:etieen parallel plates. This study concerns was achieved at 1455' zoalence occurred. A 1 izrbulence which strc' heresults indicate a h‘ :czrrence of strong velC teases in boundary layers Byneans of an on-li as closely investigated. S‘itemittent between a laeansof condition san ‘lrae acteristic of the A theoretical bou ned Wcharacteristic ”We e°f ReYnolds s l sindicated 90' ABSTRACT AN INVESTIGATION OF INCOMPRESSIBLE CHANNEL FLOW By Norbert Anthony Feliss This study concerns the mechanism of the transition process of flow between parallel plates. An experimental critical Reynolds number of 7280 was achieved at which the flow became unstable and transition to turbulence occurred. A wave of 63 to 75 Hz. was amplified at the onset of turbulence which strongly suggests its role in the transition process. The results indicate a highly localized transition process involving the occurrence of strong velocity spikes which are similar to transition pro- cesses in boundary layers. By means of an on-line data acquisition system the bursting process was closely investigated. The shape of the velocity profile when the flow is intermittent between a turbulent slug and a laminar flow was analyzed by means of condition sampling. The major portion of the burst profile is characteristic of the common "one-seventh" turbulent shape. A theoretical boundary layer analysis was utilized in order to elu- cidate the characteristics of the developing velocity profile in the con- verging entranceregion of the channel. Entrance lengths were calculated over a range of Reynolds numbers and compared with eXperimental values. The results indicated good correlation between theory and experiment. . Mi 1" Partial DEpartme AN INVESTIGATION OF . INCOMPRESSIBLE CHANNEL FLOW By Norbert Anthony Feliss A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1973 The author wishes to it. Smith for assistar "inks are also due to C' tseior serving as nenf. Scaciai appreciatior estrag .eht throughout 'he financial suppor‘ i‘i‘tering Research mads C ACKNOWLEDGEMENTS The author wishes to thank his advisors, Drs. M. C. Potter and M. C. Smith for assistance throughout the course of this study. Thanks are also due to Dr. J. V. Beck, Dr. N. L. Hills, and Dr. G. E. Mase for serving as members of the guidance committee. Special appreciation is extended to Jim for his support and encouragement throughout the study. The financial support which the author received from Division of Engineering Research made the continuation of graduate study possible. ii F ‘l' was” "' .ISTCIFTIELES. . . . . .IS'OFFIGMS . . . . a 'q.? I); b all! .55 l 2 INTRODUCTION 1. Des! 2. Sat THEORETICAL l 1. DEV! Bour 2. Soh' 3 Res; EXPERIMENTAL I. Cha Ins Dat EXPERIMENTAL I-Et LIST OF TABLES .......................... LIST OF FIGURES CHAPTER I 2 TABLE OF CONTENTS INTRODUCTION ..................... l. Description of Problem ........... 2. Background ................. THEORETICAL ANALYSIS FOR DEVELOPING CHANNEL FLON . . . l. Development of the Potential Flow and Boundary Layer Equations .......... 2. Solution of the Boundary Layer Equations 3. Results of the Boundary Layer Analysis EXPERIMENTAL FACILITY AND EQUIPMENT ......... 1. Channel Assembly .............. 2. Instrumentation .............. 3. Data Acquisition .............. EXPERIMENTAL RESULTS ................. l. Entrance and Transition Regions ....... Laminar Entrance Lengths ........ Critical Reynolds Number ........ Curve-Fitting to Velocity Profiles . . . Examination of the Laminar and Variation in Reynolds Number and Variation in Pressure Drop and HICDI‘I'I l'l'l DOW) iii Turbulent Regions ........... Probe Position . . .......... Probe Position ............ Side Wall Boundary Layer ........ Laminar Fluctuations Before Transition . Spectral Analysis ........... Page vi Page 2. Mechanism of the Burst Process ....... 82 A. Span-Wise and Length-Wise Variation of the Burst Process .......... 82 B. Location of Burst Generation ..... . 9l C. Periodicity of Burst Process ...... 91 D. Velocity Profiles During Transition. . . 99 5 DISCUSSION OF RESULTS ................. lO9 l. Conclusions ................. llS 2. Recommendations ............... ll7 APPENDICES A - SYMBOLS FOR BOUNDARY LAYER ANALYSIS ..... 118 B - SYMBOLS USED IN EXPERIMENTAL RESULTS . . . .lZl REFERENCES ...................... l23 iv 1. Variation of : channel . 1 Variation of t . . channel ‘7‘) . Periodicity of LIST OF TABLES Page Table 1. Variation of burst process along length of channel ...................... 86 2. Variation of burst process along width of channel ...................... 89 3. Periodicity of burst process ............ 96 Two-dimension:- lllustration : Calculated dis for various Fe Calculated fre x-coordinate f Calculated ski x-coordinate f CoriDarison of lengths . . , Photograph of a Schematic draw Photograph of y SChematic draw- Photograph of ; aYer Probe . SCheiTotic draw‘ Photograph of 1 SChemth drawi Variation of cc function of the ComDarison of t ”perimental er Inter'SItY Varia function 0f the Figure 10. ll. l2. l3. T4. 15. l6. 17. LIST OF FIGURES Page Two-dimensional inlet configuration .......... 8 Illustration of notation for boundary layer analysis . lO Calculated displacement thickness versus x-coordinate for various Reynolds numbers ............. 22 Calculated free stream velocity distribution versus . . 23 x-coordinate for various Reynolds numbers ....... Calculated skin friction coefficient versus x-coordinate for various Reynolds numbers ....... 24 Comparison of theoretical and experimental entrance lengths ........................ 25 Photograph of experimental channel assembly ...... 32 Schematic drawing of eXperimental channel assembly . . 33 Photograph of motorized y-coordinate Probe ...... 34 Schematic drawing of motorized y-coordinate Probe . . . 35 Photograph of z-coordinate side wall boundary layer Probe ...................... 35 Schematic drawing of z-coordinate side wall boundary layer Probe ................. 37 Photograph of IBM l800 computer facility ....... 38 Schematic drawing of computer interface ........ 39 Variation of computer sampling rates as a function of the number of channels used ........ 40 Comparison of theoretical (Schlichting; 1952) and experimental entrance lengths ............. 42 Intensity variation (with the use of grids) as a function of the Reynolds number ............ 44 vi {H [router prijt‘ velocity pros‘. Cor-outer print Cmouter print: velocity profi Corouter print Variation of F» laminar, burst Variation of F Cocparison of a drop as a func Side wall tour Intensity of 1 wall boundary Fr‘~‘Q‘JenCy Spec VATIOUS distan 0.340 inches , IntenSltY of n ReVholds numte Intenmy vari Intenmy vari eynOIdS numbe IntenSltY vari Reynolds numb‘ liltehgj ty Vat“ y va ReVhol r d5 numb Mosaic of int range 3800 to l‘dllge 6400 t J 18. T9. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. Page Computer printout of calculated and experimental velocity profiles (parabolic velocity) ......... 48 Computer printout of residuals ............. 49 Computer printout of calculated and experimental velocity profiles (non-parabolic profile) ....... 51 Computer printout of residuals ............. 52 Variation of Reynolds number with Probe position; laminar, bursting, and fully turbulent regions ..... 54 Variation of Pressure with Probe position ....... 58 Comparison of theoretical and experimental Pressure dr0p as a function of Probe position .......... 6l Side wall boundary layer thickness ........... 62 Intensity of laminar fluctuations for the side wall boundary layer .................. 63 Frequency spectra for the side wall boundary layer; at various distances from the wall: 0.0l0, 0.l50, and 0.340.inches ...................... 64 Intensity of natural disturbances as a function of Reynolds number; at X/d = 0 .............. 66 Intensity variation as a function of X/d; for the Reynolds number range 3800 to 7300 ........... 69 Intensity variation as a function of X/d; for the Reynolds number range 4200 to 7300 ........... 70 Intensity variation as a function of X/d; for the Reynolds number range 4800 to 7l50 ........... 7l Intensity variation as a function of X/d; for the Reynolds number range 5000 to 7000 ........... 72 Intensity variation as a function of X/d; for the Reynolds number range 5400 to 6500 and 6400 to 7600 . . 73 Mosaic of intensity data: a) for the Reynolds number range 3800 to 6800, and b) for the Reynolds number range 6400 to 7600 ................... 74 Frequency Spectrum of 60 Hz. noise associated with the hot wire instrumentation .............. 78 vii La Frequency spec? of 6800 . . . . Frequency spec‘ of 7300 . . . Frequency spec‘ of 6800 . . . Schematic draw together with ' Comparison bet. sampling lnlt‘. U' portion of Z = -7.0 in. a U. portion 0f 1 2‘ l5.5 in. a U! DOFLTOn 0f 1 Z = ”.0 in. a U' portion of Z=l4.51n_ a U. p0rtl0n of 2 =14.5 In, a Mecham‘ sm of 1 Mosaic 0f Vela Compute" Dri n velocity Prof Computer Prin VQlOfity Char “ring the b; Perm-bah.“c 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 49. 50. 51. Page Frequency spectra at X/d of 336; for a Reynolds number of 6800 ........................ 79 Frequency spectra at X/d of 336; for a Reynolds number of 7300 ........................ 80 Frequency spectra at X/d of 144; for a Reynolds number of 6800 ........................ 81 Schematic drawing of Schmitt trigger circuit together with hot wire instrumentation ........ 83 Comparison between automatic and manual computer sampling initiation .................. 84 u' portion of computer sampled burst process at z = -7.0 in. and z = 10.5 in. ............ 87 u' portion of computer sampled burst process at z = 15.5 in. and z = 2.0 in. ............ 90 u' portion of computer sampled burst process at z = 12.0 in. and z = 9.0 in.; at X/d of 96 ...... 92 u' portion of computer sampled burst process at g z = 14.5 in. and z = 1.0 in.; at X/d of 120 ...... 93 u' portion of computer sampled burst process at z = 14.5 in. and z = 1.0 in.; at X/d of 144 ...... 94 Mechanism of the bursting process ........... 97 Mosaic of velocity profile during burst process . . . . 100 Computer printout of calculated and experimental velocity profiles (1/7 power law profile) ....... 103 Computer printout of residuals ............. 104 Velocity changes near the wall and the centerline during the burst process ............... 106 Perturbations during the laminar portion of the 107 burst process ..................... viii ‘m' '. Sescription of the F The transition pro: it". :f the experimenta' I-‘ZJZtt-d at Michigan Si iesotect ratio Channe E"Weater than previc '5’e’ria‘ental for the 1 “£09? 0 «my SINCE the ful' Etta] ‘ - . 1nl’95t196tions a ?:'3F‘h r ' xii-‘3? "‘9 velocity pro ’F‘ . me than encompa Ethic: '\’ V CHAPTER 1 INTRODUCTION 1. Description of the Problem The transition process of laminar flow between parallel plates is part of the experimental research effort on Plane Poiseuille flow being conducted at Michigan State University. The current study deals with a high aspect ratio channel facility which previously yielded Reynolds num- bers greater than previously reported. The phenomenon of transition which is fundamental for the science of fluid mechanics has been studied eXper- imentally since the full mechanism has not yet been established. Theo- retical investigations of the development of the boundary layer in the entrance region of the channel can elucidate the characteristics of the developing velocity profile. The thesis encompasses an extensive and explicit study of the following: 1) A detailed eXperimental study of the entrance region, fully de- veloped region and the side wall region for laminar and turbu- lent flow. The approximate region of transition is located in this investigation. 2) An examination of the turbulent slug peculiar to channels. The phenomenon of the bursting process is explored with the aid of an on-line data acquisition computer facility. 3) A survey of the entrance flow region elucidating both experimen— tal and theoretical investigations of the developing velocity profile. '. Eazizrcund To date, stability t' ':""'ear process by whic t't'e‘location" of tra 912'; which profiles ar fett'ysnall. It also c i __L‘-* 'z‘mst reSponsive an 3:a'areters such as we Ice the Schubauer-SL-rar LA asset-ed that in all ca 725! 05 the laminar f1 The theoretical WO‘ ’az-etveen parallel p azrztatic exPallSlo“S Y wage velocity and C 52rbalence. L- “- icritical Reynolds I‘3aerfcomet) by H. C .e ‘oblitUde in 33% .‘ Ca} 0 “Chm que Tara . ' x i :3 CH) that a re 2. Background To date, stability theory can predict neither the details of the nonlinear process by whiCh the flow changes from laminar to turbulent, nor the "location" of transition. What it can do is determine approx- imately which profiles are unstable providing the disturbances are suffi- ciently small. It also can identify those frequencies for which the sys- tem is most responsive and provides insight into how a change in the sys- tem parameters such as wave numbers or wave Speeds can affect transition. Since the Schubauer-Skramstad (1943) experiments it has been generally accepeted that in all cases turbulence does indeed arise from an insta- bility of the laminar flow for parallel shear flows. The theoretical work of predicting the instability of Poiseuille flow between parallel plates was conducted by C. C. Lin (1945). Using asymptotic expansions he obtained a Reynolds number of 7,100 based on average velocity and channel height for the critical number at the onset of turbulence. L. H. Thomas (1953) using a numerical method calculated a critical Reynolds number of 7,700. A theoretical stability analysis was performed by N. C. Reynolds and M. C. Potter (1967) for a wave of finite amplitude in a parallel shear flow. Using a more sophisticated numerical technique they also determined a critical number of 7,700 and concluded that a relatively weak but finite disturbance markedly reduced this number. The process of formation, growth, and coallescing of turbulent spots in a boundary layer has been studied in detail by Emmons (1951), Schubauer and Klebanoff (1955), Elder (1960), and spangenberg and Rowland (1961). The' hot wire measurements of Klebanoff et. all. (1962) indicated that “l V )1 'TMSIX.“ I .l ’ entices generated by t eddies") are 1' ":73 unstable and bre an: irto random fluct Tani (l967) descri :eatsence of large di 7'12: stages in the f 0‘. 57.55, (2) further nonl Tract of high shear la lfia'iect randomness . 35:55 aDoearance of we :3"! 0f laminar inste Previous work in E i'e'li ”(1960): pBEET i vortices generated by the high shear layer at the wall (which they termed Vhairpin eddies") are indicative of turbulent flow. These vortices are highly unstable and break down into smaller vortices which eventually de- velop into random fluctuations typical of turbulent flow. Tani (1967) described the transition process in boundary layers, in the absence of large disturbuing influences, as a sequence of four dis- tinct stages in the following order: (1) amplification of weak disturb- ances, (2) further nonlinear development of the disturbances, (3) devel- opment of high shear layer disturbances, and finally (4) development of turbulent randomness. Tani noted that the transition process is preceded by the appearance of weak oscillations of the type predicted by linearized theory of laminar instability. Previous work in a rectangular channel such as those conducted by Sherlin (1960), Patel and Head (1969), and Breslin (1970) demonstrated that the transition process can take place at a critical Reynolds number of 3,000 or less. Sherlin determined the growth and propagation Speed of turbulent slugs in a rectangular duct. Narayanan and Narayana (1968) ex- tended his work and determined the characteristics of the bursting process at different downstream positions. The stability of laminar flow in a rectangular channel was investi- gated experimentally by Kao and Park (1970) with and without artificial excitation. The critical Reynolds number was found to be 2,600 using water as the fluid. In particular, the progress of growing disturbances was followed and breaking was found to be the ultimate fate of a growing wave. A thebretical analysis of a nonlinear instability burst in plane age] flow has recentl. gwfjm). The object revehcn of transition :1 of the evolution 0‘: iygranski and Chart; Z’IESS in flow through Fiz’iary layer. They e7: xiii-acted the occurre mess. They demonstra itzath the leading and The develoment of 30 a plane char :‘try in :ytheory, It has a‘ 5'35"“ features of vi hm” flow in th i‘r; Schlichti "g. S Dc i"Eniquations by ex. ratarces Iran the In =.,'iith the DEESSurE Hg; parallel flow has recently been achieved by Stewartson, Stuart, and Brown (1972). The object of this investigation was to understand the phenomenon of transition by an extension of the linearized theory to the study of the evolution of arbitrary infinitesimal disturbances. Nygnanski and Champagne (1970) found the behavior of the transition process in flow through circular tubes similar to that occurring in a boundary layer. They experimentally investigated the transition process and detected the occurrence of strong velocity Spikes during the bursting process. They demonstrated for a pipe flow that the fluid was entrained at both the leading and trailing edges of the burst. The development of a parabolic Poiseuille profile downstream of entry into a plane channel is one of the standard problems in laminar flow theory. It has attracted attention because it exemplifies certain general features of viscous flow. A determination of the entrance length for laminar flow in the inlet of a straight channel was initially treat- ed in Schlichting's now classical paper (1934). He solved the boundary layer equations by expanding the stream function in a series for small distances from the inlet, starting with the Blassius solution, and match- ing with the pressure distribution for a uniform accelerating flow. Most approximate analyses of the developing flow problem involve some form of Prandtl's boundary layer approximation. Four general methods of solution may be discerned in the literature: (1) numerical finite- difference solution of the boundary layer equations initiated by Bodoia and Osterle (1961), (2) linearization of the inertia terms, Boussinesq (1891) and Langhaar (1942), (3) integral expansions, Schiller (1922), and (4) series eXpansions, Schlichting (1934). An assumption common to 1‘. fast two methods is n: :i'eells together with .‘EESES downstream to sat The way in which the State parabolic veloci e m-unifomity of as; esstacied, ““50” (197)) re-exa .es of modern boundary 1 .‘e ertrance. The model 5721 it pa.es was found to larinar flow develO" ate paralle plate chanr 7m - r5- In this work ti PE: The . “I“ Phase of 5~rwt .3 h and deveh1F>'“en method b S Her e the last two methods is that the flow consists of boundary layers near the Walls together with a central inviscid core in which the velocity in- creases downstream to satisfy continuity: The way in which the boundary layers eventually merge to form the ultimate parabolic velocity distribution was considered by Van Dyke (1970). The non-uniformity of asymptotic expansions at large downstream locations was studied. Wilson (1971) re-examined the work of Van Dyke (1970). The techni- ques of modern boundary layer theory were used to examine the region near the entrance. The model of uniform flow into an infinite cascade of par- allel plates was found to be the most satisfactory. Laminar flow develOpment in the entrance region of a circular tube and a paralle plate channel was analyzed by Chen (1973) at low Reynolds numbers. (In this work the momentum integral method was utilized without the use of boundary layer assumptions. The first phase of this thesis includes a study of the boundary lay- er growth and development in the entrance region of a channel. Finite difference methods were utilized for solving the coupled partial differ- ence equations of mass and momentum. The three finite difference boun- dary layer methods commonly used are those of: (l) Spalding and Pantankar (1968), (2) Cebeci and Smith (1968), and (3) Mellor and Herring (1969). The method of Mellor and Herring (1968) was chosen because of its accur- acy, physical soundness, and adaptability to the particular application problem. THEORETICAL A ‘. levelepfient of the indies of boundar he‘s the practical ne -‘: other is the de Tear of boundary lay T73 thesis were design. Ptal techniques empll "'3'“ Herring (195; In order to solve i t‘e ee stream vel oc 1' t1 isolation of the DH fretidn was obtained 1 "iitno distinct ”391'0’ E's? plate channel. 11 1936 until well into t? :tthe law of conserve rear. vel oci U- "a :onti nuity CHAPTER 2 THEORETICAL ANALYSIS FOR DEVELOPING CHANNEL FLON 1. Development of the Potential Flow and Boundary Layer Equations Studies of boundary layer flows have been made for two reasons: one is the practical need for boundary layer solutions in design problems and the other is the desire to achieve a better understanding of the phe- nomenon of boundary layer flows. The calculation methods described in this thesis were designed to expedite both of these objectives. The nu- merical techniques employed in the analysis make use of the method of Mellor and Herring (1968). In order to solve the boundary layer equations a necessary input is the free stream velocity. Ordinarily, this is obtained from a potential flow solution of the problem. For the present channel facility this in- formation was obtained in the following manner. The channel was divided into two distinct regions, the entrance section and the horizontal par- allel plate channel. It was assumed that the boundary layers did not con- verge until well into the constant gap channel. For the contracting re- gion the law of conservation of mass could be utilized to obtain the free stream velocity. From continuity f a (do) dA = o _ (2.1) or -pU.lA1 + pUzA2 = 0 Therefore, U1 = (AZ/A1)U2 (2.2) where U1 is the free stream velocity in the contracting region and U2 is the free stream velocity at the entrance to the parallel plate region. The shape of the contraction is a cantilever-cantilever beam 6 faction curve and is 3 e‘ttions y at any poi' rereVis the force at ' t‘i‘of'eft of inertia, L “'2" constant. The hot. deflection curve and is Shown in Figure l. the general formula for the deflections y at any point of a beam fixed at both ends is y = ”3‘:— (3L - 2x) + C (2.3) 12EI where N is the force at the fixed end, E the modulus of elasticity, I the moment of inertia, L the length of the contraction, and C an arbi- trary constant. The boundary conditions are x=0, y=h and dy/dx = 0 x=L, y=d* and dy/dx = o where h is half the gap width in the contraction at x = -6 ft., and d* is d/2, half the gap width in the channel. Employing these relations in (2.3) the following equation is obtained W _ d*-h __ _ (2.4) lZEI —L§—. The general equation, that describes the variation of y as a function of x in the contracting region is y = gigh-xz (3L-2x) + h (2.5) L Employing continuity (2.2) the potential flow velocity in the contraction, U], can be expressed as a function of x and the potential flow velocity in the channel, U2. * U = d. u l . . h - 3:9? x2 (3L-2x) L3 2 (2.6) A computer program developed by H. J. Herring and G. L. Mellor (1970) was modified to include the calculation of the potential flow solution for the present analysis.‘ The equations governing the flow of an incom- pressible, two dimensional, boundary layer illustrated in Figure 2, b Shape Figure 1. Two di ‘ r Shape of Controcii o n d/2 Figure l. Two-dimensional inlet configuration Iii": “it” = “fin + 3 trifert boundary layer ‘- ': :6an by g L 1'?“ I l . .3r aminar flow‘ ,.: ggynrjary conditions limit I; y " 0° 0 ...:.icns (2.7) and (2 lHott transfori T'Rscas e of turbule "ath' 15‘hansformat are given by ____3(:Ul + .1431”) = o (2.7) 3x 3y u.2% + v 9%: = U 9%: + l. §LE_E222. (2.8) 8x 3y 3x r 3y where r(§}y) = rw(x) + y cos a(§). The equations apply to laminar or turbulent boundary layer flow if the definition of T/p is take to be T/p = v-§% - u'v' - (2.9) 3y where ~u‘v' is the kinematic Reynolds stress. The effective viscosity is defined by T/p = v (Bu/333 ' (2.10) e where for laminar flow ve is equal to v, the kinematic viscosity. The boundary conditions are u(3‘<‘,0) = o (2.11a) v(7,0) = o (2.1m) limit I7 [U60 - u(7,e;)] dE; is bounded (2.11c) y a.m Equations (2.7) and (2.8) are transformed with a variation of the Prob- stein-Elliott transformation. x=7 ‘ (and {)7 r(x',E)/rw d5 ' (2.12o) V In the case of turbulent flow u is interpreted as a time average quantity. Using this transformation and the resulting relations =i+§x_§_ 3x 3x 3y 0) xlo) 2'5 2:79” by (,g‘ C 1)‘ >(x) = 35. x= 53- /I7o (2.19a.b) V In the two layer model, there is a region where the layers overlap and both expressions for \é apply simultaneously. v = v¢ = 6*U ¢ = Kn6* V T/p The value K = 0.41 is the van Karman constant and is chosen to predict the experimentally observed logarithmic law of the wall. The composite function is expressed as the sum of the inner and outer functions minus their common asymptote. Thus, the nondimensional effective viscosity, T, can be written for turbulent flow over the whole layer as T=¢(X) + R,o(—’i) - x (2.20a) 5 R6,, T=,—,—‘- 41R,.x) + (x) -x 12.20:») 6* . _ uo* . _ where R.* - -—73 For lam1nar flow T .‘ l/R6*. 6 3. Solution of the Boo" Equation (2.16) wt "tear parabolic partia' ‘igasolution is: com nation using finite d4 “siting equation by a in adaption of tilt x-zerivatives by finite {says stable. The err! 2.7-5) is written in te exposition of the k we: ”L‘T‘d’ xi 0 n _.n u A I 13 2. Solution of the Boundary Layer Equations Equation (2.16) which describes the boundary layer flow is a non- linear parabolic partial differential equation. The procedure of obtain- ing a solution is: conversion to an ordinary differential-difference equation using finite differences for the x-derivatives; and solving the resulting equation by a fourth order Runge-Kutta method. An adaption of the Crank-Nicholson scheme is used to represent the x-derivatives by finite differences. This is an implicit method and is always stable. The error is of second order in the x-step size. Equation (2.16) is written in terms of average functions at a point halfway between the x position of the known profile, xi_], and that of the profile to be calculated, xi. [(1 + Can) No + [(a; + F + nun-f) - Vw/Ujr" + [me—2))?" = _* 1 _ _* — is: “'f'Wi ‘ lei-1))r is: 1‘" (fr- fin) (2.23) where the relation f“ = l/2(f% + f%_]) is used. Equation (2.23) can be written in terms of functions at position x1 as follows ‘[(1+Ca") Tf"]'i = ‘16 + C1(fi I fi-1) * C2(fi + fi—1) -c3(f; +r;_,) - c4(f1 + r,_,) (2.24) where C1 = (a; + P + R1In-1/2(f,+ f,_,)] - (v"i+ VWi-1)/(Ul + U,_]) c2 = p [1/2(r; +f;_1) - 2] - (a? + 5$-1) [1 - 1/2(f% + f%_])]/AX w I c4 = (a; + a;_1) 1/2(f; + f;_1)/Ax 76 = -L(1+ Car) “e resulting fom‘. of F7 " ' : + :DSf]1 b4 C were the coefficients b1"C4 1‘: - ”2 C2 C3 h3=c1 b=-T'+Cf;' 4 b 1,- b5=‘(1 + Ca. r 1 15' .1..- imwon of (2.26) imi- The coeffi ”HIMPUS iteration . The dismaCer “‘5‘ ‘5 rem-1' tten ‘Er, .9 ‘ -_ '“qdéf‘y 14 To = ‘[(1 + Ca”) Tf i—l (2.25a,b,c,d,e) The resulting form of the nonlinear ordinary differential equation is [b5f"]i = b4 + b3fi + bzfi + blfi (2'26) where the coefficients are 61 = -c4 b2 = c2 ‘ C3 b3 = a1 b4 = 'Tb + C1f¥_] + (c2 + C3)f1!_1 + C4f1._1 b5 ‘ ‘(1 + Ca. ”) Ti (2.27a,b,c,d,e) 1 The solution of (2.26) is carried out iteratively because of its non- linearity. The coefficients b1 to b5 are evaluated using the result of the previous iteration. The resulting linear equation is solved for f' and f“. The displacement thickness 6* is adjusted so that f(w) = l to some specified accuracy. The parameters P,Q,R, and Ca are recalculated and the effective viscosity function, T, is also recalculated. Equation (2.26) is solved with a Runge-Kutta method which is a pro- cedure used for solving first order ordinary differential equations. The equation is rewritten as a set of first order equations as follows. f(]) = f: f(2) = f', f(3) = beH (1) (2) Th , 3f 2 3f 3 en E; = f( )’ if? = f( )lbs 3 3%} l = b4 + b3( f(3)/bs) + b2f(2) + blf(1) (2.28a,b,c) The boundary conditions (2.17a) and (2.17b) are applied at the wall and EL __ u , a . F . x“ H‘ I‘K' La 37:} is applied at t" .::a ‘etnod requires t'r :er of the aperationa‘ rdaparticular solutic :cxdary :onditions at t are articular solu‘ no) is reset to f" xii-“:5 progressively c‘ :sir.1al but it does re :1? the l» ‘ "3'“09eneous sol. 15 (2.17c) is applied at the outer edge of the layer. Since the Runge- Kutta method requires three boundary conditions at the wall advantage is taken of the operational linearity of equation (2.26). Both a homogeneous and a particular solution are calculated which satisfy the following boundary conditions at the wall: For the particular solution f3 (x,0) = f"(x,0), (from the previous iteration) (2.29a) f6 (x,0) = 1.0 (2.29b) fp (x,0) = 0.0 (2.29c) f3 (x,0) is reset to f"(x,0) each time so that the particular solution becomes progressively closer to the complete solution. This step is not essential but it does result in an increase in accuracy. For the homogeneous solution fh (x,0) = 1.0 (2.30a) fh (x,0) = 0.0 (2.30b) fh (x,0) = 0.0 (2.30c) The a composite numerical solution is constructed according to the relation f' = f6 + Afffi (2.31) Af is a free parameter which is determined by matching the outer boundary condition. The outer boundary condition is the solution of equation (2.16) at large n which is an improved outer boundary condition of equation (2.17c). With the appropriate terms omitted (f' small, f = 1) equation (2.16) is [(1 + Can)Tmf"]' + (a + R)(n-l)f“ - 2Pf' = 5* 3;} (2.32) A.solution of (2.32) can be written (ma-l)2 - (n-l)2 f'(n) = f'(na) epr 25(x) (1+Can) ] (2.33) 'l V. I. h , ~. A" u unl- .f‘ ; E. remix) is a soluti< :E‘s 1.0 for a two-dir aid-a is the value of :ieasynetotic solutim "it 2.34), replaces station (2.33), at scr flea - » 'esarareter s(x) is ‘ 2.24) 16 where s(x) is a solution of the equation ds _ 2 _ (o*/2) 32' + cs(a + R)S — TmCs (2.34) CS is 1.0 for a two-dimensional flow and 0.5 for an axisymmetric flow, and na is the value of n at which the numerical solution is matched to the asymptotic solution. Therefore, equation (2.33), where s is obtained from (2.34), replaces (2.17c) as the outer boundary condition. From equation (2.33), at some point ”a’ near the edge of the layer r" (ha) = -Asi'(h,) (2.35a) where ( ) _ (n-l) “" Ca As ‘ s(x) (1+Canl [ ‘ ‘ 211163313 (2'35”) The parameter s(x) is obtained from the approximate solution of equation (2.34) S 2(aX/ar)i;c§ + 51,1 [1 - (nan) cS (Ax/5‘1] 1 ;.._ __ (2.36) 1 + (Q+R) CS (AX/6*) Inserting equation (2.35a) into (2.31) and rearranging yields, Af = IS (”a) + Asfp (”a) (2.37) f; (na) + Asffi (na) In the range (0 com mmmcxuwgu pcmsmumpammv umumpzuqu .m mgzmmm 3.: 82455 x m d N o . N- v- o. q l. . _ . _ _ q _ a 22 consum. . commem. . . ooomhma . - o m commuma .ommmumo . 23 .mcmne:c mupoczum maowcm> com xuwuopm> seesaw mmcm empmpsupmo EL: mozflrma x v N o N... v. .e «gamed — _ — q i o 4 0 O O O O I lay maul ha I. u 4 4 4 4 4 4 0. ON on ('038 I '15) fl 24 .mcmne:c mvpocxum maowgm> cow anymoremmou cowuuwcm :wxm umumpzupmu .m wgzmwu EL: woz1. . ... r 'a' C. r- 43 84 channel widths (3.5 ft.) along the abscissa. Therefore, a 6 ft. con- verging entrance region has developed the flow an equivalent length of 84 channel widths for a Reynolds number of 4800. For the present channel assembly the laminar entrance length 1e can be related as 2e = (1.42 x 10'3) R - 1.86 ft. (4.1) and a similar relationship can be derived from the analytical solution “e = (1.67 x 10'3) R - 3.00 ft. (4.2) 8. Critical Reynolds Number The burst Reynolds number of the flow in the present facility can be influenced by controlling the intensity level of the free stream velo- city fluctuations. Grids of different diameters and mesh sizes were placed in the entrance region and the fluctuation levels were measured at X/d = 168 before the onset of turbulence. A plot of intensity level ver- sus critical Reynolds number is presented in Figure 17. The curve mono- tonically decreases to an intensity of 0.13% for a Reynolds number of 7280. This is the natural intensity level for the channel assembly. At the Reynolds number of 7280 no grid was utilized to disturb the flow. In- spection of the curve reveals that at large Reynolds numbers it becomes almost parallel to the abscissa. Since the theoretical Reynolds number for transition is 7700 a sudden decrease at this point in the curve is indeed a possibility. This has been predicted by Reynolds and Potter (1967). Unfortunately, the present channel assembly does not provide flow stability at Reynolds numbers above 7280. C. Curve-Fitting to the Velocity Profiles A statistical treatment of the data has been prepared to give infor- mation about the develOping profiles, the nature of the fully developed PERCENT INTENSITY 44 X/d =168 0 1 1 1 1 M 0 1 2 3 4 5 6 7 8 CRITICAL REYNOLDS NUMBER (103) Figure 17, Intensity variation (with the use Of grids) as a function Of Reynolds number. ;. ...w 293?. '1. M ..‘s‘ a 1W“ '0' I V-I IZ’E' neg .1.“ ”hi, _' I ...: he: “E‘s- y C 1 $14 ‘I 45 profile and an accurate estimate Of the channel gap width. This analysis is based upon standard curve-fitting and parameter estimation techniques. Although the horizontal plates Of the channel can be accurately positioned tO give a channel gap width Of 0.500 t 0.010 in. by means Of a special gapping instrument, changes in temperature and humidity from day to day can change these measurements by more than 0.015 in. Therefore, it was considered desirable tO Obtain an estimation Of the gap width by other means. Velocity profile data were collected at different X/d sta- tions at z = 0 along the length Of the channel and a curve-fitting tech- nique was employed to Obtain information not only about the nature Of the profile but also Of the channel width itself. In the case Of two-dimensional, steady, fully developed flow in a channel with two parallel flat walls a very simple equation is Obtained from a solution Of the Navier-Stokes equations and continuity. . 2 dP d u = u (4.43a) a?” E;2' 35' = o (4.43a) For the present study the coordinate system was established at the bottom plate Of the channel. Thus the boundary conditions are u(y=0) = 0 (4.44a) u(y=d) = 0 (4.448) where'u(y=d/2) = umax‘ The solution to equation (4.43a) is u = 4 umaXy/d (1-y/d) (4.5) In parameter estimation the model _ 2 or- M " y. bud. 46 can be studied by the method Of least squares, Beck (1966). Comparison to equation (4.5) allows vi = ui B1 = 4 umax/d B2 _ '4 umax/d2 Xi = yi (4.7a,b,c,d) Ordinary least squares is used to find the estimates Of B] and 82, namely, b1 and b2. A sum Of the squares is defined by n 2 " 2 s = 2 £1. = 2: [v]. - E(Y1.)] (4.8a) i=1 i=1 " 2 By minimizing S with respect to B] and 82 the estimates b] and b2 are Obtained. These are defined by the following equations 2x? zxiv1 -w 2x? zxgvi b‘ = 2 4 3 2 (4'9) 2x, 2x, - ( 2x, ) 2x? zxgvi - zxiv1 2x? h2 = (4.10) 2 4 3 2 2x, 2x, - ( 2x, ) _ _ 1 _ _ l and Y - 'fi 2Y1 , X - n 2X1. (4.11a,b) The residual e, is the measured value Of Yi minus the predicted value ei = Yi - Y1 (4.12) where the predicted value Of Yi is denoted Yi x? (4.13) Y, = blxi + b2 ‘E‘i .. 1-1 -HL‘L‘I-n . ' .. I'- ‘l . l f I I did a .ut l I'C’Jl C‘ .334 . CO... v . ....-.c \ “- 'uy ..‘1 . egr- o... fr . v til-“fl ‘ u 47 In this manner both the gap width Of the channel and the maximum velocity could be estimated. Knowing the value of the maximum velocity the gap width Of the channel could then be established. In practice a saphisti- cated computer program written by Nicely and Dye (1971) was utilized to estimate these parameters. This program handles problems which are either linear or non—linear in the adjustable parameters. The residual e. is minimized with respect t0 the parameters by the 1 curve-fitting routine. This is an important parameter which indicates : "a inn-J I the goodness Of fit Of the model to the data. If the residuals are large and do not give a mean Of zero then the model is either incorrect or Iii-Iiimam .m ..“l. . i ‘l .— several parameters in the model exhibit linear dependency. The criterion Of minimum residuals is used to establish when the flow is fully developed. A parabolic velocity profile is determined when the sum Of the residuals is less than 10.8. In Figure 18 a typical computer printout is presented Of calculated and experimental values. The experimental values fit very well to the theoretical curve. A standard residual plot of this same data is presented in Figure 19. The sum of the residuals for this particular experiment is -0.5161. This curve reveals a random scatter Of residuals which indicates a good fit. The run test (also called the Nald-Nolfowitz test) can be applied to residuals and other data to investigate the independence Of the measure- ments” This type Of test was utilized to determine the randomness Of the residuals in Figure 19. ‘This eXperiment gave rise to a 95% confidence arrangement Of residuals. Therefore, the results show a gOOd fit tO equation (4.6). This data and analysis is accepted as defining the flow parabolic and therefore fully develOped. 1.1 I] In.- all- 1 n! I It... .4-‘ I... u 0 In. I"I"O .lv'tl I 40‘s. five. 5' ‘4. d.) D:- f .1 I: a. II «I 0 § .01 ub-n I uzquyovVe-o-fluo-I-I I beat-ox .94. .11 04-4()0 1° I 5‘54 ’4. -‘ UQc'l‘) 04.0:(xa‘u! 5|: 0.- ...)..uzu' 48 b3’— 50 Stu ~—In 335 muJ—bozz >p—UOJU) topaz ~h\0\~— oooooooooooooooooooooooo Ola uflII'lfi0000m0000m0000m----WI 0-0m----m0000m00I0m----m0000m0000m----m0000m0000w----m..-.fi----m----m-'..fl m on _ _ _ — _ _ _ _ m m _ .oouooco_. Noeuncns_. ~o.u9~o0.. ~0.uo-.‘. o_ ~ _ .oeuoooo‘. ~o.u.cn_~. ~o.u0-—~. ...uoomo_. s. _ _ .oeuoooo_. Noouoonm~. ~oeu’nnmN. oneuoonoo. a. _ _ goeuooooa. NoeumoNoN. NoouoonON. ooeuoo_~’. m. _ m .oeuoooo—. No.u_.:_n. ~o.uo~o_n. ooouooaon. 0. m _ .oouoooo.. No.uon~nn. Noouomonn. ooeaoono~. n. a n _ .o.uoooo.. ~o.um~oqn. Noouo~ocne oo.uoo_c~. ~— — _ .oouoooo.. Noewnoonq. Noeusnomn. ooeuoosoo. _. n _ .o.uoooo~. ~oeuo~mnne ~oou0nonn. oo.uooo_o. o. _ m .o.uoooo.. ~o.u~oman. Noeuoooan. oo.uoocaa. o m _ .o.uoooo.. ~o.uom~nn. NoeuONnaq. oo.uoo)~:. s — _ .o.uoooo.. ~o.uoooon. ~o.uo:n_n. oo.uoomo:. s - _ .o.uoooo_. ~o.u~n-~. ~o.u9.na~. ooeuoo~:n. o n _ _ .o.uoooo.. ~o.uooo:~. ~oeuoson~e oo.uooc-. m a u m .oeuoooo_. No.u~:~o.. ~o.uo:~,.. ooeuootoN. c m _ .o.uoooo—. ~o.u0non_. ~o.uonma.. ooeuoo~:_. n u _ goouoooo.. .o.u~_coo. .o.u~,~m:. .ouuooooo. N _ .. .oeuoooo.. .o .o .o . _ _ pro—u: 442.; o_muz . ._.~.: ._.~.‘ ...—.n 10:13: — m m _ o n _ n u _ _ _ _ m m _ a _ _ ~ _ o - _ _ m m _ — _ _ _ o _ _ u u m a m — u _ o — — I 1 — _ _ m o m _ o n _ o o — _ n o o — _ 4 4 _ .nuceom n 1 n n) 11w w u n n n in m n n u in) u in u » 4—4ua >0 n 4—4uo utcm u:— z- ua4 pz_oc ou~4aauacu 084 acpzuzucutuu 84 nz4u: a .ou—poas m. o_mux aux: sum: m— 0 >420. pz—oa au—caauadu 4 azaux o .px—Ot achzux_zutnu :4 mzdux x oo.u-s. - pzuxuauz_. .o o toppam u:— —4 u344>e~0ouomne a so» at» —« u=4¢> .o—muz to .—.~.x m— J¢u_p¢w> .ocu~__. - pzwxuauz—._oou.._e a —:a_: at» —4 wade». .o a puma u!» .4 u34¢> .._.¢¢>x.x so ._.—.n m— mmoauc nd experimental iles (parabolic profile). Computer printout Of calculated a velocity prof Figure 18. 49 ba‘: 50 EU N-In ~55 mug-g v—nuoag Count -\0\N— oooooooooooooooooooooooo gm .m----h u u u in 1 n u a a u a a w a 411.1. m a a _ e _ _ . — I — _ _ m a _ _ _ . _ _ _ . . m m _ _ _ e . _ _ _ _ m e n _ . _ n U I I - _ . _ . e _ m em _ _ _ e . _ . _ _ _ m m _ _ _ _ _ . .e _ m ...uoooo.. o...e~q_n. :. a _ ...uoooo.. ..-uo’hxmeo ._ _ _ ...uoooo.. ...acaoonee c. e _ _ ...uoooo.. ...aocee... m. _ _ ...uooo... ...uaaee~.. .. _ m ...uoooo.. ..-uee~me. a. n _ ...uooo... oo.u.°o._.- ~_ . _ ...uoooo_. ..-umnnen.e .3 . _ ...uooo... ...uo.~.... .. _ _ ...uoooo.. ..-uo..~o.o o _ m ...uoooo.. ..-u~mco~.- c n _ ...u.ooo.. ...uoeoxeeo . _ _ ...u...o.. o..u~m~eo.e c . _ ...uooo... .o.uomcoe. m . _ ...uooo... ...un-mo.u o _ m ...u...... ...ueoeoee n n _ ...»...... ..-»oae... ~ . _ ...uooo... .. . _ _ mpzo.ua 44:9.au2 _ _ . .m n n i n in n u . Computer printout Of residuals. Figure 19. > (has >8 1 (~48 in U: 8- 08¢ :38 Ou—dguiu g J‘uzruxgu 8‘ air I .aUppo.‘ m— o-nuz 8*.) 3.33 m— I pg. :39. Gapdgudcu C as, O opt-Os intruxrdu 84 93! n - .zuguzoz...o.uooo.. o no..oo u:— p. a:4<».o..uaao. . go— at. p. nag.» .o_nu¢ so ._.~.n m. 4¢u.—¢u, . .zuxuzuz_._o.u-.. - :3... U.» 3 3.13.222. .- :3 V.— = 3.!» £733.: co 3...: 2 $284 ~°0uonno —°0u~:. pr» 50 A problem arises when the velocity profile is not fully developed. The model in equation (4.6) cannot be utilized to give information about the gap width. Figure 20 is a typical example Of this case. The Reynolds number is 6800 and the X/d station is 96. At this Reynolds number a parabolic velocity profile is not realized until X/d is 192. The resid- uals for this experiment are plotted in Figure 21. The sum Of the resid- uals is 426.0 and the run test does not indicate a random arrangement Of '} residuals. “4! Estimation Of the gap width for a developing profile is obtained 1 from polynomial regression. An nth ‘ degree polynomial is fitted tO the ". data in the same manner as described previously. The linear regression of ii y upon a single variable x can be extended to the multiple regression _ 2 n Yi - 80 + 81x, + ezxi + ... + enxi + a. (4.14) In the model, Xi are known precisely and Yi are subject tO random error which is normally distributed about the regression line with constant 2 variance 0 . The procedure to estimate the parameters Bi is again to minimize the sum Of the square " 2 " 2 S = X 81 = 2 [Vi ' E(Yi)] i=1 i=1 n _ n 2 S ‘ 2 [Y1 ‘ B0 ‘ B1X1 ' ‘ ani] i=1 The result is a system Of n simultaneous linear normal equations which are conveniently solved by inversion Of an (n x p) coefficient matrix T X equation (4.14) can be expressed as ‘X, where p represents the number Of parameters. In matrix notation v = 28. + E_ (4.16) and the least squares linear unbiased estimate Of _B_, designated _b_, is 51 patlu so can II: 305 mUJu 88a >h~9°4u> Gaul! N~\O\- IIIIIIIIIIIIIIIIIIIIIIII OZU ~m0000m O -0 -m0000m 0-00W0000‘--- 0m--0In0000“-...m---.“-i0000m0000m0---“-i:%&0000mi$0000fl m In _ _ . _ . _ _ _ m m _ .°.uoooo_. ~o.uoonm_. ~o.uooom_. ...uo~.... o. . _ .o.uoooo.. ~°.u.o~o_. ~oeuomco~. ...uo.mo.. - — _ .o.uoooo_. ~o.uco-~. ~o.uompn~. oo.uoonoo. o. _ . .o.uoooo_. ~o.uamom~. ~o.u9ono~. ...uo°.~o. m. _ m .o.uoooo_. ~o.uoo~.~. ~o.uo~no~. oo.uoooos. 6. m _ .o.uoooo_. ~oeuoo~o~. ~o.u::oo~. ...uoomsa. a" o _ _ ...uoooo_. ~o.um_.on. ~..uomnon. oo.uoo_c.. u. _ _ .o.ueooo.. ~¢.un~ooa. ~o.uoooen. ...uoosoo. _. _ _ .o.uoooo_. ~o.unonon. ~o.uoooon. o..uooo.a. o. e _ m .o.uoooo.. ~°.uoooo~. ~o.uoacon. o..uooonh. o a _ ...ucoo... ~o.u_nme~. ~..uoooe~. ...uooe.e. : . . .o.uooo¢.. ~o.u¢e~e~. ~o.ue.~s~. o..uoomo:. s _ _ .o.ucooo.. ~o.umn~n~. ~..uo-o~. ...uoo~on. o . . ...uooo... ~o.uoamo~. ~o.uaooo~. ...uoo:.~. m o _ m ...uoooo.. ~o.uoooo.. ~o.uonoo~. oo.uooao~. : m _ .o.uoooo_. ~o.ua.o__. ~o.uon_n.. o..uoo~c_. a _ _ .o.uoooo.. .o.u~:on.. ...useose. ..-uoooae. ~ _ . .o.uoooo.. .o .o .a _ . .3 _:a_un ace—u o_mua . ...~.u ._.~.q ...... xuxxaz « _ an a _ o _ _ . _ u _ _ m a . _ _ o n _ . _ . _ n o n . _ _ n o — _ x . _ . m o m _ o _ _ x n — _ _ . . n o n m _ o _ . n o a _ a I I I n _ . .h.. n h u n n u n n n in u n n in: in u u . 90.3.4.0. FOIUN— - o I pin-3392.. I 3.089.087.0603.- » (has >0 a dud! t6N0oun2... I 3.9—8 U.» .4 34¢). I to» !— p4 v.3: .33.. ..o 2.3: 2 ecu—2.2. a pen; us» pa nag.» ....c4»3.n so .....u a. mmoaua les (non-parabolic profile). i intout Of calculated and experimental Computer pr velocity prOf Figure 20. III- olv’Io-54o33001 I [...-44- III. I! 2.40). a. I 430- VII .0 3.10) ...-III III» Inn-tel I. 31.4.1.1 52 —--—-fi———I—fi—I———mI.—I-I-m—I-I-I—m—I--~m-—I—I—fi————fi————m———.Im .oauI—I. d I“ ‘Oouoooo—o acouoooo~o «Cowoooo-e acouoaoo—e n.6uoooo—o ~0ouoooo—o quouooooao —.ouoooo-o .06uoooono ..ouoooo_o acouoooo-o ..ouoooo—o ~06uoooo—o .oouoooo-o ..ouoeoo-o ~0.uoooo~. ..ouoooo—o _.ouoooo~. mptonux I plolatutuo-OI .3606- _oou\‘-o I _l.tda.v:-o—0o .\V—o H‘ 5 U on UN ooouoo—QIII .o.uoo_~_oo .oeunmsa—ou ooowom_oooo .oounmoogoc oceuohooeou oo.uoono~oo ..-uomm0—on oo.u»mo~.on .oownnoo-oo ...u.oo—.en .oeuc—NINoo .o.um:~o~oo ...uocooneu ...u.ooon.u .ooucmONnoI .o.uo°_m_ol co 4433-mua \- u ~~HOI€|OP~OO u “ CI: 805 “NJ—LOCQ rb—UOJUD OohIl ~h\0\- IIIIIIIIIIIII “ “ watts so clu III-III-mwuunfl—n-I-m-I-I-I-m-n-I—mnun—Inm————m—-—-m-—--m----W ’0 0 I 0‘- -0 0'- 0 I 0"---M-0000’---0"-0"-00-m----*----m----m--:“----fi----mg--m----r----m--'$-00". --— » c.4ua .c ‘ 4.4uo stem at. 2. ~24 pz.o; capcaauaau oz. acpzut.tusnu cc «tau: . .au..oas a. :_a4: zeta can: a. 6 .420. .2.98 ou.44:ua¢u c azauI o .px.oa aortas—xuaau ca «2.3: . I to..:o us. .4 I .33.: at» .4 a:44>.o..u.... a34¢)o I to. u!- p. ~34¢> .o—aut to ._.~.¢ a. adu_-I.. I use; exp p1 wadc» ..—.a¢.¢.n 2° ._...- o— 1:.Iuc Computer printout Of residuals. Figure 21. 53 T .518) = (l +1.3.) (1 '- X._B_) . (4.17) The well known least squares estimator is the result Of the minimization procedure (Beck, 1966). _ T -1 1 2,, - (a 20 a. x. (4.181 The (n x n) symmetrical variance-covariance matrix Of the estimates is cov(b_LS) = (X? X)-1 02 (4.19) where 02 is estimated from TR TT 2 Yr - b X 0 = —X ——-Y" (4.20) "'P In practice, a computer routine was utilized that employs a least squares fit Of the data by five successive polynomials, n = l, 2, 3, 4, 5 and the standard deviation 0 about the regression line was examined in each case. It was Observed that 0 decreased fairly rapidly with increasing n until a good fit was Obtained. Further increases in n actually increased O (again due to the reduction Of the degrees Of freedom in the denominator Of (4.20). The best fit to the experimental data occurred for n = 3. 0. Examination Of Laminar and Turbulent Regions The procedure Of parameter estimation was utilized to establish the gap width in those regions where the velocity profile was not fully devel- Oped. The average velocity which is a function Of the gap width was Ob- tained by a numerical procedure. For a parabolic profile the average ve- locity was simply 2/3 "max' At each X/d station a Reynolds number based upon the estimated gap width and the average velocity was calculated. A collection Of curves representing the Reynolds number as a function Of X/d and different flow rates is presented in Figure 22. Examination Of Figure 22 reveals that the channel Reynolds number 54 .cowpwmon mace; sue; Logan: mupoczmm we compmwcm> .NN acumen mmomm mo muz 30, 0+ = 5.5-+ 2.5 1ny+ (4.25a,b,c) where u+ = U/U*. 11* = ”070 y+ = yu*/v 4* = du*/v ‘(4.26a,b,c,d) 58 3 0.15 IN ”6 3 0.10 .1: (J .5. 9. 0.05 Figure 23, 96 I92 288 384 X/d Variation Of pressure with Probe position. 59 The friction factor is derived employing equation (4.25c) to evaluate a bulk mean velocity. Equation (4.25c) should not be used for y+<30 but since little Of the mass flow passes inside Of y+ = 30 little error is introduced. The bulk mean velocity is defined by the relation 4. _ d _ + d + + Uave - l/d.g u dy — uf/d 6 u dy (4.27a,b) The friction factor is, by definition T0 f. _7__ . 20*2/Uive (4.28a.b) pUave/z The result Of the integration is a relation for the turbulent friction factor * (2/1=)”2 = 3.0 + 2.51n [g— (172)1/2] (4.29) The shear stress at the wall is a linear function Of the pressure gradi- ent and is related by the following equation _ d The pressure gradient is evaluated by substituting equation (4.30) into (4.28a) to give -dP/dX = g- pUgve (4.31) and in terms Of the Reynolds number 2 -dP/dX = f —P—§’—R*2 (4.32) 4 d By apprOpriate substitution Of the laminar and turbulent friction factor the pressure drop can be evaluated and compared to the experimental val- ues. The following model for the pressure drOp as a function Of the probe's distance is utilized -AP = -AP1am(X+2) - 0.75 APturb(24-X-2) (4.33) 60 Aplam and Apturb represent the laminar and turbulent pressure drops. The turbulent wake generated by the probe fills approximately 75% Of the total channel area. Therefore, the model encompasses the laminar region in front Of the probe and 75% Of the turbulent region behind the probe. Figure 24wdepicts the experimental and theoretical pressure changes as a function Of the probe's position and Reynolds number. The analyti- cal results correlate well with the experimental values except at the high Reynolds numbers. At the high Reynolds numbers for small X/d loca- tions the flow is not fully develOped. Since the velocity profiles used to develop the relations for the friction factors are based upon fully developed profiles the friction coefficients are not valid for the high- er Reynolds numbers. 6. Side Hall Boundary Layer The influence Of the side wall boundary layer on the mean flow was determined by Observing the growth Of the boundary layer and the intensity Of the fluctuations in the boundary layer as a function Of the Reynolds number. The growth Of the boundary layer can be compared with the Blassius solution for flow over a flat plate. For u = 0.99 U0° the boundary layer thickness is defined as 6 = NW N” (4.34) This relation predicts a thickness Of 0.77 in. for a length Of 24 ft. and an average velocity Of 24 ft./sec. corresponding to a Reynolds number Of 6100. Figure 25 presents the experimental values Of the boundary layer thickness as a function Of the Reynolds number. A thickness Of 0.670 in. is Observed for a Reynolds number of 6200. However, a minimum thickness is Observed for a Reynolds number Of 2500 which increases to a maximum 201 -AP (inches of H 61 oExperimental ATheoreiical “Nab l J J I 96 I92 288 384 480 Figure 24. X/d Comparison Of theoretical and experimental pressure drOp as a function of Probe position. 9 m ' .0 A .0 r0 Boundary Layer Thickness (111.) 62 X/d8576 l l I 2000 4000 6000 R Figure 25. Variation Of the side wall boundary layer thickness as a function Of the Reynolds number. °/o INTENSITY .0 01 .0 .rs .0 m 63 O 0010"" e A 0.040 CI 0.070“ r- I 0.l48u X/d8576 . AI 1. II at .. {I II at 01“ II II 2219’ III ‘00. .000 R Figure 26. Intensity of laminar fluctuations for the side wall boundary layer as a function Of Reynolds number. \/ I? (F1./sec.-10'2) 64 5'0 z=O.0101n. ) R=7IOO 2.5» 0.O:\t W MAMA/1AM 20 40 60 80 5'0 z=0.1501n. 2.5- 0.0 ()0 1 m 1110-14.11 20 4O 60 80 5'0 Z=0.3401n. 2.5- Figure 27, 11,113.. ' Frequency (Hz.) Frequency spectra for the side wall boundary layer. 65 laminar fluctuations in the boundary layer. Bursting occurs simultan- eously in the boundary layer and the mean flow when the critical Reynolds number is reached (one probe located at 0.010 in. from the side wall and another at Z = 0 determined the simultaneity Of the burst process). H. Laminar Fluctuations before Transition Investigation of the growth and decay Of laminar disturbances with increasing Reynolds number was made possible with a wave analyzer and an RMS voltmeter. Intensity data were Obtained along the centerline at var- ious X/d stations. ‘The intensity is defined as 141 = 100 7 ET? / uéwe (4.35) where u' is the fluctuation about the mean and Uave is the average chan- nel velocity. The disturbances generated naturally in the developing region of the channel appear to damp as they move downstream. However, this effect ave as the probe is pulled downstream. The decrease ’72“ in intensity is a result Of increasing Uave and not a decrease in u is due to increasing U Damping does occur whenever the Reynolds number is increased. Figure 28 reveals this decrease in intensity Of the natural disturbances at X/d = 0 for a range Of Reynolds numbers. For the experimentally critical Reynolds number Of 7300 the intensity at the inlet is approximately 0.12%. In Figures 29 through 33 the intensity Of the laminar fluctuations is presented as a function Of X/d. The data is shown at three y-locations representing the position Of the probe at y = d/12, the closest position Of the hot wire to the wall, y = d/4, a position midway between the cen- terline and the wall, and y = d/2, the centerline Of the channel. The symbol d represents the gap width Of the channel which in this case is °/. INTENSITY (10') 66 X/d = O 3.0 ~ 2.5 - I 2.0 - 1.5 - I.O _. I l 1 2000 4000 6000 R Figure 28'. Intensity of natural disturbances as a function Of the Reynolds number. X/d = 0 67 taken to be 0.500 in. Figures 29 and 30 show the intensity variation with Reynolds number and X/d before the onset Of transition. The range of Reynolds numbers in each figure represents the changing flow rate due to the positioning of the probe at various X/d stations. The intensity of the disturbances versus X/d for the Reynolds num- ber ranges 3800 to 6800, and 4200 to 7300 are depicted in Figures 29 and 30 respectively. All three y-locations appear to have approximately the same intensity with small variations between each curve. In both Figures the intensity at the centerline, y = d/2, crosses over the inten- sity curve represented by the region close to the wall, y = d/12, at X/d of 330. After 330 gap widths the centerline region appears to have the maximum intensity. Figure 31 represents the range of Reynolds numbers for which tran- sition will occur. When the probe is pulled downstream to X/d of 384 a burst is observed. It is interesting to note that the centerline contains the most intense fluctuations at all X/d locations. The regions close to the wall, y = d/12 and y = d/4, represent a lower intensity level, but only slightly SO. A similar behavior is exhibited in Figure 32 for the Reynolds number range 5000 to 7000. The maximum intensity is located at the centerline while the minimum is at y = d/4, the region midway between the centerline and the wall. The location near the wall, y = d/12, represents the next largest intensity. A linear decrease in intensity occurs for all three locations until X/d Of 312 at which point transition to turbulence occurs. A strikingly dissimilar intensity curve is recognized in Figure 33 for the Reynolds number range of 5400 to 6500. At X/d Of 96 the regions 68 y = d/12 and y = d/4 suddenly increase in intensity whereas the center- line continues to linearly decrease in intensity. The maximum intensity appears to be at the wall region before transition to turbulence occurs. A similar behavior is indicated for the final Reynolds number range Of 6400 to 7600. The increase in intensity at the location near the wall, y = d/12, for X/d Of 96 is more pronounced. The centerline appears to be the most stable region at these high Reynolds numbers. The two dissimilar sets of intensity data represent two different mechanisms for transition to turbulence. In the lower Reynolds number ranges the wall region appears tO contain the maximum intensity. At large X/d locations the centerline becomes the region of maximum intensity. Linear stability theory predicts that the location near the wall permits a particular frequency Of disturbance to grow in amplitude. Transition occurs becauSe of this instability close to the wall. In the present channel at the critical Reynolds number Of 7300 the flow becomes unstable and a burst originates.near the point of instability. This first mechanism for transition is based upon the fact that the velocity profile becomes unstable due to a growing instability. At the larger Reynolds numbers the second mechanism for transition is apparent. It was Observed that the regions y = d/12 and y = d/4 sud- denly become very unstable at X/d of 96, whereas the centerline remained very stable. At this specific value of X/d the developing boundary layers have not converged. Therefore, it appears that these boundary layers be- come unstable in the developing region. Turbulence is generated due to the growth Of disturbances in the boundary layer. A mosaic Of the instantaneous intensity distributions is presented .1“ '1 "lo INTENSITY (l0 ) 2.4 I" m E“ o in 01 L4 69 )W. F‘ \\ 1:1Y=d/I2 R 3800 “'6800 AY=d/2 0 Y=dl4 )— I JJ 1 I I. 96 I92 288 384 X/d Figure 29. Intensity variation as a function Of X/d; for the Reynolds number range 3800 to 6800. %1NTENSITY(IO") 7O R 4200‘7300 2.2 “ \ AY= d/2 ; 0 Y = d/4 2 0 b \: DY = d/IZ \A 1.8 ~— \- 1.6 - '\- 1.4 - '.\ J I I ~ I 96 I92 288 384 X/d Figure 30. Intensity variation as a function Of X/d; for the Reynolds number range 4200 to 7300. in '9. >_ 1.6 t: (D Z mIA. ... 2 4 ° 1.2 71 R 4800'7I50 AY=d/2 0Y=dl4 nYédNZ l J Figure 31. 1 96 I92 288 384 X/d Intensity variation as a function of X/d; for the Reynolds number range 4800 to 7150. '01 °/. INTENSITY (10") id 1? 72 R 5000'7000 A AY=d/2 “ix; CY = d/4 '. . nY= d/12 _ ° : Btfrst 1 1 1 96 192 288 X/d Figure 32. Intensity variation as a function of X/d; for the Reynolds number range 5000 to 7000. °/.. INTENSITY (10") '01 21:. i0 73 AY=d/2 oY=d/4 uY=dN2 Burst I R 5400' 6500 R 6400'7600 I I 96 I 92 288 X/d Figure 33. Intensity variation as a function of X/d; for the Reynolds number range 5400 to 6500 and 6400 to 7600. 74 Figure 34a. Mosaic Of intensity data; for the Reynolds number range 3800 to 6800. FigLne 34b. Mosaic of intensity data; for the Reynolds number range 6400 to 7600. 75 in Figure 34. The first mosaic represents a typical intensity distribu- tion for Reynolds numbers before transition such as shown in Figure 29. The large fluctuations close to the walls and the growing fluctuations at the centerline can be clearly Observed. In the developing region the maximum fluctuations are localized at y = d/12. The second mosaic repre- sents the intensity distribution at Reynolds numbers large enough to pro- duce an instability in the boundary layer such as shown in Figure 33. Thus, the fluctuations near the wall are clearly being amplified in the last two frames Of the mosaic. I. Spectral Analysis The disturbances at the wall and the centerline have been studied by Observing the intensity Of the u' fluctuations. A spectral analyzer also was used in conjunction with the RMS voltmeter to Observe the appear- ance Of the dominating waves before transition. In all spectral exper- iments a strong signal is Observed at the frequencies Of 55 to 65 Hz. This signal is associated with the 60 Hz. noise level of the hot wire instrumentation. The amplitude of this signal is 0.2 millivolts which is large enough to be detected by the analyzer. This is evident in Figure 35 which 158 spectral analysis of the output from the linearizer with no signal from the hotwire. The Spectral investigations were performed at three ranges Of Rey- nolds numbers, at X/d locations of 336 and 144, and at three y-locations. Typical spectra before the onset of transition for the Reynolds number range of 4000 to 7100 are presented in Figure 36. ‘The X/d location is 336 and the Reynolds number at this location is 6800. As indicated by the three curves the number Of dominating waves is a maximum at the region 76 near the wall, y = d/12, decreasing in number to the centerline, y = d/2. The dominating frequencies near the wall occur up to 80 Hz. The location midway between the wall and the centerline is associated with frequencies up to 40 Hz. I In all spectral investigations the low frequencies occuring between 10 and 30 Hz. determine the maximum amplitude Of disturbances. Therefore, the intensity at the centerline is greater than the intensity at the wall since the maximum amplitude Of disturbances occur between 10 and 30 Hz. The spectra presented in Figure 37 are realized at the onset Of transition for the Reynolds number range of 5130 to 7300. The X/d loca— ‘ fl.“ tion is 336 and the Reynolds number at this location is 7300. As indi- I; ‘ cated previously the maximum number Of dominating waves are localized near the wall, y = d/12. Karnitz (1971), in a previous investigation, has ascer- tained that the flow disturbance oscillates between 25 and 35 Hz. to 50 and 70 Hz. Because the period Of oscillation between these two signals is small compared with the analyzer integration time the instrument averages the signal and shows all frequencies. Near the centerline the 63 to 80 Hz. waves disappear. The Reynolds number range of 5800 to 700 associated with X/d loca- tions of 0 to 192 represents the range for which disturbances will grow until transition is reached. The spectra for the X/d location of 144 and a Reynolds number of 6800 are presented in Figure 38. A very large dis- turbance is observed near the wall, y = d/12, at a frequency of 63 to 75 Hz. The amplitude Of this disturbance is attenuated at y = d/4 and is entirely missing at the centerline. The spectral investigation presented at this location can be compared with the intensity data in Figure 33, 77 The intensity at X/d of l44 is a maximum for the region near the wall, y = d/lZ, decreasing to the centerline, y = d/2. The large intensity is probably associated with the 63 to 75 Hz. signal. The large signal of 63 to 75 Hz. observed in Figure 38 is absent at locations upstream of X/d = T44. Since the initiation of turbulence is observed at X/d = l92 the large growth of the 63 to 75 Hz. signal must occur between these two locations. I": The large growth of the 63 to 75 Hz. wave strongly suggests its role I -J in the transition process. This wave is present at the smaller Reynolds numbers before the onset of transition. Linear stability theory determines ,,~:‘ that at the minimum critical Reynolds number of 7700 the frequency of the 4} wave that amplifies as it progresses into unstable conditions is 92.5 Hz. Although a 25% difference exists between the predicted and the experimen- tally observed frequency the 63 to 75 Hz. signal does possess the char- acteristics of a perturbed wave being amplified at the onset of transition. RMS (milivolts) 9‘ 0 L5 .0 o 78 I , MK I 20 40 so so Frequenc y (Hz) Figure 35. Frequency spectrum of 60 Hz. noise associated with the hot wire instrumentation. (FL/Secs! '2) 2 U 79 5.0F l R-GBOO Y=d/l2 X/d=336 2L5“ JMJ'| ‘I‘J‘ 1|l1‘_lilllhnc L 0'01 ‘0‘“"10 “ .c .0 7 ' 5.0 Y=d/4 2.5” 0.0] ZJA‘MJ ‘Aaflzm l so so 5.0 [ Y=d/2 2.5 ..L. 4-1.1- . 0'0 . 40 so so Freq uency (Hz.) Figure 36. Frequency spectra at X/d of 336; for a Reynolds number of 6800. 80 5° R=7300 Y=dxm 25W X/d=336 °'°' 20 4o ' so so A 50F (NI -9 Y d/4 L; 2.5 GD . (I) “~ EITDII TALLAI A_z.1\l‘1 - . A - e 20 40 so so IQ! :3 so vac/2 2.5 0‘0 20 40 so so Frequency Figure 37. Frequency spectra at X/d of 336; for a Reynolds number of 7300. u'2 (Ft/Sec: I 0'2) 8l 5C) R=6800 r Y: d/IZ X/dtl44 ELSF 0.0 ‘ ' Aha-£4 20 40 6O 80 Y= d/4 so ’ 51> 2L5* 0&2 Figure 38. Frequency(HzJ Frequency spectra at X/d of l44; for a Reynolds number of 6800. 82 2. Mechanism of the Burst Process A. Span-Wise and Length-Wise Variation of the Burst The turbulent burst was studied in a qualitative manner elucidating its.role in the transition process. Before the bursting process occurs there is a parabolic velocity profile in the fully developed region of the channel at a sufficiently low Reynolds number. At the critical Rey- rualds number the flow becomes unstable and a burst develops and grows as P? it washes from the channel. Karnitz (197l), in a previous investigation, de- termined that the burst was uniformly distributed across the width of the cihannel and did not vary with the z-coordinate. J l‘. The nature of the bursting process was analyzed by means of two hot vvire probes and an on-line data acquisition system with the aid of an IBM l800 analog-digital computer. An effective trigger mechanism was util- ‘ized to enable immediate data sampling at the onset of a turbulent burst. The use of a Schmitt trigger circuit is illustrated in Figure 39. In this design the triggering circuit executes on the basis of a large negative voltage. Because a large negative spike is produced at the initiation of a turbulent burst the system utilizes this information to initiate data sampling. Figure 40 presents two graphs which depict a typical burst, one sampled automatically and the other manually. The automatic sampled data made use of the Schmitt circuit. The computer can also be made to sample data by sending a 0 to l0 volt signal to the A-D converter and this is defined as a manual operation. Using the Schmitt trigger the time lag between the exact instant of the hot wire voltage change and the initiation of computer sampling is less than l00 microseconds. In Figure 40 the AC component of the signal 83 Schmitt Trio-er, Probe Assembly 1 0-500 Hz. Low pass Filter» l :10 Volts . l DC signafl LAC signafl Process Interru t Computer l:Probe Assembly zj—1 o-soo Hz. ‘ Amp _ Low pass Filter 2 [DC signal] [AC signal] Com uter Figure 39. Illustration of Schmitt trigger circuit together with hot wire instrumentation. ul (ft/sec.) ui (ft/sec.) 84 0L if T! I' "’ 1M1 ‘l‘l'u‘I ..|().- ..EEC).. 1 l 1' ICC 200 300 Time (msec.) 0L 'il {.1 "ll-'1', ["1‘ Y, i‘ TH" -.O h I IV) 0 l I ICC 200 300 Time -(msec.) Figure 40. Comparison between automatic and manual computer sampling initiation. 85 is shown with the DC portion removed. Characteristic of all bursts is an initial large spike deflected either positively or negatively depend- ing upon the z-location of the hot wire. A similar large spike is appar- ent at the end of the burst process and the deflection is opposite to that observed initially. The amplitude of the spike varies with X/d lo- cation. In order to determine the streamwise variation of the burst two “1 probes were located at the downstream end of the channel (previous inves- It‘d tigations determined that the burst initiates upstream of X/d = 360). The probes were placed near the centerline, y = d/Z, approXimately 19 in. J apart (Probe l at z = -7 in. and Probe 2 at z = 12 in.). Probe 2 remain- sj ed fixed while Probe l was pulled downstream. The trigger mechanism was applied to Probe 2. If any span-wise or streamwise variation in the turb- ulent slug existed as it washed downstream it would be detected as a time delay for burst detection with respect to Probe 2. For a Reynolds number of 7300 the results of this investigation are presented in Table l. The bursting process did not appear to be well defined or continuous across the width of the channel but isolated to particular regions. At some z-1ocations a number of bursts were observed at Probe l. The char- acterization of these bursts could be described as "mini-bursts". An ex- ample of this unusual phenomenon is presented in Figure 41 for the loca- tion X/d = 288 with Probe l at z = -7.0 in. and Probe 2 at z = 10.5 in. After an initial time delay of 52.2 msec. the first burst appears and quickly dies with the generation of a secondzshort lived burst and then another much longer burst. The number of short lived bursts and their occurence at particular locations appears to be random. Inspection of Variation of Burst Process Along Length of Channel 86 TABLE 1 Expt. 1 D00 QmVOfiU'I 10 X/d Probe l z=-7 in. y=d72 264~ 288 312 336 360 384 408 432 480 528 X/d Probe 2 z=12 in. y=d/2 288 288 288 288 288 288 288 288 288 288 # of Bursts observed at Probe l 3 3 2 turbulent Time Delay (msec.) with respect to Probe 2 52.2, 177.0, 290.4 75.8, 194.6, 251.2 35.2, 131.0 J L 62.8 96.6 l00.4 l55.4 160.0 9.0, 86.6, l75.4 87 0' (ft/sec.) 1 L 100 200 300 Time (msec.) O Z=005HL r7 1 8 -10- U) \. 55 -2o 7: -3° 1 1 ICC 200 300 Time(msec.l Figure 41. u' portion of computer sampled burst process at z = -7.0 in. and z = 10.5 in. 88 Table 1 reveals that this phenomenon is evident at 4 different locations. The time delay between the appearance of the bursts at the two probes increases with X/d. The fact that there is a delay at Probe 1 sug- gests that the bursts originate at the right half of the channel, up- stream of both probes. At X/d of 528 no delay is apparent since the flow is fully turbulent at this point due to the wake effects from Probe 2. In order to characterize the Span-wise variation of the bursting process Probe 2 was placed near the right side wall at z = 10.5 in. and the loca- tion of Probe l was varied with respect to the z-coordinate. The results of the experiments at X/d of 288 are presented in Table 2. The observed time delay is With respect to Probe 2. The results indicate that the burst process is not two dimensional across the width of the channel but isolated near the right hand side. At distances greater than 2 = -l0.0 in. the burst does not appear to sweep past this region. Moving Probe 1 towards the right hand side gave rise to smaller delay times. The neg- ative delay times in experiments 6, 7, and 8 indicate that the burst gen- eration upstream occurs randomly at regions close to the right hand side wall. Figure 42 depicts the burst process over 3 seconds at X/d of 288. Probe 2 is located at z = 15.5 in. while Probe l is at z = 2.0 in. The initial Spikes observed at the beginning of every burst are.different at the two.z-locations. A large negative deflection is apparent at the ex- treme right hand side of the channel, whereas near the centerline a posi- tive deflection followed by a larger negative spike is observed. This characteristic is apparent at other X/d locations.‘ Variation of Burst Process Across Width of Channel 89 TABLE 2 Expt. 1 \OCDVO‘U'l-wa _a o d u—‘ —.| N _.a (A) ...: p Z (in.) Probe 1 X/d=288 y:dgz -15.5 -9. l . N 01 O O O O O O O O O O P-hbb-h-b-DN u—l d 11.5 (X/d=312) 11.5 (X/d=336) Z (in.) Probe 2 X/d=288 EL 10. 10. 10. 10. 10. 10. 10. 10. 10. 15. 15. 15. 15. 15. 5 mmmmmmmmmmmm AZ in. 26. mmmmmoom oommmmmmmmmmo 11. ll. 4.0 Time Delay (msec.) with respect to Probe 2 no burst on Probe 1 no burst on Probe 1 62.8 125.0 9.9 -9.4 -6.3 3.1 7.9 11.0 8.6 10.2 48.0 turbulent 90 loi- 2:15.51n. § 0 -1 1'1L111'1‘J ‘ ."||'L‘ \. 3 '10 7: ‘20 J l ' J l 2 3 Timeisec.) 5 z-2.01n. 1 . l ’7 O L J ill L W o [T I"! 1‘ 0 Q \. z '5 3 l V .1. O L 1 2 Time (sec.) Figure 42. u' portion of computer sampled burst process at z = 15.5 in. and z = 2.0 in. 91 B. Location of Burst Generation In order to establish the location of burst generation the probes were moved upstream with Probe 2 near the right side wall, 2 t 14.5 in.,. and Probe 1 near the centerline, z = 1.0 in. It was observed that at X/d of 96, a region where the velocity profile was still developing, the burst process was isolated to a narrow region near the right hand side Iwall. The Reynolds number for transition at this location was 7300. Large F.1 scale velocity spikes were first observed at z = 10.0 in., growing to a ' ‘"A maximum at z = 13.5 in., and decreasing to the right side wall. The width 3 of this unstable region is approximately 7 in. Figure 43 presents the E , “7 burst process at z-locations of 12.0 in. and 9.0 in. if! In order to observe the behavior of the burst process as a function of increasing X/d the two probes were pulled downstream. Figure 44 de- picts the bursting at X/d of 120 for z = 14.5 in. and 1.0 in. The burst- ing from the right hand side wall region is associated with large velocity spikes which resemble the beginning and end of a burst observed at down- stream locations. The frequency of the spikes is approximately 6 to 8 Hz. The centerline region exhibits velocity fluctuations whenever large nega- tive spikes are produced at z = 14.5 in. The bursting process at X/d of 144 is diSplayed in Figure 45. The bursting is more defined at the side wall region with a larger burst duration time. The turbulent core region of the burst appears to be establishing itself. C. Periodicity of the Burst Process By placing the probes at regions close to the right and left side walls and successively measuring the burst process at every X/d location a mapping of the burst process was established. The initial burst and u' (ft/sec.) u' (ft/sec.) 92 E z=12.om. s 1 4 1 () a ., _ 11 , L I. ‘1 .1111 11” L. L 11' f 1 '1' 11 '5 -15_- 1 ’ ‘ . l 21 l l 2 3 4 Time(sec.) Z='9.0‘n £5_' 1 0'. ’ A! ' '1' 1'1”“ 1' ..551_ ~10.- “15+- " 1 1 1 l 2 3 4 Time(sec.) Figure 43. u' portion of computer sampled burst process at z = 12.0 in. and z = 9.0 in.; at X/d of 96. 0' (ft/sec.) u' (ft/sec.) ~20~1 1 1 J1 93 Z=l4.5in. .. 11 1 I 0| 1 Time (sec.) 2 Time (sec.) Figure 44.. u' portion of computer sampled burst process at z = 14.5 in. and z = 1.0 in.; at X/d of 120 u' (ft/sec.) u' (ft/sec.) 1 1 2 3" 2:1.01n 1' -2- " i l I 2 Time (sec.) Figure 45. u' portion of computer sampled burst process at z = 14.5 in. and z a 1.0 in.; at X/d of 144. 95 laminar duration times including the sUccessive burst duration times are . presented in Table 3.‘ The two probes were located at z = $13.5 in. The duration times of the bursts and of the laminar interval between each burst increases with increasing X/d. However, this linear increase is a result of the interaction of the probes with the flow. Pulling the probes downstream causes an increase in the flow rate for a constant set- ting of the fan speed. Furthermore, the duration time of the first burst f”? is always less than the time observed by the next successive burst. It .Etvw‘ is approximately 80 to 85% of the duration time of the next succeeding bursts. The burst process generated at X/d of 96 is localized at the right he] side wall. From Table 3 it is observed that this burst does not sweep past Probe 1 which is situated at the far left hand side wall until X/d of 312. Utilizing this information a simple model of the burst process has been devised. This mechanism is illustrated in Figure 46. The angle of the burst wake is approximately 30 degrees. The wake fills the entire width of the channel at X/d of 336. The wavey line indicates a region of the burst wake that is dissimilar from the core of the wake. It has already been established that in this region the burst process is initi- ated by a positive spike, whereas the region close to the right side wall experiences a large negative spike. The two dimensional character of the burst process is not observed unti1 X/d of approximately 450. In this region the burst process is observed to initiate with a large nega- tive spike and end with a large positive spike. The leading edge of the burst travels at a maximum speed of 30 to 33 ft./sec. The region surrounding the burst is laminar and the maximum 96 TABLE 3 Periodicity of Burst Process Initial Burst Duration Time (msec.) Z (in.) 13.5 -l3.5 160 no burst 156 no burst 160 no burst 232 no burst 664 no burst 720 228. 536 264 832 528 936 994 1096 1112 1256 1240 Initial Laminar Duration Time (msec.) Z (in.) 13.5 -l3.5 168 no burst 164 no burst 200 no burst 320 no burst 616 no burst 568 736 712 992 528 824 520 488 552 520 464 480 Successive Burst Duration Time (msec.) Z (in.) 13.5 -l3.5 328 no burst 440 no burst 488 no burst 376 no burst 752 no burst 920 584 1280 648 1112 1072 1240 1096 1312 1320 1400 1272 Successive Laminar Duration Time (msec.) Z (in.) 13.5 -l3.5 360 no burst 352 Inc burst 496 no burst 648 no burst 440 no burst 368 684 448 744 448 488 480 592 496 504 352 464 97 1111 Model of Burst Process \ l4° Laminar Zone Figure 46. 1. 1 O — ,43 Initiation of E Bursi .{ f "'96 P —144 __|92 Turbulent Zone —240 i.— ~288 _ X/d ' —336 -384 -—432 e480 r528 576 Mechanism of bursting process. 98 velocity experienced in this region is 40 to 42 ft./sec. Nevertheless, the average speed in both the turbulent and laminar regions is 28 ft./sec. The velocity of the edge of the burst wake in the z-direction is observed to be 8 to 9 ft./sec. This description characterizes the highly localized mechanism of the burst process. In order to elucidate the cause of the localized burst generation the intensity level across the width of the channel at X/d of 0 was inves- tigated. For a Reynolds number of 7100 the intensity level at the center- line, X = Y = Z = 0, was observed to be 0.12%. By variable placement of. the probe with respect to the z-coordinate a high intensity region was observed at z = 13 to 15 in. A 0.23% intensity level was established in this region. Locating the probe inside the contracting region, X = -12 in., established that the high intensity was generated by the filter de- vice. At this point in the investigation the settling chamber was rebuilt, the straws removed and replaced, and the various screens aligned properly. The previous experiment was repeated and the effect of rebuilding the fil- ter region was the elimination of the high intensity isolated at the far right hand side. Unfortunately, another high intensity region was gener- ated on the far left side. For a Reynolds number of 5000 the intensity level at the centerline was observed to be 0.15%. Proceeding towards the left side wall the intensity level increased to a maximum of 0.19% at z = -16 in. The bursting process was examined after the settling chamber was rebuilt and it was observed that the bursts were generated at the far left side wall, specifically at X/d of 84 and z a -13.5 in. The Reynolds num- ber for transition was 7100. Therefore, the assymetrical intensity across 99 the width of the channel which was generated in the settling chamber affected transition downstream of the location of maximum intensity. If the location of maximum intensity was moved, the location of transition moved accordingly. 0. Velocity Profiles During Transition When the Reynolds number is increased until the critical number of 7300 is achieved the flow is intermittent between a turbulent burst and a laminar flow. The intermittency of the burst process has already been established. However, the velocity profile at transition also varies be- tween a parabola and an unknown turbulent profile ("unknowni in the sense it has never been eXperimentally determined). The velocity profile in the turbulent region of the burst may or may not possess the characteris- tics of the "one-seventh" profile. By means of conditional sampling the velocity profile was eXperimentally realized as the flow oscillated from a laminar condition to that of the burst condition. For the velocity profile burst investigation the probes were placed at X/d of 360 with Probe 1 near the centerline, z = -l.0 in., and Probe 2 near the right side wall, 2 ' 15.0 in. The velocity profile was obtained from measurements of the centerline probe. In order to initiate each experiment the Schmitt trigger was applied to Probe 2 near the right side wall. Instantaneous on-line velocity measurements were obtained by tra- versing Probe 1 in the y-direction at the onset of every burst. Seven- teen experiments, each representing a different y-location, were performed in order to realize an entire profile. A mosaic of the velocity profiles during the bursting process is presented in Figure 47."Each'profile is the result of 68 individual 100 Figure 47. Mosaic of velocity profile during burst process. 101 experimentsz‘ a)'4 experiments at each y-location; and b) 17 experiments to cover the full height of the Channel. The 4 experiments at each y- location were averaged to obtain a single point. The on-line instantane- ous velocity measurements were time averaged over 60 msec. Therefore, each point represented an average over time (60 msec.) and an average over 4 individual experiments. The sampling rate of the computer was initial- ized for 6 msec./sample. The total number of samples for each experiment was 500 which represents a total time of 3.0 seconds. The 60 msec. time average represents 10 samples. The final task of the investigation was the registration of 17 experiments in order to create the full velocity profile. Inspection of the mosaic in Figure 47 reveals that a parabolic vel- ocity profile is apparent at t = 0 and t = 1140 msec. This represents the beginning and end of the turbulent burst. At the onset of the burst, t = 60 msec., the first portion of the profile to encounter a deformation is the centerline, y = d/2. The total time in which the profile changes from a parabola to a turbulent profile is achieved within 120 msec. The turbulent profile is fully established at t = 180 msec. This profile re- mains unchanged until t = 720 msec., whereupon the change to the parabolic profile is initiated. The rate at which the turbulent profile changes to that of a parabola is a much slower process and takes approximately 360 msec. This is 3 times slower than the change from a parabola to a turb- ulent profile. Therefore, the mechanism that distorts the profile occurs at a faster rate at the beginning of the burst process than during the termination of the burst. The One seventh power law is often used instead of the logarithmic 102 law to express the turbulent velocity profile if the Reynolds number is not very large. The relationship is eXpressed as u = umax (2y/d)”7 (4 36) Substituting this one-seventh power velocity profile in equation (4.27a) a relation is obtained between the centerline velocity and the mean vel- ocity. u = 0.967) (4.37) ave umax ( Equation (4.36) represents the profile for one half of the channel. In order to determine whether the velocity profiles generated by the burst- ing process obey a one-seventh velocity profile the following equation was employed u = umax [ 1 - ABS( 31—5—9) 11/7 (4.38) This equation describes a one-seventh profile over the entire width of the channel. The profiles represented in the mosaic of Figure 47 were curve-fitted with respect to equation (4.38). A curve—fitting routine described earlier was employed for this purpose. The results of the investigation determin- ed that the profiles represented by times of 180 to 720 msec. obeyed the one-seventh power law profile. Experimental and theoretical values are plotted with respect to the y-coordinate in Figure 48. The residuals gen- erated by subtracting the experimental and theoretical values are plotted in Figure 49. This plot indicates a random distribution of residuals. A standard ”run" test;was.empJeyed in order to test for randomness of the residuals. The experiment gave rise to a 95% confidence or random arrange- ment of residuals. Theref0re, the result of this investigation . 3a:— 3 3.5 mu>¢39 U—pm—CUhu¢z¢tu dJ-bOCt xbld>am U26 n~\0\o IIIIIIIIIIIIIIIIIOOOIIIO aru ‘I'II‘ I ‘ ‘ ~ “I I I I I .- 1mental ile) (1/7 power law prof 1(13 m .m _ — _ _ _ u _ — m m - — _ u — _ u _ m m u a u _ _ .o.ueooo_. ~o.uoooo~. ~o.umo¢o~. ...u‘ooco. .. _ _ .e.uoooo.. ~..um‘on~. ~..uoeoo~. ...uoom~o. .- _ m ...ueoee_. ~..u.~om~. ~o.uonno~. o..uo...o. a. m _ .o.uoooo.. ~o.uoeoo~. ~o.ue.c-. ...uomeen. m. _ " ...usooo.. ~o.uneo.~. ~..u-.e~. ...uo.~mn. o. “ ...ueooo.. «o.u.ooe~. ~..um~m-. ...uom.~n. n. . .o.uooo... ~o.un_.o~. ~..uoooo~. ...useoon. ~. . m .o.ueeoo_. ~o.uooco~. ~o.uon~o~. ...umno.~. .. m _ .e.ueooo.. ~o.uoomo~. ~..unmeo~. o..u.~oc~. a. _ _ .o.uo°oo.. ~o.u.n.o~. ~o.uc.no~. o..u.~m-. o _ “ .o.uoeoo.. ~o.ucaa-. ~o.uooo-. o..umooo.. o “ .o.ueooo.. ~..u_~.e~. ~..mnn.-. o..u.oc~.. e m ...ueeeo_. ~o.u~_oo~. ~o.uo~n-. ...uaooo.. a m “ .o.ueeeo.. ~..uoo~m~. ~..uo.on~. ...uo~.~.. n M . ...ueeo... «..uoneou. ~o.uo~mn~. .oiu..o~o. o . _ ...uooo... ~..u~n.n~. ~a.uo~oo~. .oiu..noo. n . m ...uooe... ~o.uoao.~. ~..uo.-~. ..iu-o..o. ~ a ...uooe... .. .o .. _ “ _xe_ua 4.2.5 o.mu¢ . ...~.n ._.~.n ...... amass: M _ . _ _ m a . _ _ a — _ _ _ _ m o h — I I n u I O u _ o n _ - I i — a o o h u a I n I n n - O I O I n _ . a u . _ n . nb-0093-00-5---0"---fl0IIIm0900fiI0-l-IIIII”-III”---'“I--.‘:--m---0m----m000-m000-?-0.“--00m---0w---0- » «:8 2. n :43 utcm U.— x— at: 3:9. 3:533 93 denly—awe: .3 «tea: a 333: a. ...-mu: cu:- aum: m— a 32°. ...—0.. capcguacu c 92!. a £1.00 Jesus—rue: .3 «:1... e 00. Hg .5. Nolucwco I to. .3: pd 84: oO—nuz 3° .-.~.n m. ecu—pad. o .5335. .... a :3 u... pa *3: £733.: a: ...—... «— inoxua I 7.8.3:. .60. uemo. I tau-pox r— pd gg>o~OoUctNo I 3.3—! V; pd 344). .0 Computer printout of calculated and exper velocity profiles Figure 48. 104 o- In I I I .15 u! 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Figure 49. 105 substantiated that the velocity profiles over 70% of the turbulent burst can be described by a one-seventh power law profile. As observed in Figure 47 the laminar to turbulent profile is quickly achieved. Regions close to the wall must undergo a sudden increase in velocity in order to maintain the one-seventh profile. Similarly, a sud- den decrease in the velocity must be evident at the centerline. A sche- matic of this mechanism is illustrated below. F1} \ I 1 "I\ . i b. I 11 / Figure 50 presents the velocity changes at two y-locations; near the wall, y = d/12, and at the centerline, y = d/2. During the bursting process the velocity increases by 17% near the wall, y = d/12, and de- creases by 67% at the centerline, y = d/2. The laminar portion of the bursting process is associated with small scale fluctuations. An indication of the level of intensity that may exist during this time is the quantity 072 . Figure 51 presents the computed fluctuations based upon the first laminar poriton of the signal in Figure 42. u'2 and 07?. are plotted versus time for the regions y = d/12 and y = d/2. It is interesting to note that at the beginning por- tion of the laminar signal the fluctuations decrease at a slow rate. The end of the laminar signal is associated with a very rapid increase in U (ft/sec.) U (ft/sec.) 106 3O 25 YBd/IZ 20’ .5 10 m I I 2 p 3 Time (seal 1? Y: d/Z 35- 30 2 j I I 2 3 Time (sec.) Figure 50. Velocity changes near the wall and the centerline during the burst process. 107 CGF~ ‘ ‘ '2 5!Y=d/12 1 “$1" 4 r%’ 3‘ (\I‘h .3 2... 111', 11 1. 1111 ’.‘e_-‘1IIV'1'I"")}W '\ ., ,4 E 0.2 0.4 0.6 0.8 T i me (sea) d? Y=d/2 '0 l0 1 '.' : 11 “s 11, 19'; 5' 11,1 “-1.. 3 '1 1} «1 "\ - 'I‘Dinmg" '/ 1"" I 11” ~111 027 O4. 06 08 Time(sec.l Figure 51. Perturbations during the laminar portion of the burst process. 108 the fluctuation level. A similar observation has been previously noted for the changing velocity profile as it moves in and out of the burst. Comparison between the region near the wall, y = d/12, and the centerline, y = d/2, shows a much larger scale of fluctuations present at the center- line. CHAPTER 5 DISCUSSION OF RESULTS The results of the boundary layer analysis in predicting the devel- opment of the velocity profile in the entrance region of the channel agree well with the experimental observations. Entrance lengths for the fully developed profile have been calculated for a range of Reynolds numbers. A linear relationship is observed between the entrance length and the Rey- nolds number. A free stream velocity distribution was calculated by means of continuity and provided a basis for the boundary layer calculations. It was observed that the free stream velocity was accurate enough to pro- vide a good comparison between theory and experiment. The skin friction coefficient was a significant parameter that was generated by means of the boundary layer analysis. The results indicated a minimum displacement thickness and skin friction coefficient 3 in. before entry to the parallel plate channel. The experimental results indicate that a laminar flow exists up to a critical Reynolds number of 7300 if the channel is sufficiently purified of disturbances. Developing entrance lengths have been obtained which compare well with the theoretical prediction established by Schlichting (1962). For a Reynolds number of 7300 the velocity profile is observed to be fully developed by X/d of 192. It is noted that the 6 ft. contrac- ting region develops the flow an equivalent length of 84 channel widths for a Reynolds number of 4800. A statistical treatment of the data was prepared to give informa- tion about the developing profile, the nature of the fully developed pro- file and an estimate of the channel gap width. A curve-fitting routine 109 110 was utilized to give information about deviations from a parabolic velo- city profile. Investigations in the regions of developing and fully developed flow have resulted in the eXperimental evidence that the wake-effects of the probe assembly result in the flow being fully turbulent in the down- stream direction of the probe. The wake spreads across the entire width of the channel 2 ft. downstream of the probe assembly. This effect re- sults in an increase of the shear stress in the turbulent region behind the probe. Since the pressure drop is a direct function of the shear stress the positioning of the probe in either the downstream or upstream direction acts as a flow meter. An equation was developed that predicted the change in the average velocity as a function of the position of the probe (for a constant setting of the fan). A comprehensive mapping of velocity profiles in the channel assem- bly allowed identification of the regions of bursting and fully turbulent flow. As the probe is positioned sequentially downstream the first burst is observed at X/d of 480 for a Reynolds number of 7300. Increasing the Reynolds number resulted in the bursts being detected farther upstream. For Reynolds numbers 8000 and above the bursting region was isolated to a small region at X/d of 192. Upstream of this region the flow was lam- inar and downstream it was fully turbulent. The influence of the side wall boundary layer on the mean flow was determined experimentally. The thickness of the boundary layer was mea- sured at the centerline, y = d/2, and found to be less than 0.750 in. thick at the exit of the channel for a Reynolds number of 7300. Based on this maximum boundary layer thickness the exiting flow is two 111 dimensional over 96% of the width of the channel. Therefore, the influ- ence of the boundary layer on the mean flow is minimal. It was interes- ting to observe that the laminar fluctuations in the side wall boundary layer increased as a function of proximity to the wall and Reynolds num- ber. The thickness was observed to decrease and reach a minimum for a Reynolds number of 4000. However, as the Reynolds number was increased above 4000 the boundary layer increased in thickness as a result of lam- inar fluctuations which were induced by the high flow rates. However, the transition process did not appear to be affected by these fluctuations since bursting occurred simultaneously in the boundary layer and the mean flow. 9 To further establish the eXtent of the transition region the in- tensity of the finite disturbances and the frequencies of the disturbing waves were measured along the x and y-coordinates. Investigations were carried out along the centerline and locations near the wall in order to determine the symmetry of the disturbances. The results indicated that the disturbances damp as the Reynolds number is increased. For a Rey- nolds number range of 3000 to 6000 the fluctuation intensity is a mini- mum at the centerline and grows to a maximum near the wall, y = d/12. Increasing the Reynolds number range, 4000 to 7300, causes the intensity at the centerline to grow. The intensity for this range of Reynolds num- bers is a maximum at y = d/12 unti1 X/d of 336. At this location the maximum intensity is shifted to the centerline. Above the critical Rey- nolds number the disturbances in the regions between X/d of 0 and 192 exhibit amplification at y = d/12 and y = d/4. The centerline is most stable at this point since it has the minimum intenSity. 112 Spectral analysis of the disturbances generated naturally in the channel showed a series of dominant waves that amplify as the flow pro- gresses into unstable conditions. The number of these dominating waves was observed to be a maximum near the wall, y = d/12, decreasing in num- ber to the centerline. A wave 0f 63 to 75 Hz. became amplified for a Reynolds number of 6800 at X/d of 144. The sudden growth of this wave strongly suggests its role in the natural transition process. The max- [Hal imum intensity of the 63 to 75 Hz. wave was detected at y = d/12 and t'" X/d of 144 for the Reynolds number range 5800 to 7000. Transition was observed to occur at X/d of 192 which implies that this particular wave E i must grow and amplify in 2 or 3 channel widths. ' E The investigation of the nature of the bursting process, analyzed by an on-line data acquisition system, gave rise to some unexpected re- sults. It was determined that the burst process was not two dimensional across the width of the channel at the burst Reynolds number of 7300. The burst occurred in isolated regions at upstream locations where the free stream distrubance level was highest. A small region, 7 in. in width, at X/d of 96, near the right hand side wall was characterized by large ve- locity spikes for a burst Reynolds number of 7300. The transition occur- ring in this particular region was similar to the transition in a boun- dary layer, in that transition in a boundary layer also is a highly loca- lized three dimensional process involving the occurrence of strong velo- city Spikes. Spangenberg and Rowland (1960) reported from optical studies that the turbulent spots grow rapidly during the first microseconds and appear to eXplode from the smooth outline of the laminar layer. Thus, the transition process is considered to be Similar to that observed by 113 Spangenberg and Rowland. They observed a turbulent breakdown as an in- termittent appearanceof ripples on the outer surface of the boundary layer. As the ripples moved downstream, each divided into several seg- ments. Each of the segments then became the source of a "shock wave". In a matter of microseconds after the appearance of these disturbances boundary layer air was "belched" from the disturbance area, and the erup- ted spot then grew into a turbulent spot. In the current study the strong velocity Spikes generated at a fre- Fig.1. quency of 6 to 8 Hz. at X/d of 96 appear to grow as they progress down- stream. The turbulent core region between the initial large negative . .~~= spike and the final positive Spike becomes progressively larger at greater 031 downstream locations. From a chaotic state of bursting at X/d of 96 the process deve10ps into a ordered periodic structure by X/d of 384. A description by Tani (1967) of the transition process indicates that turbulence is initiated in small localized regions in the form of "turbulence spots". These Spots grow as they move downstream until they merge to form the fully developed turbulent boundary layer. The process observed at a burst Reynolds number of 7300 in the channel can be compared quite well to Tani's description. However, the dimensions are greatly in- creased in x and z. The final investigation of the bursting process concerned the shape of the profile when the flow was intermittent between a turbulent Slug and a laminar flow. It was observed that the mechanism distorting the para- bolic profile into a turbulent one at the burst leading edge proceeded at a faster rate than the turbulent profile into a parabola at the burst trailing edge. Close examination revealed that the centerline region was 114 the first to be affected by the burst process. Comparison to a one-sev- enth power profile substantiated that the burst profile is characteristic of the common turbulent shape over 70% of the burst process. 5'» ‘e'u. I. n . Eim‘ 3r" . .,_I .. ; -“_ 115 1. Conclusions From the present study the following conclusions can be drawn: a) b) C) d) e) f) a linear relationship has been obtained between the entrance length ( the distance that is needed to achieve a parabolic velocity profile ) and the Reynolds number. The experimental data was observed to correlate well with analytical predictions. For sufficiently high Reynolds numbers regions of bursting and fully turbulent flow have been identified by a comprehensive mapping of velocity profiles. The developing region between X/d of 0 to 192 was shown to be laminar at all Reynolds numbers. An experimental burst Reynolds number of 7300 has been achieved for the present channel assembly. The transition Reynolds num- ber can be monotonically increased by decreasing the fluctuation intensity. A minimal effect of the side wall boundary layer on the mean flow was determined experimentally. For a Reynolds number of 7300 the boundary layer thickness was determined to be less than 0.750 in. at the exit of the channel. A wave of 63 to 75 Hz. was amplified as the Reynolds number was increased to 6800 at X/d of 144. The sudden growth of this wave influenced the fluctuation intensity level for regions near the wall, y = d/12 and y = d/4. The wave was not present at the cen- terline, y = d/2. Linear stability theory predicts a wave of 92.5 Hz. at the minimum critical Reynolds number of 7700. A close examination of the burst process revealed a three-dimen- sional highly localized bursting phenomenon. Strong negative 9) 116 velocity Spikes were observed to originate at X/d of 96 for a burst Reynolds number of 7300. A non-uniform fluctuation inten- sity level at the inlet of the channel influenced the origin of the burst. The location of the burst coincided with the max- imum disturbance level at the inlet of the channel. The Shape of the turbulent profile achieved during the bursting process was observed to be the one-seventh power law profile over 70% of the burst. The mechanism distorting the parabolic ‘3 “A profile into a turbulent one proceeded at a faster rate than the turbulent profile into a parabola. .. 117 2. .Recomnendations ' In order to study the transition mechanism and evaluate the burst process in better detail the following reCOmmendations are made: a) The settling chamber should be redesigned. Specifically, the straw chamber should be rebuilt in order to provide an even distribution of straws across the inlet. The entrance region is an important factor Since the intensity level at the inlet to the channel strongly influences the origin of the burst pro- cess. b) Further investigations are necessary in order to elucidate the highly localized transition process. If the intensity is uni- form across the width of the channel would the point of transi- tion become random or would it remain localized? c) It would be of interest to traverse the boundary layer along the side walls with respect to the y and z-coordinates. This would provide a mapping of the velocity profile. The influence of high Reynolds numbers which induce large fluctuations in the boundary layer should be further investigated. d) Finally, another measuring device or transducer besides the hot-wire anemometry system (such as a sensitive piezo-resistive accelerometer) could be utilized in order to investigate the bursting phenomenon without grossly affecting the flow. fij f. 1 APPENDIX A SYMBOLS FOR BOUNDARY LAYER ANALYSIS APPENDIX A SYMBOLS FOR BOUNDARY LAYER ANALYSIS Af constant used to match numerical to asymptotic solution (2.31) b]...b5 coefficients in linearized momentum equation (2.26) C = 2(6*/r ) cosa , parameter related to axisymmetric flow appearing in a Equation (2.23) 1w, pU2/2, coefficient of skin friction ('5 ll c1...c4 coefficients in equation (2.24) d gap width in the parallel plate channel dt = d/2 half the channel gap width E modulus of elasticity f1k) variables used in the Runge Kutta method, k=l,3 f' = (U-u)/U, velocity defect variable f'p particular solution of equation (2.26) f'h homogeneous solution of equation (2.26) h half the gap width of the contraction at x = -6 ft. (see Fig. l) I moment of inertia L length of contraction (see Fig. 1) P = [6*dU/dx]/U, parameter in equation (2.16) 0 = [d(U6*)/dx]/U, parameter in equation (2.16) R = [6*drw/dx]/rw, parameter in equation (2.16) rw radius of curvature R6* = 6*U/v, Reynolds number based on displacement thiCkness r = rw(§) + ycos 0(x), radius of a point in the boundary layer' 118 119 s parameter in asymptotic outer solution (2.33) 6,7' time average velocities in x and y directions respectively '557r Reynolds stress vw , wall tranSpiration velocity x streamwise coordinate (see Fig. 2) y c00rdinate normal to wall (see Fig. 2) Ax = xi+1 - x1, numerical integration step in the streamwiSe direction 0 angle of the tangent to the surface with respect to the axis of symmetry 6 boundary layer thickness 6* =.6m (U-u)/U dy, displacement thickness n = y/5* nonedimensional coordinate normal to the wall 6 =J‘°°'u(U-u)/U2 dy, momentum thickness K = 8.41 von Karman constant in the effective viscosity function v molecular kinematic viscosity ‘ v effective kinematic viscosity p density T local shear stress T6 = 3(r tlrw pU2)/3n, non-dimensional shear stress gradient (2.24) T a non-dimensional effective viscosity (2.20) 0,0 inner and outer effective viscosity functions x coordinate normal to wall in outer effective viscosity function x coordinate normal to wall in inner effective viscosity function Subscripts ( )a variable evaluated at asymptotic matching point ( )1 index of variable in the x direction ( )5 variable evaluated at point where recalculation begins 120 ( )0° variable evaluated at the edge of the boundary layer Superscripts ( )' differentiation with respect to n = y/d* (‘7 denotes untransformed coordinates. Also used with functions of x only, denotes average value, [ ( )1+1 + ( )i]/2 “APPENDIX B SYMBOLS USED IN EXPERIMENTAL RESULTS APPENDIX B SYMBOLS USED IN EXPERIMENTAL RESULTS b]...bn predicted value of parameters used in least squares analysis (4.9, 4.10) d channel gap width d+ = du*/v, normalized channel gap width e1 = Yi - Vi, residual (observed minus predicted values) (4.12) E(Yi) expected value of Y1 f friction coefficient for laminar or turbulent flow I = “5;: / Uave’ intensity of disturbances (4.35) 1e entrance length (distance needed to achieve a parabolic velocity profile) P Pressure AP change in pressure (4.33) dP/dx pressure gradient (4.32) R* = 2dUave/v, Reynolds number based on twice the channel's gap width R = dUave/v, Reynolds number based on full channel gap width S sum of squares used in least squares analysis (4.15) t time Uave = l/d.6d u(y) dy, average velocity UO average velocity at probe position x = 0 ft. U23 average velocity at probe position x = 23 ft. u x-component of velocity u+ = u/ut, normalized velocity C I- II “1079’ friction velocity 121 xi’ i x,y,z 122 time-averaged velocity perturbation in x-component of velocity maXimum velocity independent and dependent variables used in least squares analysis (4.6) coordinate axes 3+ = yU*/ , normalized y-component B]...Bn 8i u parameters used in least squares analysis (4.6) error in estimating Yi molecular viscosity kinematic viscosity density shear stress evaluated at the wall “"7”";uum! 01.th VOCDNOI 11. 12. l3. 14. 15. l6. 17. 18. 19. 20. LIST OF REFERENCES Beck, J. V., Proc. of the Third Internationalgfleat Transfer Confer- ence Vol. IV. pp. 74-80 (1966), and ResearCEEProposal Michigan State University. Bodia, J. R., and Osterle, J. F., Appl. Sci. Res. 19, 265, (1961). Boussinesq, J., Comptes RenduS, 113, 9 (1891). Breslin, J. A., Lehigh_University Technical'Rep.,"Ng;_2g_(1970). Cebeci, T., and Smith, A. M. 0., Proc. 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