THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Aeronautical and Astronautical Engineering Aerodynamics Laboratory Technical Report MEASUREMENTS OF THE CORRELATION BETWEEN THE FLUCTUATING VELOCITIES AND THE FLUCTUATING WALL PRESSURE IN A THICK TURBULENT BOUNDARY LAYER C. E. Wooldridge W. W. Willmarth ORA Project 02920 under contract with: DEPARTMENT OF THE NAVY OFFICE OF NAVAL RESEARCH CONTRACT NO. Nonr-1224(350) WASHINGTON, D. C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR April 1962

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1962.

TABLE OF CONTENTS LIST OF ILLUSTRATIONS NOMENCLATURE ABSTRACT CHAPTER Page v viii xi I. INTRODUCTION II. EXPERIMENTAL APPARATUS AND PROCEDURE A. Wind Tunnel Facility B. Instrumentation III. EXPERIMENTAL ENVIRONMENT A. Extraneous Disturbances B. Properties of the Turbulent Boundary Layer Used in the Investigation 1. Mean velocity profiles 2. Wall pressure spectrum 5. Velocity intensity profiles 4. Velocity spectra IV. SPACE-TIME CORRELATION OF PRESSURE-VELOCITY A. Errors in the Measurements B. Results of the Measurements V. SPACE-TIME CORRELATION OF PRESSURE WITH THE TIME DERIVATIVE OF THE VELOCITY FLUCTUATION NORMAL TO THE WALL A. Errors in the Measurements B. Results of the Measurements VI. CONTOURS OF CONSTANT CORRELATION A. Errors Caused by the Application of the Convection Hypothesis B. The Structure of the Correlation Fields VII. VECTOR FIELD OF THE CORRELATIONS [III. CONCLUDING REMARKS 1 4 4 4 8 8 9 10 11 11 13 17 17 20 23 25 25 27 28 50 54 36 ~ iii

TABLE OF CONTENTS (Concluded) Page APPENDIX. LOW-FREQUENCY FILTER EFFECT 38 REFERENCES 44 iv

LIST OF ILLUSTRATIONS Table Page I. Comparison of Boundary Layer Parameters with Coles' Ideal Turbulent Boundary Layer Parameters 10 Figure 1. Scale drawing of wind tunnel test section and massive vibration isolation mounting for the transducers. 45 2. Pressure transducer and hot wire installation. Hot wire shown at closest spacing to plate, C.050 inch. 46 3. Mean velocity profiles in the turbulent boundary layer. Refer to Table I for other boundary layer parameters. 47 4. Dimensionless power spectrum of the wall pressure. Vertical dashed line shows the frequency below which signals were rejected in the subsequent measurements. 48 5. Turbulent velocity intensity profiles in the frequency band used in the subsequent experiments. 49 6. Intensity profile of longitudinal derivative of velocity normal to wall. 50 7. Dimensionless normalized power spectra of the longitudinal velocity components at various heights in the boundary layer. Spectra are normalized to have unit area in the frequency band of the subsequent experiments. 51 8. Dimensionless normalized power spectra of the velocity components normal to the wall at various heights in the boundary layer. Spectra are normalized to have unit area in the frequency band of the subsequent experiments. 52 9-26. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure. 53 27. Measured values of the spatial correlation of fluctuating longitudinal velocity with fluctuating wall pressure showing symmetry in the transverse direction. 71 v

LIST OF ILLUSTRATIONS (Continued) Figure Page 28. Measured values of the space-time correlations of fluctuating transverse velocity with fluctuating wall pressure showing the effect of extraneous large-scale flow disturbance probably caused by Taylor-Goertler vortices and density stratification. 72 29-31. Measured values of the space-time correlation of the time derivative of fluctuating velocity normal to the wall with fluctuating wall pressure. 73 32. Correlation contours of PRu in the xl-x2 plane. Correlation normalized on the value of the velocity fluctuation at x2/8* = 0.51. Origin of coordinate system at pressure transducer. 76 33. Correlation contours of Rpu in the X2-x3 plane. Correlation normalized on the value of the velocity fluctuation at x2/5* = 0.51. Origin of coordinate system at pressure transducer. 77 34. Correlation contours of R'u in the Xi-x3 plane. Correlation normalized on the value of the velocity fluctuation at x2/6* = 0.51. Origin of coordinate system at pressure transducer. 78 35. Correlation contours of Rpv in the xl-x2 plane. Correlation normalized on the value of the velocity fluctuation at x2/5* = 0.51. Origin of coordinate system at pressure transducer. 79 36. Correlation contours of Rpv in the x2-x3 plane. Correlation normalized on the value of the velocity fluctuation at X2/8* = 0.51. Origin of coordinate system at pressure transducer. 80 37. Correlation contours of Rpv in the xi-x3 plane. Correlation normalized on the value of the velocity fluctuation at x2/5* = 0.51. Origin of coordinate system at pressure transducer. 81 vi

LIST OF ILLUSTRATIONS (Concluded) Figure Page 38. Correlation contours of RI in the xi-x2 plane. Correlation normalized on the value of the velocity fluctuation derivative at x2/8* = 0.51. Origin of coordinate system at pressure transducer. 82 39. Correlation contours of R'. in the X2-X3 plane. Correlation normalized on the value of the velocity fluctuation derivative at x2/8* = 0.51. Origin of coordinate system at pressure transducer. 83 40. Correlation contours of Rp. in the xl-x3 plane. Correlation normalized on the value of the velocity fluctuation derivative at x2/8* = 0.51. Origin of coordinate system at pressure transducer. 41. Vector Field of' Correlation,. Magnitude of the vector at any point is Ru + RPv Direction of the vector at any point as measured from the positive xi-axis is given by tan1 RPv. 85 Rpu vii

NOMENCLATURE E E E Pu'Rp^ I(cf) power spectrum of the wall pressure.cju() power spectrum of longitudinal velocity fluctuation W.v(a) power spectrum of velocity fluctuation normal to plate f frequency k wave number; k = - U PIP total pressure, mean plus fluctuating; see Eq. (7) p fluctuating wall pressure R Reynolds number based on distance from virtual origin of turbulent boundary layer RE Reynolds number based on momentum thickness,Rpw normalized wall pressure-velocity correlations; see, e.g., Eqs. (4) and (5) R v normalized correlation of wall pressure with time derivative of velocity normal to wall; see Eq. (11) Rpvx normalized correlation of wall pressure with longitudinal spatial derivative of velocity normal to wall, see Eq. (9)',Rw standardized correlations normalized on the velocity field at y/w* = 0.51p R normalized wall pressure correlation,R w normalized velocity-velocity correlations U mean velocity in the boundary layer in the stream direction UW free-stream velocity UT wall friction velocity Uc convection speed of the fluctuating velocity field which is correlated with fluctuating wall pressure viii R'u R pu' pi Rul

NOMENCLATURE (Continued) _?9 total velocity, mean plus fluctuating; see Eq. (7) u,v,w fluctuating velocities x,y.z directions rv time derivative of velocity fluctuation normal to wall vx longitudinal spatial derivative of velocity fluctuation normal to wall xl.X2,X3 spatial separations of pressure and velocity transducers in x,y,z directions x distance parallel to wall, increasing in the stream direction y distance normal to wall, increasing away from wall z distance parallel to wall and perpendicular to stream, forming a right-hand Cartesian coordinate system with x and y 5 boundary layer thickness )* boundary layer displacement thickness ~ boundary layer momentum thickness (Apu)i,(Apv)i integral scales of Rpu and Rpv in xi-direction; i = 1,2,3 (Ap+)i integral scale of Rpv in xi-direction; i = 1,235 (Auu)i integral scale of Ruu in xi-direction; i = 1,2,3 (App)i integral scale of Rpp in xi-direction; i = 1,3 v kinematic viscosity p density T time delay fT t time delay equal to transducer spacing divided by local speed ix

NOMENCLATURE (Concluded) JD circular frequency; co = 2itf ( ) time average x

ABSTRACT The purpose of this investigation was to gain additional knowledge about the structure of the turbulent boundary layer flow. Turbulent motion in a thick (5 inch) boundary layer with zero longitudinal pressure gradient which is produced by natural transition on a smooth surface has been studied. Measurements have been made of the space-time correlations between the fluctuating wall pressure and the fluctuating velocities in the layer, and between the fluctuating wall pressure and the time derivative of the fluctuating velocity normal to the wallo A study of the data of other investigators as compared to the data obtained in the present experiments shows that some features of the turbulent structure obtained in many of the previous investigations in tripped boundary layers were not always universal but were a function of the experimental conditions imposed by the tripping device. At any height in the boundary layer the convection speed of the velocity disturbance which was correlated with the wall pressure disturbances was the local mean speed. The correlation of the wall pressure with the longitudinal velocity was opposite in sign to the correlation of the pressure with the velocity normal to the wallo This implies that the part of the velocity field which is correlated with the wall pressure also acts to produce turbulent shear. The integral scales of the pressure-velocity correlation extend approximately one boundary layer thickness along the stream and 0.2 of a boundary layer thickness in the other two perpendicular directionso The wall pressure is an integral of the longitudinal spatial derivative of the velocity normal to the wall over all space This spatial derivative can be related to the time derivative through the convection speed. The measurement of the pressure-velocity time-derivative correlation shows that most of the wall pressure is produced by velocity fluctuations near the surface. The correlation has a measurable value only in the inner 0.4 of the boundary layer. This is in agreement with the fact that the final value of the wall pressure convection speed which is reached at large spacing between the points of measurement, as measured by many investigators, is only about 0.8 of the free-stream speedo This speed occurs at approximately 0.2 of the boundary layer thickness. The integral scales of the pressure-velocity derivative correlation are smaller than those of the pressure-velocity correlationso These scales are approximately 0.2 of the boundary layer thickness along the stream and 0.15 of the boundary layer thickness in the other two perpendicular directions. xi

A vector field of the pressure-velocity correlation data is presented. The hypothesis is made that this field represents the velocity field generated by a superposition of eddies of all sizes whose axes lie parallel to the wall and perpendicular to the mean velocity which are in contact with a layer of fluid near the wall. xii

I. INTRODUCTION The structure of wall turbulence is not yet well understood from either a theoretical or an experimental point of view. Further development of the theory will depend upon use of the knowledge gained from experimental measurements. Previous experimental investigations of the structure of the turbulence, some of which are summarized below, have been concerned with measurements of either the fluctuating velocity field or the fluctuating wall pressure field. The present investigation is concerned with the relation between the velocity field and the wall pressure. One of the first extensive investigations of velocity fluctuations in a turbulent boundary layer was made by Townsend. He found that the principal energy-containing eddies were attached to the wall and lost their energy through direct viscous dissipation near the wall. Later Grant2 measured the spatial correlations of the three fluctuating turbulent velocities with themselves. Evidence was found of jet-like stress relieving.motions originating over a large range of distances from the wall and squirting outward and downstream away from the wall. Favre, Gaviglio, and Dumas3'4 were the first to make measurements with time delay. Exhaustive measurements of the correlation between longitudinal velocity fluctuations in the turbulent boundary layer were made. It was discovered that the lines of maximum correlation at optimum time delay traveled outward from the surface at an angle of approximately 1

2 2~ with the streamlines. Grant2 has pointed out that such lines may represent the average particle paths, as viewed in a laboratory frame of reference, in the mixing jets suggested by himself. However, Favre found the longitudinal scale of the correlation at zero time delay to be three to four times the scale found by Grant. The cause of this discrepancy is unknown at the present time. These measurements by Townsend, Grant, and Favre have made it clear that there is a certain amount of order in the eddy structure of the turbulent boundary layer. More recently, the study of wall pressure fluctuations beneath the turbulent boundary layer has been undertaken. Experimental measurements of wall pressure intensity, power spectral density, and correlation have been obtained by many investigators. The results are summarized and compared in the paper by Willmarth and Wooldridge5 which gives an account of a recent experimental investigation of the wall pressure beneath a thick turbulent boundary layer. This investigation established that the longitudinal and transverse scales of the pressure disturbance on the wall are nearly equal. The pressure fluctuations are found to be convected downstream at 0.85 Uo and it is found that a disturbance of a given wavelength decays after traveling approximately six wavelengths. In view of the previous velocity and wall pressure measurements described above, it was decided to measure the correlation between the fluctuating velocity and the fluctuating wall pressure to obtain additional knowledge about the structure of the flow.

3 The only previous pressure-velocity correlation measurements reported in the literature are those of Kawamura. Kawamura attempted to measure the correlation between the wall pressure fluctuation and the longitudinal velocity fluctuation at zero time delay and zero longitudinal spacing between the points of measurement. The results do not agree with the present work for reasons which are not definitely known. The discrepancy may be due to the low signal-to-noise ratio- reported by Kawamura, to the limitations of his pressure-measuring device, or to the influence of his tripo The present investigation was carried out in a 5-in. thick turbulent boundary layer flow which was assumed to be incompressible. The ratios of pressure transducer diameter and hot wire length to boundary layer thickness were approximately 1:50 and 1:100 respectively, allowing a study of the detailed structure of the fluctuations in the layer.

II. EXPERIMENTAL APPARATUS AND PROCEDURE A, WIND TUNNEL FACILITY The experiments were carried out in the boundary layer on the floor of the 5 by 7 ft low speed wind tunnel facility at the Aeronautical Engineering Laboratories, The University of Michigano A schematic diagram showing the general lay-out is given in Fig. 1. Natural transition took place in the contraction section and no tripping device was required to make the boundary layer fully turbulent. A varnished and waxed sheet of masonite extending 14 ft upstream from the point of measurement was installed to make the wall aerodynamically smooth. In order to eliminate the undesirable effects of wind tunnel vibration, measurements were made on a 1-in. thick smooth (oil-lapped) steel plate, 20 in. in diameter and mounted on a heavy pedestal, which was inserted flush with the floor. The 1/16 in. gap which was allowed between the plate and the wind tunnel floor was sealed on the outside of the tunnel with a strip of rubber. The mounting pedestal was vibrationisolated from the floor by means of rubber shock pads. Holes were drilled in the plate to accept the pressure crystal assembly; holes not in use at any given time were filled with brass plugs which were fitted within ~ 0.001 in. of the surface. B. INSTRUMENTATION The fluctuating wall pressure was measured with a 0.163 ino diam lead-zirconate crystal mounted in a brass body which was supported in the 4

5 steel plate by rubber 0-rings. A schematic diagram of the transducer installation is shown in Fig. 2. The transducer was connected through a low-noise cable to a low-noise preamplifier having a cathode follower input with an input impedance of 1.2 x 108 ohms. The crystal capacity was 285 micro-microfarads, allowing a low frequency response down to approximately 5 cps. The gain of the preamplifier was approximately 50; it was followed by a two-stage amplifier which gave the entire system a maximum gain of 100,000. The bandwidth of the amplifier circuitry was adjustable between 1 cps and 160 kc. The fluctuating velocities were measured with constant current hot wire equipment manufactured by Shapiro and Edwards. The frequency response of the uncompensated amplifier was flat from 0.13 cycles to 320 kilocycles. Adjustable filters in the amplifier were used to cut out frequencies above and below the range needed in the experiments to eliminate unnecessary noiseo Tungsten wires 0.0002 in. in diam and approximately 0.050 in. long were attached to the supports by the usual method of copper plating and soldering. The time constant of the wires was approximately 0.40 msec. The compensation required to correct for the time lag of the wires was determined by the square wave method and accomplished by a resistance-capacitance network in the amplifier. No wire length corrections were applied to any of the data since the microscale of the turbulence near the wall was calculated from the measured spectra and found to be approximately twice the hot wire length. Moreover, the

6 upper frequency limit on the tape recorder described below limits the smallest eddy which can be studied to a scale of approximately twice the hot wire length. In the inner 0,2 of the boundary layer a micrometer screw was used to position the hot wire probe. For the closest spacing to the wall, 0.050 in*., the probe was located by measuring the distance between itself and its surface reflection for greater accuracy. At y/S > 0.2 the probe position was measured with an ordinary scale. A drawing of the hot wire probe for the closest spacing to the pressure transducer is also shown in Fig. 2. The signals from the pressure and velocity transducers were recorded on a three-channel Ampex Model FR-ll00 magnetic tape recorder which utilized 1/2-ino wide tape traveling at 60 in./sec. This recorder has a bandwidth extending from d-c to 10 kc which provided the limiting upper frequency for the measurements. A special play-back unit was designed for the recorder with one movable head to allow the introduction of time delay between a pair of signals. The smallest incremental time delay which could be set was 0.017 msec. The correlator used in the measurement of the space-time correlations worked on the mean-square principle. A thermocouple was used to measure the mean-square values of each of the two signals to be correlated, of their sum. and of their difference. The sum was obtained from a simple resistive summing circuit built into the apparatus, and the difference was obtained by first sending one signal through an electronic phase in

7 verter and then into the summing circuit. A provision for external filtering of the signal going to the thermocouple allowed the measurement of correlations in adjustable frequency bands; a Krohn-Hite Model 310-AB filter was used for this purpose. The output from the thermocouple was measured on a Sensitive Research d-c millivoltmeter having a time constant of 3 sec. The spectral measurements were made using a fixed band-pass General Radio Model 736-A wave analyzer of the heterodyne type. The thermocouple circuitry in the correlator described above was used to obtain the meansquare value of the fluctuating output from the wave analyzer. Rootmean-square measurements were made with a Ballantine Model 520 true rms meter. Electrical signals were monitored with a DuMont Model 522-A dualbeam oscilloscope. For the measurements of the correlation of pressure with velocity derivative, the velocity signal from the tape recorder was fed into a simple resistance-capacitance differentiating network having a linear response up to 10 kc. The attenuated output was amplified and fed into the correlator.

III. EXPERIMENTAL ENVIRONMENT A. EXTRANEOUS DISTURBANCES The accuracy of the measurements to be reported herein will be affected by the extraneous disturbances which are present. These include the vibration of the measuring apparatus, the sound field in the wind tunnel, the free stream turbulence level, and the mean flow conditions in the boundary layer. A comprehensive discussion of the sound measurements and flow visualization studies which were carried out is given in Ref. 5. A short summary of the results will be given here. A check on the effectiveness of the vibration isolation of the mounting showed that the energy which was present because of vibration amounted to less than 1/100 of the energy in the turbulent pressure fluctuations and less than 1/100 of the energy in the turbulent velocity fluctuations near the edge of the boundary layer. The sound field in the test section was first measured by a pressure transducer located on the stagnation line of an airfoil-shaped body exposed to the free stream. The spectrum of the stagnation pressure fluctuations had peaks at 135 and 200 cps. The wall pressure correlation measurements described in Ref. 5 which were made later showed a small peak at negative time delay which was caused by sound propagating upstream. From these data it was finally determined that the sound energy amounted to approximately 1/20 of the energy in the turbulent wall pressure fluctuations. The mean square velocity fluctuations associated with 8

9 the sound would amount to less than 5/1000 of the mean-square turbulent fluctuations near the edge of the boundary layer. The settling chamber of the wind tunnel contains four turbulence damping screens. The free-stream turbulence level in the test section increases with velocity and has been measured at 50, 100, and 150 ft/sec. Extrapolation of the data to the 200 ft/sec speed used in this experiment u2 -4 results in a value of - = 6 x 10 for the turbulence level of the axial U06 velocity component. The level of the transverse velocity component is approximately three times the level of the axial component. Large-scale flow disturbances in the test section boundary layer were first discovered during the measurements of the wall pressure spectra which are described below. The entire wind tunnel, with the exception of the test section, is exposed to the weather. Heat transfer through the steel walls from sunshine produced a density stratification near the walls which in turn produced vorticity when accelerated into the test section. Observations of the streamlines on the concave surface of the contraction section showed large scale oscillations which were swept into the test section. It is believed that the large-scale disturbances observed in the test section are caused by a combination of the TaylorGoertler boundary layer instability on the concave walls of the contraction and the density stratification. B. PROPERTIES OF THE TURBULENT BOUNDARY LAYER USED IN TIE INVESTIGATION All measurements were made with natural transition in the boundary

10 layer which occurred approximately 24 ft ahead of the measuring point. 1. Mean Velocity Profiles.-The mean velocity profiles were measured with a total head tube; the static pressure was measured on the wall. No correction was made for the effect of turbulence on the mean values. Since the data presented in this report were taken over a period of time when the ambient temperature changed by some 200, velocity profiles were taken at the extremes of temperature. The results were plotted on a non-dimensional basis in Fig. 3. The wall friction velocity was obtained from a measurement of the wall shear stress with a Stanton tube using the calibration results reported by Gadd. Also shown in Fig. 5 is the ideal turbulent boundary layer profile of Coles.9 The properties of the measured boundary layer profile are tabulated in Table I together with the properties of Coles ideal boundary layer at the same value of Re, the Reynolds number based on momentum thickness. Figure 3 and Table I show that there is satisfactory agreement with the ideal case. TABLE I COMPARISON OF BOUNDARY LAYER PARAMETERS WITH COLES' IDEAL TURBULENT BOUNDARY LAYER PARAMETERS T UOO Re 5' e Q */e UT/Uoo R Remarks 67 204 8,00ooo 0.42 0.041 0.0315 1.50 0.0326 3.1 x 107 Present investigation. 38,000 1.30 O. 0318 3.2 x 107 Coles' ideal boundary layer based on same Re~ 45 203 43,000 0.42 0.041 0.0315 1.30 0.0325 3.85 x 107 Present investigation. 43,000 1.295 O. 0315 4.0 x 107 Coles' ideal boundary layer based on same Re.. ~~~~~~~)

11 2. Wall Pressure Spectrum. -The non-dimensional spectrum of the wall pressure is shown in Fig. 4o The data were always repeatable for CaZ* caE* U-> 0.13, but were not repeatable for U-. < 013 (f < 105 cps). The results varied with the amount of heat transfer to the tunnel by sunlight. This variation is believed to be caused by the large-scale disturbances which arise from the density stratification discussed earlier in Section III-A. In the measurements of mean-square pressure and velocity and spacetime correlation of pressure with velocity the Krohn-Hite filter has been used to reject all frequencies below -- = 0513. The upper limit on the U00 frequency, - = 12.5, was provided by the tape recorder response. It is shown by the pressure-velocity correlation measurements reported in Section IV that the disturbances at any height in the boundary layer are convected at the local mean speed. At the closest spacing of the hot wire to the wall, 0.050 in., the local speed is approximately 0.58 UO. Since the boundary layer thickness is 0.42 ft, disturbances convected at this speed which produce pressure fluctuations at - = 0.13 Uco have a wavelength of 2.75. Hence, even with filtering information can be obtained about eddies whose scale is at least 2-1/2 times the boundary layer thickness. The root-mean-square value of the wall pressure fluctuation was 2.19 times the wall shear stress and 4.7 x 10 times the free stream dynamic pressure. 3. Velocity Intensity Profiles.-The intensity profiles of the ve

12 locity fluctuations measured in the frequency band 0.13 < 5 - < 12.5 are U00 shown in Fig. 5. A comparison with the work of Klebanoff10 shows that in the present case there are higher intensity levels in the inner half of the boundary layer and lower levels in the outer half of the layer for all three turbulence components. However, there are two main differences in the experimental conditionso In the present investigation there was natural transition of the boundary layer whereas Klebanoff used sandpaper roughness on the, leading edge of his plate to trip the boundary layer. Also, the Reynolds number based on momentum thickness in the present experiment is approximately six times as large as Klebanoff s value. It should be noted that Klebanoff and Diehl11 measured the u intensity profile in 1952 under the same operating conditions as those of Klebanoff10 in 1955 and found values as much as 305 higher. Accordingly, it is believed that the intensity profiles of the turbulence in the boundary layer which are measured in any given case will be.affected by the tripping device, if any, which is used and will depend upon the value of the Reynolds number in some unknown way. This is supported by the measurements of Ref. 5 which showed that either wall roughness or a boundary layer trip acted to increase the wall pressure intensity. The intensity profile of the longitudinal derivative of the velocity fluctuation normal to the wall is shown for the inner half of the bound10 ary layer in Fig. 6. A comparison with the data of Klebanoff shows that the present results are approximately 2-1/2 times as large near the wall and 1-1/2 times as large near the center of the layer. It is believed

that this discrepancy is also due to either the boundary layer trip used by Klebanoff or the difference in Reynolds number. 4. Velocity Spectra. -The frequency spectra of the u and v velocity fluctuations which were measured directly with the wave analyzer have been converted to wave number spectra by assuming that the disturbance at any height is convected at the local mean speed and decays slowly. In this case the wave number is defined by k = - (1) and the relation E(k) = UE(n) (2) holds. These spectra which are shown in Figs. 7 and 8 are normalized to have unit area above - = 0.13; the values of U corresponding to this are marked by the flagged point on each curve. The non-dimensional u spectra, Eu() scale with the local nondimensional wave number, kS* = -U except for the spectrum at the closest spacing to the wall at large wave number. This result implies that the disturbances in the frequency band under consideration are convected at the local mean speed. Moreover, the disturbances must be decaying slowly with time in this frequency band since only in the case of a completely frozen flow with no decay would the u spectra collapse perfectly into one universal curve. The spectrum near the wall at large wave number deviates from the other spectra because the small scale disturbances which are

relatively abundant near the wall decay relatively rapidly. These data can be compared with the velocity-velocity correlation data obtained by others. The wave number spectral density, Eu(k), is the transform of the longitudinal spatial correlation of the u velocity component; i.e., Eu(k) = - J RUU(Xi) cos kxidx. (35) If EU(k) is a function of k only, as the present data shows, then Ruu(xl) must be a universal function which is independent of the height in the boundary layer. Hinze7 reports that the shape of the correlation curves Ruu(xi) measured by many investigators (see, e.go, Grant2) is independent of height except near xi = 0. The shape of Ruu(xl) near xl = 0 is governed by the behavior of the spectrum at large wave number (see Hinze7) and the non-uniform behavior of Ruu in this region is believed to be caused by the decay of the large wave number (small scale) fluctuations. For the same reason the spectral curves of Eu(k) must separate at high wave number. The upper frequency in the present investigation was limited by the tape recorder response, and the data does not extend to a high enough wave number to show this effecto However, the separation at high wave number is shown by the data obtained by Klebanoff who obtained measurements at frequencies higher than those studied in the. present investigation. In the outer half of the boundary layer, Klebanoff' s u spectra in the frequency band corresponding to the present measurements also scaled

15 with the wave number based on the local velocity. In the inner half of the layer, however, the spectral curves were separated. The normalized curves showed relatively more high frequency energy near the wall. It is not known at this time whether this disagreement with the present data is an effect of the different Reynolds number or of the trip which was used by Klebanoff. However, the power dependence on non-dimensional wave number of the present spectral measurements agrees well with the power dependence found by Klebanoffo The normalized v spectra shown in Fig. 7 do not scale with the wave number based on local velocity. Equivalent behavior is found by.other investigators (see, e.g., Grant2) who have measured the longitudinal correlation of the u velocity and found the shape of Ruu(xi) to depend upon the height in the layer.* Therefore, it cannot be determined from the spectral curves whether or not the v disturbance is also convected at the local speed. However, the measurements of correlation between pressure and velocity which are presented in Section IV establish that this is the case o The v spectra at the two points farthest from the wall exhibit humps near -U = O4o It is believed that these anomalies in the v spectra are caused by the large-scale extraneous disturbances discussed in Section III-Ao Evidence to support this conclusion was obtained during *These results imply that the v velocity fluctuation near the wall decays at a faster rate than the u velocity fluctuation in this region. This finding agrees with the results of the pressure-velocity correlation measurements described in Section IV which show that near the wall the Rpv correlation decays faster than the Rpu correlation.

16 measurements of the pressure-velocity correlations and is reported in Section IV.

IV. SPACE-TIME CORRELATION OF PRESSURE-VELOCITY The fluctuating wall pressure and the fluctuating velocities in the boundary layer are stationary random functions of time and position. The pressure-velocity cross-correlation coefficients which have been measured are defined by Rpu( x, x2,3;T) = p (x, yz;t)u(x+x, y+x2 z+X3; t+T) (4) p2(x,y,z;t) /u2 (x+x1, y+x2,z+x3;t) and Rpv(Xl, x2,x; *T) = p(x,y,z;t)v(x+xl, y+x2,z+X3;t+T) () Rpv(X1,X2, X3;rT) = j _ - - i \ p2(x,y, z;t)j v2(x+xl,y+x2, z+x3; t) The experimental measurements are shown in Figs. 9 through 26 as U.(T-T1) functions of the non-dimensional time delay --.- The shift of the U' origin on the time delay axis, given by -,, is computed at each value of - by dividing the longitudinal spacing i by the non-dimensional local mean speed f-. A vertical line on each curve shows the true origin U T A. ERRORS IN THE MEASUREMENTS The averages involved in the computation of the correlation coefficients were obtained by visual observation of the output meter reading. Since the meter had a time constant of 3 sec, the fluctuations in the needle reading were small and the probable error in the correlation arising from this source is estimated to be + 5% of the maximum value. 17

18 It was pointed out earlier in Section III-B that it was necessary to filter out the low frequencies, U — < 0.13. Such filtering will change the cross-correlation coefficient from that which would be measured if there were no filtering. The correlation will be changed in two ways: 1. the filter acts directly to destroy the correlation in the frequency band below its cut-off point, and 2. the filter acts indirectly to boost the value of the correlation coefficient at all values of time delay by reducing the magnitude of the denominator. To estimate the amount of correlation caused by convected eddies which is destroyed by the filter, it is assumed that the wave number cross-spectral density is a slowly-varying function of time in a reference frame moving at the convection speed. In this frame the value of the spectrum below the wave number corresponding to the filter cut-off frequency is assumed to be constant and equal to the value at the cut-off frequency. An analysis made by proceeding on this basis is given in the appendix. The results of this analysis show that the direct effect of the filtering changes the correlation curves most near their tails. This is to be expected since at large time delay the high frequency (small scale) velocity fluctuations have decayed, making the low frequency (large scale) fluctuations relatively more important. The filter effect also increases with distance from the wall since low frequency fluctuations become increasingly important in the correlation measurement as this distance increases. In no case, however, is the amount of correlation coefficient destroyed by

19 the filter larger than 1Co of the peak value. The indirect effect of the filtering is to increase the measured values of R by approximately 15% and those of R by approximately 8% above the values which would have been measured with no filtering. The boundary layer should be homogeneous in the z-direction; one would expect Rpu and Rpv to be symmetric about x3 = 0 and Rpw to be zero. Symmetry about x3 = 0 was found in the measurements; see, for example, Fig. 27. The cross-correlation coefficient Rpw5 defined in a similar manner to Rpu and Rpv, was measured at various values of - in the two frequency bands 0,15 < 1 < 12.5 (the usual band) and 0.38 < - < 12.5. Some typical results are shown in Fig. 280 The measured correlation is small but definitely not zero, and has approximately the same scale throughout the boundary layer. The transverse scale across the stream is only about 0.1 of the longitudinal scale along the stream. Filtering of the frequencies below 300 cps (- < 0.538) destroys most of the correlation. The reason for the lack of symmetry UcOT about i. = 0 is unknown at the present time. It is believed that this Rpw correlation is caused by a vortex whose axis is in the longitudinal direction and near the edge of the boundary layer. This vortex is believed to be a result of the Taylor-Goertler instability and the density stratification which were discussed earlier in Section III-A. Unfortunately, no measurements of Rpw were made at different cross-stream locations; such measurements might have shed more light on the nature of the ihhomogeneity.

20 The filter cut-off frequency U- = 0.38 above which the Rpw correlation is nearly gone corresponds closely to the frequency at which the v spectra exhibited a peak. It is believed that both phenomena are connected with the extraneous vortex disturbance. In this case the value of the Rpv correlation will also be affected near the tails of the curve where the low frequencies make a sizable contribution, but this effect is believed to be small. In view of these results the measurements are believed to present a valid description of the fluctuating flow field which is produced by the convected disturbances. B. RESULTS OF THE MEASUREMENTS The convection speed, Uc, of the disturbance which gives rise to the pressure-velocity correlation is defined as the speed of travel of the center of the disturbance where the correlation is zero. Examination of Figs. 9 through 26 shows that at any height in the boundary layer this convection speed is equal to the local mean speed; i.e., Uc = U. (6) When normalized on local root-mean-square values of the fluctuating velocity, the pressure-velocity correlation coefficients near the edge of the boundary layer have approximately the same magnitude as those near the wall. When the correlation coefficients are normalized on the values of the fluctuating velocities at one point, however, the magnitude of the

21 correlation near the edge of the layer is only 0.01 of the magnitude near the wall. It is shown in Section V that the wall pressure fluctuations are produced by the interaction of the longitudinal derivative of the velocity fluctuation normal to the wall with the mean shear in a region close to the wall. It is believed that the wall pressure is correlated with the fluctuating velocities far from the wall because these velocities are correlated with the velocities near the wall which do produce the pressure. One might hypothesize that the fluctuating velocities near the wall which are involved in the production of the wall pressure are caused by those eddies of various sizes which are in contact with a layer of fluid very close to the wallo These eddies extend outward into the stream varying distances and the largest scale of such eddies is probably of the order of the boundary layer thickness. The values of the correlation coefficients based on the root-mean-square velocity fluctuations at one point decrease with increasing distance from the wall because less and less of the eddy field is seen by the hot wire as this distance increases. A further discussion of this hypothesis is given in Section VII. Figures 9 through 26 also show that the pressure-velocity correlations Rpu and Rpv at any height have nearly equal magnitude and opposite algebraic signs except at large values of negative time delay where both are small and negative. This result implies that the velocity disturbance which is correlated with the wall pressure also acts to create at least part of the turbulent shear. Further investigation is needed, however,

22 before the exact relationship between the turbulent shear and the fluctuating wall pressure will be known.

V. SPACE-TIME CORRELATION OF PRESSURE WITH THE TIME DERIVATIVE OF THE VELOCITY FLUCTUATION NORMAL TO THE WALL In incompressible flow the equation of continuity can be combined with the equations of momentum to give a Poisson equation for the pressure, v2 = - Pxj Jx (7) in tensor notation, F and 2 represent the total pressure and total velocity respectively. The equation for the fluctuating pressure can be obtained by setting each flow quantity equal to a mean value plus a fluctuation, expanding the derivative, and subtracting the equation for the mean pressure from the total equation. Kraichnan12 has shown that the dominant term in the equation for the fluctuating pressure is 2 = -2p. (8) Since the wall pressure p is produced by the values of the deriva6v tive - which exist throughout the velocity field, a measurement of the ax correlation of p with - will show which regions of the fluctuating veax locity field do not contribute to the wall pressure. The cross-correlation coefficient Rpv, where vx = x-, is defined by p(x,y, z; t )vx(x+xl,y+x2, +X3; t +T) Rpvx - _ _ (9) p2(xyz;t) v (x+xl y+X2 z+x3;t) 23

This correlation is difficult to measure experimentally. However, if the velocity disturbance is convected past the hot wire with little distortion (see Section VI), the equation V = v- lv (10) ax Uc at can be used to relate space and time derivatives. In this case R = - R av where v-= t. The cross-correlation coefficient Rp - p(xy,z;t)v(x+xl,y+x2,z+x3;t+T) (11) p2 (x,y,z;t)j 2 (x+xl,y+x2,z+X3;t) can be easily measured by differentiating the velocity signal electrically and correlating the result with the wall pressure. The measured values of this correlation coefficient are shown in Figs. 29 through 31, again plotted versus the non-dimensional time delay Uo(r-T'). A vertical line on each curve for 0- > 0 shows the true origin U00T -O, = 0. These measurements again illustrate the fact that the convection speed of the velocity disturbance which is correlated with the wall pressure is equal to the local mean speed. A very simple mathematical relationship exists between the correlation coefficients Rp and RpvB R = a R V- (12) Rp~ = ~-T Rpv

25 The decrease in the slope of Rpv as the distance from the wall increases means that velocity fluctuations far from the wall have comparatively little effect in producing pressure fluctuations at the wall. However, these velocity fluctuations are correlated with the wall pressure because they are correlated with the velocity fluctuations near the surface which do produce the pressure (see Section IV-B). A. ERRORS IN THE MEASUREMENTS As in the measurement of the pressure-velocity correlations, the probable errors which arise from the averaging process are estimated to be + 5% of the maximum value of the correlation. During the measurement of Rp~ the frequencies corresponding to < 0.13 were filtered out as before. An analysis of the low-frequency U00 filter effect, based on the assumption of a constant wave number spectral density below the filter cut-off point as before, is given in the appendix. This direct filter effect of destroying the correlation at low frequencies is found to be negligible. The indirect effect of the filter reduces the size of the normalization factors at all values of time delay. The measured values of Rpr are increased approximately 6% by the filtering. This error is comparable to those occurring in the pressure and velocity measurements. B. RESULTS OF THE MEASUREMENTS The curves of Rp+ given in Figs. 29 through 31 are normalized on the local values of the fluctuating velocity field. When normalized on

26 the velocity at one point, the data exhibits the same general behavior. A maximum in the peak value of the correlation is reached at a short distance from the wall; the peak value then decreases with increasing distance from the wall and becomes negligible for - > 4.08. The convection speed of the disturbance which gives rise to Rp is the local mean speed in the boundary layer as pointed out at the beginning of this section. On the other hand the convection speed of the wall pressure disturbance has been determined in Ref. 5 to vary from 0.56 U0 to 0.85 Uo, as the spacing between the measuring points is increased. Since the wall pressure fluctuation, p, is an integral over all space of the velocity fluctuation derivative, v-, the final value of the convection speed of the wall pressure disturbance which is attained at large transducer spacing should not exceed the local speed in the boundary layer where Rpv becomes negligible. At = 4.08 the local speed is approximately 0.9 of the free-stream speed and the restriction on the convection speed is satisfied. The speed 0.83 UN occurs as the local speed in the boundary layer at A- = 2.25 (y - 0.22).

VI. CONTOURS OF CONSTANT CORRELATION The statistical structure of the turbulent velocity field which creates the turbulent wall pressure is shown by the spatial correlation contours of Rpu, Rpv, and Rpvo A direct measurement of enough data points to allow the construction of these contours would be extremely time consuming; furthermore, it was found during the experimental measurements that interference effects between the hot wire probe and the pressure transducer made it impossible to obtain reliable data with the hot wire ahead of the pressure transducer, xl < 0. The measurements for xl > 0 show that the rate of decay is slow and that the convection speed is the local speed as pointed out in Section IV. Hence, it is possible to obtain approximate spatial correlation contours from the measured time-delay correlations at zero longitudinal separation by using the convection transformation x = X U (13) a5, 5a* U0O The correlation contours obtained by the use of this transformation are shown in Figso 52 through 40 where the origin of the coordinate system is located at the point where the wall pressure was measured. The values of the correlation coefficients which are shown have been standardized using the root-mean-square values of the velocity fluctuations at y/5* = 0.51 as normalization factors; ioe.o at any position 27

28 R = R local U0.51 R' = R vIocal (14) p pv pv - _ 0. 51 vlocal R'. = R. - pv Pp v V0.51 The ratios of the root-mean-square velocities and velocity derivatives are obtained from the intensity profiles in Figs..5 and 6. A. ERRORS CAUSED BY THE CONVECTION HYPOTHESIS The validity of the convection hypothesis clearly depends upon the spatial rate of change (growth or decay) of the velocity disturbance. The velocity spectral measurements reported in Section III-B-4 above show that the u velocity disturbance decays slowly and lend support to this hypothesis. The amount of error introduced can be determined at those values of X1 E- for which data were taken by comparing the measured values of the spatial correlation at zero time delay with the values predicted by the convection hypothesis. These data show that the use of the transformation is most in error near the wall. This is to be expected since the high frequency (small scale) velocity fluctuations which make a measurable contribution to the correlation near the wall decay more rapidly than the lower frequency fluctuations farther from the wall. The u velocity spectra discussed

29 previously in Section III-B also tended to show that the hypothesis of convection is most in error near the wall at the high frequency end of the spectrum. The measurements also show that the error introduced in Rpv by the convection transformation near the wall is greater than that introduced in Rpu. This effect is also apparent in the velocity correlation data obtained by Grant2 which shows that correlation of the velocity component normal to the surface decays much faster than the correlation of the longitudinal component parallel to the surface (see also Section III-B-4). The reason for this behavior is not known at the present time. The maximum error in the correlation near the peak value of Rpv is approximately + 20% of the peak value near the wall; for y/k* > 0.2 the error is + lOo or lesso The errors in Rpu near the peak values are ~ 10o or less of the peak value for all heights in the boundary layer. A few points measured at large values of A- show that the errors in the tails of the curves for large time delay are also approximately ~ 10% of the peak value. Since Rp is the derivative of Rpv, the error in Rpv near its peak value can be obtained by extrapolating the Rpv data which were measured at various values of x to ~ = 0 and comparing the slope obtained to the slope predicted by the convection hypothesis. The results show that x2 the error in Rpv is a maximum near the wall at 3- = 0.1 and is at most + 350. At all other spacings the error in Rp is less than ~ 10 l.

30 B. THE STRUCTURE OF THE CORRELATION FIELDS A few general remarks can be made about the shapes of the correlation contours. The contour curves of Rpu and Rpv in the Xl-X2 plane (Figs. 32 and 35) show that the scale perpendicular to the wall is slightly larger downstream from the pressure transducer than upstream. This is also shown in the contours in the x2-X3 plane (Figs. 5533 and 36). The Rpv contour curves in the xl-x3 plane (Fig. 37) show that near the wall the contours bend upstream. The Rp' contours (Figs. 38 through 40) are related to the Rpv contours through the differentiation noted previously. The region of negative Rp, (corresponding to positive RpvX) has a slightly larger scale in the transverse than in the longitudinal direction. The spatial extent of the turbulent velocity field which is correlated with the wall pressure can be put on a quantitative basis by defining integral scales. The integral scales of Rpu and Rpv in the xidirection are defined to be''maxi(15) (APV)i = IR1 v Imax) For the scale normal to the wall the integration extends from zero to infinity. Calculation of this basis shows that: 1. The longitudinal scales of R and R are approximately equal to the boundary layer thickness, b, being slightly smaller near

31 the wall and slightly larger near the middle of the layer. 2. The transverse scales parallel to the wall and perpendicular to the mean velocity increase with distance from the wall and reach a maximum value of approximately 0258 at a height of approximately 0.25 in the layero Beyond this height the scales decreaseo 35. The scales normal to the wall are a maximum at x3 = 0. The scale of R is approximately 0.25 and that of R is 0.53 at xl A- = - 1.25; these scales are approximately l1o larger at 5- = + 1o25. These results demonstrate that the region of the velocity field which is correlated with the wall pressure is comparatively long in the longitudinal direction and narrow in the other two perpendicular directions. A spatial scale for the velocity correlations measured by Grant and Favre3'4 can be defined by (Au)i = 2 f Ruu Qi d ( > (16) Using this equation, Grant's longitudinal scale is calculated to be approximately 0.45 and Favre's to be approximately 1.05o In the transverse direction both investigators find a scale of approximately 0536. The reason for the discrepancy in the longitudinal scale is unknown at the present time but probably involves the differences in the tripping devices used in the two experimentso These results show that even though a fundamentally different measurement is involved, the velocity-velocity correlation scales are of the

32 same order as the pressure-velocity correlation scales. The spatial scales of Rpv are defined in a similar way to those of Rpu and Rpv by.1 | r1 f IR|p (- I d(-) (17) These scales are smaller than the corresponding scales of the pressurevelocity correlations. Calculation shows that 1. The longitudinal scale is approximately 0.25. 2. The maximum transverse scale is approximately 0.155 at a height of 0.1&. 3. The scale normal to the wall, a maximum at x3 = 0O is approximately 0.155 at xl = 0. A scale for the wall pressure correlation reported in Ref. 5 can be defined by ( A f Rpp x ( (18) The longitudinal scale of the pressure-pressure correlation calculated in this way is approximately 0.255; the transverse scale is slightly larger than this. Hence, the integral scales of Rpp and Rp+ are nearly the same. Since the wall pressure is an integral of the velocity derivative over all space, this result shows that the present data is consistent with the data obtained in Ref. 5. The scale of R. normal to the wall is only about one-half of the

33 scale of R normal to the wall. This lends support to the hypothesis (Section IV-B) that even though only the derivatives of the velocity fluctuations near the wall create the wall pressure, these velocity fluctuations are part of an eddy system which extends farther outward into the flow.

VII. VECTOR FIELD OF THE CORRELATIONS The correlation vector field. shown in Fig. 41 has been constructed with the origin at the pressure transducer such that: 1. the length of the vector at any point is given by Rpu+R pv, where the correlations are based on the local root-mean-square values of the velocity fluctuations, and 2. the direction of each vector, measured from the positive x1-axis, is given by tan1 Rpv where the wall pressure pulse is assumed Rpu to be negativeo It is known that part of the velocity field which produces the fluctuating wall pressure is convected at the local speed in the boundary layer with a slow rate of decay. The wall pressure p depends upon the 6v values of -x in the vicinity of the wall. The velocity fluctuations are caused by a collection of vortex filaments with random orientation. It is hypothesized that those filaments which lie along the x3-direction make the major contribution to Rpv as the velocity field is convected along the wall, but it is admitted that further experiments are needed to clarify this point. Furthermore, it is known from the measurements of Rpv reported in Section V that only those filaments whose influence extends to the vicinity of the wall act to generate the wall pressure. On this basis it is further hypothesized that the velocity field which is correlated with the wall pressure is generated by a system of eddies of all sizes which are in contact with a layer of fluid very close 54

to the wall (see also Section IV-B). The correlation vector field shown in Fig. 41 is believed to represent the velocity field which is generated by a superposition of these eddies. As pointed out above, the field was mapped by assuming the pressure pulse to be negative. The rotation of the vector field in Fig. 41 is then clockwise which is reasonable from a consideration of the mean shear. There is some evidence in the Rpv and Rp correlation contours in the xl-x3 plane (Figs. 37 and 40) that near the wall the vortex filaments which generate the vector field bend upstream. Further experimental investigation is needed to positively confirm or deny the hypothesis of the hierarchy of eddies which is set forth here, but all of the present data support it.

VIII. CONCLUDING REMARKS The main results of this experimental investigation can be summarized as follows: 1o Evidence has been accumulated that a boundary layer trip has a rather severe effect on the properties of turbulence in the boundary layer. From a study of the data of other investigators and a comparison with the present investigation, it is evident that a trip acts to increase the wall pressure intensity and the velocity intensity near the wall. The experiments of Grant2 and Favre3'4 show that the integral scale of the longitudinal velocity correlation is also affected by the trip used. It now appears that some features of the turbulent structure obtained in many of the previous investigations in tripped boundary layers were not always universal but were a function of the experimental conditions imposed by the tripping device. The results of the present investigation with natural transition of the boundary layer are believed to be universal because no tripping device was required to make the boundary layer fully turbulent. 2o The convection speed of the velocity disturbance which is correlated with the wall pressure disturbance is the local mean speed. 35, The R and R correlation coefficients are of opposite sign. This implies that the part of the velocity field which is correlated with the wall pressure also acts to produce turbulent shear stress in the boundary layer. 36

37 4. The measurement of Rpv shows that most of the wall pressure is produced by velocity fluctuations near the surface. 5. Velocity fluctuations far from the wall are correlated with the wall pressure because they in turn are correlated with the velocity fluctuations near the wall which do produce the wall pressure. 6. Present measurements of the Rpv correlation show that the convection speed of the wall pressure disturbance should not exceed 0.9 Um, the local speed at y/& - 0.4. The measurements of Ref. 5 showed the value of this convection speed to be 0.83 Uo at large transducer spacing and the restriction is satisfied. The speed 0.83 U, occurs in the boundary layer profile at y/6 - 0.22. 7. The integral scales of Rpu and Rpv are approximately the same as the integral scales of Ruu measured by other investigators. The longitudinal scales of Rpu and Rpv are approximately equal to the boundary layer thickness, 5. The transverse scales are approximately 0.28 and the scales normal to the plate are approximately 0.2& and 0.35 for Rpu and Rpv respectively. 8. A vector field of the pressure-velocity correlation data is presented. The hypothesis is made that this field represents the velocity field generated by a superposition of eddies of all sizes whose axes lie parallel to the wall and perpendicular to the mean velocity which are in contact with a layer of fluid near the wall.

APPENDIX LOW-FREQUENCY FILTER EFFECT

LOW-FREQUENCY FILTER EFFECT During the analysis of the experimental data it was found that it was necessary to filter out the low frequency energy, U- < 0.13. Here an expression will be derived for the amount of correlation in the frequency band below the cut-off point which is destroyed by the filtering. It is assumed that the wave number cross-spectral density in a frame of reference moving at the convection speed is a slowly varying function of time so that it can be approximated by a constant below the wave number corresponding to the filter cut-off frequency. Take the Fourier transform of the cross-correlation coefficient Rpv, say, in one dimension in the fixed coordinate system. Fpv(kl,x2,X3;T) = 1 fR p(Xl,X2,X3;T)e-ik dx 2i- -oo pv dx1 (1) The inverse relation is =(kX2X3; ikl l Rpv(xl,X2,X3;T) = / Fpv(ki,x2,x3;T)e ildkl -00 (2) Let the coordinates in a rE vection speed be denoted by (xi are related by X1 X2 X3 In terms of the new coordinates eference frame moving at the local con-,x~,x3). Then the two coordinate systems = Xl + UCT = X2 = X3 (3) 39

40 Rpv (' +UC TX2, ) FpV3( k, = X2 x 3; T) e klXldkl (4) RP(xi+UcT,X2,x3;T) = rf~2pTklekXcelekicdkl (4) -00 The last equation serves to define a new spectral function Gp which is the transform of Rpv in the moving frame. Let Gpv = Upv + i Vpv (6) Since Rpv is a real-valued function, Upv and Vpv must be respectively even and odd in ki, allowing the correlation function to be written as Rpv(xi+Uct,XI,X3;T) = 2 f'Upv(ki,x2,x3;f) cos klXldkl 0o -~,. (7) 2 f 7 Vp(kl,x2,x3;T) sin klxldkl The filter acts to truncate the spectrum at some wave number k = K1. If it is assumed that Upv and Vpv are constant below this wave number, one can write ARpv (x+UcT,X2,X3;T) = 2 Upv(Kl,x2,x3;T) f 1 cos klxjdkl 0 - 2 V p(Ki,,,x,;T) f sin klxldki (8) 0 = 2 7U sin Klx _ 2 V 1-cos Kjx{ pv pv Xi xi This equation is the expression for that part of the cross-correlation coefficient which has been destroyed by the filtering process, under the assumption that the cross-spectral density is constant for wave numbers below K1.

41 The formulas for the calculation of Upv and Vpv at the wave number K1 are Upv(Kl,x2,x3;T) = I 0 JfRP\TXx+UCTXX;T) cos Kixidxl (9) Vpv(Kl,x2,X3;T) = - l2 - Rpv(xi+UcT,X2,X3;T) sin Klxldxl (10) 2r-CO Since the correlation Rpv has been measured in the time domain, the computation of the spectral densities can be simplified by transforming Eqs. (9) and (10) into the time domain. If xl, the distance between the transducers in the fixed frame of reference, is taken as a parametric constant, the wave number in the moving frame of reference can be related to the frequency by kl = Uf If l, is the cut-off frequency of the filter, Cl1 K1 = - Uc (11) The coordinate transformation relating xl and xi is [from Eq. (3) ] X1= Xj Uc T (12) for xl constant, dx' = - UcdT (13) Using Eqs. (11), (12), and (15), the expressions for Upv and Vpv can now be written in terms of fixed coordinates and time delay as

42 Upv(xl,X2,X3;Wl) = Uc fJRPV(XlX2X3s;T) cos -x ( - T) dT Vpv(xi,x2,x3;Clu) = - -Uc Rpv(xl,X2,X3;T) sin )1 - T) dT 271 -oo C (14) (15) The expression for the increment in the correlation coefficient becomes sin u (x~-T ARpV(x1,X2,xs;T) = XU Upv(Xl,X2,X3;1) X - 1 - — T OS ( T Uc 2 uc - - V (x1,x2,x3;al)- co —-(6 Uc A similar analysis leads to an equation for ARpu with Upv and Vpv replaced by Upu and Vpu respectively. Note that the hypothesis of convection which has been used leads to the conclusion that a constant wave number spectrum for k, < K1 implies that the frequency spectrum is a constant for w) < D1. The results of the calculation using (16) have been reported in Section IV-A. The effect of the filtering on Rpv can be obtained by differentiation. One can write AR = g4 V2 d AR ARp = -H ARp AT_ x- _ C - Uc v~ - c ) UU c Uc COS Cl -r X1_ _T Uc M:}~~~~~~~~~~~~~~~~~~~~ (17)

43 A direct calculation shows that ARpv is negligible in all cases.

REFERENCES 1. Townsend, A. A., "The Structure of the Turbulent Boundary Layer," Proc. Camb. Phil, Soc., 47, 1950, 375-395. 2. Grant, H. L., "The Large Eddies of Turbulent Motion," J. Fluid Mech., 4, 1958, 149-190. 35. Favre, A. J., J. J. Gaviglio, and R. Dumas, "Space-Time Double Correlations and Spectra in a Turbulent Boundary Layer," J. Fluid Mech., 2, 1957, 3513-342. 4. Favre, A. J., J. J. Gaviglio, and R. Dumas, "Further Space-Time Correlations of Velocity in a Turbulent Boundary Layer," J. Fluid Mech., 3_, 1958, 344-356. 5. Willmarth, W. W., and C. E. Wooldridge, "Measurements of the Fluctuating Pressure at the Wall Beneath a Thick Turbulent Boundary Layer," Univ. of Mich. ORA Report for ONR Contract No. Nonr 1224(30). Also submitted to J. Fluid Mech. 60 Kawamura, Mo, "Pressure-Velocity Correlation and Double Velocity Correlation in a Turbulent Boundary Layer Along a Flat Plate," J. Sci. Hiroshima Univ. Sero A, 24, No. 2, 1960, 403-416. 7. Hinze, J. 0., Turbulence, New York: McGraw-Hill Book Company, 1959, Chapter lo 8. Gadd, G. E., "A Note on the Theory of the Stanton Tube," Brit. R. and M. 3147, 1960. 9. Coles, D., "Measurements in the Boundary Layer on a Smooth Flat Plate in Supersonic Flow. I. The Problem of the Turbulent Boundary Layer," Jet Prop. Lab Rep 20-69, 1953, or Z.A.M.P., V, 1954, 181. 10. Klebanoff, P. S., "Characteristics of Turbulence in a Boundary Layer With Zero Pressure Gradient," NACA Rep. 1247, 1955. 11. Klebanoff, P. S., and Z. W. Diehl, "Some Features of Artificially Thickened Fully Developed Turbulent Boundary Layers with Zero Pressure Gradient," NACA Rep. 1110. 1952. 12. Kraichnan, R. H., "Pressure Fluctuations in Turbulent Flow Over a Flat Plate," J. Acoust. Soc. Am., 28, No. 3, 1956, 378-390. 44

5 __ 10 15 20 _ 25_ 30 35 40ft....'II I' I I I I' Airflow <n I^ v ___ 4- 1 r' ////// 15 ft. Fig. 1. Scale drawing of wind tunnel test section and massive vibration isolation mounting for the transducers.

I//////// I'4.' \4 0CHORD AIRFOIL I. PHANTOM VIEW OF / I I PROBE HOLDER. USED I I \. I/ I' I. \s^' — RUBBER O-ReilMG /^ -^' I G / r-_-_ ——:'- -- - - I W —---- --—' —"" I / L —_ ____- - --— ______-_____-4_-__-J, ~/ ^ HOT WIRE PROBE 1 OLDER CLAY. FAIRING -1 HOT WIRE PRO' ~E I r i I i 1; f4I PRESSURE TRANSDUCER STEEL PLATE CLA' AIR SEAL Fig. 2. Pressure transducer and hot wire installation. Hot wire shown at closest spacing to plate, 0.050 inch.

l 30 25 20 0 U Ur -p 15 10 5 - Coles Ideal Turbulent Boundary Layer o Re8-38,000 Uoo=204 ft/sec T=67~F o Re= 43,000 I I I I I I I I Uoo=203 ft/sec T=45~F I l I I I I II. 0 I I I I I I I v 10 100 1000 10000 UTy v Fig. 3. Mean velocity profiles in the turbulent boundary layer. Refer to Table I for other boundary layer parameters.

UcpEp(w) U-o qZ- 8* qcOD<~~ ( | 206 ft/sec 10 I I I I i0-s.01..1 I 10 Uao Fig. 4. Dimensionless power spectrum of the wall pressure. Vertical dashed line shows the frequency below which signals were rejected in the subsequent measurements.

0.10 - uw ~0.08 -— ^< —--— ~ —- ^ ----- U co 0.04 00,,. I^ B.L. Edge 2 4 6 8 10 12 y Fig. 5, Turbulent velocity intensity profiles in the frequency band used in the subsequent experiments.

0.6 Uco 0.3 0.2 0.1B.L.Edge 2 4 6 8 10 12 y 8* Fig. 6. Intensity profile of longitudinal derivative of velocity normal to wall. 50

10 I I 10 Eu(w)U' B I v) o 0 VV OA 0o0 0 Ai 0 % o 12 10 y 8* 0.10 o 0.51 0 2.04 A 4.08. 6.12 v Flagged Points Indicate Filter Cut-off Frequency 0 X o x~ rd~; Lzr 0 o0 o 0.01 0.01 I I I I 0.1 1.0 W83* U Fig. 7. Dimensionless normalized power spectra of the longitudinal velocity components at various heights in the boundary layer. Spectra are normalized to have unit area in the frequency band of the subsequent experiments. 51

101 e A 0 o v A 100 Ev(w)U 161 8 1 1o2 v o A'o O0A V0 OA v Do V 0 so v y 8* 0.10 o 0.51 o 2.04 A 4.08 * 6.12 v Flagged Points Indicate Filter Cut-off Frequency 0 3~ 0 V 0 0o oA 6001 0.01 0.1 1.0 wBm U Fig. 8. Dimensionless normalized power spectra of the velocity components normal to the wall at various heights in the boundary layer. Spectra are normalized to have unit area in the frequency band of the subsequent experiments. 52

Rpu 0.1 Fig. 9. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

- d Rpv 8*0 _ 0 X|=0.10.o I O U)(r 0 j/'.~ 8*8 20 -18 -1 -i6 -14 "-1? " ~ n12 4.=0.51 =0 =0.88 -:= 1.02 = 0 = 1.76 =11.22 =0 =19.36 =0 =0.20 = =0 = 0.41 ~,^:, ^~ fo. I I Io * -- - - o ^ - u - -- o L o - ^ ^ irl) \J1 -p~ Fig. 10. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

X*=.~ 8 - =0. ZOo ~ U2(o(-12 -10 -30 -28 -26 -24 -22 -20 -18 -16 6-86 -6 -8 -2 10 12 - 416 18 =0.51 =0 2o^^o ^-c>^^ ^ ^=1.02 =0 1.61 s^^^"^~~~~~~~o ^~ / =11.22 =0 avn.' Io = =0.20' \ \ ~-'A_'' " 1' X', 1 -,-O.-~._.^,_.=1.02 Fig. 11. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure. rl)

X- =0.20 8* XI X$ 8* a* Us=0 U,,(T-T8 ) 8 _, ~cb~-c~-L~-t -4 -2 10 1'8 -Ort _ =0.51 =0 = 0.80 A i = 1.02 0 = 1.61 = 11.22 =0 = 17.71 =0 =0 =0.20 =0 =0 = 0.41 0 =0 1.02 Fig. 12. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

Rpu X3 =0 a*-' =0 Fig. 13. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

Rpv 0.1 2 = 0.51 8* X8 0 8* -30 00'&U~~~~~~~~~~~~~~~~~~~~~~~~ =0.20 =0 Fg.-' e v tt Fig. 14. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure. =0.41 l

X2 - =:1.02 X = X3 = Ur8* 8* UaDT = y -vUD (T- T') -18 -16 - 12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14 I0.51, = 0.67 =2.04 =0 Measured values of the space-time correlation of fluctuating with fluctuating wall pressure. Fig. 15. velocity

X2 =1.02 8*.0.1 / 1 1 1 1 1 X, X3 =O 8* a I-~ 8* =O -24 -22 -20 -18 -16 - -8 -6 -4 -2 I I I I I I / 2 4 6 8 10 12 =O0 =O 1,, I I' 0 0 0 ~ ~~~~~~~~~~~= 01~~~~~~~~= =o = 0 I ~, ~ I I I~~= =0 Up( T-T' ) 4 8* =0.20 =0.41 = 1.02 = 1.53 Fig. 16. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

X2 =1.02 8* _~ J, I ^ XI X3 -8. = 0 - = 0O O sU-xUOD rl O- -0.I Uoo(T-T') 8* -28 -26 -24 -22 -- - - 46 -42 -I0~~ - 8\6 2 4 6 82 4 ~\ Q-^^ ~ ~~ =0.51 =0 16 _________________________.P'c,,, ^____,=0.67 = 1.02 =0 =1.34 C 2.04 =0 I 2.68 = 11.22 =0 = 14.77 Fig. 17. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

X2 X = 1.02 ~ —.O - IT, Rpv xI -0 X3 =0 8* 8= 0.1 U s8I;*:o 8 -22 -20 -18 -i6 -1 -4 -0 -8 -6 - 4 21 =0 =0.20 =0 =0.41 =O iO =1.02 =0 =1.53 =0 ______,__,_,o_____,_,_,,^-^^' __________J, ~o"^-o^..____________,__,1,____ I I= Fig. 18. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

Fig. 19. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

XL —2. Rpu X = 0 3 -22 -20 -18 -16 -14 - - 08 -6 -46 -2 2 4 6 8 10 12 14 16 = 0 = 0.20 -0 =0.41 ~ — o..~u = =0 -o_~ _L... = 0 =1.02 =O ^ --— o.0- ^=0 = 1.53 =0 10 - ~ ~ ~ ~ ~ ~~= oC\ -p Fig. 20. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

^ ^- \ v -4.08 \J1 1~~~~~~~~~~~~~~~~~~~~~~~~~~~ A4.96 =13.64 Fig. 21. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

Rpv ^- \ -aL = 0 U(~T-T-') -26 -24 -2 -8 -6 4 -2 2 4 616 0 =0.20 A S _,,_,O,^~^^^^ 0 =0.41 Cl = i s^~ ^^^ ^^^0 =1.02 Fg22 Meuevu = 0 =1.53 =0 Fig. 22. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

Xp" aI —0 ==0.51 0.1 X =3.06 0 L -12 -10 =36~o -18 I 16 -4- 2 4 6 8 10 12 14 =O =0 X -4.08 -— i — — 6-I -8 2 6=2.04 =0 =2.24 = 4.08 -O =4.49.=0 Uc(Tr-Tr) s * Fig. 23. Measured values of the space-time correlation of fluctuating.velocity with fluctuating wall pressure.

OD XI X3 8*=30 -Rpv0 - =0.51 3.06 ^r 8,6;;6 4-2'D2\< 16 -26 -24-22 -20 -18 -16 i- o - 6 -4 -2 2 4 6 -- 12 14 1E X2 =4.08 = =0 =0 J^ I I I I I I O2...,2.04 i0 =2.24 = 4.08 = O < 4.49 Am ^^^^cy-^ ^0 = 1.02 Fig. 24. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure. U*(T-T ) a*

X2 6.12 8* -0.1 X3. 3 Ua'=O a* X3.so U-(r-r') 8"* I I I I I I -30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 - 8 =0 =0 10 12 14 -1.02 16 18 20 -X28.16:= =0 - -o....-j ^ =4.08 =0 =4.13' 4.08 = o X2=10.20 _ =4.08 I* I I I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -kp Fig. 25. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

X X3 Rpv X- =6.12 U (T-r') -286 -24 -22~a. 16 -14 -6 -4 -2 6 T-8p —- [ =0 =1.02 = 2 =8.16 -Pr__cL.o/ —-o__, — o4.08 =0 > 4.13 X2=10.20 6b < = 4.08 =0 = 0 =4.08 0 0 Fig. 26. Measured values of the space-time correlation of fluctuating velocity with fluctuating wall pressure.

0.2 X=1. 12 a* 0.1,=0.51 8* X3 -0.4 -0.2 102 04 0.6 8* -0.2 Fig. 27. Measured values of the spatial correlation of fluctuating longitudinal velocity with fluctuating wall pressure showing symmetry in the transverse direction.

Rpw -0.1 8*.8*" X2 _ RI UGr -14 -12 -10 -8 -6 -4 -2. -* 2 4 6 =0 =0 =2.04 =0 =0 =8.16 [X_~~~~~~~ 1. I. ~ R' Rpw 8*- 1.12 T0.1 X8*-051 ori3'- -(.6-04-02 026 3 ).6 -04 -0,.2 0.2 0.4 0.6'~~~~~~~~~8 r) w(A o 0.13 <- < 12.5 * 0.38< - 12.5 U0, Fig. 28. Measured values of the space-time correlation of fluctuating transverse velocity with fluctuating wall pressure showing the effect of extraneous large-scale flow disturbance caused by Taylor-Goertler vortices and density stratification.

RpV 83 =0 8* L o0 X3.o 8* 8* X2 0.20 -o0. UT' -../.A _ 8* Xl.' X3so 80 8 ~ -0.1 U)'' 8" 0.10 X=0.10 B*r U8l(T-T') 8* =0.51 =0 =0.73 * A-1 1l - _ 0' 020 ( = 1.02 =0 y =1.46 =0 =0.20 =0.41 =0 =0 =0.41 =0 fjpOo-,Q-: - in ___ I_ I - I —-- t O^QLE% Fig. 29. Measured values of the space-time correlation of the time derivative of fluctuating velocity normal to the wall with fluctuating wall pressure.

X2 =1.02 8* lRpv Xi-=O =0O.0 ]0 U(T-rT') 8 -~8*' I 8~ _ ~e~=W~( ~ -8 -6 - -4 -2 X,I Z = 0.51 6 8 = 0.67 = 0 = 1.02 = 1.34 =0 = 2.04 A-^ = 2.68 =0 _r~~~~ -- b = 0 =0 =0.20 =0 = 0 = 0.41 1 -1 I 8,F=2.04 0-1 ur4 8* ~0 I,R, %-T O Po - 0 0 Xz 0.1 -10 -8 -6 - -4 -2 xor 2 =1.02 - 0 =1.24 8' ^^ "^< p:^ = 2.04 =0 = 2.48 -I0 -8 y= 4.08 = 0 =4.96 =0 =0.20 =0 =0 =0.41 =0 =0 =o...'= <'' -r') I I l _ 0~t~l \ i _____n~ ei___ i^- - /- o =1.02. =0 = 1.02 Fig. 30. Measured values of the space-time correlation of the time derivative of fluctuating velocity normal to the wall with fluctuating wall pressure.

Rp. 0.1 X8 — o -XT =0.51 X= 3.06 i- -14 -, _ _- -o-_ _ -16 -14 -12 -10 -8 -6 -4 -2' X2 8= 4.08 UGoT' U<pD(Tr -') *8ax - a8 6 8 10 12 14 16 I H r- ^ I - z 4 =0 =0 =0 -1n I -— Ii n- ~ I =2.04 =2.24 =0 v~~~~~~~~~~~~~~ 2_2 =0 =0 =1.02 I -- I — I-,'' -- l — -r — Fig. 31. Measured values of the space-time correlation of the time derivative of fluctuating velocity normal to the wall with fluctuating wall pressure.

X2 S* Edge X3 8* + 8 -r -6 -5 -, -3 -2 -1 2 3 4 __6 _ 89,,I, 12 3 -7 -6 -5 -4 -3 -2 -I 0 I 2 3 4 5 6 7 8 9 10 II 12 8* Fig. 32. Correlation contours of Rpu in the xl-x2 plane. Correlation normalized on the value of the velocity fluctuation at x2/S8 = 0.51. Origin of coordinate system at pressure transducer.

8* 8* X2 8* Edge B.L. Edge. o1 XI = -1.25 8* XI =1.25 84.8 Rp = -0.010 -6 \ \ \ \ \I \ \\ I N \ \ I \-,': I, I. I I.,',-,,;,'1f';: +0.04' -0.025 -0.045 -0.060 - 0.05C. 1`1 - x -2 -I 0 I 2 3 4 8 -2 -I 0 1 2 3 4 Fig. 33. Correlation contours of R' in the X2-X3 plane. Correlation normalized on the value of the velocity fluctuation at x2/s* = 0.51. Origin of coordinate system at pressure transducer. a*

Rpu = + 0.020 +0.060 +0.075 | \ |+0.040 -0.120 -0.050 -0.060 l,I,, /I X2 8* I I I I t %= --, - t - - - A _l In -o _Q _ _ A _ I -' — k 1IV -t -o -r -b -o -3 -2 2 -- 0 b —2- 3. 4 - 0.060.0.040 5 6 7 8 9 10 II 12 8* X3 8* X = 0.20 8* +0.040 + 0.060 +0.080 1 1 1 1 1 1 1 1 - I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -0080 -0.150 -0.120-060 I I I 00 - 0 i X, 8*....r~,Af -. ----'I +0.040 0 X3 8* -0.080 -0.040 -0.20-0.100, / -0o060 I 0.030,0.020 +0.090 +0.020 +0.040 +0.060+0.080 I I I I I I I I I i- X-2 =0.51 8* I I....I 8" UJ N) ----— z C -~ CC / - -7 - \t I (I _ J............; X3 F* Fig. 34. Correlation contours of R' in the xl-x3 plane. Correlation Pu normalized on the value of the velocity fluctuation at x2/5* = 0.51. Origin of coordinate system at pressure transducer.

X2 8 12 X3 * =0 8 Edge 8 - 0.020 /. I \ I > " - 0.040 /,, -0.060 " ~...'"'"'" —. / / I~/.~ ~~,/'7 -0.090 I I / 1 I / / -,'I \ /\. - -- - _10 -8 ~ 7 r —— 6 --— 4- —..r... -2 \ -10 - 8 — 6 — 4 — 2 -1 -12,XI 8 Fig. 35. Correlation contours of R1vin the xl-x2 plane. Correlation normalized on the value of the velocity fluctuation at x2/5* = 0.51. Origin of coordinate system at pressure transducer.

X2 8* X2 a* B.L. Edge B.L. Edge XI = -1.25 a* XI =1.25 8' 8 + o0 +0.040 0 Rpv - 0.020 4 -0040'. 0 +0.060. _ 0.060', 0 +0.090 -0.090, \ +0.1 10 \' o t +0.120 - 0. 120, i n / -2 -1 0 1 2 3 4-2 -1 0 I 2 3 4 Fig. 36. Correlation contours of R'y in the X2-X3 plane. Correlation normalized on the value of the velocity fluctuation at Xa/5* = 0.51. Origin of coordinate system at pressure transducer.

X2 +0.080 +0.020 8 I 0 9 8 7 6'"_.. —0 5 - —' —---._-: — /c 5 6 8 +01' I0 I, 6 7 094,' 11' 0 ~'X X2 ~ i +0.080 +0.020 0.20 0.0?0 -0.040 -60 /+0.10o+0.04/,, __o.o1, _-o.014, _I V"Io 1. _ I IJI~ XI 8X -\\ ~ ~ ~ ~ - -0- -o —- o —0 0C,e~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ " —----- cO i-J X3. _,-W 8 -0.080 -0.120;0.020, -0.040, -0.06O0-0.110 / +0.100 +0.040 X00 8*=0.51 +0.100 +0.040 +0.01+0 8 +ol2o/-,0 o8o/ +0.0o0_ 0o0.005/+o.o0,, _ Xi,, —-,, ~~~ —-C1' r IYI - ----- ~- - ~ - -- -- 0 -1.I ~~=(.)~S I~~CC~- ~ CCj I - -::.-, ) 01 8* X3 %* Fig. 37. Correlation contours of RIv in the xl-x3 plane. Correlation normalized on the value of the velocity fluctuation at x2/8* = 0.51. Origin of coordinate system at pressure transducer.

X2 8* Edge X3 3 =O 8* 00 n>3)^~~~~~~ j~ +-~0 ~ Rpv = + 0.005 - 0.010_ 4 XI f ^ ^ s^ O.005 I / "','! -8 -6 -4 -2 0 2 4 6 8 8 Fig. 38. Correlation contours of Rp-. in the xi-x2 plane. Correlation normalized on the value of the velocity fluctuation derivative at X2/*= = 0.51. Origin of coordinate system at pressure transducer.

X2 8* - B.L. Edge -10 Xl 8* 8" -8 -6 CD \I0 Rpv = -0.010 -4 0 < -0.025 -0.050 -0.070 -0.090 \ I \ I \ I / \ I /, / I I -O.l. — I0 X3 I I I 8 2 3 4 - -3 -2 -I 0 I Fig. 39. Correlation contours of RpT in the X2-X3 plane. Correlation normalized on the value of the velocity fluctuation derivative at X2/5* = 0.51. Origin of coordinate system at pressure transducer.

-0.01 +0.025 +0.005,R 0~.020 Xi 4 ----- II — 8* -6 -4 _ -2 o -o0.0105o' + 0.015 Z2 T +0.005 -( i0.010 8* )20,+0.015 +0.015 +0.01 +0. +o.o00o \\ o.oIo ~ X -0.20 ~ -g — XI -1 -4.005 4 6'-"- -o, ~,- _.020 -0.02+0.005 -0.050 01+0.020 -0.075 0.050* +0.005 +0.01' +0.015 \ 0.I025 > L _ i...; -! L t~-=o~s 0 XI 8* -6 -4 0' O'1' 7 +0.005 -1\ ~0 +0.010 -0.010 X83 8* 4 6 Fig. 40. Correlation contours of RT. in the xl-x3 plane. Correlation pv normalized on the value of the velocity fluctuation derivative at X2/s* = 0.51. Origin of coordinate system at pressure transducer.

f / y 0'10 / \ \ / / \ V\ k i 6 -4\ \ \\\ \ \ \ \. \ \\t 4 2 \ \ \ X, _ 4 x. \ x i 111 As\~~~~N -- A -,w r, l X x 10 8* -10 -8 -6 -4 -2 2 4 6 8 Fig. 41. roint is from the Vector Field of Correlation. Magnitude of the vector at any /Rp + R2. Direction of the vectrr at any point as measured positive xi-axis is given by tan - -lD Rpu

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