaks showed voids of vorticity in the vorticity field and local concentrations of shear strain in the shear-strain, contour plots at nominally the same streamwise locations. The evolution of the flow structures within the flow field was investigated using LSE based on modes A and B and the combination of the two modes. The analysis revealed quiescent and active periods in the flow. In addition, a pattern from the active 198 phase of the flow seemed to emerge for all modes. Between x/h = 2 - 3, a concentration of vorticity grows and reaches a height roughly equal to the step height before moving downstream. It was then seen that the coherent structure moves to x/h = 3-4 where it remains stationary while growing in size, shape, and strength. The coherent vorticity concentration then sheds downstream as another coherent structure starts to form between x/h = 2-3. Finally, a comparison between the instantaneous vorticity fields and those estimated using multi-point LSE based on the instantaneous pressure signature and the pressure signature of modes A or B was presented for two instances in time. Results revealed that the multi-point LSE provides a reasonable estimate of the large-scale organization in the vorticity field using either the instantaneous or mode pressure signature. Nonetheless, it is evident that details due to small-scale contributions are lost; and, therefore, it may be necessary to use either a higher-order, estimation technique or number of modes in the reconstruction of the flow field. It is also noted that the success of multi-point LSE in capturing the flow features is closely related to the periods when large flow organization takes place. During these times, the pressure fluctuations are well correlated with the coherent structures and therefore are able to reflect the global characteristics of these structures. On the other hand, at times where the flow is in an apparently random state and the pressure fluctuations are weak, LSE would be useless in trying to shed light on the flow state. 199 6.4 References 1 Thomas, A.S.W. and Bull, M.K., “On the role of wall-pressure fluctuations in deterministic motions in the turbulent boundary layer,” Journal of Fluid Mechanics, Vol. 128, 1983, pp. 283-322. 2 Snarski, S. and Lueptow, R.M., “Wall pressure and coherent structures in a turbulent boundary layer on a cylinder in an axial flow,” Journal of Fluid Mechanics, Vol. 286, 1995, pp. 137-171. 3 Johansson, A.V., Her J ., and Haritonidis, J .H., “On the generation of high-amplitude wall-pressure peaks in turbulent boundary layers and spots,” Journal of Fluid Mechanics, Vol. 175, pp. 119-142. 4 Lee, I. and Sung H.J., “Multiple-arrayed pressure measurement for investigation of the unsteady flow structure of a reattaching shear layer,” Journal of Fluid Mechanics, Vol. 463, 2002, pp. 377-402. 5 Adrian, R.J., “Conditional eddies in isotropic turbulence,” Physics of Fluids, Vol. 22, 1979, pp. 2065-2070. 6 Tung, T.C. and Adrian, R.J., “Higher-order estimates of conditional eddies in isotropic turbulence,” Physics of Fluids, Vol. 23, No. 7, 1980, pp. 1469-1470. 7 Guezennec, Y.G., “Stochastic Estimation of coherent structures in turbulent boundary layers,” Physics of Fluids A, Vol. 1, 1989, pp. 1054-1060. 8 Cole, DR. and Glauser, M.N., “Applications of stochastic estimation in the axisymmetric sudden expansion,” Physics of Fluids, Vol. 10, No. 11, 1998, pp. 2941- 2949. 9 Stokes, S. and Glauser M., “Multi-point measurement techniques used in the study of separated flows,” 30m AIAA Fluid Dynamics Conference, AIAA 99-3518, Norfolk, VA, June 28 — July 1, 1999. '0 Naguib, A.M., Wark, CE, and Juckenhofel, 0., “Stochastic estimation and flow sources associated with surface pressure events in a turbulent boundary layer,” Physics of Fluids, Vol. 13, No. 9, 2001, pp. 2611-2626. 1' Murray, N. and Ukeiley, L., “Estimating the Shear Layer Velocity Field Above an Open Cavity from Surface Pressure Measurements,” 32"" AIAA Fluid Dynamics Conference and Exhibit, AIAA 2002-2866, St. Louis, Missouri, June 24-26, 2002. '2 Ukeiley, L. and Murray, N., “Velocity and surface pressure measurements in an open cavity,” Experiments in Fluids, published on-line April 6, 2005. 200 '3 Heenan, AF. and Morrison, J. F ., “Passive Control of Pressure Fluctuations Generated by Separated Flow,” AIAA Journal, Vol. 36, 1998, pp. 1014-1022. 14 Driver, D.M., Seegmiller, H.L., and Marvin, J.G., “Time-dependent behavior of a reattaching shear layer,” AIAA Journal, Vol. 25, No. 7, 1986, pp. 914-919. '5 Hudy, L.M., Naguib, A.M., and Humphreys Jr., W.M., “Wall-pressure-array measurements beneath a separating/reattaching flow region,” Physics of Fluids, Vol. 15, No. 3, 2003, pp. 706-717. '6 Ho, C. and Huerre, P., “Perturbed Free Shear Layers,” Annual Review of Fluid Mechanics, Vol. 16, 1984, pp. 365-424. '7 Brown, G. L. and Roshko, A., “On density effects and large structure in turbulent mixing layers,” Journal of Fluid Mechanics, Vol. 64, No. 4, pp. 775-816, 1974. '8 Morris S. C., and Foss, J. F., “Turbulent boundary layer to single-stream shear layer: the transition region,” Journal of Fluid Mechanics, Vol. 494, 2003, pp. 187-221. '9 Raffel, M., Willert, C., and Kompenhans, J., Particle Image Velocimetry: A Practical Guide, 3rd ed., Springer-Velag, Berlin, 1998, pp. 160-163. 20 Naguib, A. M. and Koochesfahani, M. M., “On wall-pressure sources associated with the unsteady separation in a vortex-ring/wall interaction,” Physics of Fluids, Vol. 16, No. 7, 2004, pp. 2613-2622. 21 Bradshaw, P. and Koh, Y.M., “A note of Poisson’s equation for pressure in a turbulent flow,” Physics of Fluids, Vol. 24, 1981, pp. 777. 22 Gendrich, C. P., Bohl, D. G., and Koochesfahani, M. M., “Whole-field measurements of unsteady separation in a vortex ring wall interaction,” 28'” AIAA Fluid Dynamics Conference, AIAA Paper No. 97-1780, Snowmass, CO, June 29 — July 2, 1997. 23 Cherry, N. J., Hillier, R., and Latour M. E. M., “Unsteady measurements in a separated and reattaching flow,” Journal of Fluid Mechanics, Vol. 144, 1984, pp. 13-46. 24 Lumley, J. L., “The structure of inhomogeneous turbulence,” Atmosphere Turbulence and Wave Propagation, Yaglom and Tatarsky eds., Nauka, Moscow, 1967, pp. 166-178. 25 Berkooz, G., Holmes, P., and Lumley, J. L., “The proper orthogonal decomposition in the analysis of turbulent flows,” Annual Review of Fluid Mechanics, Vol. 25, 1993, pp. 539-575. 201 26 Daoud, M.I., “Stochastic Estimation of the flow structure downstream of a separating/reattaching flow region using wall-pressure array measurements,” Ph.D. Dissertation, Michigan State University, 2004. 27 Wee, D., Park, S., Yi, T., and Annaswamy A.M., “Reduced-order modeling of reacting shear flow,” 40’” Aerospace Sciences Meeting and Exhibit, AIAA 2002-0478, Reno, NV, January 14 - 17, 2002. 202 7 Conclusions with Discussion and Future Work The purpose of this chapter is to integrate the individual findings of previous chapters into a common view of the dominant flow features in the back-step flow that relates to the wall-pressure generation process. For more detailed information on the specific observations of the research, the reader is directed to the conclusion section of the end of the previous chapters. Additionally, recommendations for future follow-up investigations are provided in this chapter. 7.1 Conclusions with Discussion Overall, the experimental data presented gives a detailed look at both the wall- pressure signature and the velocity field downstream of anaxisymmetric, backward- facing step. Both the pressure and velocity measurements have been individually analyzed and found to be consistent with the results reported in the literature. The two datasets have also been analyzed together in terms of how the wall-pressure signature relates to the velocity field. Of particular significance is the first-time investigation of the dominant flow features in the BFS flow through utilization of multi-point linear stochastic estimation coupled with wall-pressure array information (Chapter 6). From this analysis, a view different from the widely accepted one seems to have emerged. From literature, it is known that at the step edge, the turbulent boundary layer separates and forms a thin, shear layer. Due to Kelvin-Helmholtz instability, the shear layer rolls up to form small vortices of a size equivalent to the thickness of the shear layer. As the vortices continue to move downstream, they start to pair, triple, or even quadruple according to Smits', who states that this appears to be the rapid growth mechanism in the shear layer. This initial roll-up of the shear layer resembles the 203 structure of a simple mixing layer as stated by Smitsl. At approximately half of the reattachment distance, the small-scale structures continue to pair and form larger-scale, coherent structures. Smitsl states that these coherent structures that are pairing continue downstream, roll around each other, and form a single, large-scale, vortical structure prior to reattachment. The view discussed by Smitsl corresponds to the widely accepted view of the BFS flow in which the growth of the shear layer structures occurs spatially; i.e., as the structures travel downstream. However, it can be seen in the present analysis that a large-scale, coherent structure actually forms while remaining stationary between x/h = 2-3. Figures 6.26- 6.28 depict this development of a large-scale coherent structure. In particular, it can be seen that a large-scale coherent structure grows in place (i.e., temporally) before reaching a height equivalent to the step height. At this point, the coherent structure sheds and by necessity accelerates to its ultimate convection speed in the downstream direction. This temporal grth of the flow structures is similar to the development of the vortex structures in the wake of a bluff body (e. g., vortex shedding from a cylinder) than it is to a simple mixing layer. Therefore, to contrast the two view points, the scenario arrived at in this study will be referred to as “wake mode” versus the traditional one, which will be termed “shear-layer mode”. It is important to realize that regardless of which of the two modes is prevalent, the flow structures ultimately grow to a scale of the order of the step height and they travel downstream with a certain convection velocity. Hence, the distinction of the two modes is likely to primarily affect the flow characteristics within the separation bubble rather than downstream of reattachment. 204 ...-um Figure 7.1 and Figure 7.2 provide an idealized sketch of the coherent structures developing in the separation bubble for the shear-layer mode and the wake mode respectively. Freestream Velocity (Um) Vortical structures in shear layer Edge Of shear layer Figure 7.1 Idealized sketch of the flow structures developing in “shear-layer mode” downstream of a backward-facing step Freestream Velocity (Ugo) Vortical structure developing in place Coherent structures accelerating % downstream Figure 7.2 Idealized sketch of the flow structures developing in “wake mode” downstream of a backward-facing step Xr 205 Only one other recent study indicates a scenario similar to that identified here. The study is conducted by Wee et al.2'3 using Random Vortex Method, numerical simulations of the flow field downstream of a sudden expansion. Wee et al.2‘3 found that large-scale vortices periodically formed at the middle of the reattachment zone before shedding downstream. More interestingly, they also linked this temporal evolution of the structures downstream of the step to the existence of an absolute instability of the flow. Subsequent linear stability analysis showed that the location of maximum instability grth rate was found in the middle of the reattachment zone (in agreement with the 1.2’3 state that absolute modes are most location of formation of the vortices). Wee et a likely to originate in a region with strongest backflow. Around x/h = 2-3 in the axisymmetric backward-facing step, the backflow is the strongest (U/Um z -0.2) as seen in the near-wall velocity plot in Figure 3.3. In‘addition, Huerre and Monkewitz4 found that a two-stream, mixing layer is absolutely unstable when the velocity ratio, defined as the velocity of one freestream divided by the velocity of the other freestream, was less than -0.l36. As shown in Figure 3.3, the ratio of the maximum, reverse velocity to the freestream in this study has a value of U/Uco z -0.2 in the x/h = 2-3 region, which is lower than the critical value found by Huerre and Monkewitz“. Thus, within this location, it is likely that an absolute instability exists, which leads to the roll-up of the large-scale vortex. The absolute instability mechanism suggested by Wee et al.2’3 and its relation to the formation of large-scale structures in the back-step flow is similar to the flow behavior in the wake of bluff bodies where the process of vortex shedding has been linked to the existence of an absolute-instability mechanism as well (Oertels). 206 It is useful to note here that wake and shear-layer modes have in fact been identified in open-cavity flows since 1987 (Gharib and Roshkob). Moreover, a link between absolute instability and the wake mode has also been proposed in this case (Rowley et al.7). Particularly interesting in the cavity flow case is that, with the exception of axisymmetric flow geometry (Gharib and Roshkoé), observations of the wake mode remain confined to computational studies (Najm and Ghoniem8 and Rowley et al.7). This has led to a recently developing belief (Naguib9) among researchers of cavity flows that the observation of a wake mode may require idealized or close to idealized two-dimensional-flow geometry. This may provide a rationale for the commonly held belief that the vortical structures within the separation bubble develop as in a simple mixing layer rather than in a wake flow. Finally, unlike the behavior of the flow in the wake of bluff bodies where vortex shedding is regular, the process seems to be highly intermittent in the back-step flow. Specifically, as noted in Chapter six, the back-step, flow field seems to alternate between phases of organization (where vortex formation and shedding takes place) and randomness. An interesting question that arises is what mechanism that is unique to the back-step flow (i.e., not found in the wake of a bluff body) “turns on/off” the vortex shedding process? The answer to this question is currently unknown and requires further research. It is noted though, that this is perhaps a case of “confused identity”, where the flow, for unknown reasons, switches between the shear-layer and wake modes. Such switching behavior has been identified in numerical computations of cavities (Rowley et al.7) under certain conditions. 207 If the mode-switching scenario were true, then the period of random inactive wall-pressure activity found in Chapter six would correspond to the existence of a shear- layer mode. This suggests that when a shear-layer mode exists, it does not have a significant, wall-pressure imprint; and, therefore, wall-pressure-based LSE cannot accurately predict the associated flow field. Figure 7.3 shows an example of a low- magnitude, pressure signature and the associated, instantaneous, vorticity field (obtained from the PIV data concurrent with the pressure signature). The instantaneous, vorticity field shows distinct coherent structures in the flow field that are, however, smaller than those found in Figure 6.30a and are located farther away from the wall. The observed features are more descriptive of a shear-layer-like mode than a wake mode (although one could not tell with certainty in the absence of temporal evolution information). xlh Figure 7.3 Instantaneous, pressure signature and associated, vorticity field The development of the large-scale structures in place provides a more logical explanation than that given in Chapter four (based on the accepted view) for the abrupt transition seen around x/h = 2 in the autocorrelation. At x/h = 2, the time scales are seen to decrease over a very short distance. In addition, beyond x,, there is a gradual transition in the length scales as the structures accelerate downstream. Figure 6.2 shows this in more detail. In the autocorrelation, the abrupt transition at x/h = 2 is associated with the in-place (“sudden”) formation of the vortex structures between (x/h = 2 — 3). Beyond x/h = 3, the large-scale structure starts to accelerate from the stationary position and continues to accelerate through x/h = 8. This acceleration, which can be seen in the average streamwise velocity at the center of the shear layer in Figure 6.2, is most likely responsible for the gradual change of the time scales in the autocorrelation in Figure 6.2 between x/h = 4 to 8. 209 "m”il :— 7.2 Future Work In closing, there are a few other suggested items that could be investigated in order to refine the present study: 1. Estimate the flow field from the pressure signature using multi-point, quadratic, stochastic estimation. As noted in Section 6.2.2, the quadratic term adds finer details to the flow field that the linear term cannot estimate (particularly in defining regions with local concentration of strain and capturing the generation and evolution of the secondary vortices arising from the interaction of the shear- layer vortices with the wall). Thus, a more detailed vorticity field could be estimated that will highlight features beyond the large-scale structures. Determine the convection velocities associated with different POD modes, which could give insight into the convection velocities determined by the cross- correlation contour plots of the wall-pressure signature. Estimate how often the dominant POD modes that are associated with organized phases of the flow field occur. This could be useful in terms of knowing at what frequency to apply flow control techniques. If flow control techniques do not need to applied often, then that can save energy in the overall operating system. Generate the time series of the convective-mode-pressure signature and determine the frequency spectrum for each microphone in the array. Early analysis of this item indicates that the strongest convective mode (mode A) dominates the energy in the spectra at reattachment; whereas, in the reattached, boundary layer, mode B dominates most of the energy in the spectra. 210 I}. 31‘ 5. Calculate the flow sources of pressure (the forcing function in Poisson’s equation) and their distribution from the instantaneous and stochastically-estimated, flow fields. 6. Obtain the turbulent-velocity-field statistics (RMS velocities, Reynolds stresses, etc.) from the stochastically-estimated-velocity field and compare it to those obtained from the instantaneous PIV shots. 211 1.1." \ “_Ifl r - 7.3 References ' Smits, A. J ., “A visual study of a separation bubble,” International Symposium on Flow Visualization, Bochum, Germany, September 9-12, 1980. 2 Wee, D., Park, S., Yi, T., Annaswamy, A. M., and Ghoniem, A. F ., “Reduced order modeling of reacting shear flow,” AIAA 40’” Aerospace Sciences Meeting and Exhibit, AIAA 2002-0478, Reno, Nevada, January 14-17, 2002. 3 Wee, D., Yi, T., Annaswamy, A. M., and Ghoniem, A. F., “Self-sustained oscillations and vortex shedding in backward-facing step flows: simulation and linear instability analysis,” Physics of Fluids, Vol. 16, No. 9, 2004, pp. 3361-3373. 4 Huerre, P. and Monkewitz, P.A., “Absolute and convective instabilities in free shear layers,” Journal of Fluid Mechanics, Vol. 159, 1985, pp. 151-168. 5 Oertel, H., “Wakes behind blunt bodies,” Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 539-562, 6 Gharib, M. and Roshko, A., “The effect of flow oscillations on cavity drag,” Journal of Fluid Mechancis, Vol. 177, pp. 501-530. 7 Rowley, C. W., Colonius, T., and Basu, A. J ., “On self-sustained oscillations in two- dimensional compressible flow over rectangular cavities,” Journal of Fluid Mechancis, Vol. 455, 2002, pp. 315-346. 8 Najm, H. N. and Ghoniem, A. F., “Numerical simulation of the convective instability in a dump combustor,” AIAA Journal, Vol. 29, No. 8, 1991, pp. 911-919. 9 . . . . Naguib, A., private communications. 212 151 :— E“--.. _ l 8 Appendix 8.1 Wind Tunnel Facility The Subsonic Basic Research Tunnel (SBRT) is an open-circuit wind tunnel located at NASA Langley Research Center in Hampton, Virginia. A schematic of the tunnel is shown in Figure 8.1. The intake side of the tunnel is surrounded on three sides with a semi-circular, entrance lip that enables the incoming flow to stay attached to the wall. The upper lip is removed due to space constraints inside the tunnel room. l Intake L 6:1 Contraction l TestSection l Diffuser and FanSeetion l I l l j I 571.5mmX819.15mm Front Rear . Bay Bay ....................................................... I........... ............ ......................................................... .77m .J 4--——-> 2.03 m 1.14 m 3.54 m 1.85 m 3.61m l Figure 8.1 Subsonic Basic Research (Wind) Tunnel at NASA Langley The inlet dimensions are 1.52 m wide by 1.82 m high. Air entering the tunnel flows through 0.2 m of aluminum honeycomb flow straigtheners that use 12.7 mm cells 213 iF‘ - can. mun-.3 designed to remove any flow swirl. The flow then moves through a double row of wire mesh turbulence screens, separated by 76.2 mm. These screens are located approximately 0.2 m downstream of the honeycomb and are 0.254 mm in diameter at about 8 meshes per 10 mm. The intake section stretches 1.14 m leading into a 6:1 contraction section that measures 3.54 m in length. Located downstream of the contraction is a 0.57 m wide by 0.82 m high by 1.85 m long newly-constructed test section. The new test section was constructed prior to the experiment in order to improve flow quality in the test area. As a result, the corresponding turbulence intensity at flow conditions used in this study was less than 1%. The walls of the new test section consist of plywood, Plexiglas, and glass as shown in Figure 8.2. The glass side was used for high-quality, flow visualization and Particle Image Velocimetry (PIV). Finally, the diffuser and fan section measure 3.6 m in length. The fan is driven by a 200 hp motor enabling flow speeds in the wind tunnel up to 70 m/s (M = 0.2, where M is the Mach number). For more details concerning the wind tunnel specifications refer to Howerton'. 214 1“.'.—~"""..——fi t . (ll-ass Model Figure 8.2 Axisymmetric, backward-facing step inside the test section of SBRT The test section of the wind tunnel had a moveable false floor that could be used to eliminate the streamwise, pressure gradient associated with flow acceleration inside the test section. Figure 8.3 shows the false floor in the test section. The height of the test section with the false floor installed was 0.62 m. 215 Figure 8.3 False floor in test section of SBRT The false floor was embedded with 18 static-pressure taps that were used to determine the pressure gradient within the test section. The pressure readings from these taps are plotted as Cp versus the streamwise distance, x/h, in Figure 8.4. Starting at the upstream end of the model, the pressure along the floor is increasing over a 20h distance. This low pressure region may be caused by separation at the leading edge of the false floor. The pressure recovers over the 20h distance upstream of the rotating section (rotator) of the model. Although within the first 18h in the rotator region, the pressure is still recovering slightly. Nonetheless, the pressure varies less than 3% in this region, making the pressure gradient in the rotator region approximately zero. At the downstream end of the rotator region, the pressure starts to increase. This is partly due to the slight flow area increase associated with the flow over the step. More significantly, there is a pressure increase associated with the flow slowing down as it approaches the shroud that covers the leading edge of the model stand. Nonetheless, it is evident by the 216 pressure distribution shown in Figure 8.4 that the boundary layer prior to separation develops under a mostly zero-pressure-gradient condition. Flow——> -0.28 -~—-———--—e— fie—r-em—w — — .— — — — — — .— —— _1- _ _ _-—.—~-—-——-——.——. +2D section» -O.3 t - Step edge *—-— Rotator region ——> . l_ __.l -1 o -80 so 40 -2o 0 '20 x / h Figure 8.4 Static pressure distribution along false floor in SBRT test section 217 8.2 Axisymmetric BFS Design The axisymmetric, backward-facing step was partially modeled after an axisymmetric-boundary-layer-model design studied by Driver and Johnstonz. There are several components that make up the axisymmetric, backward-facing-step configuration: skeleton shaft, nose, 2D section, rotator, step, instrument plates, motor and drive, tail, wire supports, and support stand. Each of these will be discussed in detail in the following sections. Some of these components are shown in Figure 8.5. Flow S direction tep fl 2D Section Rotator Motor Tail KI . V 1......- Nose Sensors Model Support Figure 8.5 Side view of the axisymmetric, backward-facing step 218 d'n‘ ’ é'uhm. fl 8.2.1 Skeleton shaft The skeleton shaft is made of solid steel ground to a 0.25 um to 0.41 pm surface finish, with a Rockwell rating of 60-65C. The skeleton shaft runs almost the entire length of the model, minus the nose and tail sections. The total length of the 25.4 mm diameter shaft is 1.94 m. This shaft provides the main support for the model, and all stationary components rest directly on the shaft. For the rotating section, bearings are placed between the rotator component and the skeleton shalt so it can rotate freely. At both ends of the skeleton shaft, a 25.4 mm deep tapped hole is used with a '/2-13 allen screw to tighten all the components along the shaft. The two screws are located in the stationary and motor components of the model. The location and function of these threaded holes will become more evident in the sections where these components are discussed. The 25.4 mm diameter solid stainless steel shaft was selected based on its deflection properties. Since the model is designed as a long cantilever beam, it was important to ensure that the amount of deflection in the model was minimal in order to ensure axisymmetry. The model was actually supported over a length of 482.6 mm at the downstream end where the support stand was connected to the model. Thus, the overall cantilever beam length was 1457.4 mm. The maximum deflection of the beam was estimated to be 0.006m using the Modulus of Elasticity for cold-rolled stainless steel equal to 190 GPa. This may seem to be a large deflection, but it is important to recognize the length and diameter of the shaft and the fact that the estimate is based on a cantilever beam analysis, where the beam is fixed only at the tip of one end. In addition, this analysis was done without considering the extra support from other model pieces, which when assembled on the skeleton shaft increase the overall moment of inertia at a given 219 “is-W: cross section and therefore provide more support. Once completed, the manufacturer of the model measured the actual, overall, vertical displacement of the model nose to be 0.381 mm. This deflection was removed by the wire support system in the front of the model, which is described in more detail below. 8.2.2 Nose The purpose of the nose is to provide a smooth transition for the flow as it approaches the model. The nose is hemi-spherical in shape as shown in Figure 8.6, which results in a step discontinuity in the radius of curvature at the junction between the nose module and the stationary 2D section. Nonetheless, a turbulent boundary layer is desired at separation, meaning that any flow disturbance caused by a discontinuity would in fact be beneficial in accelerating the development towards a turbulent state. The nose was constructed out of solid aluminum with a radius of roughly 70mm. It is positioned at the upstream end of the axisymmetric BFS and is attached by a male- female fit between the nose and the stationary section. A set screw is then used to secure the nose in place. Figure 8.6 shows the nose configuration with dimensions. 220 114i -- Sin-.... .121 Radius 5 70 mm \ Used in male-female fit Flow 9 / between nose and stationary section Figure 8.6 Schematic of nose section 8.2.3 Upstream Stationary Section The stationary section is located downstream of the nose and upstream of the rotating section. It is made from 0.35 m long aluminum tubing with an outside diameter of 0.124 m and an inside diameter of 0.11 m. Figure 8.7 shows a schematic of the stationary section. With the two donuts (0.11 m diameter cylindrical spacers, 25.4 mm thick) on either end supporting the tubing, the skeleton shaft can be threaded through the 25.4 mm diameter holes drilled in the center of each of the donuts. Thus, the stationary section rests on the non-rotating skeleton shaft. 221 Flow Donuts Wxxxxfl xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx m 2:wa Skeleton Shaft Figure 8.7 Schematic of the stationary section Located on the upstream portion of the 2D section is sandpaper, which was used to trip the boundary layer. Due to the long, development length needed to obtain a turbulent, boundary layer and the lack of space in the test section for such a length, the boundary layer was tripped with three pieces of sandpaper. The piece farthest upstream was located 70 mm downstream of the nose tip. This first piece was 38.1 mm wide and 120 grit. The second piece was 150 grit, measured 50.8 mm in width, and was located 115 mm downstream of the nose tip. Finally, the third piece was 50.8 mm in width, 150 grit, and placed 25.4 mm downstream of the second piece of sandpaper. Results showing the characteristics of the turbulent boundary layer established at separation may be found in Section 3.2. The last function of the stationary section was in the support system of the axisymmetric BFS. The upstream portion of the model was supported by four piano wires located at 45° with respect to the centerline of the model. The position of the holes 222 can be seen in Figure 8.7. Without the wire support the model acted like a cantilever beam. Holes at the 45° mark, 1 mm in diameter, were drilled in the 2D section. Piano wire (z 1 mm in diameter) were threaded through the holes and secured on the inside with a nut. The piano wire was pulled taut and fastened to the wire-support system, which is discussed in further detail in the Wire-Support System section. 8.2.4 Rotating Section The rotating section, shown in Figure 8.8, measures 0.124 m in diameter and 0.75 m in length. It was constructed out of aluminum tubing, having a wall thickness of 6.35 mm, to minimize the overall weight. Spacers, also referred to as donuts, were press fit into the tubing on either end. These donuts were annulus in shape having an outer diameter of 114 mm with a bearing press fitted into the hole drilled through the center. The main support shaft (25.4 mm diameter) was threaded through the bearing in each donut allowing the rotating section to spin freely around the main, support shaft. These bearings were self-aligning. The upstream donut housed a CSB series bearing (CSB 205-16) manufactured by Fafnir. This bearing is a combination bearing and collar with two set screws used to secure the collar to the skeleton shaft. By placing a bearing/collar at the upstream donut, the gap between the rotator and the section containing the step could be adjusted by sliding the bearing/collar along the skeleton shaft. Moving the bearing/collar in the upstream direction results in a wider gap and sliding in the downstream direction gives a smaller gap. 223 I :9! E flmuq The downstream donut in the rotator supported a Precision Steel Needle-Roller Bearing (supplier: McMaster-Carr 7929K24). The bearing was made for a 25.4 mm shaft diameter and had an outside diameter of 38.1 mm with 19.05 mm width. A needle bearing was used in order to conserve space for the three bolts that were used to connect the drive shaft to the downstream rotator donut. By connecting the drive shaft to the donut, the rotator spins with the drive shaft. A flange bracket was used to make the connection as seen in Figure 8.8. The three bolts connected the bracket to the donut and two more were secured to the actual drive shaft. The three bolts actually made a loose connection in the donut. In other words, the three holes in the donut were drilled oversize so the bolts did not make a rigid connection, but still forced the rotator to spin. Through trial and error it was discovered that running the rotator with the loose, bolt connection reduced the vibration in the model compared to running the rotator with a rigid connection. FlowE with CSB bearing Flange bracket and Its Donut with needle bearing Skeleton Shaft Drive shaft Figure 8.8 Rotator schematic 8.2.5 Step Section and Step Ring The step portion of the model is the “area of interest” in the project. This is where the boundary layer separates and forms an unstable, shear layer that eventually reattaches 224 downstream of the step. Figure 8.9 shows a schematic of the step region of the model. The step section is composed of several different pieces including the skeleton shaft that runs through the center of the whole model. The steel, drive shaft has a wall thickness of 7.13 mm and fits over the skeleton shalt leaving a 5.56 mm gap all the way around. A Double-Row Steel Ball Bearing (supplier: McMaster—Carr 5885K22) supports the drive shaft at the upstream end of the step section while still allowing the shaft to spin. This bearing has an outside diameter of 90 mm and an inside diameter of 50 mm with a maximum rpm rate of 4,000. A 1At-20 set screw was used to secure the double-row ball bearing in place. At the downstream end of the step section, a needle bearing (supplier: McMaster-Carr - 7929K24) was placed between the skeleton and drive shaft. On the outer side of the drive shaft, the CMT pulley (manufacturer: CMT 42-XL-037-3-A-312) with a pitch diameter of 67.9 mm was used to rotate the shaft. The pulley was connected to a timing belt, which wrapped around the same type of pulley that was press fit on the motor shaft. Details of the motor and timing belt are given in Motor and Drive section. Finally a thrust bearing (supplier: McMaster-Carr 5909K36) and two thrust washers (supplier: McMaster-Carr 5909K63) were placed at the downstream end of the drive shaft to absorb any axial loading caused by the rotating drive shaft. 225 71—!— 4¥ .L“-- H! l Flow ——> Bearing CMT 5885K22 Step Needle bearing I: :: Jacket D1112 shaft Skeleton Shaft 7929Kg4 Pul'ley W I - “.111?!” ...-x-vrxtnnxfuvxumx" I txu-;:7j-;;1;- _[\ I’d . .. ...... . ..... . ........................................................... A I / . % Spacers Thrlcrist beetr‘rrngs l/4-20 threaded hole for jacket an was ers for set screw Figure 8.9 Schematic of step section The instrument plates that housed the static pressure taps and the microphones were placed in the step section immediately downstream of the step. These plates are explained in further detail in Section 2.3. A step ring measuring 0.124 m in outside diameter and 50.8 mm in width was secured around the step section of the model as shown in Figure 8.10. 226 Slop Ring Figure 8.10 Picture of step ring on model The step ring was constructed out of two halves so the ring could be easily taken off and put on the model if the instrument plates needed to be accessed. The gap between the step ring and the step section was less than 0.254 mm. The step ring was anodized to reduce surface glare in the Particle Image Velocimetry images. A step ring was used in order to keep flexibility in the model design. By using the concept of a step ring the distance between the rotating section and the edge of the step could be adjusted by machining additional step rings of varying width. Changing the width of the step ring or removing it all together varies the length over which the 3D boundary layer relaxes. This affects the strength of the cross flow at the separating edge. The height of the step was a key parameter in the design of the model. Out of the three classic separating/reattaching geometries (backward-facing step, splitter plate, and splitter-plate-with-fence) the backward-facing step has the smallest separation bubble as 227 noted in Hudy3. Eaton and Johnston4 reported that for a planar BF S the reattachment distance varied from 4.9 to 8.2 step heights for approximately 23 experiments reviewed by the authors. Generally, the reattachment length is around 6 step heights downstream of the separation edge. When designing the model, careful consideration was given to the reattachment length because this length would dictate the size of the separation bubble. It was necessary to have a “good sized” separation bubble so a series of microphones and static pressure taps would fit under the bubble and give good spatial resolution. In addition, the separated-flow region needed to be large enough so it could be resolved well using PIV. If the bubble were too small then that would restrict the number of microphones and taps along with the number of velocity vectors that could be captured beneath or inside the bubble. If the bubble were too large then, in terms of PIV, multiple views would need to be set-up to capture the entire separation bubble region, which is very time consuming. Besides separation bubble size, blockage ratio was another limiting factor in terms of determining the maximum diameter of the model. Blockage ratio is defined as the frontal area as seen by the flow divided by the total area of the test section perpendicular to the flow direction. Because a section of the model needed to rotate and sensors were to be embedded in the model, space inside the model was necessary and a limiting factor regarding how small in diameter the model could be in order to fit all the components. The other issues were the overall cost of the materials and manufacturing the model. Based on these design constraints, it was decided that a 101.6 mm outside diameter tubing with 6.35 mm wall thickness could house the 25.4 mm diameter skeleton shaft with the 50.8 mm diameter drive shaft and still have a 12.7 mm clearance for the 228 electronics. In addition, the step height was optimized to be 12.7 mm, which gives a reattachment length of 76.2 mm based on the xr = 6h and a blockage ratio of 3.6%. The thirty-two microphones, each with a diameter of 2.54 mm, were spaced 2.2 mm apart, edge to edge. The microphone spacing resulted in a total measurement distance along the centerline of approximately 150 mm, which ensured that the reattachment point would be contained within the extent of the microphone array. Once the step height was determined, the rest of the model’s dimensions could be calculated based on the information presented by Driver and Johnstonz. Driver and Johnston2 used an axisymmetric model to develop a three-dimensional, shear-driven, turbulent, boundary layer. The total axisymmetric BFS model length was 187h. The long length of the model used in this study not only ensured a reattachment point on the model but also aided in reducing the effects of downstream disturbances that could propagate upstream, i.e., acoustically, and change flow conditions in the measurement region. Farabee and Casarella5 observed, in their backward-facing-step study, energized structures from the shear layer up to 72 step heights downstream of the step. The point where the wake develops is so far downstream of the measurement region that the probability of the acoustics from the weak disturbances convecting upstream to the region of interest is small. One other important item to mention in this section is the jacket shown in Figure 8.9. The jacket was used to separate the tubing and wiring from the drive shaft in the step region. The jacket was made out of a 0.8 mm thick aluminum sheet that was rolled to make a cylinder. It was secured to the inside walls of the step section using spacers as shown in Figure 8.9. 229 8.2.6 Instrument Plates (I-plates) Both static pressure taps and the Emkay microphones were housed in the north instrument plate seen in Figure 2.5. The north instrument plate is beneath the main area of measurement. PIV measurements were taken above this plate in a plane perpendicular to the microphones. The details surrounding the construction and the instrumentation of the I-plates are described in Section 2.3. 8.2.7 Motor Section The motor section of the model stretches 0.41 m in length with a 101.6 mm outside diameter. This section is made out of 15.9 mm thick steel tubing. Two donuts at either end support the skeleton shaft. In the downstream end a 1/2-13 allen screw is used in the threaded hole at the end of the skeleton shaft to tighten all the model components. The clamp collar (supplier: McMaster-Carr 6435Kl8) prevents any axial shift of the pulley. The motor section fits into the step section using a male-female fitting as shown in Figure 8.11. 230 Clamp collar Flow 5 Threaded holes to attach model to stand Figure 8.11 Motor section schematic 8.2.8 Drive Shaft and Motor The drive shaft was constructed out of aluminum. It measures 557.2 mm long and 50.8 mm in diameter. Figure 8.12 shows a schematic of the drive shaft. On the upstream edge, two tapped holes are shown. These two holes were used to attach the flange bracket (Figure 8.8) to the donut inside the rotator. By making this connection, the rotator spins when the drive shaft spins. Farther upstream, there is a change in diameter from roughly 50 mm to 50.8 mm. The junction between the two diameter sizes is a shoulder for the double-row steel ball bearing (5585K22). The upstream portion of the shaft was machined to fit the bearing. On the downstream edge, there is another shoulder 17.6 mm long that was machined to fit the pulley. Inside the drive shaft, two needle bearings were used to support the drive shaft on the skeleton shaft while still allowing the drive shaft to rotate. 231 Shoulder for Double- Row Steel Ball Bearing Shoulder Flow (588 51(22) for Pulley L ' l u I v ' I I 9 I I t u ' I I I I t I o . r | I I t - l . I I I I . I I r I I u I l I I ‘ I r I I r I r l I u ' r I I a u I t t . r I v t r a I r l I I I a I I I t I I I I ' I l l l l I I D . l I I I u a I o r l . I | l r r I I r r I - I I 1 a l t I l I I O ‘ I I I o I I I I r I ' l r I I l r t n . I I I I l I I I I I I I o I I l l c I I I l ' I I I r l I a l l I ' r u r I I I l I , I I I I I l I l I I I - l l I t I l I I I I I ' I t r I I I r I I u ' I n I I I n l I I u l I . i l .L 1 Tapped holes for mounting flange bracket for connecting drive shaft to rotator Figure 8.12 Schematic of drive shaft A dc motor (M1110015) from Leeson was used to drive the shaft that spun the rotator. The motor was a sub-fractional-horsepower, DC, permanent-magnet motor. It was designed for use with full-wave, nonfiltered, SCR controls for adjustable-speed applications requiring dynamic braking and constant torque throughout the speed range. (Information based on catalog.) It had a full load RPM of 3500 at 90 Vdc. The motor was powered using a direct-current, power supply from Instek (Model GPR-11H30D). The power supply had a 0.01% high regulation, constant voltage and current operation, internal select for continuous or dynamic load, and low ripple and noise. It was capable of output 0 to 110 volts over 0 to 3 amps. The motor was connected to the same pulley described in the Step Section using a timing belt, thus resulting in a one-to-one ration between the motor and the drive shaft. An optical sensor (TPU-HSA) from Dart Controls was used to monitor the pulley and thus, the rotator in order to determine the rotations per minute. 232 8.2.9 Tail The tail section was used to reduce possible downstream disturbances by gradually tapering the model thickness to a point. This would moderate the step-like feature at the end of the model that may create disturbances. These disturbances could propagate upstream, i.e. acoustically, and change the flow conditions in the separating/reattaching zone of interest. The tail of the model was attached to the motor section using a male-female fitting. It was constructed out of solid aluminum and conical in shape. The height of the cone is 266.7 mm with a 101.6 mm base diameter. The tip of the tail was fastened on by a 10-24 screw. This allowed for the option of removing the tip to screw on a turning mirror for positioning the laser beam upstream during PIV measurements. A sketch of the tail is shown in Figure 8.13. w r r Removable 1 1 tail tlp Male-female fitting Figure 8.13 Sketch of tail section 233 8.2.10 Wire-Support System The wire support system was used to support the most upstream end of the model. It consisted of four cylinders, 0.99 mm diameter piano wire, four swivels, four hanger screws, and some washers and nuts. The piano wire was strung through each of the 1 mm holes drilled at 45° angles with respect to the normal and spanwise planes in the stationary section of the model (Figure 8.7). These holes were located 38.1 mm downstream of the end of the nose section. Nuts were tied at the end of the piano wire so the wire would not slip through the holes. The piano wire was then stretched outside the tunnel at the 45° angles and fitted around the swivels, which were directly connected to the hanger screws. Each hanger screw was threaded through the cylinder that had a flat top with a centered hole. Washers and nuts were used to secure the hanger screw and tighten the wire. Figure 8.14 shows a picture of one corner of the wire support system and Figure 8.15 shows one side of the support system in use at SBRT. The actual system itself was connected to perforate square bar using threaded rod and nuts. The advantage of this system was its versatility. The angle of the cylinders could be adjusted as well as the length of the piano wire and the amount of tension applied to the model. The nuts on top of each of the cylinders were adjusted until the model spun freely and was level with respect to the tunnel floor. 234 ‘I’I’ 1;! 'I ) ,. ) .i I I I Figure 8.15 View of the wire-support system in use in SBRT 235 8.2.11 Support System In order to support the axisymmetric backward-facing step, a support system made almost entirely out of steel was constructed. Steel was used to ensure that the weight of the support system could counter the weight of the model and prevent it from tipping over. There are two parts to the model, support system: the stand and the traverse table. The stand is shown in Figure 8.16. It stands 0.52 m tall and is constructed from two 19.05 mm thick steel sheets welded together to form an upside-down “T.” The base of the stand measures 0.50 m wide in the spanwise direction and 0.70 m long in the streamwise direction. The stand is bolted directly to the motor section of the model using 8 bolts. It houses the motor, as seen in Figure 8.16, used to drive the shafi that spins the rotator. A shroud was placed at the upstream edge of the stand to round the leading edge, reducing the possibility of disturbances that could propagate upstream into the measurement region. Inside the shroud, the microphone wiring and static-pressure tap tubing were hidden from the free stream and directed outside the model to be connected to the appropriate instrument. 236 it E] [‘3 i] E] 1 Motor Figure 8.16 Model support stand The second part of the model, support system is the traverse table shown in Figure 8.17. The traverse table is made almost entirely out of steel. Its purpose is to vertically and horizontally traverse, rotate, and/or change the pitch angle to align the model in the test section. The pitch angle is defined as the angle between the streamwise axis that is parallel to the floor and the centerline of the model in a vertical plane. For example, if the nose is higher than the tail, the model is at a positive pitch angle. The model stand was bolted directly on the traverse table, which was leveled with respect to gravity before starting to align the model in the test section. 237 Figure 8.17 Picture of the stand used to support the model At NASA Langley the model, support stand was raised 177.8 mm in order to center the stand in the test section. To accomplish this, a 177.8 mm tall by 503.8 mm thick by 762 mm long piece of steel was bolt to the stand. Figure 8.18 shows the extension added to the model stand at Langley. 238 Figure 8.18 Model-stand set-up in SBRT at NASA Langely 239 8.2.12 Model Assembly axisymmetric, backward-facing step. Disassemble the model 1. 9‘95”!” >19 9. 10. 11. 12. 13. Re-assemble the model 9989999391“ The following is a rough procedure for disassembly/assembly of the Remove the set-screw - located on the outside of the model at the upstream end of the 2D section. It is used to secure the nose in place. Remove the nose - this may require applying carefial pressure (tight fit) Loosen the 1/2" bolt inside the model (exposed by removing the nose) Slide the 2D Section off the center shafi Remove the set-screws (there should be 3) inside the upstream end of the rotator, which are used to secure the donut inside the rotator to the center shaft Slide the rotator off using LIGHT up and down pressure (may have a tight fit) The three bolts used to connect the rotator with the drive shaft should not make a rigid connection. These should slide in and out of oversized holes. To remove the step module for access to the pulley and timing belt, this should require first matching the red lines between drive shaft and step module Remove the set-screw at the upstream part of the step module (might need slide hammer to remove set-screw) Use slide hammer to take out 3 set-screws in the downstream part of the step module There should be one bolt to remove At this point, there should be easy access to the pulley. The tail tip can also be removed see Figure 8.13 Reverse disassembly steps Replace step, bolt and set-screws Replace rotator module Make sure to use a 0.015" shim between the step and the rotator modules Tighten the 3 set-screws inside the upstream end of the rotator Slide the 2D Section into place along the center shaft Tighten the '/2" bolt inside the model at the upstream end of the 2D Section Replace the nose Tighten the set-screw used to hold the nose in place 240 8.3 Azimuthal Uniformity To verify the alignment of the axisymmetric BFS in SBRT, the azimuthal uniformity of the flow was checked using a hotwire probe. The sensor was placed approximately 6.35 mm upstream of the step at five different heights: 6.35, 9.53, 12.7, 19.1, and 25.4 mm. This translates to y/0 = 3.4, 5.1, 6.8, 10.2, and 13.6 for the Re}1 = 8081 case, which was the only case used in this test. To position the sensor, a second ring designed similar to the step ring was used to hold and rotate the probe around the model. The inner diameter of the ring was equal to the outer diameter of the model downstream of the step. Figure 8.19 shows an image of the hotwire set-up for the azimuthal uniformity test. A pitot probe was also placed in the free stream at the same streamwise position as the hotwire probe and was rotated with the hotwire sensor. The pitot probe provided total pressure data at the same streamwise location as the probe was rotated around the model. The two probes were rotated through 16 angles of 225° increments for a total of 360°. 241 Figure 8.19 Set-up for azimuthal, uniformity test Prior to running the azimuthal, uniformity test, the azimuthal uniformity associated with the rotation of the ring holding the hotwire probe was checked. This was done using a dial gauge indicator secured to the probe-holding ring. Figure 8.20 shows the dial gauge test set-up. The results are plotted in Figure 8.21, where the reading of the dial gauge was set to zero at the north position ((1) = 0°). As seen from the figure, the azimuthal traversing uniformity is within i 30% of the momentum thickness for the Reh = 8081 case. This variation could be due to the lack of concentricity of the first and/or second step ring, which is difficult to decouple. The height differences resemble that of a sine wave with peak high variations occurring at angles 50 and 200 degrees and a low peak near 125 degrees. 242 Figure 8.20 Dial-gauge-test set-up 0 $5 225:? o. o 2%: Angle (degrees) Figure 8.21 Results from the dial-gauge test 243 The results for the mean-streamwise-velocity, azimuthal-uniformity test are shown in Figure 8.22 for all 5 heights tested. The variations measured in the azimuthal symmetry are within i 10% of the azimuthal-average value of the measurements. This variation could be due to the change in position of the probe within the boundary layer associated with the inaccuracies of the azimuthal traversing mechanism used. Nonetheless, the variation is reasonably small. 80 i 1 , 1 i fl .i.____-.. ..i__._ ___. i __.__ ._ i._.' 50 100 1 200 250 300 350 Angle (degrees) Figure 8.22 Azimuthal symmetry of the flow “at” separation for Re;I = 8081 244 8.4 Additional Microphone Information 8.4.1 Microphone Specifications gaskets iIiI (no Senslttvlty In as relative to 1.01m Pa (NImZ) do o 2 3 456789 2 3 4667 Open circuit sensitivity with 1.3VDC supply § Figure 8.23 Typical FG-3629-P16 Frequency-Response, Curve - Emkay data sheet 8.4.2 Circuitry Figure 8.24 shows a picture of the wiring for the microphone array that was embedded in the I-plate located downstream of the axisymmetric BF S. Each microphone had three 25.4 mm leads (for more microphone detail, see Section 2.3.2): power (V cc), ground, and output. Copper tape was laid on the underside of the I-plate and was used as a ground bus. In addition, a printed-circuit board was cut to the appropriate length and width so it would fit inside the I-plate slot on the model. On the copper board there was a Vcc bus and individual plugs used to connect the output from each microphone. The three leads from each microphone were soldered to the appropriate bus or pin (Figure 8.24). There was one common ground and Vcc on the I-plate and 32 individual output signals. 245 Therefore, there were 34 wires coming from the I-plate to the circuit box, which was used to power the microphones and break out the output cables for individual microphones. ’4. V“. bus Output signal Ground 9 “(.0 ”mm Wire Color Legend Gray = Ground Figure 8.24 Wiring for the I-plate The circuitry to operate the microphones required a 1.5V voltage regulator to supply power to all 32 microphones via the single supply (Vcc) line. The output signal lines from the microphones were AC coupled with BNCs using a 0.1 uF capacitor. These BNCs were then directly connected to the anti-aliasing low-pass filters and digitizers inside the control room. In the original design, the signal outputs were sent through an amplifier board before connecting to the BNC outputs. However, there were grounding issues with the circuitry; and, therefore, the direct connection was made instead since the lengths of the coax cables were not too long and onboard amplification was not essential. 246 8.4.3 Calibration A bench calibration was performed on each microphone before placing the microphones in the model in order to determine the individual sensitivity and phase reponse. The microphones were calibrated using a B&K 4226 multifunction calibrator at NASA Langley. The calibration assembly is shown in Figure 8.25. The calibration frequencies extend from 20 Hz - 15 kHz, using a frequency step size of 5 Hz over the range of 20 - 100 Hz and a step size of 250 Hz over the range of 100 - 15000 Hz. Results for each microphone showed sensitivity values similar to that plotted in Figure 8.26, plus a ISO-degree phase shift between the calibrator and the microphone output. The phase is constant over all frequencies with an RMS deviation of 5° or less for all microphones. The individual sensitivity values for each microphone used in the data analysis are plotted in Figure 8.27. One other point to mention is that 20 Hz was the lowest frequency that calibration data could be acquired reliably using the B&K 4226. Nonetheless, the low-end of the measured frequency was below the frequency response range of the microphones. A similar situation was seen by Hudy3. Because the microphone sensitivity below 20 Hz decreases, a reduction in the amount of energy recorded in the signal from the surface pressure fluctuations below 20 Hz is expected. This primarily affects the energy level for the microphones closest to the step, where the low-frequency fluctuations are dominant. However, a very small effect, if any, is expected on the shape of the wall-pressure spectra seen in Section 4.2.3. Because the A/D channels were multiplexed, there is an average time delay between each sampling microphone, which results in a systematic error in the sampled 247 data. The time delay was estimated from the phase calibration data to be 336 us at the frequency where the wall-pressure spectra exhibited a peak (ill/U.o = 0.03 9). This value is compared to the convective time scale over the length of the entire array of microphones, which is 15 ms. This is two orders of magnitude larger than the time delay of the microphones. 248 7. 18- 2003 Figure 8.25 Microphone, calibration assembly 249 L l N I N L (SPme) - (SPLSW), dB 6: l l l l l I l l I I l I I l I l l l l 1 1 l 1 l 100 1000 10000 180 Q 8 A 0’! -45 Phase + 180°, deg C 100 1000 10000 Frequency, Hz Figure 8.26 Sample of calibration results for the FG-3629 microphone 250 Microphone number Figure 8.27 Microphone sensitivities from bench calibration 8.4.4 Preliminary tests Several preliminary tests were completed in effort to establish the proper, microphone, experimental procedure. These tests included cross talk check, laser noise, and mirror interference. Below is a brief description of each test along with the results. 8.4.4.1 Cross talk check Because of the multi-channel nature of the microphone measurements, the microphones had to be checked for cross talk. Cross talk is defined as the ‘leakage’ of the signal from one microphone into the neighboring channel of another microphone by electric means (e.g., electromagnetic or capacitive coupling). In order to test for cross talk, the hand-held B&K 4226 multifimction calibrator was used to generate a harmonic 251 sound signal at 250 and 1000 Hz. The signal was fed to a single microphone using the calibrator shown in Figure 8.25. Data were acquired for all 32 channels, and the auto- spectrum of each channel was calculated and used to estimate the relative strength of the cross talk signal at the channels other than where the sound was applied. The acquisition rate per channel used was 6000 kHz. The results showed that the maximum amount of cross talk observed was about 35-40 dB lower than the input signal at 1 kHz. That is, if one channel receives 1 kHz sound signal and the spectrum obtained at this channel was normalized so that the spectral peak at 1 kHz is 0 dB, then the other 31 channels would read no higher than —35-40 dB at 1 kHz. This means that there is almost two orders of magnitude difference between the input signal recorded on a single microphone and leakage recorded by the other microphones. Thus, the cross-talk that may occur is very small and not a concern. 8.4.4.2 Laser Noise Because microphone and PIV measurements were conducted simultaneously, it was important to check that the laser did not induce a noise signal in the microphone at the laser repetition frequency. Such a signal could be produced by electromagnetic coupling since the laser sheet was immediately above the centerline microphones or by the audible sound heard when the lasers were pulsed. Figure 8.28a, b, and 0 show the surface-pressure power spectrum obtained at microphones 2, 11, and 26 respectively. Inspection of Figure 8.28a, b, and c clearly shows that the measured surface-pressure auto-spectra contain no peaks at the laser frequency (10 Hz) or its harmonics. Therefore, no special filtering needed to be done for the laser noise. 252 f(Hz) 253 x10 9 7 I l I l l‘rll‘ 7 ‘7 :I'lll 1 s ‘7 2027‘ a, . {1771777717 Y7 , i l I i H Ii; 5” ‘1! l i l i l i l i l 7. , . ‘ 7111117.);21' Li .‘ _ " “ L ‘M l l l 7 77‘ M . 7 7 7 l, ,, “ 1;) ,7 1:171:3111‘, , J I. , ,7 ,, l ‘i i lilllil‘ ‘9 ll l i ‘ I ‘ lll hill 1 I ., , ‘ . . ,lgl - . ‘ ‘. - - N 5‘ ll ll ' fl‘ l 77 l 8 l l I Ill‘l l i ii i‘ i i l i i ‘l l ‘7. .1117: H .:1 “ 37“ 7 "fir”m 5:777? l . ‘7‘77‘ 1 f ' ”1,7197 1111‘77‘l 21* ‘ *7 7‘ 7 77 *‘7 ‘7‘i-777‘7'7r'7777'l-{7‘7 ‘ {k I ‘l i l‘ l J *lTh l l l l "i . 77 77I ;’7 i . ‘72 '7; 1 T A f 1 ’1’ l T 1’71 "7 ‘(r ‘F'TT (’7 'lT l 7 7‘ Nb : 1‘::‘7‘l 0L L7 1,117.1; L 7;; ‘ggyrnzmmbn‘ LLL‘JJ 10° 10‘ 102 1o3 10‘ f(Hz) Figure 8.28 Surface-pressure power spectra at microphones: a) 2, b) ll,andc)26 8.4.4.3 Mirror Interference As discussed in Section 2.3.3.1, the region of the PIV measurements was illuminated with the laser sheet using a mirror that was mounted to the tail of the model. Figure 8.29 shows a picture of the mounted mirror. Due to the abrupt shape of the mirror and mount, it was desired to check if any disturbances produced by the protruding mirror affected the surface pressure measurements. Additionally, the mean pressure distribution was checked to verify that there was no effect on the mean flow as well. To this end, the microphone as well as the static pressure tap data were checked before and after the mirror was mounted on the model in the tunnel. Figure 8.30 shows the static pressure data for the two conditions for the Re.1 = 8081. Although there was a slight increase in the static pressure along the surface when the mirror was placed in the tunnel, there was 254 no change in the characteristics of the flow. In addition, reattachment remained at practically the same position for both conditions. l ' .3.. With mirror I -0.15 ‘0.1[ iii’nfifi r? ‘ +? ra” H-e- Without mirrorl l l l l r -0.2i 0° ' -0. 25 x/h Figure 8.30 Static pressure data with and without mirror in tunnel for the Re.I = 8081 255 Surface-pressure, spectra data were compared with and without the mirror in the tunnel for the microphone closest to the step and the microphone nearest to reattachment, Figure 8.31a—b respectively. The data shown are for Reh = 8081. Wall-pressure data were acquired for 60 seconds at a 12 kHz sample rate independent of PIV measurements. As seen from Figure 8.31, there is a small increase in the overall level of pressure fluctuations (corresponding to 12% in the RMS level). Nonetheless, the shape and dominant peaks of the spectra remain unchanged. Only the magnitudes of the peaks increase with the addition of the mirror, which could be due to the fact that the pressure signatures were acquired on different days. The percent change in RMS level between two different days with the mirror installed was 5% over all microphones, suggesting that the microphones may have been affected by such outside factors as relative humidity between the two days. This systematic influence seems to have affected the magnitude and not the frequency content. 256 3 _7 -L777n .......... _7-7.~77Li-- - - w 4 ,7 llkllil, llllllllll 7 llllllll r— lllllllll .),ii t .................... Liii iii ib_ lllllllllllll kl lllllllll _ilii—Illl , 'rnor i — Without rm — With mirror 7.4 1.8 Figure 8.31 Auto spectra of microphone data with and without mirror: a) microphone closest to the step and b) microphone nearest to reattachment 257 8.5 Additional PIV Information 8.5.1 Dot Card The dot card, shown in Figure 8.32, was used in the post-processing of the images. The image shown in Figure 8.32 is reversed in the streamwise direction compared to how the results are presented in this paper. All the PIV imaging was done with the flow moving from right to left; however, the results are presented with the flow moving from left to right, which is the typical way to display such results. The region of interest in the flow stretched about 8.4h downstream of the step and -O.5h upstream of the step. In order to have good resolution in the PIV images, especially in the high shear regions of the separating, boundary layer, the field of view for one camera stretched only 4.5h. Thus, two cameras were needed to capture a large portion of the region of interest. With the two cameras, the flow field captured stretched from roughly -O.5h to 8.4h in the streamwise direction as shown in Figure 8.33. 258 Figure 8.32 Example of dot card image from camera #1 z r: ——————————— ‘I y l | ' I . i .: ' X ; Field of I I Field of I I view for l I view for I i camera #1 | l camera #2 : I I 1 u : g —>: i<—— Overlap | 7 <—— 8.4h —> <— Streamwise distance of microphone array — 12.5h —> Figure 8.33 Flow regions captured by the two cameras used Because two cameras were used, the two images needed to be merged in order to show the full region of interest. The images were merged before processing the vector 259 fields. Prior to merging the PIV images, a piecewise, bilinear, dewarping technique was employed to remove perspective and optical distortions. This technique of “straightening” the image was developed for Doppler Global Velocimetry by Meyers6. The dewarping technique required dot card images in order to set-up the correct transformation between the old image and the new dewarped image. The images were then merged by removing the pixel columns that overlapped. The number of columns to be removed was determined from the grid card, which is explained in the next section. 8.5.2 Grid card The grid cards, shown in Figure 8.34, were used in two ways: 1) to determine the number of pixel columns to drop in order to merge the images, and 2) to calibrate the field of view of the cameras in order to determine the physical units per pixel. First, the grid cards were dewarped as described in the previous section. Then overlapping columns were removed from each image and a new matrix describing the image was formed. Figure 8.35 shows an example of a merged, grid-card image. The settings used to remove the columns for the grid card images were used on the PIV images afier dewarping and prior to processing the vector fields. The grid card was also used to calibrate the field of view. The calibration was done on a merged, image grid-card, similar to the one shown in Figure 8.35. To calibrate the field of view, the number of pixels was counted between grid lines. The spacing between each grid line was 10 mm. The number of pixels was divided into 10 mm to determine the calibration constant of mm/pixel. The calibration constant was determined to be 0.0569 mm per pixel. 260 Figure 8.34 Grid cards from cameras 1 and 2 - after dewarping, prior to merging Figure 8.35 Example of merged, grid card from PIV set-up 261 8.6 Stochastic Estimation The low-dimensional, mathematical model used in this study to estimate the velocity field from surface pressure measurements was stochastic estimation (SE). The technique was first proposed by Adrian7 as a tool for extracting coherent motion in turbulent flows in the 19708. Since then, the low-dimensional model has been used to estimate the conditional average in a variety of flows including isotropic turbulence (Adrian7), homogeneous shear flows (Adrian and Moins), separated flows (Stokes and Glauserg), and turbulent wall-bounded flows (Adrian et al. 10, Guezennecl 1, and Naguib et al. 12). Stochastic estimation is a conditional, averaging technique that uses unconditional statistics, namely two-point correlations, to estimate a variable at a particular point in space or time based on the information of a known variable at the same or other point. In the case of this study, the known variable is the pressure signature on the surface, p'(xo,y0=0,zo=0,t), and the unknown variable is in the turbulent velocity field above the surface, u'(x0+Ax,Ay,t) and v'(xo+Ax,Ay,t). Thus, the turbulent flow field above the wall was estimated from a known pressure event or events located on the wall. SE is based on Taylor series expansion of the unknown variable in terms of the known variable: 5’06. “my. +Ay,t)= A..(Ax,Ay;xo,yo)p'(xo,yo,t) +Bu(Ax9Ay;xo’yo)p'2 (xo’yo9t)+0(p'3) (8°13) i7"(x0 + Ax,y0 + Ay,t) = Au(Ax,Ay;x0,y0)p'(xo,yo,t) + Bu (Ax, Ay; meo )p'2 (x0,yo,t) + 0(p'3 ) (8.1b) 262 where 17' and I7. is the estimated streamwise and normal fluctuating velocity in the flow field, respectively, x0 is the streamwise position of the known variable, Ax is the streamwise distance between the known and unknown variables, yo is the normal position of the known variable, Ay is the normal distance between the known and unknown variables, t is the instance in time the pressure event occurs, p' is the measured fluctuating pressure on the wall, A., is the linear estimation coefficient term, and Bu is the quadratic estimation coefficient term. From Equation (1), the estimation coefficient terms can be determined by minimizing the average squared error between the estimated flow field and the measured velocity field at each node in the flow field. E{ez} =< [17'(x0 + Ax,y0 + Ay,t) — u'(x0 + Ax, yo + Ay,t)]2 > (3,2a) E {e2} =< ['17'(x0 + Ax, yo + Ay,t) — v'(x0 + Ax, yo + Ay,t)]2 > (3.21)) where u' is the measured fluctuating velocity component in the flow field, e is the error between the estimated and measured fluctuating velocity components, and E is the expectation or average of the square of the error. Minimization requires setting the derivative of the expectation to zero and solving for the coefficients Au and Bu, etc. 8E{ez} _ 6E{ez} _ 6A 63 ' II II -- = 0 (8.3a) 6E{e2} _ 8E{ez} _ 6A, _ aB _' V -~ = 0 (8.3b) 263 The number of estimation coefficients to use in the SE of a velocity field was first investigated by Tung and Adrian13 who looked at higher-order estimates of conditional eddies in isotropic turbulence. The researchers compared first to fourth-order expansions of the SE equation using velocity measurements at one point to estimate the conditional eddies in the flow field. Based on their findings, Tung and Adrian'3 concluded that linear, stochastic estimation (LSE) was the highest order required to satisfactorily describe large-scale features of the two-point conditional eddy field. Guezennecll also investigated using linear versus non-linear estimates and found little improvement when using the non-linear estimates to predict the flow field in a turbulent boundary layer. Although the author felt that the quadratic term did convey additional information that would be important when the probability density fimction (pdf) of the velocity was highly skewed. Guezennec” also investigated making time-dependent estimations of the turbulent boundary layer using multiple velocity reference points as proposed by Adrian et al.10 These multi-point, conditional averages were computed in order to force a length scale on the estimated flow field, which allowed the author to focus on structures of a particular length scale in the boundary layer. As a result of the author’s findings, Guezennecl ' suggested that using multi-point, conditional estimates coupled with measured velocity time sequences could lead to more “realistic” estimations of the evolution of the instantaneous velocity field. Cole and Glauser14 reaffirmed the idea of using multi-point SE by investigating utilizing multiple reference velocity points in their estimation of the dominant structures in a jet mixing layer. They presented their results in terms of percentage of energy captured by the multi-point estimates. The authors found that single-point estimates did not adequately estimate the instantaneous flow field 264 and that at least two reference points located on opposite sides of the shear layer did represent the flow field well. Thus, from this discussion, it can be concluded that multi- point, linear, stochastic estimation can satisfactorily estimate the velocity field. However, the estimations in these studies used a velocity point within the flow field to estimate the entire flow field. In the present study, the wall-pressure signature is used to estimate the flow field. The Naguib et al. '2 study was the first that investigated using stochastic estimation to determine the flow structures associated with surface pressure events in a turbulent boundary layer. In their study, they used the pressure signal that was acquired simultaneously with hot-wire measurements to estimate the flow field above the surface. The group concluded that the quadratic term was needed to accurately represent the conditional velocity field and attributed this to the influence of the turbulent-turbulent (non-linear) pressure source term. In the present study, single- and multi-point, linear estimation has been utilized for the estimation of the flow field. In addition, single-point quadratic SE, as suggested by Naguib et al. '2, has been computed to assess the sufficiency of the linear estimations in capturing the details of the flow features. Future work includes expanding the single-point QSE to a multi-point estimation. Thus, linear, stochastic estimation was used to estimate the u' and v' components of the velocity based on the wall-pressure signature according to 17'(x0 + Ax»). + Ay, t) = A. (Ax, Ay;xo,yo)p'(xo,yoat) (8-4a) '17'(x0 + Ax, yo + Ay,t) = A,(Ax,Ay;x,,y,)p'(xo,yo,t) (8.4b) 265 where 17' and '5' is the estimated fluctuating velocity components, and Au and Av are the linear terms for the estimation of u' and v', respectively. Employing Equations 8.2 and 8.3, and after some algebra, Au and Av terms are found as shown below ~, u p, I (8.53) P iws ~' V. I p. i 8.5b P iws ( ) where, p'RMs is defined as the root-mean-squared of the fluctuating pressure. Both linear coefficients are a function of the cross-correlation between the fluctuating velocity components and the fluctuating pressure on the surface. These two-point correlations are computed at zero time delay between the pressure signature and the flow field; and, therefore, it is necessary to acquire both the pressure and velocity data simultaneously. An important point to recognize is that with stochastic estimation if there is no correlation between the flow field and the pressure signature, the estimation has no significant physical value. The technique provides information about the flow structures that are directly related to the pressure signature. Thus, SE gives a physical interpretation of the dominant the flow structures associated with the pressure generation on the surface. In addition to single-point, multi-point LSE was used to estimate the velocity field in the present study since a coherent motion, depending on its size and strength, will most likely affect more than just one microphone. The single-point, linear, stochastic-estimation equation given above can easily be extended to the multi-point LSE. 266 17': Auop'(xo,t) + Amp'(x1,t)+ Auzp'(x2,t) +...+ Amp'(xn,t) ‘7': Avop'(xo’t) + Avrp'Ixrat) + szp'(x2,t) + ...+ I‘an'mat) (8.6a) (8.6b) where, Aug-Au“ and Avg-Ayn are the linear estimation coefficients for each pressure point used in the LSE. These are obtained from I- “ F- Auo p'o [9'0 [9'0 19'. [9'0 19'; [9'0 19'. u'p'o Aul F1 F0 P'r p'r P'I F2 17'] p'" u'p', A02 17.2.p'o P'zop'I p'2.P'2 17.2.10." u'op'z (8.7a) LAW- Lp'. [2'0 19'. p'. p'. p'z p'. p'. _u'p'n_ _ — I- I I I I I I I I _-l — I I .- AvO popo pop] 1’on PoPn Vpo Av! P'I 17.0 17.1 17.1 P'I P'z p'r p'n v'p'l I2 [2'2 [2'0 12'; 19'. [9'2 [9'2 10': p'. V'p'z (3.71» LAW _p'. p'o p'. p" p'. p'z p'. p'._ _V'p'._ In general, flow structures will imprint a pressure signature on multiple microphones. Thus, when estimating the flow field above the surface from the wall- pressure field, the multi-point, linear, stochastic estimation uses the pressure signature from an array of two or more microphones as the condition on which to estimate the flow field. The number of microphones to use in the estimate depends on the correlation of In some cases a flow structure may affect the the velocity with the pressure field. 267 pressure seen by five or six microphones only; and, therefore, only those microphones would be needed in the estimation. However, it is difficult to know exactly how many microphones are appropriate for reconstructing the flow field based on the pressure distribution. This problem can be remedied by using a large array of microphones extending beyond the spatial range of interest for the estimation. Overall, using multi-point LSE can increase the accuracy of the estimate compared to single-point LSE as explained by Cole and Glauser”. With the single-point LSE, the flow field is being estimated based on a single condition at one point at the wall. Thus, it is unknown in the estimate how the pressure varies spatially up and downstream of the location of the pressure event at the single, streamwise position. Thus, the spatial pressure pattern associated with the single-point-pressure value is generally random. By using the multi-point, LSE technique, the flow field is estimated based on a particular, spatial-pressure, distribution condition. Thus, the condition for the estimate is stricter; and, therefore, the estimate is more likely not to deviate substantially from the instantaneous flow field (especially for flow fields where the pressure generation is dominated by coherent motion). Being able to impose a strict, spatial condition for estimating the average, flow field is easier to do using multi-point, stochastic estimation than using conventional, conditional averaging. The difficulty with the conventional technique is the amount of data that must be acquired to ensure being able to obtain a converged average of the flow field based on the spatial pressure condition. With multi-point SE, the same spatial pressure events can be used as the condition, but the average can be found with smaller data sets since all the samples are used in the two-point correlations. This can be 268 advantageous when handling large, two- or three-dimensional data sets as a majority of the calculation only needs to be performed once, regardless of the number of conditions or the complexity of the conditions to be analyzed. Nonetheless, it is necessary to check the accuracy of the stochastic estimation by comparing the estimated velocity field results with the conditionally averaged flow field since the stochastic estimation should converge to the conditional average. One way to check if the estimation does converge is by performing the conditional average of the flow field. However, as explained previously, this requires a large data set that may not be easy to acquire. Therefore, another method for checking the accuracy of the stochastic estimation is to do both the linear SE and the quadratic SE. If the quadratic SE shows very little deviation from the linear-term estimation, then this provides some measure of confidence that the LSE converged to the conditional average. Thus, quadratic, stochastic estimation was used to estimate the u' and v' components of the velocity based on the wall-pressure signature according to 17'(x0 + Ax, y. + Ay,t) = A.,quaamx, Ay;x0,y.)p'(xoayoit) + 19.0376,Ajax..,y.,)p'2 (xoiyoit) (8.8a) 3'06. + Ax, y. + Ay, t) = A.,quaamx, Ay; xoiy.)p'(xaiyoit)° +BV(AX,Ay;x0,y0)p'2(xo,yo,t) (8.8b) where 17' and "7' is the estimated fluctuating velocity components, and Amquad and Amquad are the linear terms and Bu and Bv are the quadratic terms for the estimation of u' and v', 269 respectively. Employing Equations 8.2 and 8.3, and after some algebra, Aumad, Av,quad, B.,, and BV terms are found as shown below A., quaa= u p _-p—f" "‘0 (8.9a) __'v— p'— —B p'3 v ,=qaaa pa (8.9b) B = flaw)— 5'30?) a (PXTD‘Zk (FT (8.10a) B_ P—'V"pi- 17072 (p Xp—‘l— W “3“") Both quadratic coefficients are a function of the cross-correlation between the fluctuating, velocity components and the fluctuating pressure on the surface similar to the linear coefficients. In addition, however, the quadratic coefficients are also a function of the cross-correlation between the fluctuating, velocity components and the square of the fluctuating pressure on the surface. More information regarding QSE is provided by Naguib et al.12 and Daoud's. Another advantage of SE that is pertinent to the present study is the ability to compare the estimated flow field to the instantaneous measurements. Particle Image Velocimetry (PIV) was used to acquire the velocity field above the surface. Details about the technique are described in Section 2.3.3. PIV allows for acquisition of the velocity 270 over the “entire” flow field instantaneously. This enables a comparison between the estimated flow field at a particular time instant and the instantaneous flow field acquired at the same instance in time. Finally, one other powerful tool that comes out of using stochastic estimation is that the evolution of the estimated, flow structure can be time-resolved even if the flow field was not sampled at a rate that is fast enough to resolve the frequency of the dominant structures in the flow. With current PIV capability, it is difficult to sample the velocity information at a fast enough rate to time-resolve the dominant structures in air flows. By using stochastic estimation, flow features can be time-resolved because the estimation is based on the pressure signature, which can be sampled at a fast enough rate to resolve the coherent motions. This has the advantage of not only being able to examine the time evolution of the flow structures, but also to focus attention on the organized motion that is directly linked to the wall-pressure generation. In other words, even if one is able to resolve the flow-field temporally, instantaneous, flow fields are likely to contain influences of random, small-scale turbulence and other details that may obscure the nature of the flow organization as well as the motions that are relevant to the wall-pressure generation. 27] 8.7 Shear Strain Angular distortion or deformation results in a change in the shape of a fluid element and is referred to as shear strain. Shear strain is used to describe the local angular distortion of fluid particles and is calculated from the velocity field. For this study, velocity measurements were acquired in the x-y coordinate plane, and therefore only the in-plane (éxy), out-of-plane (fizz), and linear (éxx and éyy) strain components can be calculated. These components are given by 5 —l 9Z+a_v. xy 2 6y ax (8.11) 4: —_1 gang .2 2 6x 6y (8.12) 6u eta-'5; (8.13) 0v 5W=— (8.14) where, fixy is the in-plane, shear-strain component, fin is the out-of-plane, shear-strain component or the normal strain, and fixx and E,” are the linear strain rates in the streamwise and normal directions respectively. From equations (8.11) and (8.12), it is clear that large shear strain is associated with regions containing large velocity gradients, meaning regions of high shear. Therefore, the shear strain distribution would be valuable in highlighting the flow within the separated, shear layer. In the case of this study, only the in-plane component is calculated. 272 As with the vorticity, the shear strain downstream of the axisyrmnetric BFS was calculated using the PIV data. An integral method was used to calculate vorticity as highlighted in Section 5.2.4. A related method16 is used for determining shear strain from the velocity gradients in the flow field PIV data. The shear-strain estimate is calculated based on eight neighboring points as shown in Figure 8.36. H .—>O——‘>.i+1 >~< ‘— ax Figure 8.36 Path of integration used to calculate the shear strain at point (i, j). Copied from Raffel et (IL16 The equations used to calculate the shear strain from the PIV data are from Raffel I. '6 and are given below. 5 _ [6U + 6V) _ _ (Ui—l,j—l + 2Ui,j-1+ Ui+l,j—i) .. 61/ 8X ,1. 8AY eta + (Ui+l,j+l + 2Ui.j+l + Ui—l,j+l) 8AY _ (Vi—l,j+l + 2Vi-Lj + Vi—l,j-l) 8AX (8.15) + (V... + 2V... + V...) 8AX 273 8.8 Appendix References ' Howerton, B. M., “Development of a passive horseshoe vortex control device as applied to semi-span wind tunnel testing,” Masters thesis. North Carolina State University, 1998. 2 Driver, D. M. and Johnston, J. 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'0 Adrian, R.J., Moin, P., and Moser, R.D., “Stochastic estimation of conditional eddies in channel flow,” Report CTR-887, Center for Turbulence Research, NASA Ames/Stanford, 1987. ” Guezennec, Y.G., “Stochastic Estimation of coherent structures in turbulent boundary layers,” Physics of Fluids A, Vol. 1, 1989, pp. 1054-1060. '2 Naguib, A.M., Wark, CE, and Juckenhofel, 0., “Stochastic estimation and flow sources associated with surface pressure events in a turbulent boundary layer,” Physics of Fluids, Vol. 13, No. 9, 2001, pp. 2611-2626. 274 '3 Tung, T.C. and Adrian, R.J., “Higher-order estimates of conditional eddies in isotropic turbulence,” Physics of Fluids, Vol. 23, No. 7, 1980, pp. 1469-1470. '4 Cole, DR. and Glauser, M.N., “Applications of stochastic estimation in the axisymmetric sudden expansion,” Physics of Fluids, Vol. 10, No. 11, 1998, pp. 2941- 2949. '5 Daoud, M.I., “Stochastic Estimation of the flow structure downstream of a separating/reattaching flow region using wall-pressure array measurements,” Ph.D. Dissertation, Michigan State University, 2004. ‘6 Raffel, M., Willert, C., and Kompenhans, J., Particle Image Velocimetry: A Practical Guide, 3rd ed., Springer-Velag, Berlin, 1998, pp. 160-163. 275 SSSSSSSSSSSSS IIIIIIIIII1IIIIIIIIIIIIII 3 1293 02736 765