I .1! 'rfifip. '3 1h} I'fif Hg”: u; wall. U ”Ii n ‘ "r-_ I " ""Ill" .5 2' . ' ‘I 1 f, ‘ . ‘. W h H ; I 1 “hr: A'n‘. ’I" I I. :. ' ‘ ‘-‘ .I . I I" "I” ‘ ‘ I ' I. t'. I" 5 IRWIN!" {I‘E \fl'fiqgeu :' I“ t ‘ ‘H' ‘ ‘V‘l I“ I": i" f" I'U'I'H‘. Iv'e'x '-‘ ‘ ‘5 lift 3 I. J. I 4 ‘ ' OI up, 1: I ‘ ’11 u' " “" I."I"' " " 'HfHP. - - - 3 .I H VII :~ “ . “I - VI - n5 ”,1 " ‘ “‘o‘u; -| ‘I ‘ ‘.' "ML! ”1 “ifl -1” f' ”I ‘ .‘l r": I I‘ 3‘; I' '1: $.10 II || 135“”. I. III. ;:I , ”I ' 't‘llIlu‘lufiv‘HWVI . I ‘ ‘ ‘I ‘ " In ‘ .. F' 4,, I H. I" H HWW'I‘; W“- m ’m’ “Woman" LIBRARY .‘l 3:1 “.6 ,c MiChig_; 7‘ ”ate University 2 This is to certify that the thesis entitled AN EXPERIMENTAL STUDY OF THE TRANSPORT OF A NON-DIFFUSIVE SCALAR CONTAMINANT IN THE DECAYING TURBULENCE FIELD OF AN ENCLOSED CHAMBER presented by Kenneth Charles Cornelius has been accepted towards fulfillment of the requirements for Ph . D . degree inMECh . Engr‘ . @3451 54 / Major professor Date Sept. 7, 1978 0-7639 AN EXPERIMENTAL STUDY OF THE TRANSPORT OF A NON-DIFFUSIVE SCALAR CONTAMINANT IN THE DECAYING TURBULENCE FIELD OF AN ENCLOSED CHAMBER BY Kenneth Charles Cornelius A DISSERTATION Submitted To Michigan State University in partial fullfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1978 .A ‘9/t%87123 ABSTRACT AN EXPERIMENTAL STUDY OF THE TRANSPORT OF A NON-DIFFUSIVE SCALAR CONTAMINANT IN THE DECAYING TURBULENCE FIELD OF AN ENCLOSED CHAMBER BY Kenneth C. Cornelius An experimental investigation of the convective properties of a turbulence field within a closed mixing chamber has been executed; the investigation is supported by apprOpriate theo- retical analyses. The purposes of the study were (i) to develop an experimental technique which could ascertain the convective, or dispersive, properties of such a turbulence field (i.e., with the constraint of "zero" molecular diffusivity) and (ii) to document the influence of the Reynolds number on the convective transport properties of a representative field. The derived technique involves the introduction of sub- micron particulate matter into the lower half-volume (as defined by a horizontal splitter plate which is rapidly removed from the chamber at the initiation of an experiment) and the sub- sequent monitoring of the presence/absence of this contaminant along a vertical line centered at the plane of the splitter plate. An optical/mechanical/electrica1 technique was used to scan 205 locations along a 7.4 cm portion of an incident, focused (d z 0.1 mm) , laser beam and the scattered light intensity was used to identify the presence or absence of the fluid elements which originated in the lower half-volume. That is, since the particulate matter is shown to be too large to respond to the diffusion effects of the Browman motion of the gas, and too small to move independently of the fluid element of its origin, it is inferred that it faithfully marks the lower half-volume Kenneth Charles Cornelius fluid over the duration of the experiment (0.6 sec). The 7.4 cm scan length covered 2.92 mesh lengths of the biplane, round rod, turbulence generating grid. Air was used for 69 experiments; air = 5'avls‘reon 12)’ A principal result of this investigation is that the Freon 12 was used for 100 experiments (v particle dispersion is dependent upon the mesh Reynolds number (UOM/v); the width of the mixing field was found to decrease as the Reynolds number was increased. A theoretical model of the dispersion was developed using the suggestion by Batchelor and Townsend [1956] that the Lagrangian velocity auto corre- lation function can be made stationary, in a decaying turbulence field,by an apprOpriate stretching of the time coordinate. The present model utilizes both the Lagrangian micro and macro scales to describe this correlation function; numerical values for these two scales are inferred from the experimental data. The observed decrease in the particle dispersion with increasing Reynolds number is supported by a previously suggested re- lationship between the Lagrangian micro scale and the mesh Reynolds number, a new value for the coefficient in this re- lationship is proposed. Analytical considerations are presented to support the use of a (spatial) Gaussian cumulative distribution function for the description of the ensemble average concentration; viz, F(z,t) = [2n om(t)2]-l/2 [2 exp - [(d - )/Zom(t)2]d&. o Satisfactory agreement between this distribution and the experi- mental data was obtained; om(t)2 was used to evaluate the mean square particle dispersion, zpz. Two properties of the instan- taneous scans were defined: i) an ensemble average measure of the instantaneous width of the mixing field, 01, and an ensemble average measure of the migration of the instantaneous center lines, OAz’ The instantaneous scans are shown to have essentially the same width as that of the ensemble average field (II S 0. 80m whereas CA2 exhibits very little growth with respect to time following the establishment of its nominal value at Uot/M = 8. In Memory of Joan M. Grotewohl Administrative Assistant of the Division of Engineering Research who unselfishly assisted the Engineering Graduate Students of my generation. ii ACKNOWLEDGEMENTS The author wishes to express his sincere appreciation to Professor John F. Foss who suggested the problem and gave encouragement, advice and support throughout the research effort as well as guidance through the author's entire graduate program. Special thanks are also due to Professor David Yen for his advice and discussion of the theoretical content. The author is also grateful to many of his colleagues including Payton Fuller for sharing his thoughts and helping him evaluate parts of the experimental program. Special gratitude is due to the following group of dedicated personnel for their unequivocal time and effort: To Lynn Foth for her excellent programming efforts To Joan Rohr for her aid with the mini-computer To David Sigmon who designed and built the entire electronics package To Charlie Hummel and Curtis Monroe for their aid in the data reduction To Don Childs and the shop crew for their endurance and machining efforts throughout the design phase. The author is indebted to the Division of Engineering Research for their financial assistance during the initial stage of the dissertation work and also the financial assistance from Project Squid contracts, without which the graduate work would not have been possible. Finally, the author is most grateful to his wife Kay whose unlimited patience and endless support brought him to the completion of the final chapter. iii TABLE OF CONTENTS List of Tables List of Figures 1. 2. INTRODUCTION EXPERIMENTAL FACILITY AND TECHNIQUES 2.1. 2.2 NNN .3 .4 .5 Introduction Details of the Experimental Facility Optical System Particle Generators Signal Processing ANALYTICAL AND BACKGROUND CONSIDERATIONS 3.1. 3.2. Description of the Problem Scalar Concentration Field 3.2.1. The Ensemble Average Concentration Distribution 3.2.2. The Ensemble Average Fluctuation Intensity Distribution 3.2.3. Dispersion Considerations 3.2.4. Fourier Transform Considerations DYNAMIC BEHAVIOR OF THE MARKING PARTICLES 4.1. Aerodynamic Drag 4.2. Gravitational Effects 4.3. Particle Diffusion by Molecular Agitation 4.4. Influence of Particles on Carrier Gas Properties ‘ RESULTS 5.1. Flow Visualization 5.2. Data Base for the Particle Experiments 5.3. Ensemble Average Distributions 5.4. Measures of the Instantaneous Scans DISCUSSION OF RESULTS 6.1. 6.2. 6 3. The Ensemble Average Concentration Distributions Measures of the Instantaneous Scans Inference of the Lagrangian Scales in the Present Experiment 6.3.1. A Priori Estimation of the Lagrangian Microscale 6.3.2. Evaluation of Parameters which Describe the Turbulent Dispersion iv Page 35 35 38 39 41 45 45 47 48 49 50 6.3.2.1. A Posteriori Evaluation of the Microscale and At 6.3.2.2. A Posteriori EvalEation of the Macroscale 7. SUMMARY AND CONCLUSIONS REFERENCES APPENDIX A: GAS HANDLING SYSTEM APPENDIX B: THE DESIGN OF THE GRID/SPLITTER PLATE ACTUATION SYSTEM APPENDIX C: REPRESENTATIVE UNSTABLE, NEUTRALLY STABLE AND STABLE, NON-DIFFUSIVE MIXING FIELDS AS REVEALED BY THE PRESENCE($). ABSENCE ( ) OF THE MARKING CONTAMINANT IN THE SCATTERING VOLUME APPENDIX D: DISPERSION IN A DECAYING TURBULENCE FIELD APPENDIX E: A REVIEW AND REEVALUATION OF SHLIEN"S EXPERIMENTAL RESULTS IN HOMOGENEOUS DECAYING TURBULENCE Page 51 54 55 60 101 108 115 120 128 TABLES LIST OF TABLES Page Table I. Slit and Scattering Volume Dimensions from Selected Locations 64 Table II. Evaluation of Dispersion Parameters 52 vi Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 10. 11. 12. List of Figures Figure Schematic Representation of the Mixing Chamber Detail of Turbulence Generating Grid Schematic Representation of the Acceleration/Deceleration System Velocity of Grid as a Function of Time Schematic Representation of the Optical and Data Acquisition Systems Detailed Representation of Disc and Slits Representative cross-section of Light Transmission Slit in the Rotating Disc Particle Generator suitable for use with air Electron Microscope Photograph of Soy Bean Oil Droplets Particle Generator to create 0.5 u crystals of Acetic Acid Electron Microscope Photograph of Acetic Acid Particles Ink Tracings of Oscillosc0pe Photographs Showing: a) an instantaneous scan of the photomultiplier tube voltage (Epmt) for the second scan following the passage of the grid b) a 3-second time exposure of the scope traces for the fully mixed condition of large time. vii Page Number 65 66 66 67 68 69 70 71 72 73 74 75 List of Figures Page 2 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 13. 14. 15. 16a 16b 17. 18. 19. 20. 21. 22. 23. 24. Figure Page Number Data Acquisition System Circuit Diagram for the Comparator Circuit Representation of the non-diffusing mixing field Representative length and velocity scales for homogeneous turbulence taken from Wyatt (1955) Schematic representation of the quantities used in the Lagrangian dispersion analysis Amplitude response of spherical particles to an oscillatory air flow from equation 4.1.7. Close-up of the initial mixing process to show the periodic structures developed from the instability of the vortex sheet Schematic representation of the parameters associated with the initial mixing process Close-up of the mixing phenomena Large view of the mixing phenomena Representative Distribu- tions for the Gas Mixing Experiments Presented on Standard and Probability Coordinates Comparison of the measure- ments for the Freon 12 and air data sets UOt/M = 12 Sample Least Squares Fits, to the Ensemble Average Data for the Freon 12 and Air Carrier Gases, on Probability Coordinates viii 76 77 78 79 80 81 82 83 84 85 86 87 88 List of Figures Page 3 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 25. 26. 27a 27b 28. 29. 30a 30b 31. 32. 33. 34. Figure Probability Coordinates Repre- sentation of Ensemble Average Concentration for Freon Probability Coordinates Repre- sentation of Ensemble Average Concentration for Air Width Measure (0 ) of the m . ensemble average concentration distribution - short time results Width Measure (O ) of the ensemble m C I O 0 average concentration distribution Long time results 2 Values of Physical Displacement from Centerline Corresponding to = 0.15 and 0.85 Definitions of Instantaneous Scan Measures Measures of the Mixing Region Width as Determined from the Conventional and the Instantaneous Scans . . . Freon 12 Carrier Gas Measures of the Mixing Region Width as Determined from the Conventional and the Instantaneous Scans . . . Freon 12 Carrier Gas Evaluation of the Lagrangran Microscale (T*) and the Effective Release time (tr)' Long Time Resultfi for the Disper- sion Parameter zp Theoretical values of 2&2 (At/At :I ,n) r * Theoretical values of z 2 (At/Atr3A ,Atj,Atr,n) ix Page Number 89 90 91 92 93 94 95 96 97 98 99 100 List of Figures Page 4 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure A.l.b A.l.c A.l.d A.2. Figures Page Number Schematic of Freon Gas Handling System Subassembly I Subassembly II Subassembly III Thermodynamic States for the Freon 12 from the heated storage tank to to the gas mixing Chamber Working sketch for the analysis of the acceleration/decelleration mechanism Final design of decellerating mechanism Representation of an unstable, non- diffusive mixing field Representation of a neutrally stable, non-diffusive mixing field Representation of a stable, non- diffusive mixing field Representation of long time, non- diffusive neutral density mixing Schematic representation of the quantities used in the lagrangian dispersion analyses Comparision of theory and experiment for small At/Atr values *2 d 8 2 d At/Atr as a function of Two time scale theory compared with the data of Shlien [1972] 103 104 105 106 107 113 114 116 117 118 119 123 137 138 139 NOMENCLATURE radius of particle angstrom 10-10 area of exit discharge nozzle experimental constant in energy decay law, see 3.1.7 area of piston cross sectional area of grid constant Fourier transform of F(x,y,z) complex conjugate of B amplitude spectra of <72> concentration of scalar contaminant constant 7.14 cm/radian constant defined in (6.3.1) drag coefficient strength of irrotational vortex see (3.2.24) cA/(l-n/z) see 3.2.5 Cf/(l-n/Z) diameter of biplanar grid elements minimum diameter of focused laser beam diameter of laser beam at (l/ez) intensity level molecular diffusivity turbulent diffusivity particle diffusivity focal length of lens xi drag force of grid Fourier transform of u(x,y,z) gravitational constant lateral dimension of chamber distance between concentrated vorticity centers intensity of segregation indicial notation Boltzman constant wavenumber variable length of cylinder (hydraulic) scale of segregation length scale for the dispersion analysis, see (3.2.21) mass of particle total mass which is attached to piston rod mesh length of grid m = (l-n/Z), m is used in the dispersion analysis number of experiments exponent of decay law, see (3.1.7) pressure (static) pressure acting on piston 1/2 turbulent energy intensity (62+62+&2) radial coordinate inner radius of helical slit 6.99 cm eddy of KolmOgoroff size Lagrangian correlation coefficient, see (3.2.19) Reynolds number (UoM/v) Schmidt Number (v/D) Strouhal number xii J. (x.y.z) time variable time increment equal to the transform, the microscale TL apparent origin of kinetic energy decay (3.2.24), of t-to from the transformation of the juncture point, see (3.2.25) temperature (absolute) total elapsed time for passage of grid through chamber see (3.2.21), release time of the contaminant transfer function velocity in x direction component of velocity (cylindrical coordinates) speed of grid velocity in y direction radial component of velocity (cylindrical coordinates) velocity in z direction r.m.s. of the 2 component velocity fluctuation total work done on fluid by action of grid mean square displacement of a fluid particle cartesian coordinates, x in direction of U0, 2 in direction of mean gradient of r length of focused laser beam in which do is constant centerline of mixing chamber normalized root mean square of the particle displacement z 2/L2 = 2*2 P P center of ensemble average distribution mean square displacement of particle in the z direction see figure 29 for definition of special symbols xiii ) GREEK SYMBOLS 3V2 /a dummy variable of integration 39/(29p+p) concentration of non diffusive contaminant fluctuating value of P; (r—.) incremental value boundary layer thickness dissipation of turbulent energy frequency (wavenumber) radians (angular) unit vector in 6 direction wavelength of disturbance Taylor microscale (9: spatial, t: wavelength of laser radiation Lagrangian macroscale A/C2 absolute viscosity kinematic viscosity density of fluid density of particle time) standard deviation of the spatial distribution for (a Gaussian distribution for is assumed, see section 5), see 3.2.12 see 5.4.1 see 5.4.2 stretched time coordinate, r = C 1n At/Atr,see (3.2.24,5) Lagrangian microscale in the stationary coordinates TL / c1 Lagrangian velocity xiv 1 . INTRODUCTION The efficient mixing of one fluid, or a contaminant sub- stance, with the host fluid is an intrinsic feature of a tur- bulent flow. The term "mixing" is often used to describe the net effects of turbulent dispersion, in which the eddying motions provide a convective transport of the material to be mixed, and molecular diffusion, in which the concentration of the material in a given fluid element is Changed by the trans- port associated with the agitation at the molecular level. As noted by Brodkey [1975] the interpretation of the degree of mixing is related to the scale of the observation, i.e., to the size of the volume used to define the concentration. Two measures of mixing are commonly employed, Brodkey [1975], the intensity of segregation Is and the scale of segregation Ls' For a binary mixture, where C is the instantaneous concentration (54c) of the contaminant, the intensity of the segretation (IS) is given as C(1-E) (1.1) and the scale of the segregation (L5) is given as L8 = f c(§)c(§+§)dr (1.2) The turbulent motions play a dual role in the mixing pro— cess. They are responsible for the convective transport of the material to be mixed and they can greatly enhance the molecular diffusion by straining the concentration field and hence steepening the spatial gradient of the transported material. This dual role, and the influence of the initial events on the subsequent dis- tributions, result in turbulent mixing phenomena which are of fundamental interest. The present investigation was conceived and executed as the first stage of a research program to examine these basic phenomena of turbulent mixing. The total prOgram is to involve measurements of the velocity, the molecular species concentration 2 and the presence or absence of a non-diffusive contaminant in a simple mixing field which may be characterized as a decaying, homogeneous turbulence field. The initial condition for the mixing is similarly simple; the fluids to be mixed are segregated in two distinct volumes until the initiation of the mixing process. The series of experiments in the total program are to reveal the influence of the control variables: different states of the initial turbulence kinetic energy and scales, and ii) different densities of the gases to be mixed. The data acquisition techniques, discussed below, are to provide quite detailed representations of the mixing process. The present study is solely concerned with the development and application of the non-diffusive contaminant experimental technique. The contaminant is introduced into one-half of the mixing chamber and is convectively mixed with the other half- volume following the initiation of the experiment. Mechanical/ optical techniques are used to identify the presence/absence of the contaminant within small volumes along a segment of the mixing region. Since the half-volume of origin of the fluid element is known, the dispersive properties of the turbulence field may be inferred. The use of a closed mixing Chamber greatly simplifies the task of providing a sufficient volume of the marking contaminant. Hence, this distinctive experimental technique would be somewhat more difficult to utilize in con- ventional wind tunnel flow systems. The first analytical description of dispersion in a homogeneous, stationary turbulence field is that provided by Taylor (1921) in the Classic paper entitled "Diffusion by Continuous Movements," Townsend (1954) recognized that the straining action of the turbulence would influence the (molecular) diffusive as well as the dispersive properties of the field and suggested appropriate modifications to the Taylor analysis. Townsend's concepts were refined by Saffman (1961) and the full problem has been reconsidered most recently by Chevray (1977). 3 The physically realistic case of decaying homogeneous turbulence is, in contradistinction to the above studies, non-stationary. Batchelor and Townsend (1956) propose that an appropriate stretching of the time coordinate will cause the Lagrangian correlation function to be stationary. This hypothesis is a central element of the theoretical structure which is utilized to interpret the experimental results. The objective of these theoretical considerations is, like the prior efforts of Uberoi and Corrsin (1953) and Shlien and Corrsin (1975), to deduce Lagrangian scales from the Observed dispersion measure- ments. Tennekes [1975] has reexamined the data of Shlien and Corrsin and has derived an analytical relationship between the Eulerian and Lagrangian time microscales in decaying isotropic turbulence. Tennekes shows that the ratio of the Lagrangian microscale to the Eulerian time microscale is proportional to RA%; (RX = q A/V). Quantitative reevaluations of the Shlien and Corrsin and the Tennekes results are presented in Appendix E. An important difference between the present and former experiments is the diffusive nature of the contaminant used to mark the fluid; a line source of heat was used to thermally mark the fluid elements in the above two research efforts. A second major difference is associated with the time which is available for the dispersion. The temperature "tagging" experiments were initiated following the establishment of a homogeneous and approximately isotropic turbulence state. Consequently, a relatively limited extent of the stretched time coordinate was available for the contaminant dispersion. Conversely, the present experiment initiates the dispersion in the initially inhomogeneous field and thereby gains a substantial period of development in the stretched time coordinate. A third feature of the present experiment is that the primary data are derived from essentially instantaneous scans through the mixing field with the ensemble statistics being generated from a number of experiments. Consequently, the data base is characterized by considerable detail in terms of a given scan through the mixing region and a relatively limited number of realizations 4 with which to form any particular ensemble average. The former feature is used to evaluate the statistical features of the instantaneous scans; these interpretations are not possible with the classical wind-tunnel, single-probe techniques. The above matters are made explicit in the following sections. Section 2 presents the Experimental Facility and Techniques, including the mixing chamber, the optical data acquisition system, the contaminant generator and the signal processing units. Analytical descriptions of the various aspects of this investigation are presented in Section 3. The dynamics of small particles and the analytical considerations which support the interpretation of the present particles as being non-diffusive are presented in Section 4. Section 5 presents the results which can be inferred from the binary (presence/absence of contaminant) data from each observation volume along the sampled segment. The relationship between the theoretical considerations and the experimental results is presented in Section 6. 2. EXPERIMENTAL FACILITY AND TECHNIQUES 2.1. Introduction A turbulent motion in the absence of a mean velocity was generated by passing a bi-planar grid through an enclosed volume. Light scattering techniques were used to sense the presence or absence of the submicron particles which served as a marking agent for the gas originally contained in the lower half volume, subsequently referred to as gas B. A rapid scan of the mixing region was made possible by an optical-mechanical system. Light scattering intensity measurements were sequentially recorded from short (0.36mm), small diameter (0.2 mm) portions of a 74 mm segment of the incident laser beam. The physical apparatus to create the mixing field and to execute the measurements is de- scribed in detail in the following sub-sections. It should be noted that the rapid scan represents a unique experimental technique. The gas handling system that was required for intro- ducing superheated Freon 12 and 22 into the mixing chamber is described in Appendix A. 2.2. Details of the Experimental Facility A schematic representation of the mixing chamber is shown in Figure l. The mixing chamber was constructed of 1.25 cm plate stock of dimensions (0.46x0.46x0.61)m; the large I-beams provided a rigid support for the apparatus. The incident light ports were covered with 17.8 cm diameter coated glass discs; the viewing port was a 12.7 cm hole which accommodated the collection lens. A 0.159-cm thick divider plate was located at the midsection of the chamber and separated the two gas volumes (A and B) prior to the initiation of the experiment. The plate extended through the end wall of the chamber which was fitted with four Teflon guide surfaces to inhibit the flexing of the plate during ac- celeration and deceleration. The grid/splitter plate system was supported on four linear ball bearings which provided a smooth but rigid transport of the plate and grid to the forward section of the chamber at the initiation of the experiment. The square mesh, biplanar, round rod grid was constructed of aluminum 6 tubing pinned at the bar cross-sections; it was permanently fixed at the end of the plate. The diameter and mesh size were selected to provide a solidity ratio of 0.34; the tubing diameter was 0.476 cm with a mesh width of 2.54 cm. Figure 2 provides a photograph of the opened mixing chamber. The splitter plate has been removed; the grid and the ends of the four guide ways are visible. A schematic representation of the acceleration/deceleration system is shown in Figure 3. The quick acting solenoid valve allowed the rapid pressurization of the pneumatic side of the cylinder. When the unit is Operated, the piston and the attached grid and splitter plate accelerated until the back pressure in the hydraulic fluid was sufficiently large to create a zero net force on the piston. The deceleration was accomplished by the gradual seating of a tapered plug attached to the end of the activating shaft. This facility provided a relatively uniform acceleration from 0 to 6.22 m/sec in 8.9 cm, a "constant" speed through the chamber, and a deceleration over the final 2.5 cm of travel. This motion was measured by attaching a hot wire (d = 20um) on the moving assembly. A linearizer was used and an exponent was chosen to match the upper velocity range with the "best" linear response. A plot of the velocity versus time is shown in Figure 4. See Appendix B for the detailed design and the computer simulation considerations of the acceleration/ deceleration system. 2.3. Optical System A schematic representation of the optical system is pre- sented in Figure 5. A Coherent Radiation CR-S Argon Ion laser was used as the light source. The laser was tuned for the 4880 A line with an output power level of 0.4 watts. The 2-mm diameter laser beam was focused with a 31.75-cm focal length spherical lens and directed by front surface mirrors from the top to the bottom of the chamber through two transparent windows. The collection optics consisted of two aerial photography camera lenses (focal number of 4.0, focal length of 30.5 cm) which were placed in close proximity in order to achieve a the len tryi; disk. linea attem by p1. slit ‘ edQES taken a gamp from t 7 1.36 magnification of the original image. This represented the maximum magnification and consequently the minimum scan length with the available optics. These conditions were necessary since the smallest scales of the turbulent motion were estimated to be 0.1 mm. The resulting scan in the mixing chamber covered 7.38 cm. The first lens was placed flush with the inside surface of the 12.7-cm hole in the side of the chamber. This eliminated the need for a finite thickness window which would have caused spherical aberrations of the scattered light. The second lens was coaxial with the first and was moved axially such that the image of the incident laser beam could be focused onto the face of a rotating disk. The high strength aluminum disk served as an optical-mechanical gate by only allowing the transmission of light from a scattering volume defined by the intersection of the recreated beam and a slit cut in the face of the disk. The slits were machined following the contour r = r0 + csei where cS = 7.14 cm/radian, r0 = 6.99 cm, 0 i 6‘: 800 and 6i = 0 + % (i-l). The disk was rotated at constant speed (4286 rpm) by a 1/2 h.p. motor. Since dG/dt = constant, dr/dt is also constant and the optical-mechanical gate provides a constant rate scan of the recreated laser line over the radial span ro :_r §.ro + 10 cm. The scan is centered at the midplane of the mixing field. Each scan was completed in 3.11 msec; the scans were repeated each 3.5 msec. Figure 6 presents a detailed representation of the disc and slits. Some difficulty was encountered when trying to cut a slit width of 0.025 cm on the face of the disk. The machining operation left a groove which varied linearly from 0.06 cm to 0.025 cm. Following numerous attempts to rectify this situation, a solution was achieved .by placing Mylar tape over the machined groove and cutting a slit with two razor blades displaced by 0.025 cm. One of the edges on the disk was used as a guide. Microphotographs were 'taken of the slit variation after the data had been accumulated; a: sample is shown in Figure 7. The slit widths determined :from these photographs are presented in Table l. The height of 8 the scattering volume is a function of the magnification ratio (1.36), the diameter of the laser beam and the intersection angle between the slit and the recreated laser beam. Table 1 also presents the scattering volume heights for representative locations. The diameter of the scattering volume was established by the focusing lens. Johnson [1971] provides an analytical description of the minimum diameter, do’ and the length, Azo, over which the diameter is maintained. The minimum diameter dO is located at A20 from either end of the "constant" diameter region. 2 A f -4 L .~. dO — fi-—a— ’— 0.1 mm (2.3.1) L 0 where f = focal length of lens (31.75 cm), ALaser = 4880A, dL = 2 mm. Az=4(£—)2A 25cm (232) o dL L ° ' The total scan length was 7.38 cm and 205 data points were ac- quired along the scan. The distance between centers of the sample volumes was slightly greater than the nominal length of the volumes, namely 0.36 versus 0.3 mm; hence no known spatial overlapping between samples was incurred. The alignment procedure was quite critical and great care was taken to align the laser beam parallel to the disk and then to focus a point source onto the face of the rotating disk. The latter was accomplished by adjusting the focal length of the second lens until the pulse width from the photomultiplier tube was a minimum. This alignment procedure allowed less than 0.1 degrees alignment difference between the incident and focused laser beams. 2.4. Particle Generators Two particle generators were developed in the course of this investigation. The first, which was quite satisfactory for use with (at least) air was not suitable for use with Freon. Both generators will be described in this section. The generator was required to produce approximately 6 . . . . . 10 particles/cc of carrier gas; this concentration is necessary to ac Furtk The 1 show: duce< catO: crea‘ were drOp. and high resu diam Unfo evap isti was it i USe Subs Part_ a lic When a Cry tiCle ProVi faCe. [1892‘ *ACet; 9 to achieve the nominal value of 20 particles/scattering volume. Further requirements include: 1) particle sizes: 0.5: = 0.3 microns (um) ii) non-toxic particles iii) method of generation should be independent of carrier gas iv) particle dimensions should be stable over the lifetime of the experiment. The first generator satisfied requirements (1) and (ii); it is shown in Figure 8. The particles were formed from soybean oil which was intro- duced as droplets into the stream of carrier gas. An oil lubri- cator, previously used in a compressed air system, was used to create the oil drops. The particle droplets for the experiment were created by the successive impacts of the initially (large) drops on the pressed wire screens (of diminishing open area) and the subsequent straining of the liquid by the shear of the higher speed gas. An electron microscope photograph of the resulting droplets (see Figure 9) indicated that droplet diameters of 0.2 — 0.4 microns were achieved with this generator. Unfortunately, the droplets so produced in Freon appeared to evaporate (or at least to Change their surface charge character- istics). NO particle scattering was observed when this scheme was used with Freon as the carrier gas. The second particle generator satisfied all four criteria; it is shown schematically in Figure 10. This generator makes use of the Rayleigh instability of a capillary jet and the subsequent formation of droplets to create the desired marking ‘particles. The stability of the particle was ensured by using a liquid solution of 10 percent acetic acid* in distilled water. Iflhen the liquid water evaporated following the droplet formation a crystalline residue was left to serve as the scattering par- ticle. It was inferred that the 0.45-um micro porous filter Iprovided a capillary jet of the same diameter at its downstream face. A capillary jet is unstable; the analysis of Lord Rayleigh [1892] was used to infer that 0.9-um droplets were formed. An *Acetic acid: CH COOH, m.w. 60.03 3 10 unambiguous measure of the particle sizes was not obtained. An electron microscope* photograph Of the particles which settled from the mixing chamber is shown in Figure 11. A long time (24 hr.) was required for the settling and some agglomeration of the initially distinct particles is suspected. This tech- nique is vulnerable to oversize droplets which impinge upon the upstream face of the filter; the apparent difficulty is the merging of the capillary jets and hence the formation of droplets of an excessive size. The generator system appeared to operate the same for the different carrier gasses....Freon 12 and 22 and Air. An improvement in the efficiency of the system would be desirable since an asymptotic state is reached when the particle contaminated gas is recycled through the compressor and particle generator. 2.5. Signal Processing The scattered light, which is collected and focused onto the photomultiplier tube by the optical system, results in a voltage signal which can be used to signify the presence or absence of the marking particles. The Characteristics of the photomultiplier tube and the signal processing electronics are described in this section. An RCA 4518 photomultiplier tube was used for the experiments. This tube has a 20 cm2 photosensitive area and a quantum efficiency of 18 percent. The 10K ohm load resistor and the associated capacitance of the output circuit resulted in a S-usec time con- stant. The tube was Operated with 900 volts across the dynode Chain; the corresponding cathode sensitivity and current ampli- fication were 0.06 amps/watt and 8 x 103 respectively. The particulate seeding was nominally controlled to pro- vide 20 particles per scattering volume for the marked fluid. With this seeding density and for an incident laser power of 0.4 watts, the optical/electronics system delivered a current of approximately one micro amp. This current level corresponds to approximately 104 photoelectrons for each S-usec sample period. This relatively large number guarantees that the photomultiplier *A 1% solution of Phototungstic acid (PTA) was added to the distilled water to facilitate the preparation of the slide. mm by ' tat 10 par noi eff sid hen val par the Sig vol mat tio ave the cri vol the ple 20 . tiO. dis and the ll tube and associated circuitry will not be adversely influenced by the dark current and that a statistically reliable represen- tation of the particles' presence can be expected. The photomultiplier tube signal was amplified (PAR 101) by 102 and input to an electronic comparator circuit. The com- parator level of 0.06 volts was sufficiently above the "system noise/Rayleigh (molecular) Scattering/dark current" level of 0.02 volts such that no spurious readings were attributed to these effects. Voltage levels of greater than 0.06 volts were con- sidered to represent the presence of particulate matter and hence the presence of gas B in the scattering volume. A minimum value of approximately 0.2 volts was observed for the region of particulate matter when the experiment was initiated; hence, the signal-to-noise level was approximately 10 and the minimum signal level exceeded the threshold by approximately 3. The voltage amplitude associated with the presence of the particulate matter did not exhibit a unique value; the amplitude fluctua- tions were inferred to be approximately :50 percent of the average amplitude for those scattering volumes which contained the marked fluid. Consequently, it was not possible to set a criterion for the discrimination of a partially filled scattering volume. There are two principal factors which contribute to the amplitude fluctuation when the scattering volume is com- pletely filled with marked fluid: (i) the nominal seeding of 20 particles per scattering volume results in a standard devia- tion of i4.5 particles or 123 percent (assuming a Poisson (iistribution for the number of particles in a scattering volume) .and (ii) the variation in the scattering volume dimensions over 'the four scans was approximately :25 percent. Figure 12 presents an oscilloscope trace of the photo- Inultiplier tube output immediately following the passage of tflne grid and splitter plate. The relevant voltage levels are ruated on this trace. Figure 13 shows a block diagram of the ciata.processing circuitry which achieves the analOg to digital (nonversion and which synchronizes the data from the P.M. tube arui rotating disk. The comparator curcuit which was used to proce show] Sign pass scan perc the the 12 process the analog signal from the photomultiplier tube is shown in Figure 14. Each scan of the mixing field is initiated by the strobe signal from the phototransistor. The phototransistor output is activated by the incident HeNe (low power) laser beam passage through the slit. This scheme for initiating the data scan is estimated to provide repeatability of 0.05 mm (or 14 percent of one scattering volume) in the spatial locations for the separate traverses. Figure 15 shows the binary output of the mixing field for one experimental run. 3. ANALYTICAL AND BACKGROUND CONSIDERATIONS 3.1 Description of the Problem It is useful to consider the present study in terms of an idealized experiment and the differences between it and the actual experiments which were executed. The idealized experiment involves a homogeneous, isotropic turbulence field (of indefinite extent) at the initiation of the experiment (t=0). The initial concentration field is specified by the conditions F(x,y,z,0) l for z < 0 (3 1 1) 0 for z 3 0 1"(X.y.z,0) where F = 1 indicates the presence of the non-diffusive contaminant in the differential sized volume located at the (x,y,z) point. The subsequent F(x,y,z,t) distributions result from the convectiVe transport of the turbulence field. The actual experiment is rather different in certain of its detailed features. The passage of the splitter plate trailing edge provides an extremely rapid mixing of the upper and lower fluid regions over a scale several times the plate thickness (tplate = 1.56 mm). This provides an actual "initial condition" which is a well-mixed region for 17mm. Detailed considerations of the initial condition are presented in Section 5. The balance of the initial turbulence field is that which results from the passage of the biplanar grid. The x-component, ensemble mean velocity will be positive (i.e., the fluid motion will be toward the receding grid) in 'the wake regions of the grid and splitter plate. The velocity u .in the open regions will be negative. The closed mixing chamber :results in the integral constraint: 1 f u dy dz '= 0 (3.1.2) for all y and z in the interior of the chamber. The differential forms of the mass and momentum conservation enquations are instructive in the interpretation of the initial xrxgion behavior. Since the subject gases behave as incompressible, Newtonian fluids, these equations are written in their ensemble averaged form as Bu. — — 5;:- = o a“ 3" 3‘” (3.1.3) 14 and 8—. _. — —. . . ui + 5. ani = _ 3p + vaui _ afiifij (3.1.4) 3t 3 ij 03xi axiaxj ij The turbulent stresses of (3.1.4) will serve to accelerate (g; > 0) and to decelerate (g; < 0) the fluid behind the grid members and in the open regions respectively. The natural symmetry of the grid and plate leads to planes of zero V and w and the conservation of mass equation can be used to infer non-zero V and w values between these planes in response to the observed afi/Bx values. For example, for the y = 0 plane between two grid members (i.e., V = 0 for all z), w is zero at the mid-plane of a horizontal rod, it is negative in the region between the plane of the rod and the center plane, and it is zero at the midplane between the rods. The presence of this mean motion following the passage of the grid suggests that convective transport effects are present in the actual experiment that would not be present if the idealized initial condition of homogeneous tur- bulence were obtained. The physical extent of the inhomogeneous region can be estimated from the experimental results of Sato and Yamamoto [1974]. Their hot-wire traverses downstream of a biplanar, round rod grid of solidity 0.44 (vs. 0.34 for the present study) indicate that homogeneity is obtained in approxi- mately 3 mesh lengths. This corresponds to approximately the fourth scan for the present study. The turbulence kinetic energy equation with q§|= BEBE/2 2 ——- ___ 29, 8 — 2 = _ 3 2 _ Bu. at + ax‘. “j ‘1 W. 153“} +p/p) 6.6. 1 3 J 1 3 5xj as. as. as. Bu. an. 3 i j i i j + v axj ui(3xj + Bxi) - v §xj (ij + axi) (3'1'5) Inakes explicit that the turbulence kinetic energy is generated iJi this initial inhomogeneous region and decays (balance of first and last terms) as a result of dissipation effects thereafter. fiflne maximum magnitude of the turbulence kinetic energy can be 15 established in terms of the work done by the grid/splitter plate movement through the chamber. Let'CD be the drag coefficient for the physical structure used to create the turbulence field. The total work done is W = I E - 33 Since FD = CD p Vo2A + G and d3 = VO dt 2 W = pCDAG ftm3 2 0 VO dt here tm is the time required to move the assembly across the chamber. Since the grid movement is controlled by the hydraulic actuator and since this motion is independent of the drag force acting on the grid, the initial kinetic energy states for the air and Freon gases will be in the ratio of their CD values. Assuming that the CD value is set by the grid elements, the data of Friehe and Schwarz [1970] can be used to establish the ratio _ 4 —T CD [Re - 6 x 10 1 _ qavg (t=0, Freon) : 1 2 (3 1 6) _ 4 “—2 - ° — ° ' ' It is well known that in a parallel flow 32 is the turbulence velocity component which is first and predominantly energized by the destruction of the mean flow energy; see Corrsin [1957]. It is expected that this anisotropic condition is present in the current experiment. A very considerable data base is available for the flow behind biplanar grids. Comte-Bellot and Corrsin [1966] have summarized these results including those of Wyatt [1955] wahich are for a configuration very similar to that of the present study. Following the establishment of a homogeneous turbulence :Eield and prior to the final period of decay, the available experimental data suggest that a power law solution to (3.1.5) is appropriate. Comte-Bellot and Corrsin [1966] organize the ex- perimental results in terms of 2 O U t U t n (3.1.7) = A o o o (:7— {T'T} which for t data turbt pared origf and I [1961 for a rele‘ from well where 16 which includes the presence of an apparent origin to account for the initial period of inhomogeneous turbulence. Using the data of Wyatt [1955] a graphical representation of the pertinent turbulence quantities in the present experiment has been pre- pared, see Figure 16. Data for round rod, biplanar, grids show that the apparent origin (to) shifts to smaller values as the velocity is increased and mesh size is kept constant; see Comte-Bellot and Corrsin [1966]. Wyatt's data suggests that Uoto decreases from 5 to 4 M for a corresponding RM range of 104 to 6x104. These are the relevant RM for the present experiment. However, it is clear, from other data reported by Comte-Bellot and Corrsin, that well defined values of Uoto do not exist. M In their study of the decay relations behind a biplanar grid, Comte-Bellot and Corrsin have reported that there was no apparent change in the decay laws for g = 9 and 3.: 18, where H represents the lateral dimension of the wind tunnel. = 18. Uzkan SHE The corresponding value for this experiment is and Reynolds [1967] have reported an experiment in which the decaying turbulence field was bounded (on one side) by a wall ‘which.moved at the free stream velocity U0. The contamination of the interior turbulence by the wall effects is seen to pro- ceed at a reduced rate for this condition. Their data show 'that it would require 54 seconds for the boundary influence ‘to migrate 5 cm into the free stream at RM = 104. Since times (of less than 0.8 seconds are of interest in the present ex- gxeriment, it is inferred that there is no influence of the txoundaries on the measurements reported herein. Bradshaw (1977) lmas reported that experiments of this type are sensitive to the Iueynolds number; however, the influence of the stationary wall Vlill not influence either the Freon or the air results. :3.2. Scalar Concentration Field 13.2.1. The Ensemble Average Concentration Distribution The transport equation for a scaler contaminant, C, is 9.9+___8.u c=D—8C (3.2.1) winere C is presumed to follow a Fickian law of diffusion. For the cond expe the Sim the [3101 mea to : tur tio Wit Net Ens 17 the present discussion, it is useful to consider the idealized condition in which D 2 0; this is closely approximated in the 'experiment as shown by the analyses in Section 4. For the condition that D = 0, the mean concentration of the nondiffusive contaminant P(i.e., fris given by a _ _ a _ a at - axj axj (3.2.2) Since E 0, except for the initial period of inhomogeneity, the expression for this restricted condition is of interest. a _ _ a _ a a 3t ‘ ax. <fijY> - 3x. {DT ij } (3.2.3) 3 J In 3.2.3 the turbulent transport term is written as a product of the turbulent diffusivity and the gradient of the mean concentration field. This is based on an ad hoc analogy to molecular transport modeling, Corrsin [1974]. For the problem under consideration, namely, homogeneous turbulence, DT can be assumed to be independent of the direc- tion of diffusion; consequently, (3.2.3) can be written as 2 = DT(t) 3 2r) (3.2.4.a) 32 3

3t ‘with boundary conditions 0 , 2:0 (3.2.4.b) l , z>0 The general solution of this equation in a semi-infinite :region (-w = t f exp t 7% F(z'0)dz' {4H J D (t')dt'} ’” {4 f D (t)dt'} 0 '1' O ’1’ (3.2.5) where I‘(z'0) -is the initial concentration distribution. f bnote that both (-) and <( )> will be used to designate an ensemble average of ( ). 18 The same result for can be Obtained by an inde- pendent analysis which is due to Batchelor [1956]. This alternative development is considered in order to clarify the assumptions and restrictions which are used in each analysis. For the conditions of homogeneous turbulence, without mean velocity gradients, it is reasonable to assume that the three-dimensional particle displacements follow a Gaussian distribution with standard deviations (xp, yp, zp). The probability density function of the particle displacements is therefore l 1 (x-x')2 Q‘X'Y'z't’ = (21% 2 2 2 35 exp ' ‘2' {——2— + (x 4 y + Z ) X P P P P 0 2 '2 (2'! ) + (z-z )} (3.2.6) y 2 z 2 P P For a scalar contaminant which indelibly marks the fluid element and for one whose presence is identified by P = l (and absence by F = 0), the ensemble average concentration distribution of the contaminant is directly related to the displacement of the original particle distribution as = III Q(X-X':Y-Y':Z-Z':t) F(X':Y':Z':0) dx'dy'dz' —m (3.2.7) where I(x',y',z',0) is the initial concentration distribution. For the experiment under consideration, the initial distribu- tion of the contaminant occupies the volume (x, y, z>0); therefore, = and °° l 1 (z-z')2 =f ——1 exp - —{—2——}I‘(z',0) dz' (3.2.8) -w (anpz) 2 2p This result suggests that the mean square displacement <3f a fluid particle, which represents the dispersive property 19 of the turbulence field, governs the behavior of the ensemble average concentration distribution. The mean square displace- ment can be developed using Lagrangian considerations; this »development is presented in Appendix D. The above equation is similar to (3.2.5), where by iden- tity, t z 2 = 2 I D (t')dt' (3.2.9.a) P o T or 2 D (t) = .1; d_z_P__ (3 2 9 b) T 2 dt ' ° ' This result was observed by Hinze [1975]. Hence the turbulent diffusivity can be calculated from the time rate of change of the particle mean square displacement. The mean concentration equation, (3.2.8), can be written in terms of the error function by defining new independent variables; see Ozisik [1968]. Let A 3 = Ygfilgig- and dz' = (2 zp2)3 do (3.2.10) 2 Utilizing the boundary conditions from (3.2.4.a), the follow- ing expression is obtained for the ensemble average distribu— tion; viz., ‘ = %-(1 + erf(z)) (3.2.11)+ For the experimental data under consideration a Gaussian curmulative distribution function which is identical to (3.2.8) (will be used to describe F(z,t), namely, 2 2 1 z (a-) A = J exp - ——————7—- da (3.2.12) (2H)gom o 2'om The separate symbols: cm and 2p, will be maintained to +The error function of (3.2.11) is defined as 2 erf(Z) = T%' f exp(-82) d8 n o 20 respectively represent the best fit of a Gaussian C.D.F.+ to the experimental data for and the standard deviation of the particle displacements in (the assumed Gaussian dis- tribution of) the flow field. The rate of transport of the marking contaminant across the midplane of the mixing chamber can be expressed in terms of the variables used in the above considerations. The ensemble average flux term of the contaminant across the midplane is z=o and, by using (3.2.3), (3.2.5) with the P(z',o) given by (3.2.4.b), and (3.2.9.b), the expression for z=o becomes d z (w) _ = _1_,_ —E (3.2.13) The total flux of during the elapsed time period oitfit may be expressed as the integral of (3.2.13); viz., t . _ 1 £z=odt - m zp (3.2.14) 3.2.2. The Ensemble Average Fluctuation Intensity Distribution 0 O 2 O The mean square concentration fluctuation, , equation is 2 2 3 2 _ _ 3 _ 3 2 _ 3C BC _ a c T; (C >' “13°93? 33?].— ‘C “3" D ‘37:; 3323." D (W) (3.2.15) where c = C - C'in which the turbulent effects serve to pro- duce and convectively transport the concentration fluctuations (i.e., the first two terms on the right side of (3.2.15)) and the molecular effects destroy and diffuse the by the action of the second two terms. In the absence of diffusion, i.e., for D = 0, a balance is possible if the production and transport are in balance. This condition occurs at the mid- plane Of the mixing region as shown by the following analysis. If D = 0, then the probability density function for P, Pr(a, t), can be represented by two Dirac functions at a = 0 + Cumulative Distribution Function We is Sen ‘1 (z with Ta“ tran 21 and a = 1 such that the mean value of F is obtained as 1 = i Pr (a, t) a d a = Pr(l, t) (3.2.16) 0 . The mean square fluctuation is l = [ (a-)2 PP (a,t) d a O = 2 P (o t) - )1 -)2 P (1 t) r ' r ' (3.2.17) = (1-) 2 Hence, at the centerline, <7 > is constant for all time (since

= 0.5); its value is 0.25. Using the results of (3.2.17), <72> can be written as = 0.25 {1 - erf2 ———E———} (3.2.18) Measurements by Foss, Schlien and Corrsin [1972] in the mixing field downstream of a half-heated grid showed that the above estimate for the temperature fluctuation was 250% larger than the experimental data. Hence, the molecular diffusivity plays an important role in such flows even though the Gaussian C.D.F. of equation (3.2.8) was closely approximated by the experimental data for the mean temperature distribution. Analytical descriptions of the fluctuation intensity for the diffusive (condition have been developed by Libby [1975] and the above investigators. 3.2.3. Dispersion Considerations The P(z,t) distributions which have been recorded in the Ixresent work represent the effects of turbulent transport. It .is interesting to note that the non-diffusive nature of the sensed contaminant results in a direct correspondence between 1 and the turbulent transport. This is in contrast vwith studies which use thermal tracers as the marking element. {Baylor [1935] was the first to analytically describe such izransport by the application of Lagrangian concepts. Using 22 vi as the velocity of a fluid dynamic particle, the Lagrangian auto correlation for stationary turbulence can be expressed as = (3.2.19) RL (v2) and the mean square displacement of a fluid element from its point of origin in a stationary, homogeneous turbulence field is 2 2 t t' ._ l — 2 ] dt 1 RL(a)da (3.2.20) 0 o for stationary turbulence, the limiting cases of short time: T+0,RL(a) = 1 and = 2 t2 and of long time: t+m, ” 2 _ 2 . f0 RL(a)da = A and — 2 At, are of interest. A turbulence field without mean shear will, of course, decay. If a tagged particle is released in such a field, its dispersion as a function of time will be influenced by the decay of the turbulence kinetic energy. A theoretical descrip- tion of this physically realistic condition would require that the Lagrangian correlation function would specifically account for the decaying turbulence kinetic energy. Falco [1972] made direct optical measurements of the velocity of small particles in the decaying turbulence field behind a biplanar grid in an enclosed water tank facility. From these measurements he obtained a (spherical) Lagrangian correlation at Uot = 89 and 156. It is apparent from these M :measurements that there is a temporal growth of the Lagrangian scales as Uot increases; however, due to the sparse nature M [of the data (8 points for the correlation) it would be dif- ficult to determine the quantitative features of this growth. Several authors, including Batchelor and Townsend [1956] and Shlien and Corrsin [1974], have formulated analyses to account for the decay of the turbulence energy and its relationship to the dispersion of a contaminant. A new version of such 23 an analysis is presented in Appendix D. The common feature of these analyses is the assumption of a self preserving state of the Lagrangian correlation function, RL(T), by the use of a stretched time coordinate, T.+ The equivalent results, for the mean square displacement, zpz, of a fluid particle for the short and long time special cases, are given below. The dispersion measure in the z direction of the present experi- ment, zpz, has been made nondimensional using the scale length L where P _ 2 _ _ 2 L — (tr to) . (3.2.21) 2 Atr tr is the "release time" for the tagged particle, to is the apparent temporal origin of the decaying turbulence and v is the Lagrangian velocity of the tagged particle. The short time result is, for L P . Atzu- 3) (1- n) 2*2 = ——2-——2 1% -(—T) 2%—)(( t) 2 +1)}+ (fl: 7 (1- g) 1- E “—17" 11'12(§:)-2(2'- (T) 2m At. At for IE? 5 IE: . (3.2.22) ‘Note, tj and I* are defined below; (3.2.22) is also equation D.2.3. The long time result is + This assumption was introduced by Batchelor and Townsend [1956]. 24 At. 22(fi—E— ) - {Zea-El) = r A* -1 A* 1 .2 m; (1-—)( -—,,—— —) ( ’2‘ .' ——2—2 [(Arf—)[(A——t:) A - (lg—t?) A 1+ (1- —) A 1 r %(—.—)[(At 2‘1 '2" — (f—lfi‘l‘ 2’11 A +1 Atr At At At. for -A—t—r- > FE: ° (3.2.23) The symbol n represents the decay exponent Of the turbulence kinetic energy; see equation (3.1.7). The variables I* and T A* are defined as 1* = EE-and A* = %—-where TL and A are the l 2 Lagrangian micro and macroscales in the stationary reference frame. Theoretical curves of 3.2.22 and 3.2.23 for various and A values are presented and discussed in Article 6. The constants C1 and C2 are related to the temporal trans- .formation in which the Lagrangian velocity correlation (3.2.19) is stationary; i.e., %E(RL(t,T)) = 0.” The temporal trans- formation which yields a stationary condition for RL(T) for ‘TL TTj is D At _.t_.= exp (.8...) (3.2.25) r f where CI = (l-%)C2. The symbols At, Atr and Atj are the times referenced to the temporal origin as At = t-to, ‘Atr'= tr-to, and Atj = tj-to. The symbol Tj represents the 25 .juncture point between the short and long time formulations. Figure 16b shows a graphical representation of these concepts. The following relationship defines the juncture point in the At ‘ AE_ reference frame: r n 2 1"'2’ n - b—-4 1- ‘2- 2 At . At . 11* 1 - (—;;) 1J1 (XE?) - (AB?) = 0 (3.2.26) The Lagrangian macroscale, A, is a property of the turbulence field and is clearly related to the dispersive character of a turbulent flow. Previous experimental efforts to determine the dispersion have made use of a passive scalar contaminant such as a low overheat thermal marker. In these studies, temperature tagging wires mark a spot or line of the fluid and downstream temperature measurements define the mean temperature as a function of the transverse distance. The value 2: is inferred from the latter. Such studies have been performed by Uberoi and Corrsin [1953], Townsend [1956], and Shlien and Corrsin [1974]. The last authors provide a well developed scheme for interpreting the influence of the de- caying non-stationary condition of the turbulence. The in- fluence of the non-zero diffusivity of the thermal energy in such experiments has been theoretically examined by Saffman [1961]. Saffman's analysis, which is limited to small times from the initiation of the mixing, provides a corrective interpretation to the earlier Townsend [1956] analysis which had.served a similar purpose for the original work of Taylor [1935]. Chevray [1977] has developed an alternate analytical structure which extends the prior results of Saffman. Quali- ‘tatively, the present investigation is as if the marking con- 'taminant were impressed upon a region, instead of a line or spot, of the host fluid. .3.2.4. Fourier Transform Considerations A useful description of the scalar concentration field can be gained by the use of Fourier transforms. Let the 26 function B(k) represent the amplitude of the wave which is related to the spatial distribution of P(§) as B(k) = (211)’3 [If r(x) exp(-i k°x) dx (3.2.27) -0 Similarly, the velocity component un can be transformed to a wave description as + ++ Fn(k) = (2n)’3 ff] un(x) exp(-ik°x) d x. (3.2.28) Operating on the conservation equation for P results in (see, eg. Yeh and Van Atta[l973]) lo; [§¢(k)] = -i kj [If Tn (k, k') dk' - D k2¢n(k) (3.2.29) 0) t where 0 = < BB* > Ak3 which represents the intensity of the concentration fluctuation at wave numbers ki 1 Aki/2. (B* is the complex conjugate of B.) The function Tn(k, k') represents the transfer of the fluctuation intensity to higher wave numbers; formally, it can be expressed as: Tn(k) = §i(Ak)3 ki (3.2.30) This transfer term is responsible for enriching or transfer- ring energy (¢(k)) to large values of k. Its effect is limited by the presence of the last term in (3.2.11) in which the Concentration fluctuations are "smoothed" by the diffusive action of the molecular diffusivity. For the idealized ex- periment involving D = 0, the striations of the mixed regions 'will become of finer and finer scale. Batchelor [1959] has developed a theoretical description of the scalar field for the condition of small conductivity (diffusivity). His work shows that the 0 (k) spectra falls off at the lesser slope of k-1 beyond the break where Fn(k) changes from a -5/3 to a region of more rapid decay. Gibson and Schwarz [1963], Nye and Brodkey [1967], and Grant & Nisbet [1957] have observed the k- jbehavior in : 1) salt concentration and temperature measure- 1 27 ments, ii) dye, and iii) temperature measurements reSpectively. Interestingly, there was no significant difference between the salt and heated water results in the Gibson and Schwarz experiment. 4. DYNAMIC BEHAVIOR OF THE MARKING PARTICLES The previous sections have identified the basic premise Of the experimental technique: that a marking contaminant may be found which faithfully moves with the same continuum fluid element from the initiation to the termination of the experiment.* The contaminant particle selected must be sufficiently large that it is not displaced from the continuum element by the random agitation of the molecular motion and it must be suffi- ciently small that it does not: i) follow an inertia dominated trajectory and thereby define a pathline which is different from that of the continuum element of its origin, ii) influence the density or stability properties of the fluid, or iii) fall away from the continuum element under the action of the gravi- tational force. The following subsections consider these effects based upon air as the host fluid for submicron particles (a = 0.250m) of specific gravity 1.0. 4.1. Aerodynamic Drag The equation which describes the motion of the fluid element and the corresponding response of the particle has been analyzed by Tchen [1947], Hinze [1975], Corrsin and Lumley [1956] and 800 [1967]. The following equation was extracted from Khosla and Lederman [1973]. du pi _ 30 [dui _ 2 V 32u 1 ' + — a .8 . dtp 20p 0 dt 3 x3 x3 Bu. 2 9U i [ (u--u )+ p(u -u )—1 p+pp 2a2 1 pi k pk axk (4.1 1) t d + 9 (L11? [P 37(ui-upi) d T (ZOP+D)a fl /Ef:—f; tr p (where up is the vector velocity of the particle, ui is the i ‘velocity of the gas, V the coefficient of kinematic viscosity, p the coefficient of viscosity, pp the density of the solid *Becker [1977] identified this technique as "marker nephelometry" 28 29 material and a is the radius of the particle. Tchen assumes that the particle radius is small and u F1 2 >> 1 (4.1.2) a2 3 ui F‘TF‘. x3 X: The most restrictive assumption made by Tchen is that during the motion of the particle, the same fluid element remains in its neighborhood. Thus he postulates 3%; 5 3E . (4.1.3) This assumption is required for a solution to be realized so that one can achieve an estimate of the phase lag as well as amplitude decrease. Since this equation is linear, a one- dimensional velocity field will be considered and the effects of self-diffusion and gravity can be Obtained by a separate analysis. The resulting expression for the particle motion is d du t (u - u ) EEE'= B 3% + 08 (u - u ) + 8 («i-3)”2 f E? 1_ p dT p tr t-T (4.1.4) = 22 3o where a a2 and B (20p + p) This equation was solved by Chao [1964] using Fourier transform techniques, which allows the time dependent character of the velocity field to be expressed as a frequency fluctuation (w). The Fourier transform of the velocity is no u(w) = f u(t) exp (-i wt) dt. (4.1.5) -m The solution for the particle velocity as a function of the fluctuation frequency is u (w) = {l +[1.5w/OL]é + i(w/a +[l.5w/a]§)} u(m) (4.1.5) 9 {1 +11.Sw/al* + 1(m/ae+(1.50/a1*)} 30 By multiplying the velocities up and u by their respective vcomplex conjugates and introducing the energy spectrum density function F(w), a relation is obtained between the intensity of the particle motion and that of the surrounding fluid. 2 2 I” 0(1) < > = F(w) dw' “p OT 2) 0(1) = (91)2 + ./'6"‘(3)3/2 + 3(9) + @793)”2 + 1 0‘ a 0‘ 0‘ (4.1.7) 82 a + B The ratio {/} can be used to infer the ability of the particle to move with an accelerating fluid element. This ratio is presented in Figure 17 as a function of the oscillation 9(2) = 1 (g)2 /E’ (%)3/2 + 3(3) + /g— (3)1/2 + 1 frequency w for the conditions: d = 0.25, 0.5, 1, 2pm), pp = lgm/cc, p = pair (STP)). One can infer from the above results that a 0.5 mm particle will follow the flow with a time scale of the order of one micro-second. Using the data of wyatt at x/leO, the Kolmogoroff time scale (V/e:)l/2 has a minimum value of 10-3sec. Hence, the .Sum particle is expected to follow the most rapid fluctuations to be found in the physical flow field. Further insight into the aerodynamic drag effects can be gained by considering the behavior of a particle in a swirling flow which simulates the motion of the small scale eddies. .Hirschkron and Echrich [1964] have analyzed the motion of a particle in an irrotational vortex (u6 = cV/r). The salient features of this analysis are presented in the following. A spherical particle is assumed and the analysis is re- :stricted to the low Reynolds number, "Stokes Flow" regime. The drag coefficient CD is given as 12 (4.1.8) pIIE-Ylla CD: where 3 is the fluid velocity, 3 is the particle velocity, and a firs the particle radius. The expression for the drag force .F is 31 P=§ch(E-$) - (E-$)na2 d; + + (4.1.9) mp <3; — 6nua [u - v] The fluid velocity field is assumed to be that of an irrotational vortex: {I = e CV/r (4.1.10) The acceleration of the particle can be written in terms of the spatial variations of the velocity field as vr dv - v8 = -Pvr dr r (4.1.11) dv v v C v 0 r 6 _ v _ r dr + r _ P[E_ v0] where P = (6hua/mp). These equations were solved numerically and the results were plotted in normalized form. The radial position (r/ro) of the particle as a function of time (t Cv/rez) is of interest for the small scales of the motion. Choosing an eddy of the Kolmogoroff size r = (03/e)1/4 = 0.123mm (4.1.12) 8 where e = 14.7 m2/sec3 at t = 0.04 sec for v = 1.5 x lO-SmZ/sec. The e value is evaluated from the data of wyatt [1955]. The kinetic energy of the dissipative eddies is approximated by q = (ve)1/2: consequently, a characteristic value of u6 is given by u6(r€)2 0.122 m/sec. The abovezvalue for ue(r€) allows Cv to be evaluated as: Cv = 15 mm /sec. Making use of the Hirschkron and Echrich solution for a 0.5-um particle in air, r/rE does not exceed 1.2 for an elapsed time of 1.0 second. The experimental times of interest are shorter than this; it is again inferred that the particle suitably follows the fluid motion. 32 4.2. Gravitational Effects The terminal velocity is calculated using the Stokes drag relationship giving 0 .— 3. _E 2 ' For a 0.5-um particle with specific gravity 1.0, VT = 0.05 mm/sec. The distance travelled by the particle in the elapsed time of the experiment is an order of magnitude smaller than the Kolmogoroff scales. 4.3. Particle Diffusion by Molecular Agitation The random molecular motion in the neighborhood of the particle will cause it to move away from the spatial volume of its origin. In the presence of a fluid velocity (i.e., a drift velocity of the molecules) this motion would be with respect to the convected spatial volume which is defined at the initiation of the experiment. The characteristics of the motion can be assessed by kinetic theory considerations as shown in the following. The root mean square velocity of a particle of mass m is given in terms of the Boltzman constant K and the absolute B temperature T as 1/2 3K T For a 0.5-um particle of specific gravity 1.0 and for 300°K, a = 4.6 mm/sec. This velocity is small (5 1 percent) with respect to the rms continuum fluid velocities of interest in the present experiment. ‘ The displacement of the fluid particle can be evaluated from the diffusion coefficient, Dp. It is defined by the ex- preSSion KBT D (4.3.2) p - 6nua iFor the conditions of a 0.5-um particle (a = 0.25 um) with 300°K air as the host fluid, 11 Dp = 6.7 x 10' mz/sec 33 In order to relate Dp to the displacement of the marking particles, 1consider a field of particles in which the particle concentration 'c is initially given as c(z,o) = Co for z i o C(z,o) = O for z > o and in which the particle concentration c follows the Fickian law of diffusion: 2 8c 8 c __,= D ___. (4.3.3) at p 322 The resulting concentration field for D = constant is given as g— = 1 - erf (4.3.4) 0 /4D t P A width measure for the migration of the particles can be specified as 5= f— .: p 4Dpt (435) For the conditions used above, 0p = 0.047 mm for an elapsed u t time of 1.0 sec (—%— Z 250). Hence the migration of the particles in one second is 0.134 times the scattering volume dimension. This gives credence to the assumption that the contaminant is "non-diffusive". 4.4. Influence of Particles on Carrier Gas Properties The experimental technique of marking a given spatial volume requires a substantial number (=20)* of marking particles. The influence of these particles on the carrier (or host) gas is considered in the following. For a concentration of 106, 0.5-um diameter particles/cc, the density increase with respect to the uncontaminated air is 0.1 percent. The influence on the viscosity can be assessed from the Einstein equation; (see Hirschfelder, Curtiss and Bird [1954]) u = u(l + 0) where 0 is the volume ratio. The *The particles may be assumed to follow a Poisson distribution as regards their distribution in the spatial volumes. Hence, thg standard deviation of the particle number per element is N where N is the average value. Consequently, N should be large to maintain an unambiguOUS scattering signal. 34 change in viscosity as a result of the additional particles is 10-4 percent. The influence of the particles on the hydrodynamic sta- bility of the host fluid has been considered by Saffman [1961]. His results can be used to infer that there is a negligible influence on the stability resulting from the presence of these 0.5-um diameter particles in air. 5. RESULTS 5.1 Flow Visualization The motion of the gas inside the mixing chamber was photo- graphically recorded; a significant feature of the initial motion was revealed by this procedure. Marking particles, with a concentration of 106 particles/cc, were initially intro- duced in the upper volume;+they were illuminated by a light sheet formed with a cylindrical and spherical lens system. The motion pictures were taken at 50 frames/sec; the exposure time was 150-1 sec. The first visible frame cOuld not be precisely correlated with the passage of the grid; it repre- sents an experimental time of 0-0.02 seconds. Representative frames are shown in Figure 18. The black/white (or marble cake) mixing character of the particle gas is apparent albeit some resolution is undoubtedly lost with the grain of the film and the relatively long exposure time. The first frame of Figure 18 is considered to be quite significant.’ Specifically, the "fuzzy" end of the marked particles in the 0-0.02 sec. time span was not anticipated; it suggests a rapid mixing in the neighborhood of z = O which precedes the large scale mixing induced by the grids. The source and character of this initial mixing process has been determined. The boundary layers which grow upon and which are shed from the splitter plate form a wake of positive and negative vortex sheets. In addition, the regular vortex structures shed from the horizontal cylinders of the biplane grid will create an organized perturbation of the vortex sheets. The antisymmetric part of the perturbation leads to an instability of the vortex sheets with a subsequent roll up or amalgamation of the vorticity into vortex clouds. Abernathy and Kronauer [1962] have computed the behavior of these vorticities. Their non- linear analysis shows that the vortex sheets are unstable to any wavelength disturbance and that the clouds may be comprised of opposite sense vorticity with a consequent reduction in the net strength of the vortex cloud. T The particles were introduced into the lower half volume for the quantitative experiments. 35 36 The spacing of the regular vortex structures shed from the horizontal rods can be determined from the data of Roshko [1954] and the conditions of the present experiment. The pertinent Reynolds number is Uod ~ Rn = “6‘ ~ 2000 (5.1.1) and, from Roshko, the corresponding Strouhal number is fAd ST = 6:)— = ‘" 0.2 (5.1.2) The wavelength of the disturbance can be calculated from fA as A = Uo/fA d/0.2 II 2.54 cm (5.1.3) From Schlicting (1968), the impulsively started boundary layer 0 grows as 5 = 4/UE (5.1.4) The elapsed time t is the time required for the grid to move one half of its total travel across the box; the resulting value for 6 is ”2.54 mm. The splitter plate is approximately 1.56 mm thick and assuming that the laminar boundary layer vorticity is concentrated at (*0/3), the h value is 23.26 mm. Hence h/1=0.13.’r Abernathy and Kronauer present the formation of the vortex clouds for h/A:0.12. As shown by the figure, the "mixing" is quite advanced for a unit value of the dimensionless time tU'/A, where U' is the velocity parameter introduced by Abernathy and Kronauer. (U'= plate velocity/2.) Consequently, the time to achieve significant mixing (or dispersion of the two gases in the neighborhood of y = 0) is given by ~ t = A/U' ~ 0.007 sec. (5.1.5) Figure 19 shows a schematic representation of the parameters in the initial mixing field. 37 Significantly, this elapsed time is quite small with re- spect to the time scales which characterize the subsequent mixing; specifically, the minimum Kolmogorov time in the homogeneous turbulence field is ” 5 msec. Also, the 7 msec to establish the initial mixing layer is comparable with the elapsed time (3.5 msec) between two successive scans. Conse- quently, the experimental technique will adequately identify the structure of this mixed region once it is formed, but it will not be able to document its evolution. The analysis of Abernathy and Kronauer show that the mixed region is four to five times the original separation distance h. Consequently, the initial conditions for the quantitative evaluation of the mixing--as controlled by the large-scale motions--will be a lower region of contaminated gas and an upper, uncontaminated gas, region separated by a well stirred region of height z0.5M. The smaller kinematic viscosity of the Freon gas suggests that the initial displacement of the two vortex sheets will be smaller than the displacement for the air case. The impul- sively started boundary layers will be reduced by the multi- 1/2 1 ] ~ ~ r 2.4. This suggests that the initial width measure for the Freon results should be smaller; plicative factor [VF/Vai this point is addressed in the next section. The later times in Figure 20 and 21 show the mixing for longer time periods. It is evident from these pictures that the interface is highly convoluted and stretched by the turbu- lent strain field. 38 5.2 Data Base for the Particle Experiments The "non-diffusive mixing" experiments are achieved by contaminating the lower volume (gas B) with a cloud of submi- cron particles whose dimensions are: (l) sufficiently large that they are not significantly influenced by the Brownian motion of the surrounding gas molecules and (2) sufficiently small that their "slip velocity" with respect to the continuum motion is negligible. The nominal diameter of the particles, viz., 0.5um satisfies these joint criteria. The turbulent transport in the chamber is revealed by the presence of B in the upper half volume and that of A in the lower volume. Par- anthetically, it can be noted that the incompressible nature of the mixing experiment requires that the volumes of the two gases are invarient with respect to time; however, this fact cannot be readily used in the interpretation of the data since information from only one spatial line is available and because sub-scattering volume mixing may be present which would yield an excessive estimate for the presence of B. In principal, the presence of B in the scattering volume is easily detected. The light scattering from the particles (of B) creates a photo- multiplier tube output signal which is many times that for the noncontaminated gas. A comparator circuit readily distinguishes the two states (a nominal signal/noise ratio :_10 was achieved) and creates a binary pulse train with a "0" indicating the presence of A and a "1" indicating the presence of B. The electronics processing circuitry converted the pulse train, from the comparator circuit, into data words which were compatible with the TI minicomputer. The data was then trans- ferred to digital magnetic tape for further processing on the CDC 6500 central computer. The experiments were carried out with air as the working medium for 69 realizations and with Freon 12 for 100 realiza- tions. The marking particles were placed in the lower half‘ volume for all cases. A-74 mm scan was provided by the mimi- fication of the optical system and there were 205 readings per scan; a 0.3 nomin slit inter Speci The a small of th above overl of th preci relat there the g be up press of ga by "5 a "bl locat dimen USe 0 39 scan; hence, each data point was sampled from some fraction of a 0.36 mm section of the laser beam in the mixing Chamber. The nominal scattering volume height (h) may be estimated from the slit width (w), the magnification ratio (M) and the angle of intersection between the slit and the vertical laser beam (0). Specifically, h = [w/M cos 0] 5 [0.25/1.36 cos(H/4)] = 0.26 mm. The actual probe volumes were both somewhat larger and smaller than this nominal value; however, less than 1 percent of the scattering volumes exceeded the 0.36 mm spacing noted above and consequently they can be considered to be non- overlapping. The scans were executed in 3.11 ms; the period of the scans was 3.5 ms. The designated scan number is precisely known from the data record; however, its time, relative to the passage of the grid, is uncertain since _ there was no coupling between the initiation of the scan and the grid motion. Consequently, the designated time should U t be understood as —9— = 0.433. The data baseMwhich results from these experiments is re- presented by the $/b1ank map shown in Figure 15. The presence of gas B, as marked by the particulate matter, is represented by "$", the presence of the unmarked fluid A, is signified by a "blank" space. The map represents data for every other location (up to location 200) in keeping with the lateral dimension from the line printer. The quantitative results make use of the complete set of 205 locations. The interpretation of the experiments requires a careful examination of numerous aspects of the physical processes in the chamber and the details of the data acquisition process; these matters are dealt with in the following sections. Similar maps for stable and un- stable cases involving Freon 12 and air are presented in .Appendix C. A.map showing a complete set of 240 scans for an air/air case is also presented in this appendix. 5.3 Ensemble Average Distributions The experimental results for the ensembles of 69 and 100 :realizations of the air and the Freon 12 gases are considered .in this section. The instantaneous concentration of gas A or £3 at a given location and at a given time from the initiation 40 of the experiment is termed I‘(z,t); for the non—diffusive ex- periments its possible values are 0 or 1. The ensemble average values of F are identified as and such values have been evaluated for a discrete set of times corresponding to 'scans (l, 2, 4, 6, 8, ll, 14, 17, 21, 25, 29, 33, 37, 41, 60, 80, 100, 120, 140, 160, 180, 200, 220). Since the mixing field for the neutral density condition is at least expected to be symmetric, and since the turbulent transport effects in a homogeneous turbulence field are known to be readily approximated as a Gaussian process, a normal distribution was examined as a way to characterize the lateral distribution of the ensemble average values. Analytical con- siderations regarding the use of the normal distribution have ‘been presented in Section 3.2. The data were found to follow such a distribution, at least for a first approximation, and the subsequent evaluation of the data will rely upon the para- meters of the normal distribution to provide reference quantities for the spatial distributions of the average concentrations. A representative data set is presented in Figure 22. The dimensional plot is shown in part (a); the solid curve of this figure is the normal distribution found by a linear least squares'fit to the representation of the data on the "probability coordinates" of part (c). The probability coordinates are "defined" in part (b). The standardized normal cumulative distribution function, defined as A A(E) ”(i—:33 [5 exp (- 9‘7) do (5.3.1) —¢D see Dubes [1968], is shown on part b. The coordinate transformation 5 = cm -1 (z - 2c) (5.3.2) related the standard Gaussian distribution to the ensemble average concentration distribution, . The symbol zc 41 represents the centerline of this distribution; i.e., : = 0.5. Om represents the standard deviation of the spatial distribution; i.e.,= 0.5:0.34. The stan- .dard deviation Om represents a convenient width measure of the distribution. Figure 23 represents the ensemble average data and Figure 24 shows the resultant transformation (onto probability coordinates for Freon and air at Dot/M = 12. Note, that 2c is different from the centerline of the mixing chamber (2m). Apparently, this is a result of large scale gross transport of the mixing region. It is assumed that this transport does not directly effect the turbulent mixing near the centerline from which Om is calculated. The essential features Of the ensemble average data are represented by straight lines on probability coordinates; see Figures 25 and 26 for a composite representation of the air and Freon 12 results. The standard deviation between the cumulative dis- tribution function and the experimental data are also shown in the tables included with these plots. The width measures, Om, for the Freon and air data are shown in Figure 27. This figure also shows the physical size of the round rod and the mesh size of the biplane grid. The evolution of the mixing field within the chamber is 0.15 and

= 0.85 are used to mark the ”boundaries” of the mixing shown in Figure 28. The spatial locations of

field as a function of the elapsed time from the initiation of the experiment (Dot/M). 5.4 Measures of the Instantaneous Scans The individual measurements of F(z,t) in the ”instanta- neous" scans of the turbulent mixing field represent a Very considerable amount of information which is only partially characterized by the ensemble distributions. Insight into the detailed structure and processes of the mixing field can be gained if suitable measures of the instantaneous scans can be formulated. One such formulation has been achieved; it is described below. 42 Consider two extreme cases for the instantaneous concen- tration field; both are capable of yielding the observed ‘following the ensemble average process. The two cases are introduced in the form of statements. ”The instantaneous concentration field is such that an instantaneous scan re- veals: Case 1. a well mixed field with transverse dimension and con- centration P(z,t) ”similar" to that of ." Case 2. a relatively thin mixed region (i.e., one which is small, for example, with respect to Om(t)) which is laterally transported by the convective action of the large scale turbulent motion to yield the observed distribution." It is not expected that either of these models will be supported in their extreme condition; however, they do serve as a guide for the definition of measures to evaluate the character of the instantaneous scans. Specifically, defini- tions for i) the center of the instantaneous scan and ii) the width of the mixed region about the instantaneous center are required. The two definitions are separately considered in the following; the center of the instantaneous scan is defined by the following operations. a.) Determine the number of blanks (or absences - A) which signify the presence of fluid A in the scattering volumes and designate this number as NA. Similarly, identify the number of filled regions and designate this number as NB (NBBZOS-NA). b.) The center of the instantaneous scan will be designated as the position along the scan where the number of scattering volumes between the upper- most position of the scan and the instantaneous center is equal to NA. Note that this definition implicitly assumes that the B fluid does not extend beyond the upper limit of the scan and 43 that the A fluid does not extend beyond its lower limit. Also note that the upper and lower designations are understood in terms of the physical experiment; in terms of the $(E B) map of Figure 15, ”uppermost" corresponds to "leftmost". A schematic representation of the above operations is provided in Figure 29. The position of the instantaneous center for the ith scan Ci)j. The lo- cation of the average of the instantaneous centerlines and for jth experiment will be designated as 2 and its displacement from the ensemble average centerline, - (see Figure 29), are of interest. The statistical characteristics of the z ) - values allow one feature Ci j Ci of the two cases to be assessed; namely, if the variation of the centerline position is large (with respect to Om for example) then support for case 2 is inferred. The standard deviation of the zci)j - population is defined as {1 I; I 12}; (541) o = — 2 -<2 > .0 Az)l N-l j=1 c1)j Ci where N is the total number of experiments. The quantity UAZ is compared with Om in Figure 30a and 30b; the interpreta- tion of these results is discussed in the next section. The width of the mixed region about the instantaneous scan can be assessed in a manner quite similar to that of the width measure for the ensemble averages. (This will offer the subsequent advantage of a basis for the comparison of their relative magnitudes.) Specifically, the emsemble average of .). was formed as the F field with respect £2 zc1 J IKZ - z .).) (5.4.2) = 3'- n ___.1 c1 3 C]. LLMJ where an and n represents the number of experiments which possess a measured value of P at the 2 location in question. The so evaluated were then fit with the normal 44 distribution as described by equations 5.3.1 and 5.3.2. The resulting value of the width measure is termed OI)i; the values of OI)i for the 69(air/air) and the 100(Freon lZ/Freon 12) experiments are compared with the width measure of the en- semble average scans, Om)i, in Figures 30a and 30b. These. results are discussed in the next section. 6. DISCUSSION OF RESULTS 6.1. The Ensemble Average Concentration Distributions Figures 25, 26, and 27 represent the primary documenta- tion of the experimental results for the ensemble average concentration distributions. These figures include a tabula- tion of the standard deviation (S.D.) values which indicate the validity of representing the ensemble average data with. a Gaussian Cumulative Distribution Function (C.D.F.).I The S.D. values are computed from the relationship between and the C.D.F. in the range 0.03 §$F(z,t)>: 0.98. These limits reflect the relatively small sample size, and hence the limited resolution, of the experimental data (1:69 air and 1:100 Freon 12) and are selected such that the Om and 2c values are "minimally" effected by the selection of an arbitrary segment of the experimental data. ,(Trial runs on one data set show only a nominal influence of the cut-off values on Om; speci- fically, Om for Dot/M = 18 did not change by more than 2% when cut-offs representing 0.014 - 0.986, 0.028 - 0.972, and 0.043 - 0.957 were examined.) The relatively small values of S.D. (compared to unity) suggest that the Gaussian C.D.F. provides, at least, a satisfactory representation of the experimental data. Since thisanalytical form is also sup- ported by: (i) the (assumed) Gaussian displacement of fluid elements, see [3.2.6], and (ii) (independently) by an assumed "DT = constant" model of the turbulent transport, see [3.2.5], this compact presentation of the results is both efficient and apparently justified. The experimental data reveal that the Om values for the air experiments are larger than the Om values for the Freon 12 experiments at a given Uot/M. This result was not expected; +The experimental data, , are used to define the Gaussian distribution by the selection of z and O which minimizes (least squares) the difference between thg experimental data values and the C.D.F. values. The latter is defined as a-z 1— : 12 exp! - lI—ilz do: (2h) 0m 2 Om 45 46 it is considered to be quite significant. If the large scale, energetic motions in the turbulence field are assumed to be independent of the Reynolds number, then the smaller Om values for the Freon 12 experiments suggests that the transport is not solely controlled by the large scale motions. Additional interpretations of this result are made possible by the ana- lytical considerations of Articles 6.2 and 6.3. The spatial evolution of the mixing region, which is suggested by these results, is presented in Figure 28. It is of interest to note that the boundaries of the ensemble average mixing region are different for the air and Freon 12 experiments and that gross differences are observed for Uot/M 3 50. It is inferred that subtle differences in the two sets of experiments are probably responsible for the differences in the spatial evolution. It is also inferred, from the relatively self-consistent behavior of am, UI and 0A2 , that these differences are not significant in the in- terpretation of the experimental results. The Freon 12 results 29:: = 70-85, where the entire mixing field appears to be transported to small 2 show a distinct trend, beginning at approximately values. Since the extent of the field that is available to be fit by the Gaussian C.D.F. is reduced below a marginal value, the Om for Uot/M>110 are not presented for the Freon 12 carrier gas. An interesting comparison can be made between the tur- bulent diffusivity represented by DI and the molecular dif- fusivity D of the carrier gas. Equation (3.2.9b) relates the turbulent diffusivity to the slope of the zp2(t) curve. The following results are obtained. DT(air) = 14.6 cmz/sec for 5: Uot/M i 20 (6.1.la) DT(Freon) = 12.6 cm2/sec D1(air and = 29.2 cmz/sec for 20< Dot/M <200 (6.1.lb) Freon) . ” 7 The following ratios are obtained for the turbulent dif- fusivity to the molecular diffusivity for air in this experi- ment. 47 D _ _- D%air) = 73 5: Uot/M :20 (6.1.2a) DT - D—(air) = 146 205 ”ct/M 5.200 (6.1.2b) 6.2. Measures of the Instantaneous Scans Figures 30a and 30b, which present the results from the instantaneous scan evaluations for air and Freon respectively, strongly support the "well mixed" condition described by Case 1+ of Article 5.4. Specifically, the CI values are with- in 10 percent of the Om values at a given Dot/M. The 0A2 values for air and Freon, which provide a measure of the range covered by the instantaneous centerlines, are substan- tially smaller than those for GI slower growth following their rapidly established "initial” value. Only the instantaneous scans for Dot/M 5.35 are considered; the ensemble average results suggest that within this bound, the mixing field is confined within the 74 mm scan length. There is clearly more "scatter" in the air results than in (or Om) and they show a much the Freon results. Presumably, this is related to the number of experiments which comprise the ensemble: air: 69, Freon: 100. The Gaussian C.D.F. provides a satisfactory approximation to the data for both the Freon and the air experi- ments. This inferrence is based upon the acceptably small values of the standard deviation (S.D.) between the values computed from the appropriate C.D.F. (The latter is characterized by the 01 value described in Article 5.4. and is determined from a least squares fit to the data. Note that the S.D. is based upon the population [ - ‘ ] where data,k Gaus. C.D.Fr'k k is the index number representing the discrete sample volume. i_<_k_<_205. )> TCase l: "The instantaneous concentration field is such that an.instantaneous scan reveals a well mixed field with trans- verse dimension and concentration r(z,t) similar to that of ." 48 The 0A2 values for the Freon 12 and air experiments are essentially identical; this comparison is shown on Figure 30b. This is considered to be a most significant result. It was expected, a priori, that the experimental results which were dependent upon the large scale motions would not exhibit a dependence on the Reynolds number. The 0A2 values are con- sidered to singularly represent the large scale transport of the interfacial region; hence, they are expected to be the same for the Freon 12 and air experiments. The agreement between the two set of experiments is considered to be a con- firmation of the experimental technique and the above hypo- thesis. However, it could (naively) be assumed that Om is also singularly dependent upon the large scale motions. This (plausible) assumption is in conflict with the observations for Om and, as shown in Figure 30, for GI as well. Specifi- cally, OI(air) > UI(Freon 12). Theoretical considerations, which are based upon the Lagrangian dispersion analyses of Appendicies D and E, are presented in Article 6.3. It is shown in that article that a rational explanation of the smaller Om values can be formulated. It is presumed that the physical influence of the kinematic viscosity, which was re- sponsible for the relatively lower Om values for the Freon 12 experiments, is also responsible for the lower Freon 12 CI values. Consequently, a consideration of the 0 results will I be deferred until Article 6.3. 6.3 Inference of the Lagrangian Scales in the Present Experiment As discussed in Section 3.2., the dispersion parameter zp is considered to be numerically equal to the 0m value. For this experiment the contaminant which occupies the lower half of the chamber has a physical "release time” tr which corresponds to the instant the grid and splitter plate pass the observation plane. The theoretical developments of Appendix D, specifically, the formulation of the dispersion using two separate time scales, can be examined in the con- text of the Om experimental values. The micro and macro time 49 .scales can be estimated by the procedure outlined in Appendix B. These applications of the theoretical results are carried out in this section. 6.3.1. A Priori Estimation of the Lagrangian Microscale Utilizing Shlien's experimental data and the theoretical development by Hinze [1975], an estimate of the normalized microscale, I*, (TL/Cl) was obtained in terms of the grid Reynolds number and the parameters in the kinetic energy decay 'law, see equation (E.7.8). For the present experiment, these parameters have the following numerical values for decaying grid turbulence: AIR FREON 12 A = 31 A = 27 U0 = 6.28 m/sec Uo - 6.28 m/sec M a 2.54 cm M = 2.54 cm _ -4 2 ‘ vair - 0.15x10 m /sec vfreon 0.17 vair n = 1.27 n a 1.27 From Equation (E.7.8.) the normalized microscale can be de- fined as 1 A .25 + --- [-J ] (6.3.1.) 3)3/4 Rm . :*= (1-521) ln_[l+ CL(2 This leads to the following values: I = .26 * (air) (6.3.2.) I = .18 * (Freon) The theoretical result for 2:, presented in (3.2.22) and plotted in Figure 33, shows that zp increases for an increas- ing value of 1*. Hence, the difference in the predicted 1* values for air and Freon 12 suggests that the dispersion, using air as a carrier gas, will be greater than that for the Freon 12 carrier gas. The 0m values of Figure 27 are in IC = .17 obtained from Shlien's [1972] data fOr dispersion ik decaying homogeneous turbulence. 50 qualitative agreement with this prediction; quantitative con- siderations are presented below. For the present experiment, the mixing begins before the turbulence field becomes homogeneous. That is, there is a period, UOt/M f 5, of inhomogeneity during which significant mean velocity gradients and spatially non-uniform velocity iflUCtuation intensities exist. Sato and Yamamoto [1974] have quantitatively investigated this period in a wind tunnel in- vestigation of the near field behind a grid. The experimentally observed results of the dispersion were influenced by the inhomogeneous turbulence field for this initial time period and since these physical effects cannot be accounted for by the theoretical considerations of Appendix D, an alternative inter- pretation of the data is required. 6.3.2. Evaluation of Parameters Which Describe the Turbulent Dispersion * The analytical structure for the dispersion z 2 which was develOped in Appendix D, and extended in Appendix E, in- volves the parameters T* ' A* In principal, the available experimental data can be inter- preted in this theoretical context to infer these parameters. This inferrential process must acknowledge that the analytical structure is quite complex and that the assumptions involved in its construction may invalidate part or all of it. It was found useful to create a set of working drawings. These include: (i) the mean square particle displacement %22 as a function of U ot/M)+ data for both air and Freon 12 over the full range of Uot/M values and (ii) an expanded scale plot of %z: for Uot/M f 35; see Figures 31 and 32. Note that these working plot identify the dispersion as a function of the (normalized) laboratory time. The theoretical curves for IFrom subsection 3. 2, 22 is identical to a: for the assumptions outlined in that subsegtion. 51 %z*2(At/Atr; I*) and %z*2(At/Atr; A*) were also prepared; see Figures 33 and 34. Note that n = 1.27 for the latter two plots; this value is taken from the study by Comte-Bellot and Corrsin [1966]. A detailed examination of these results reveals that the present analytical formulation is incompatible with the ex- perimental data. Specifically, the experimental results re- 2 distribution (25 f Uot/M f 40) and an examination of Figures 33 and 34 show that a similar juncture veal a "plateau" in the 2 between the microscale and macroscale dominated distributions is not possible; that is, the present analytical formulation only provides for a juncture of the two linearly increasing 2: ' distributions. Apparently, a more accurate representation of the Lagrangian correlation function would be required to pro- vide the "plateau" observed in the experimental data (Figures 31 and 32). 6.3.2.1. A Posteriori Evaluation of the Microscale and-Atr The experimental data can be used to evaluate the Lagran- gian micro and macroscales; the procedure is to find the parameter values which provide optimal agreement between the theoretical and experimental results. This task is made ambiguous by the unknown value of Atr to use in the analysis.+ Physically, tr = 0; however, the analysis is not valid for small times because of the strong inhomogeneities of the turbulence field. Consequently, a scheme to select self- consistent parameter pairs (I*, Atr), which also yield 2: values in agreement with the data, was adopted. This scheme is described below. ' The 2* 2(At/Atr ) theoretical results from the microscale analysis (Figure 33) show that z *2 increases linearly for sufficiently large At/Atr values. (The curves are necessarily terminated at At Lr/At ). Since the experimental results for +An alternative data processing scheme which would remove this ambiguity is under consideration. 52 :z2 also reveal a region of linear growth (see Figure 31), a ‘scheme which equated the theoretical slope to the processed data slope was adopted. The experimental data for z; are made non-dimensional by the specification*of an arbitrary Atr value (see 3.2.21) and the slope: 0.5d(z 2)/d(At/Atr), is then determined. A unique value Of I* can then be found which provides the best agree- ment between the experimentally and theoretically determined slopes. (Note 3.2.22 for z*(At/Atr; I*, n); also note that n is adopted from wind tunnel data for turbulence kinetic energy decay, Comte-Bellot and Corrsin [1966].) This process yields the following results for the 2:)air data. Table II Evaluation of Dispersion Parameters (Uo/M) (Atr): 1.73 2.14 3.47 5.17 7.36 10.8 I*: 0.23 0.25 0.3 0.35 0.4 0.5 (Uo/M) (AtL): 3.25 4.24 7.91 13.5 22.08 42.55 The AtL values of this table are instructive; they provide a means of establishing an upper bound for Atr' The experimental data reveal that the plateau begins at approximately Uot/M = 20. Hence UoAtL/M must be less than 3 15 since Uoto/M 3 5. (See Comte-Bellot and Corrsin (1966].) It is apparently not pos- sible to select a lower limit for the possible combination of Atr and I* values since the 22 data could continue to grow linearly beyond the region which is dominated by the micro- scale. However, some guidance regarding the minimum value to be expected is gained from the z":2 distributions of Figure 32. The value Ti = 0.25 has been adopted as the plausible minimum. Note that the implied juncture region for this Ti value would be quite extensive since the macroscale dominated ~ region is not initiated before Uot/M ~ 30. +From the expression: At /At = exp[I*/(l-n/2)], it is seen ‘that At represents the Maxifium possible value of At for which2 the mictoscale-dominated"analySis Could be need to describe 2 ° 53 The estimate of I* = 0.26 presented in 6.3.1. is, there- fore, suspect. It is not clear whether this prediction would be expected to fail because of an improper numerical value for CL or the inappropriate use of information from Shlien's experiment. It is pertinent to note that if the I* values of (6.3.2.) are accepted as valid, it would be extremely dif- ficult to confirm them with experimental data since the AtL values would be quite small. The present experimental data will be used to reevaluate++ the constant CL in the expression (6.3.1.) for I* as a func- tion of the parameter A and the mesh Reynolds number (RM). For this, it will be assumed that Tgir = 0.35 for the air results; note that this value represents the ”largest feasible” 2 data for P Atr which can be used to fit the slope of the z 4 Uot/M :_20. For A = 31, RM = 10 , and I* = 0.35, CL is evaluated to be 0.2 and (6.3.1) can be rewritten as n A 0.25 RM This equation can be used to assess the influence of the ki- nematic viscosity on the observed lepe of the 2: distribution between the air and Freon 12 results.’ ' t The calculation of IFreon 12 +++ that Atr is increased by (Uot/M = l) to Uotr/M = 6.17 since to (apparent temporal origin of the decaying turbulence) for the larger Reynolds number is known to decrease by approximately this amount; see wyatt [1955]. The experimentally observed slope of the 2: distribution can requires a knowledge of Atr. It is reasonable to assume IMolecular diffusion effects from the temperature marker in Shlien's experiment could provide a low estimate of I*. This is discussed in Appendix E.3. IIShlien's data was used in E.7.8. to evaluate C1; a value of 0.17 was found. The present CL is approximately 18 precent larger than that inferred from the Shlien data. C (3/2)'75; its value was determined to be 4.44. The present 8 = 0.2) differs from CL found using Shlien's data (i.e., 0.17) by +18 percent. IIIThe final result is only slightly affected (5% increase) by this assumption. 54 be made dimensionless by the procedures noted above; its value O I * is 0.34. This corresponds to a IFreon 12 value of 0.24. Hence 1* Freon 12 _ 0.24 air ' The ratio of the I* values computed from (6.3.3.) using the air and Freon 12 values is 0.74. The agreement between these two results is interpreted as a strong endorsement for the theoretically inferred effect of the kinematic viscosity on the experimental results. 6.3.2.2. A Posteriori Evaluation of the Macroscale If it is assumed that the maximum value of Atr is reliably inferred by the procedure described in 6.3.2.1 and if it is assumed that Atr for the short and long time analyses are the same, then A* may be calculated from the Observed slope of the experimental data for large Uot/M values. These calcula- tions have been executed; it is found that the dimensionless .slope (see E.4.2.) is relatively large: i.e., 0.694, and that the corresponding A* is also relatively large. Figure E.3.+ provides a numerical value of 3.9 for A*; however, the figure also shows that a slight change in the evaluated slope will create a large change in the inferred A* value. Hence, the indicated A* value is rather uncertain. Figure 34 shows that * z 2 is not particularly sensitive to the value of A*. +Figure E.3. is specifically for At/At = 3.5. However, the magnitude of the slope will be only.inghtly affected by this ratio; see Figure 34. 7. SUMMARY AND CONCLUSIONS A viable experimental technique to measure the instantaneous concentration (F) of a non-diffusive scalar contaminant along a line segment parallel to the principal concentration gradient fhas been developed. This technique, which is discussed in Section 2, includes the capability to generate small (0.5 pm nominal diameter) marking particles which serve as the non- ‘diffusive contaminant. Theoretical considerations, developed in Section 4, show that the particle's response time (to an im- pulsively applied motion of the continuum fluid element which serves as the host fluid) is sufficiently small that they follow the continuum motion with satisfactory accuracy, see Figure 17. It is also shown that the particle diffusion caused by the Brownian motion of the host gas, has a negligible influence on the inferred dispersion properties of the turbulence field. The experimental facility was a closed mixing chamber fitted with a (horizontal) Splitter plate to separate the two host 'gas regions prior to their mixing. A bi-plane, square mesh, round rod grid was used to create the mixing field. Light :scattering measurements from small cylindrical volumes (2 2 0.36 mm, d = 0.1 mm) were used to sense the presence or absence of the marking contaminant. The concentration at a given location (2) and at a given time from the passage of the grid (t) will be 0 or 1 depending upon the origin of the fluid element in the scattering volume. (I - 1 if the origin was in the lower half-volume.) Ensemble averages of the F(z,t) data have been constructed (). The separate assumptions of (i) a constant turbulent diffusivity (Dt) and (ii) a Gaussian distributuion of the particle displacement lead to the same analytical result that is a Gaussian Cumulative Distribution Function (C.D.F.). The experimental data were fit with such a C.D.F. and the standard deviation of the fit was evaluated. ' ‘ Conclusion: The Gaussian C.D.F. provides a satisfactory description of the experimentally determined . The standard deviation of this 55 56 distribution (Om) provides an appropriate width measure of the ensemble averaged mixed region. The experimentally determined Om and the above noted analytical considerations allow D to be evaluated. t Conclusion: The ratio of Dt to the host gas molecular diffusivity is, for air as the host gas, Dt/D = 73 5 f Uot/M f 20 Dt/D = 146 20 < Dot/M < 200 The dispersive property of the decaying turbulence field is represented by the mean square particle displacement 2:; see (3.2.8.). [2:(t) is shown to be equal to a; (t) for the assumed Gaussian C.D.F. of the particle displacements]. An extensive theoretical development for the relationship of z: to the Lagrangian micro and macroscales which are used to characterize the Lagrangian auto correlation function, RL, has been carried out in Appendix D. This analysis has been used to reevaluate the experimental results of Shlien [1972], see Appendix E. The present theoretical work utilizes the concept of a stationary RL(T), Batchelor and Townsend [1956], where I is a stretched time coordinate (I = C ln[(t - to)/(tr - to)]); this feature is‘in common with prior analyses. The present theoretical structure is distinct in its use of separate micro and macro scales for the Lagrangian correlation function and for its relatively direct procedure to relate the experimental and theoretical results. Specific conclusions based upon the comparison of the theoretical and ex- perimental results follow. Conclusion: The experimental data of Shlien [1972] are well fit by the present analytical formulation. The Lagrangian, normalized microscale I* was determined from the 2: data for small values of At/Atr; its numerical value is 0.185. The Lagrangian normalized macro scale was determined from the 2: data for large At/Atr values; its numerical value is 0.213. 57 Precise Atr values (42 and 98) are known for the Shlien data since the thermal marker was introduced with a heated wire; a precise value of Atr cannot be established for the present experiment since the mixing begins in the inhomogeneous region immediately fOllowing the passage of the grid. However, for the present study, the I* and Atr evaluations may be jointly determined by forcing agreement between the experimental and theoretical sloPes of the 22 vs At/Atr curves. This procedure results in a range of I* and Atr values; this range can be limited by the following observation. 2: (t) initially grows in apparent agreement with the theoretical description provided by the microscale dominated region of the RL(T) distribution, it passes through a "plateau" region where its growth is relatively slow, and it continues its growth in apparent agreement with the macroscale dominated region of the RL (I) distribution. (Both the air and Freon 12 host gases exhibited this behavior.) The initiation of the plateau allows a maximum I* to be defined. Conclusion: The maximum value of I*)air is 0.35 and the corresponding UoAtr/M value is 5.17, see section 6.3.2.1. Conclusion: The macroscale, A*, calculated from the comparison of the data to the long time theoretical result, is 1.9, from Section 6.3.2.7. This value is recognized to be quite uncertain because the calculation procedure shows that A* is quite sensitive to small changes in the Atr inferred from the macroscale evaluation. The "maximum value" of Atr was used to determine A*. The Reynolds number dependence Of the Om and 2p results was not expected; it was expected (a priori) that these measures of the dispersion field would solely reflect the large scale motions and that the scale of these motions would be fixed by the grid mesh size, M. However, the theoretical considerations of the disperion analysis provide a rational explanation for the experimental observations. Specifically, it is shown that I* may 58 be expected to decrease as the Reynolds number is increased since the constant C, in the I = C 1n (At/Atr) transformation, is Reynolds number dependent. Hence the 6 fold change in V between Freon 12 and air can be expected to change 2;, the mean square particle displacement. Conclusion: Using the air results for the maximum I* and the theoretical structure of Appendix D, the smaller dispersion for the Freon 12 shown in Figure 31, is made rational both .qualitatively and quantitatively from Equations (6.3.1.) and (6.3.3.) as * * I Freon 12) = 0.69, I Freon 12) = 0.74 I*Air I* Air experimental theoretical The availability of "instantaneous" scans for P (z,t) allows additional measures of the mixing field to be evaluated as discussed in Section 5.4. Statistical measures of the instantaneous scans are examined to determine if the ensemble average mixing results can be traced to (l) a well mixed region for each realization which has the same nominal width as the ensemble average field or (2) a relatively narrow mixed region which is convected by the large scale motions to yield the observed width of the ensemble average field. The instantaneous scans were shifted (A2) to align their instantaneous centers. The width OAZ reflects the magnitude of these shifts; OI reflects a measure of the Gaussian C.D.F. which was fitted to the shifted and ensemble averaged concentration distributions. Conclusions: The approximate equality of CI and 0m (OI > .80m) and the small values of OA ~ 2 (OAz f .Som)imply that the observed mixing is strongly influenced by the relatively small motions of the turbulence field, see Figure 30. Conclusions: 59 The essential equivalence Of the OAz values for the Freon 12 and air host gases implies that 0A2 is both a measure of the large scale motions and that these large scale motions are in- sensitive to the Reynolds number. LIST OF REFERENCES 10. 11. 12. 13. 14. 15. REFERENCES Abernathy, F. H. and Kronauer, R. E., "The formation of vortex streets," Jour. Fluid Mech., v. 13, no. 1, (1962). Batchelor, G. K., Howells, I. D. and Townsend, A. A., "Small scale variations of convected quantities like temperature in turbulent fluid," Jour. Fluid Mech., v. 5, p. 134, (1959). Batchelor, G. K. and Townsend, A. A., "Turbulent Diffusion," Surveys in Mechanics, Ed. by G. K. Batchelor and R.M. Daires Cambridge Univ. Press, New York, (1956). Becker, H. A., "Mixing, concentration fluctuations and marker nephelometry", Studies in Convection, v. III, ed., B. Launder, p. 45-139 (1977). Bendat, J. S., Piersol, A. G., Random Data: Analygis and Measurement Procedures, Wiley IntersCience (1971). Bradshaw, P. "Interacting Shear Layers in Turbo Machines and Diffusers", Turbulence in Internal Flows, Ed. SNB Murthy, Hemisphere Publishing Corp., Washington (1977). Brodkey, R. 8., "Mixing in turbulent fields", Turbulence in Mixing Operations, Academic Press Inc., New York, p. 48-119 (1975). Chao, B. T., "Oesterreichischer", Ingenieur-Archiv., v. 18, p. 7 (1964). Comte-Bellot, G. and Corrsin, S. “The use of a contraction to improve the isotropy of grid generated turbulence", J. Fluid Mech., v. 25, p. 657-682 (1966). Comte-Ballot, G and Corrsin, 8., "Simple Eulerian time correlations of full and narrow band velocity signals in grid generated isotropic turbulence," Jour. Fluid Mech. Vol. 48, p. 278 (1971). Cooper, G. R., McGillem, C. D., "Methods of signal and systems analysis", Holt, Rinehart and Winston, Inc. (1967). Corrsin, S. and Lumley, J., "On the equation of motion for a particle in a turbulent fluid", App. Sci. Res., v. A 6, p. 114 (1956). Corsin, 8., Limitations of Gradient Transport Models in Random Walks and in TurEUlence Advances in Geophysics, v. 18A, Academic Press, N.Y. (1974). Corrsin, 8., "Some current problems in turbulent shear flows", Naval Hydronamics Publication 515 (1957). Chevray, R,. "Total dispersion of a scalar quantity in turbulent flow", Private communication of a manuscript (1977). 60 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 61 Dubes, R. C., The Theory of Applied Probability PRENTICE- HALL, Inc., Englewood Cliffs, N.J. (1968). Falco, R., "Measurement of intensity and correlation from the initial to final period of decaying grid generated turbulence," Ph.D. dissertation, Dept. of Aero and Mech. Sciences, Princeton Univ. Foss, J. F., Shlien, D. J. and Corrsin, 8., "Study of a thermal interface in isotropic turbulence," Bulletin of the American Physical Soc., sec. 11, v. 7, no. 11, p. 1089 (1972). Friehe, C. A. and Schwarz, "Grid generated turbulence in dilute polymer solution," Jour. Fluid Mech., v. 44, p. 173-193 (1970. Gibson, C. H. and Schwarz, W. H., "Universal equilibrium spectra of turbulent velocity and scalar fields," Jour. Fluid Mech., v. 16, p. 365, (1963). Grant, H. L. and Nisbet, I.C.T., Jour. Fluid Mech., v. 2, p. 263 (1957). ‘ Hinze, J. 0., Turbulence, McGraw Hill Book Co., New York (1975). Hirschfelder, J. O., Curtiss, C. F. and Bird, R. 3., Molecular Theory of Gases and Liquids, John Wiley and Sons, New York (1954). Hirschkron, R., and Echrich, F. F., "Entrained particles trajectories in swirling flow," ASME Paper 64-WA/FE-30, (1964). Johnson, D. A., "An investigation of the turbulent mixing between two parallel gas streams of different composition and density with a laser doppler velocimeter," Univ. of Missouri, Columbia, Ph.D. dissertation, Mechanical Engineering Dept. (1971). Khosla, P. K. and Lederman, 8., "Motion of spherical par- ticle in turbulent flow," Pibal Report NO. 73-22, Polytech. Inst. of New York (1973). Libby, P. A., "Diffusion of heat downstream of a turbulence grid," Acta Astronautica, v. 2, p. 867-878 (1975). Monin, A. S. and Yaglom, A. M., Statistical Fluid Mechanics, V. 1, English translation by J. L. Lumley, M.I.T. Press (1971). Nye, J. O. and Brodkey, R. S., "The scalar spectrum in the viscous-convective subrange," Jour. Fluid Mech., v. 29, p. 151- 163 (1967). Ozisik, M. N., Boundary Value Problems of Heat Conduction, International Textbook Co. (1968). 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 62 Potter, M. C. and Foss, J. F., Fluid Mechanics, The Ronald Press CO. (1975). Rayleigh, Lord, "On the instability of cylindrical fluid surfaces," Phil., May 5, XXXIV. 177 (1892). Roshko, A., "On the development Of turbulent wakes from vortex streets,” NACA Rep. 1191 (1954). Saffman, P. G., "Some aspects of the effects of the molecular diffusivity in turbulent diffusion," The Mechanics of Turbulence, Gordon and Breach, New York (1964). Saffman, P. G., "On the stability of laminar flow of a dusty gas," Jour. Fluid Mech., v. 13, p. 120-128 (1961). Sato, Y. and Yamamato, K., "Local structure of isotropic turbulence," Bulletin, Yokohama Nat. Univ., v. 23 (1974). Schlichting, H., Boundarnyayer Theory, translation by J. Kestin, McGraw Hill Book Co., New York 6th ed. (1968). Shlien, D. J., "Dispersion measurements of a fluid sheet in two turbulent flows," Ph.D. dissertation, Mech. Dept., Johns HOpkins Univ. (1972). Shlien, D. J. and Corrsin, S., "A measurement of lagrangian velocity auto correlation in approximately isotropic turbulence," Jour. Fluid Mech., v. 62, p. 255-271 (1974). 800, S. L., Fluid Dynamics of Multiphase Systems, Blaisdell Publication Co., Massachusetts (1967). Taylor, G. I., "Diffusion by continuous movements," Proc. London Math. Soc., ser. A, v. 20, p. 196-212 (1922). Taylor, G. 1., "Statistical theory of turbulence IV- diffusion in a turbulent air stream," Proc. Royal Soc., ser. A, vol. 151, no. 873, London, p. 465-478 (1935). Tchen, C. M., "Mean value and correlation problems con- nected with the motion of small particles suspended in a turbulent fluid," Dissertation publ. 51 of Lab for Aero- and Hydrodynamics, Tech. Univ. of Delft (1947). Townsend, A. A., "The diffusion of heat spots in iso- tropic turbulence," Proc. Royal Soc., vol. A209, no. 1098, p. 418-430 (1954). Townsend, A. A., "The diffusion behind a line source in homogeneous turbulence," Proc. Royal Soc., v. A224, no. 1159, p. 487-512 (1956). Uberoi, M. S. and Corrsin, S., "Diffusion of heat from a line source in isotropic turbulence," NACA Report 1142 (1953). Uzkan and Reynolds, "A shear-free turbulent boundary layer," J. Fluid Mech. v. 28, pt. 4, p. 803-821 (1967). 63 48. Yeh, T. T. and Van Atta, C. W., "Spectral transfer of scalar and velocity fields in heated grid turbulence," J. Fluid Mech., v. 58, no. 2 p. 233-261 (1973). 49. Wyatt, L. A., "Energy and spectra in decaying homogeneous turbulence," Ph.D. thesis, Univ. of Mass. 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H. m.o ___ @ Eo\mmH0Huumm mod ._ A m umuHHm mDOMOQOHOHE amuum comm owawaq mnficmmo : m¢.o .MHv Eu v.5 mumamouv .ch n.o¢uoH. .smumm 74 mmaofiuumm baud oaumofi mo smmuwouoam mmoomOHOHz couuomHm .HH musmHm :oflmw>HQ\En m.o .sumcwa wocwumwwm .Q meoHuHmm .m ' .fla/ I 75 ) Epmt 1 Et = 0.06 volts L_ a- -1 “44” 'sc U a. Instantaneous Scan ' Note: E = threshold voltage level for the comparatSr circuit, see Figure 13. Epmt l 0.2 volts time - I b. Fully Mixed COndition Note: No signal dropouts Figure 12. Ink Tracings of Oscilloscope Photographs Showing: a.) An instantaneous scan of the photomultiplier tube voltage (E ) for the second scan following the pgggage of the grid b.) 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U° (3.5 msec) - 0.865 M. 79 .czonu nosflo> nonuo ousmsou ou mcoNusHou Denounced no“: cons muss consumes A av N oduau aquHHOn vm.o .Uwum won vases HacsHmHn a}. 3... u on :8 EN .. a ....H u as 3302 .Hmmmau.uunhz no snow «m on» scum wousasoaso mocoHsnusu usoocmmosoc How aoHooo huwooHo> can succua o>wuoucououmom .wH ousmwm . 2 H m u.||| . no: 0H oH 0H oH n . N H o oH A OH x m\ Evu o N n N . 3 x m}... «A 3 . N N i .EE.H ’r n . cu AEEVA A oaxEEvc l LIJLI OH 80 t apparent temporal origin of o . turbulence w2 = 1 [”0t _ Uotol-n 002 A M M t apparent time at which r O I contaminant is released (I‘=l) Uo = velocity of grid a.) Schematic representation of mixing t I macroscale t8 = At/Cf microscale t8 = At/CA .- (t - to) b.) Schematic representation of Lagrangian time scales development R L . 1 solid curve rep- resents R (T) KL \ L \ T2 - - - exp (- T) 9 T2 c. Definition sketches for T (microscale), T. (juncture point) and fl (macroscale) for the Lagrangian auto corre7ation function which is assumed to be stationary in the stretched T coordinates Notes: R (t 6t) = i”(t) “(t+At)> == R (T) 6(t) fiHv one .0002 con 00.0 .3.“ no» 40.0 «H no» No.0 « on. pH.» oz» uo owoanon one soak aofiaa vunauHu .uuonn nacho» one as haHHHnuAnca on» souu voaoau>ov nunsaoshaa advoauoa one son» 0» unououa uaaxAI H-Haacd one «o annuuoHo .oH ousuHu : mN.o 83 Eu cw mum aconcmEHU HH¢ "wuoz .mmoooun "...:me .335 05 5M? voumHQOuum muouorcmuma of ..o coauMucououmou umuanzum .mH 0.500% m m m o mhv.0 .D . . ... .4 common 0338 we £03m? ..UmuOuQEQN: . r423.- Him I 3 84 0.25 K Figure 20. Close—up of the mixing phenomena. Elapsed times from the passage of the grid are 1 0.0L sec ii 0.h2 sec iii 1.68 sec iv 2.68 sec 8 v 6. see Note: The divider plate is at the bottom of the lighted region in (i). (111) 85 (1) (M (n) (V) H———u 1.0 H Figure 21. Large View of the mixing phenomena. 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H u u<\.u< mmm.o u Hm\c I HH “meoz Hm~.~.mv coHumswm scum 1100 u n Ac. uq..u<.« HmuHumuomne .vm mustm u “Huq mH oH vH NH OH m o v H H H H H HI H H HI H H H H H H H H H.o u i m~.o u < i H .H U IWII Wk. NINI HHuHH N. u H H u.o u < a. N .I m - H I s uuq Hug m~.m.m coHumsvm H HMQ A mm HH Hmmmv . HmmmHHH+IHm> mos v Honsmso mcHxHE Hommmumeoo _ H H L C Homcmsoxm vows HHoo memoo I U numb swam; no: I m Hmummn wocmumHmmH I m m>am> mumm IT HOHMHHHme IT mesa Edsom> umuHHm moouom IOHUHE Hmamumm cHsvHH 104 Vapor S tate To A I Vapor SZPregzure 4 (II) State Vapor —- state 7 (r—Freon 22‘\ 45 60 p819 ’( Freon 12 psig Freon 12 liqui state I ___f __. A VA" __w 12R, adjust- able heater Heat exchanger Water at about 135°F; 1 3°]? ...1 c011 controlled by F. temperature D sensitive c - element Figure A.l.b Subassembly I 105 Freon Subassembly I W Gate valve Gate valve Electric heater coil Electric heater coil (/’ Freon 22 fi‘\\ Vapor storage tank fireon 12 \ Vapor storage tank _l [— s 3.: Figure A.l.c Subassembly II Pressure regulator'* * flow to subassembly III 106 Solenoid valve [ ” ‘ ] Allows simultaneous filling of top and bottom of mixing—- chambers Gas \\\‘5 valves Pressure above choking rigical valves Freon 12.22 (40 psig) Temperature readout IIRoom temperature Thermocouples Mixing chamber Figure A.l.d Subassembly III 107 ..HonEoHHo 9:ng now 05 cu UHHHmH owauoun @030: 05 Scam NH ooouh ”Ha .HOH 033: uHEnEHvoEquH .N .¢ 055.; HH HoHuuoc uHHHOu can .HoHaHHHwou unannoua Auaoufi moat ouanuoum van chanson—8H3 min H. .3 m m ham.» m .H. .o Housnaoum ucflocou \ 3 $58: .8525 x 32H mm m m \ hoooH m .H. .m :53» k I H . m Homdb voumonuomao m o» .330: swdoufi ounuuokm «532.53 38 mm m m hoomH m .H. .v HuoHoHsmou 0.3:on ~ H Annoufi coup unannoum inane—0:0 ucduucoov mHum mm m m “Hook m .H. .m 050v $3.3 5. momv .HOHHSH 9.3 o2 m m hoomH m .H. .N 253 5 239.: 93 8H m m ."HoomH m .H. .H 108 APPENDIX B: The Design of the Grid/Splitter Plate Actuation System The actuation system to accelerate (in the minimum feas- ible time) the grid and splitter plate to the desired speed of 6.2 mps, and to decelerate it at the end of the travel was composed of the following major components: i) a high pressure, 173 n/m2 (250 psig), air supply which provided the motive force, ii) a piston cylinder arrangement with air as the forc- ing fluid and hydraulic oil as the reactive (speed limiting and decellerative) fluid, and iii) a receiver tank for the hydraulic fluid. A schematic representation of the system is presented in Figure 8.1. This appendix presents the analysis which was used as the basis for the system design. Figure 3.1 presents a working sketch for this analysis and defines the terms required for it. The dynamic equation for the mass of the grid and splitter plate can be written as dV _P_= _ mt dt POAo PpAp (8.1) where Vp is the piston velocity, Pp is the pressure of the hydraulic fluid, Po the air pressure and m represents the mass of all the moving elements (grid, splittertplate, piston). Note that the bearing fricton and aerodynamic drag on the moving parts are considered to be negligible with respect to the motive force, POAO. It will be assumed that PC is constant following its sudden rise after the quick acting valve is opened; that is, the pressure drop associated with the transfer of the air from the holding tank to the cylinder will be neglected. Con- sequently, (3.1) can be solved for VP if Pp(t) is known. This time dependent pressure can be solved for in terms of the assumed behavior of the hydraulic fluid as it exits from the cylinder into the receiver tank. The analysis of Pp(t) makes use of the conservation of mass and momentum for the deformable control volume defined in Figure B.1.- The analysis will be presented in two segments: i) the acceleration phase, and ii) 109 the deceleration phase. The complete expression for the conservation of mass is, see, e.g., Potter and Foss [1975], O = I pVb-n dA + f %E_p dV + f er'n dA (8.2) c.s. c.v. c.s. Since Bp/at = O (fluid is incompressible), this reduces to Vp AP = VeAe (8.3) where Ve is the average velocity at the nozzle exit. The con- trol volume form of the energy equation is o - f 2 2 + P v -A dA + f PV -A dA + d ] Vzd 4 — olV/ /p]rn b“ EE- 97" (8.) c. c.s. c.s. V. where the major assumption is the neglect of loss effects and the further assumptions of no elevation changes and no heat transfer effects are implied. The first term represents the flux through the exit plane of the nozzle, the second repre- sents the work rate term of the piston acting on the fluid and the third accounts for both the accelerative and volume change effects. (8.4) can be rewritten as _ . 2 +." 2 + ." a 2 AprPp — p [(v /2)v n dA + [(v /2) vb n dA + I 51“" /2) dV (3.5) A c.s. GOV. e where the pressure Pe is zero gage and the last term in (8.4) is expressed in terms of the last two terms in (8.5) by use of the Liebnitz rule. Assuming that the velocity is uniformly distributed over the cross sectional area, the equation 8.5 simplifies to 2 2 L+L V V Vis) A(s) av -_- _9.-_£ Pp p 2 2 + A f v A 5? ds (8.6) xp 9 p Since the [AV] product is constant, the last term can be fur- ther simplified as 110 L+L 3V L+L f 8 3(5) 2(5) g—‘é ds = —P-3t [L - xp] +1 8 3‘1 ds (13.7) xp p p L for xp EL' The integrand of the last term is most easily in- terpreted as BV/at = (3Vp/8t)(Ve/Vp)(V/Ve); with this, L+L V 3V L+L 3V e e V I e (—) ds = (—)(—E) I (—) as L at Vp 3t L Ve Ve av (0.55 _Le) (5;) (FEB) If it is assumed that L - xp 2 L for the times of interest, then (A.8) can be written as A dV v2 p A 2 [L + 0.55 XELe] ER” [(312) -1] 73 = EP' (8.8) Equation 8.8 provides the description for Pp required in (8.1). It will be convenient to write (8.1) in terms of xp instead of VP. That is, Vb = dxp/dt. With this substitution and using A.9, A.l becomes 62x EEE. dx 2 52.2 A dtz [mt + pLAp + p Ae (O.55)Le] + (a?) [(Ae) -1]p2-E = AOPO (3.9) The coefficients C1 and C2 are defined by the simplified form of 8.9 as u .2 _- where the boundary conditions are x(o) = o and x(o) = o. Normalizing the variables as: * t = C1Vot * x = Clx * V = V/V 111 * 0* . and using V = x , the differential equation becomes .* * 2 _ 2 _ ' 11 V + (V ) — [CZ/Clvo ] - C3 (8. ) with b.c. V*(o) = O and V*(o) = C3 . A linear equation results . * u from the substitution q/q = V ° Namely, q - C3 q = 0 (8.12) The solution to the equations are * q = cosh /C3 t * t V = /C3 tanh /C3 t and i t x = 1n [cosh /C3 t ] (8.13) A quantity of principal interest in the design of the actuation system is the distance required to bring V? - to 0.99Vb. From the above -1 0099 “‘53 x* = ln[cosh (tanh ( ))] (8.14) For the pr0posed physical apparatus, C321 and the distance re- quired to obtain V = 0.99 V6 is given as 2 32 4.0 [mt + pLAp + pAe 0.55 Le] (8.15) x (V é 0.99V ) = o is 2 pApHAe) - 11 The x value derived from (8.15) was quite large; a modi- fication to the physical facility was therefore suggested. An additional piston rod was mounted on the (hydraulic side) face of the piston; it is shown as a dashed line in Figure 8.1. Once the rod passes into the exit nozzle, the ratio (AP/Ae) is 112 increased and the distance is decreased according to (8.15). The deceleration of the moving parts was achieved by caus- ing a pressure rise in the hydraulic fluid. The pressure rise was caused by a restriction of the hydraulic fluid outflow as conical shaped plug moved into the exit nozzle. The plug geom- etry varied continually to provide a nearly uniform decelera- tion; the "shock loading" was therefore minimized. A numerical solution of the appropriate control volume equations was ex- ecuted using a forward time stepping procedure. The axisymr metric contraction had a fixed contour to provide a smooth but 'rapid acceleration of the hydraulic oil to the exit plane. This contour is defined in Figure (8.2). The plug geometry attached to the face of the piston had a conical shaped pro- file. The slope of the cone was varied until the piston ve- locity was sufficiently small (20.5 cm/sec) at E = 0. The final design configuration is shown in Figure 8.2. 113 .amHsmnoos :OHuouoHHmoov\ooHumu oHooom may mo mHmmHmcm on» How nouoxo onxuoz .H.m ouome o. 41.5.. - o> .mm .o< chHqucoo ume / am .a> .aH / msOHquooo moon ooude uHo vommouoaoo \“\\\‘ umuuHHom\vHum ou vou wcHuowsaoo umHoH “Hm 114 T m m mx m» th.o a a .Nv.H u x msH.oHHH~.~H mHIHH I H n N I :oHuomuucoo Ho Huumsomm Eu CH mum mconcoEHo "mmuoz EchmsowE wcHumumHoomo mo cmHmoo Hmch ~.m muswwm . .HHm wst HmUHsoo ousmmmud cmH; uHaHH UHHamHuH; aoumHa_IIIH NNNNN\\\N\N\NN\\NK\NN\\\NN\NNNNN ...—IIEAIH \D 0* N NH.m I I E1 I I I I N--NNN~N~NNNNNNNNN‘NNNNNN‘NNNN‘N‘NN‘NNN‘NNNNN N‘NNN‘NNNNb _.I.T_ 115 APPENDIX C: ' Representative unstable, neutrally stable and stable, non-diffusive mixing fields as re- vealed by the presence ($), absence ( ) of the marking contaminant in the scattering volume. The figures of this appendix present additional results for unstable and stable, non-diffusive, mixing fields. Ten experiments for each stability condition were executed; these data were not examined quantitatively. The upper gas (A) and lower gas (8) are identified on each figure. The common conditions for each scan are: S - presence of gas 8 in scattering volume Nominally every other data point in a scan is shown (2,4,6,...200); these are labeled (l,2,3,...100) For the graph horizontal locations: 1 is the 2 volume +37.1 mm and 100 is the 2 value -35.3 mm. For the graph vertical locations: 1 + 120 are the scan numbers; each scan is executed in 3.11 msec and the scans are repeated each 3.5 msec. UO(3.5 msec) = 0.865M ' 116 0 000 0 0 0 0 0 0 00 0 0 0000 0 00 0 . 0 0 0 0 0 0 000 0 0 0 . 0 0 00 0 0 0 0 0 0 00 0 00 0 0 00 00 00. 0 0 00000 u “0 0 000 0 0 00 00 0 00. I . I»..u I I.” .uu.. u... .u.. 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I ..u u u. u.»uuu.uuuuuuuuu.uuuuun 00000000000000000000000 0 0000000000000 0 00 0000000000000000000000, 000000000000000000000000000000000000000000000000000000000 0”00000000 0 000000000 0 0000000000000 0 00 0, 00000000000000000000000000000000000 000000000000000000000 0 00000 0 0 00 0 000 00 0 0 0000000000 0““ 00““00 00000000000000000000000000000000000: 000000000000000000000000000000”lu000000 0H0 000000000”“0000”00000§ 00000000000000000000000000000000000000000000000000000000 000u000n00000000u0u00000 000000 0 000 0000- 00000000000000000000.0000000000000000000000000000000000000 00000 ”0u0u0u0000000000 00 00000000000000.0000“ 00. 000000000000000000000000000000000000000000000000000000000u0 0 0u0M 0 0 0 00000 0000 00 0000000000000 0000”“00 0000000000000000000000000000000000000000000000000000000. 0 0 000 00000000 0 0n0000 000 0000000000 00000 00. 000000000000000000000000000000000000000000000000000000000000 0 000 000000u000“0 0 000000000 000000000000; 0000000000000000000000000000000000000000000000000000000000000 00 “0”0 0 0”00 00 0000 0 000 0 0000 0000 00000000000 000000000000000000000000000000000000000000000000000000000“0”00 000 0 0 0 0 00000000 000000 0000““0luuuuflulluu00 0I000000000000000000000000000000000000000000000000000000 0 000000 000 00 “000 0“00 0 0 00 0000000000 000000000. 0000000000000000000000000000000000000000000000000000000 000 0 0“ 0 0000000000 0 0 0 000 000000000000000000000. 00000000000000000000000000000000000000000000000000000000000000000 "0“0 0 0 0 0 0 00 00000 0000 00 0, 00000000000000000000000000000000000000000000000000000000000000 000 0 u ”0“ “I“ 0000“ 0 0000000000 0000000000‘ 00000000000000000000000000000000000000000000000000000000000000000000” 0 ”00lu000000 000000000000uu0000uu00I 00000000000000000000000000000000000000000000000000000000000000000000 000000 0““0000 0 000000uul0u000flu000000 00000000000000000000000000000000000000000000000000000000000000000000000000 000 0 0 00000000 000000000000000000000 00000000000000000000 0 000000000000000000000000”“0”M00000000000000000000000000u0 0 0 0 0 000 0 000 00000000000 00000000000000000000000 000000000000000000000000 0 00000000000000000000000000 0 0 0 0 00 00000 0000000000000 00000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000 0 0 0 000 0 00000000 00 00 000000000000000000000000000000000000000 00000000000000000000 00 00000 00000000 uMH““00““unfluuu00fluuuufll0ul0000000000000000000000000000000000000000000000000000000000000“ 0H0u0fllfl0u00flufluuuuuu” 00000000000000000000000 00000000000000000000000000000000000000000000000000000000000 000 ”00 “00 ”000 00 00000000000000000000000fl0000000000000000000000000000000000000000000000000000000000000000 0 0000 000 000000000000A 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00”000 00” "0“ “flufluuuflfluul0 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00 0 00” 0 00 0! 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000 00 0“”0 0”0000”00000000 11000010003IIIIIOIIIIIIIstotuau..I. nun... 0n:0.=..unuuunuuuuuuu$. nun... I. ”0.... unuunuunnuuunuu 00:“... I. usuuuu . u Itlsuulr__03u00010000..0..00. ‘ .. ' ’ ‘ . - "Huufluuuuuuunuuuuuuuuu ”0”00000000000000000000000000000000000000000000000000000000000000000 I u0u000u00nuuuuuun I 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0 0 000 00 |)\05010901230 (IIJQMMMO [230007009 00000 0 000000-00 = Freon 22 121), Gas 8 Representation of an unstable, non-diffusive, reon 12 (M.W. mixing field Figure C.l. NOTE: 117 1234.» vuonzuso 109012365610 901zanyumzassanoolzassonsénzu suuéuuumannsnuhzuunw unfit- OdeOMWWNCdOOOUV’ ' ‘” I .. "I: "r: mmwmwmwmzmwmmm m 00 0000 WC moo-cu. ~0 *0. ."NWWWOOOO OOOOMWWWWWWp—o— coo-nunomgoqgsouwSIOwwowaoooaouomzaOJWG~co«ntuwoooaounnwwoqoutmoo-aovia-WacoqouomcowuoUN-‘OOOdO'9W0" WWW I I 3223' I I‘ 0000000000 0 Figure C., 2. Representation of a neutrally stable, non- diffusive mixing field . , - ”w _ . --‘ Note: Gas A = Freon 12 (M.W. = 121), Gas B = Freon 12 (M.WI 121) 118 I 7 3 b 5 o I r v I736» monuusanomznsu an|23b50109ou3¢5018v01736501004012} sonumUhM luuullusonouu‘ush n00 5 g 3 3 I ‘III II I 3 6 II I I I ‘- 5 S 3 9 I h ‘ II 9 10 u I0 u R ‘i a: 5 ‘5 9 .3 I In 19 I 3 i! u '2 3 3 4 s a $5 $9 9 II I 2: a ’ 4 §: i" . 5? 3 II 2 2i . i: 5 I ‘ 35 is a. .. . . gs I I ’ s s s u 39 6 60 *1 ‘ :s a ‘ 8 II II C 65 g, I I I #6 I 61 on II 6. 6. IO 50 ‘ I 50 II I I II I; I III II I; 3‘ :4 33 a: g; II I 2.7, “ R 2 ". “ :5 I: 63 .5 . * 2‘2 63 I II I? ‘3 t s “8 ,o . - 2:» i2 5 1; II I ; A I II I 15 II I ' 3, h . fl TI 10 . . ss~ a} .: ' :3 .6 I II II I II 05 ’9 's s ‘ s 3 =0 ’ II In is II ’. I a? :% 3. “ ‘ ‘2 96 I I I :2 95 I II I 95 ,9 '. :30 0 u I s on N Q0 100 00- 18} “‘s“ ' 85 03 I I I s m l ‘ I m. §2 II‘I‘I I to): 4% ‘ m 1??» ‘3 ‘ (:3 II I I I P, I: 3 us a s ‘1 A II o 5 II I II I‘ Li? I II‘II .: II ’ :7 ‘3 “.. 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Development of Mean Square Displacement in Decaying Turbulence Consider the one-dimensional displacement A2 of a fluid element leaving the origin at t = t : r %E(% A22) = A2 §E(Az) = Az-v (D.1-l) where v is the Lagrangian velocity. 3 1 2 t EE“: Az ) = mt) £v(t')dt' (0.1.2) r Taking the ensemble average over many experiments; 2 t % = f dt' (0.1.3) t r 9_ dt The integrand can be made dimensionless by using the Lagrangian, rms intensity values at t and t'; viz, 5(t) 5(t') = RL(t,t'; tr) (0.1.4) An assumption, of critical importance, introduced by Batchelor and Townsend (1956) is that RL (T-T') is stationary in the decaying turbulence field where T is (locally) defined as _ dt dT - m . (D.1.5) S The symbol ts(t) is an intrinsic time scale which is made specific in the following considerations. If ts(t) is a linear function of the experimental time t, then a tractable mathematical formulation results. A rational approach, which leads to the desired form: t-to ts(t) = c (0.1.6) _ . 121 A is to express tS in terms of an appropriate length scale Lg (t) and velocity intensity v(t) as: L£(t) ts(t) = 5(t) (D.l.7) If the turbulence field is homogeneous and if the ergodic hy- pothesis is assumed, then the Eulerian velocity fluctuation intensity is equal to the Lagrangian intensity and w(t) = K(t--to )-n/2. It will be assumed that the appropriate length scale is propor- tional to its Eulerian counterpart, consequently, L £~(t- to )(l 2). Consequently the assumption that tS(t) is linearly related to the experimental time t is supported and will be henceforth used. The equation for T then becomes: t-to dt' ts(t$ (D.l.8) 1(t) I 0 2o :3 where tr is the reference time at which the tagged particle is released. Equivalently, (t-to)/(tr-to) = exp(T/C). Figure D.l shows a graphical representation of these concepts. The equation for ,defined as 23 for convenience, can now be written as t I = w(t) It w(t') RL (t, t; tr) dt' r d_(.'}_2 —“'t t~l l t .dt 2 2p) — w( ) It w(t ) RL (r-r )dt (0.1.9) r where the stationary character of RL is reflected in the single variable argument (T-T') which has replaced the two variable argument (1,1') and the parametric dependence of RL upon tr is included in the definition of T. Making use of the previous relationships allows (D.l.9) to be written as (1-E - E _ 2 - c(tr to) arm fl Nua N T l w(t) f exp[(§—(1-§)] RL(T-T')dt' (9.1.10) 0. ..H .. J. { . . 122 Equation (D.l.lO) involves a convolution integral. It is advantageous to alter the integrand in the manner shown below. t t g f1(x) f2(e-1)d1 = i fl(B-A) f2(l)dA )K 1(1- __ “'(t) [ golf [_T' (1-52‘4 C W exp E 0 exp E (1-3 = (tr-to) 2 ]RL(T )dT' (Dololl) Integrating (D.l.ll) with respect to time and utilizing the same change of variables yields K 2 T T' 2 - ' u u u z = (tr—t0)( n)(E) é exp[2% (1_%)]dt'£ exp[—% (1-3)]RL(T )dT .NIH 'Ulo (D.l.lZ) Note that this is a generalization of the result expressed in Hinze (1975), eq. 5.186, which iszfor n = l; the above expression is for an arbitrary n.l Since (g) (tr-toéz-n = w2[tr-to](tr-to)2 this product can be used to normalize .Zp as *2 1 22 %-z = §.a2[t -tP](t -t )2 (0.1.13) r o r o Equation (D.l.lZ) can be integrated if a suitable form for RL(T) is available. The analytical description of RL(T") must recognize that the Lagrangian time macrosclae grows more rapidly than the Lagrangian-time microscale. This difference is acknowledged by a two time scale description; viz, T1 = cxznAt/Atr and 12 = cflnAt/Atr. See Figure D.l.b. For short elapsed times from the release of the particle, i.e., for small T values, the Lagrangian correlation function in the T reference frame can be approximated by the following equation: T1 2 RL(T1) = [l - (?—) ] for 1‘: Tj L (D.l.14) 123 t apparent temporal origin of o . turbulence w2 = l [Uot _ Uotol-n U05 A M M t apparent time at which r . . contaminant is released Atr = tr-to, At=t-to = velocity of grid a.) Schematic representation of mixing ts f macroscale t8 = At/Cf b. microscale t8 = At/CA A 4_1., (t - to) ) Schematic representation of Lagrangian time scales development R L 1 solid curve rep- resents R T RL\ ' L” \ 1'2 ' ’ ‘ eXP ("K-) “ 12 c. Definition sketches for T (microscale), T. (juncture point) and'fl (macroscale) for the Lagrangian auto correlation function which is assumed to be stationary in the stretched’1.coordinates Notes: R (t 61:) s Sufi) U(t+At)> = R (I) 5(t) G Tj (D.l.lS) where A is the Lagrangian macroscale. D.2. Dispersion Results for Short Times The symbol T, in this Section, will be used to refer to T1. Making use of (D.l.l4) for small T, (D.l.lO) can be written as 1 *2 1 T 21' T' r" r" 2 7 z = ——E f exp[E——]dT' f exp[-Er][l-(?—) 1dr" Cl 0 l o l L (D.2.1). where C1 = CA/(l'%)° The symbol CA is used in place of C for 2 in (D.l.7), is the microscale. The above integration can be performed equation (D.6) since the appropriate length scale, L directly; the result is: C 2 * %z 2m = —-1n—2 [[exp(-§1) + 11 {-21- - (1‘1" 1 (21.-‘2') - l L T T 2 1 Cl 2 + eXP(E—)[(;—) -2(§ - (;—) )1] (D.2.2) l L L . At . At _ T Transforming back to the XE_ var1able where-KE- - exp(E—) or r r A _ At T — CA £n(-A—t-;) 2 2 2 2m 2 *2 2n (%—f:—)-2[-21- - ($7) 1} r m %z =l_z.{[(%§— +11[%-<%;)1+(%-) ((13?) m r r * -2 Ii“ $0211} (D.2.3) where T = TL/Cl and m = l-n‘Z. Note that the dispersion is para- metrically dependent upon T , hence, I may be inferred from a comparison of (D.2.3) with experimental data at known At/Atr values. Note also that (D.2.3) is only valid for At < Atj. 125 D.3. Dispersion Results for Long Times The symbol T in this section will represent 12. For the condition that T > Tj, the appropriate length scale in (D.l.7) is the macroscale L! which must be evaluated in the decaying turbulence; C, in (D.l.6), becomes CI and it is convenient to define C2 = CI{(1-n/2). Consequently, using (D.l.lS) for RL(T) in (D.l.lZ), z for large I may be written as T. T I 2*2(T) - 2*2(T.) = —25- f exp(%l—)dr' I exp[-T"(%— + %)]dr" 3 cf 13. 2 o 2 (D.3.l) This can be integrated directly; the result is 2*2(T) - 2*2(Tj) = *2 2 l l A l l [eXP[T-(— - )] (——;—) (exp[(T-T.)(— - —-)]-1) (l-n/2)2 3 C2 A? l-A 2 3 C2 A* 1 21. A* 2(T-T.) + -2- exp (E‘l) (-;—-) (exp(——-"—C )-1)] (D-3-2) I 2 A +1 2 Transforming back to the %%— coordinate, * * t r .2 A. may?» rash—=1» l ”(At - 1 ”(é—i) - 1 uT—A m—i) 49L) 1 2 2 FE" '2' 7‘ At ' ‘2' 2 At At r r m A -1 r r + l<—A* )[<9£—)2m- (£51me 2 A*+1 Atr Atr (13.3.3) * where A = A/C2 and At>Atj. Note that the dispersion is defined in terms of the parameter * O A , this parameter will also be estimated from exper1mental data. D.4. Analysis to Define Juncture between Micro and Macro Scales and the CA and Cl Values The juncture value in T1 coordinates, Tj, is defined as the intersection of the Lagrangian RL(T) curves for short and 126 long times; viz, T. 2 -T 1 - ($10 = eprE——%-] (D.4.l) L (—*) Cf Note that “cl/“r2 = CA/Cf which accounts for the Cx/Cf product in the denominator of the right hand side. It is now useful to transform back to the AE— coordinate system where: Atr :1 At. C1 = in(At ) T T T A C1(1-§ l A___ A _ A C ' n - C m f C2(l-§ 2 Equation (D.4.l) becomes, after simplifying, * (1-9) At. At. -(1-3)/A - 2 2 _1__.1 2 _ l 1* 2n (Atr) (Atr) - 0 (0.4.3) It will be useful to have explicit relationships between * * Cl' CI and T , A . These are developed below. The Lagrangian * time microscale can be expressed in terms of T by the following steps. At t -t * L _ L o _ T At t -t - exp( n) (D.4.4) r r o 1-7 The Lagrangian time microscale, 6tL, is defined as tL-tr; hence, dtL = tL-tr = (tL-to) - (tr—to) t -t _ L o -{[t _t 1 - 1} (tr-to) ro- * . . ={exp[ Tn] -1} (tr-to) (0.4.5) 127 The stretching factor between the experimental time and the T lcoordinate, viz, ts‘ is defined in D.l.6 as a linearly in- creasing function of time with C"1 as the constant of propor- tionality. An internally consistent definition of CA is to utilize 6tL(tr) as a particular time scale and (tr - to) as the particular time at which ts=6tL With these arbitrary selec- tions, CA becomes ) - l} (D.4.6) A similar procedure is adopted to compute Cf. Specifically, the Lagrangian time macroscale is defined as tf-tr and since At t -t * f f o = _ epr ] (D.4.7) tr tr to 1-2 2 and tf-tr = (tf-to)-(tr-to) = Gt] Gt]. = (tf-to) - (tr-to) t -t f r = {(-—:——) - l} (t -t ) tr to r o 11* V = {exp[l_£l-l }(tr-to) (D.4.8) 2 Following the pattern which was used to define Cl' C! is arbitrarily defined as the ratio: t-t c=_£_£_= At..- -1 ,, j- 6tf(tr) {expll-gl l} (D 4 9) 128 APPENDIX E A Review and Reevaluation of Shlien's Experimental Results in Homogeneous Decaying Turbulence The experimental investigation by Shlien (1972) has pro- duced the most comprehensive data from which to infer the dis- persive properties of homogeneous turbulent flows. These data have been reanalyzed in terms of the considerations presented in Appendix D. The reinterpretted results are discussed in this appendix. . Shlien measured the time averaged temperature distributions downstream of a line source of heat (i.e., a "tagging wire") in decaying homogeneous turbulence at two reference positions: Uotr/M = 42 and 98. The mean temperature values provide an indication of the dispersive properties of the turbulence: specifically, for a line source, the standard deviation de- termined from a Gaussian fit to the mean temperature measure- ments is equal to the mean square particle displacement under the assumptions leading to (3.2.8). In the analysis of this data, Shlien makes use of Batchelor and Townsend's (1956) dis- persion relation for decaying turbulence, however, his analytical structure is not as extensive as that developed in Appendix D. Specifically, he makes use of the form (D.l.6) and selects the Lfi ande values for (D.l.7) from Eulerian measurements. Consequently, his T values were defined as T(t) = t-to 2n (E:Eg—) EE7W. tr-to where LE and w are the Eulerian macroscale and turbulence in- tensity, respectively. Note that this formulation does not differentiate between the macro and microscale effects intro- duced in terms of T1 and 12 of Appendix D. Apparently, the Shlein analysis is primarily intended for small values of the dimensionless time At/Atr. In order to evaluate the experimental results, Shlien introduced a scaling factor for the dispersion measure, 2:. This scaling factor involves the velocity-time product as an appropriate length scale as was used in (D.l.13) 129 times a similar scale defined at the elapsed time for which the dispersion was measured. In this sense, the scaling involves a function of time which is more complex than the length scale introduced in (D.l.13). Shlien executed the comparison of the theoretical formulation with the experimental results in the transformed coordinates: z*z=z*2(r). Consequently, the coordi- nate stretching, as well as the description of the Lagrangian correlation function, are represented in the graphical pre- sentation and the evaluation of the Lagrangian scales of the data. The present scheme, which allows the data to be pre- sented in the normalized observational coordinates: At/Atr, is considered to offer the advantage that the coordinate stretching does not directly influence the inference of the micro and macro scales. The Shlien data, reprocessed in terms of the present for- mulation, are presented in Figure E.1. The experimental results for the two source locations: Uotr/M = 42 and 98;Zstrongly support the assumed stationarity. That is, the 2 values differ by only 5% for the two source locations over the ob- served values of At/Atr. The following subsections outline the procedures to obtain the parameters 1*, A* and Atj and they present the results of these evaluations. E.2. Summary of Shlien's Results The turbulence field data for the Shlien experiment may be represented by the expressions‘ #(t) _ _1__ {U0t _ Uoto f“ (3.2.1) 2 " l m m U 0 and n L U t U t 1-— — -‘= 0.048{—9—- - ° 0} 2 M M M where Uo = 10 mps, M = 5.04 cm, n = 1.25, Uoto/M = 3.5. The * dispersion results, i.e., z 2, were represented as 130 1 2 *2 2Cf{§ 2p - 0t} _ At 2 ’ ' T ‘ CI in A?” (E 2 3) [Atrw(Atr)] [Atw(At)] r ° ' where Atr = 0.1945 sec which corresponds to a source location at Uotr/M = 42. The micro and macro scales that were inferred by Shlien are TL é 1.5-1.85 and A = 2.4. The former corresponds to 0.069 $.5tL (tr) : 0.083 sec and the latter to 6tf(tr) = 0.110 sec. E.3. Analysis to Define Microscale Ratio (TL/Cl) From Appendix D the equation for the normalized dispersion . At At. can be wr1tten for At_ f-At as r r *2 m 2m (m2 B 2n2(Atr )- 2A) + A ((A-—t——r —) +1)] (3.3.1) NIH t z = —— [(———) m2 Atr A = % - (l/T*)2; B = (l/T*)2; m = l-n/2 Expanding the above function in series for small 6t where At ——— = 1 + 6t Atr 2 l *2 _ l 2 1 _ 3 1_ 2_ _ l_. m (l3m-1) 4 7 z — 5 6t + E (m l)6t + [4:(7m l7m+11) (T*) 12' ]6t 5 + ...0(6t ) (E.3.2) This expression reveals that 1* does not influence 2*2 signifi- cantly for 6t 5 .3. It is also apparent that there is no direct procedure to obtain 1* for small St. A procedure was adopted to find 1* which best matched the experimental data points. However, since 5:1 is not known until both 1* and A* are de- At At. termined, a firstrapproximation to A31 is given as r At * TEE = exp(T n) (0.3.3) r 1': 131 At. At where Ktl i AtL by the definition of the juncture point; ‘ r r At < AtL hence the data should be analyzed for AE—’~ AET" Figure E.1 r r shows a graph of equation (E.3.1) for various values of 1* and the corresponding data. The best choice of 1* was ascertained to be 0.192. This compares to Shlien's corresponding range of (0.132 - 0.164). Since there is no relevant theory to predict the molecular diffusion effects for decaying turbulence for small times, it is'inferred from Saffman's (1960) theory for stationary turbulence that 1* 2 .192. E.4. Analysis to Define the Macroscale Ratio (A/CZ) At. The dispersion equation for %%— > Atl can be written as r r i 2*2 (9.2—) _ l 2*2(£:j_) = 2 Atr 2 Atr A31 A31 1 A22 At' (A* ) At m(x* ) 1 A* At 2m At' 2m —[ H—l) -( ) 1+ —(—;——)((—) - (——l) 11 m2 A*2_1 Atr At; 2 A +1 Atr Atr (3.4.1) where m = l - n/2. The above equation can be readily differentiated, viz. -1 2m m( * ) ] (0.4.2) D n l l > Figure E.2 represents a plot of this function evaluated at * %E— = 3.5 for various values of A . The value of the slope r can be evaluated from Shlien's experimental results. His values 1 *2 for 7 z are presented in Figure E.3 and the derivative of the experimental data, for %E— = 3.5, is 0.32. Using this value of * the slope, the A value ig 0.213 evaluated from equation (E.4.2). The corresponding ratio, using the A and CI from Shlien's results, 132 - is A*=10.21.3 This is a surprisingly similar value considering that his results for A and CI are significantly different from the corresponding results of Section E.5. Since he arbitrarily _chooses Cf, his result perhaps represents a numerical convergence ‘on a value of A which matches the dispersion data. It is useful .to transform the result for A* back to the 92— coordinate to At define the value of At]; viz, r At! A* ' ——- = exp( ) = 1.77 = l + 6t (E.4.3) At n f r 1'3 . E.5. Analysis to Define Juncture Between Micro and Macroscales The juncture point Tj is defined as the intersection of the two approximations to the Lagrangian RL(T) curves; see _(D.4.1). The equation which defines the juncture ratio Atj in the transformed coordinates is written as Atr At. -(E;)=o ——1) A ‘ (0.5.1) 2 t. A Atr. A - E. __l _ 1 (1*) mt) ( r where m = l-%. Solving the above equation using the Newton- Raphson method leads to At. Atl = 1.46 ~ (E.5.2) r ‘— * CA is obtained from equation (D.4.6) utilizing T =0.l9; * C1 = (exp(%—) - 1)"1 5 1.45 . (0.5.3) Similarly, CI is obtained from equation (D.4.9) utilizing A*=o.213; * c = (exp(£—) - 1)’1= 1 3 (0.5.4) I m ' Note that Shlien chose CI = 4.24 (stretching parameter) based on Eulerian measurements,wh0reas the present results show CA = 1.45 Cf =5 10.3 133 It is useful to compare these results of the Lagrangian scales with the corresponding Eulerian scales. 6tL and dtf, which were defined in (D.4.5) and (D.4.8) respectively, can be compared to the local Eulerian time scales measured at U t _§_.= 42 by Comte-Bellot and Corrsin (1971). The results are 6tL(tr) = .130 SEC : At = .0062 SEC (E.5.5) 5tf(tr) = .15 SEC ; Lt = 0.084 SEC (13.5.6) Hence the local Lagrangian microscale and macroscale are 20 and 1.8 times greater than their Eulerian counterparts. The local Lagrangian macroscale is 1.12 times greater than the microscale. The present estimates of GtL and 6t] are 71 and 36 percent greater than the valves inferred by Shlien. E.6 Composite Theory and Comparison of Lagrangian to Eulerian Scales Equations (E.1.l) and (E.2.l) can be combined at the juncture 3:1 to give realistic estimates of the dispersion. Atr Figure E.3 shows a graph of the composite theory and the cor— responding data. Note that the theoretical results have a greater absolute value than the experimental data for %E— > 2.5. It is useful r to introduce Saffman's (1960) conceptual theory for the effects of molecular diffusion for long times. This theory was de- velOped for decaying turbulence and can be represented as -1 (oMz-ZDt) = z;2(l-C 151/2(%) RA ) ~ C ~ 0.23 is estimated from experimental results of Michelson (1960) D - -\)__ 1.43 fix RA: —§3- = 71.6 (from Shlien's data) Substituting these values into (E.6.l), 2 _ -2 (0M - 2Dt) — zp (l-.018) 134 The difference between the theoretical and the experimental At values in Figure E.3 at AE_' ~ r = 3.5 can be represented as (z‘Z-Ut) z;2(1-.022) Hence the two-time scale theory presented in Appendix D agrees with Shlien's data remarkably well. The effects due to molecu- lar diffusion are in agreement with Saffman's theory. A E.7 Estimation of Lagrangian Parameters from the Decay Laws in Grid Turbulence Uberoi and Corrsin [1953], Hinze [1975], and Tennekes [1975] derive an expression for 6tL for large values of R). where there is an appreciable wave number range in the Kolmogoroff spectrum? 2 _ 2 2 -1 1 a ) - C ( ) R (E.7.1) StL 1 L Ag A For isotropic turbulence, the dissipation can be represented as 2 e = 15v (¥- . (0.7.2) 9 Solving for GtL dt = 1_ a (E.7.3) L C A 174 L (es/v) For grid turbulence the decay laws can be expressed as 2 A -n 02(t) = U (39:-- 29:3) (0 7 4) o M M ’ ' 3 n U U t U t -(n+l) — -§.___2. ‘1 .2..- _2_2 c(t) — 2 M A ( M M ) (E.7.5) (E.7.4) and (0.7.5) are used in (E.7.3) to obtain the microscale growth rate as a function of time. (-l/2+1/4n) vA 1/4 Uo -) ] (t-t ) M (3/2n)3/4 o (l/4n+3/4) 1 at = [_ ( E0706 L CL ( ) +Note that the CL used by Tennekes is larger than the present CL by a factor of /2. ' 135 For Shlien's experiment, n = 1.25 and, from (E.7.6), 6tL ~ (t-to)1'065 Consequently, in a strict sense, the microscale does not grow linearly in time. If a linear growth is assumed, a closed form solution of the dispersion equation may be formulated. This mathematical convenience is considered to be sufficient justi- fication for the assumption that n = l in equation E.7.6. Hence, GtL may be written as 1 . 5tL '6'; (t-to) Note that CA is identified as the stretching parameter to render the Lagrangian correlation stationary in T coordinates. Utilizing the numerical values in Section E.2 for the correspond- ing parameters in equation (E.7.6), with n = l, GtL may be expressed as __ 0.117 _ 6tL - ___—CL (t to) Using the data of E.5.5, CL = 0.17. Using the GtL value of 0.076 sec given by Shlien and Corrsins [1974], Tennekes [1975] calculated CL (as presently defined) to be 0.32. It is note- worthy that the difference between GtL and At, which provided the stimulus for the Tennekes paper, is apparently more pro- nounced that was suggested by the previously inferred value of St Thus the ratio expressed in (D.4.4) can be written as: L. t 1/4 T 1 0A exp( _ ) = [1 + ( ) 1 (0.7.7) 1 n72 CL(3/2)3/4 UOM * T can now be expressed from (E.7.7), where CL = 0.17, as T 1/4 _ .1: -0 A. This parameter uniquely determines the dispersion for short 0 * O O t1mes. T is expressed in terms of the Reynolds number for the grid, and the decay relations for the kinetic energy. 136 There appears to be no relevant theory for the Lagrangian macroscale growth rate. Hinze (1975) suggests that dimensional analysis leads to the following result for a spectrum that is dominated by the inertial subrange. 2 6t 3 «He (E.7.9) This result can be expressed in terms of decaying turbulence as 6t! = (l/Cf)(t-to), a result which is independent of the physical constants of the turbulence decay relations which seems unlikely. An alternate approach, utilizing the results of Shlien, assumes that the Eulerian macroscale growth rate is proportional to the Lagrangian macroscale growth rate. It leads to _ E w where the Eulerian scale is represented as n LE Uot UotO (l-I) “'17" AT: 0 T) (3.7.11) Substituting this relation and the decay relation for the kinetic energy, (E.7.10) becomes _ 1/2 _ th - C1 Lo A (t to) (E.7.12) From the corresponding value from Shlien's data this quantity has the value (from (E.5.6)) of fit] - 0.77(t-to). This leads to the result _ 1/2 - 6t] - 3.5 L0 A (t to) (0.7.13) Hence the ratio (A/Cz) expressed in (D.4.ll) becomes * A :0 ‘.> E; = (1-g) £n[l+3.5 Lo Al/Zl (0.7.14) 1137 .AthHV cmaazm EoHu muse mmsHm> u<\u< Hamfim 00m ucmEHuomxm 0cm whomne mo comaudeOU .H .m musmwm 0.N ¢.H m.H h.H w.H m.H v.H N.H N.H. . H.H 1)) q q A. q u _ A d . I!).o \. . MH.~ u o\ mH.O . x mH.O x N m Na H O 2 mm. H o Hu< I Mud mm u u a x qua v as S u .H 0 (WI 8 NV US 0 mhM. "Elfin-IS F P ..HU EV N HUG NE N «I I | HI tl I I III II n l AzmA HV ”_NI A HuuA «< mo cowuocsu m mm a N N\H v .m.m whom“; 0% . . m < o m a. m o A a m. A. o. t m. s. m. N. m o 1!( d . 1. q 4 . q “,4 q no. .1 OH. 1 ON. mutt m.cmfifitm III M E N as ~\A p .H .H .H .H l . _.. TIWGIV I AulualmVA luulmv AH+¥ “‘1 20:; A72: 1 4.0 At A'tr Figure E.3. Two time scale theory compared with the data of Shlien [1972].