THE UNIVERSITY OF M I C H I G A N COLLEGE OF ENGINEERING Department of Aerospace Engineering Aerodynamics Laboratory Technical Report AN EXPERIMENTAL STUDY OF THE STRUCTIURE OF TURBULENCE NEAR THE WALL THROUGH CORRELATION MEASUREMENTS IN A THICK TURBULENT BOUNDARY LAYER Bo-Jang Tu William W. Witlmarth ORA Pkroject 02920 under contract with: DEPARTMENT OF THE NAVY OFFICE OF NAVAL RESEARCH CONTRACT NO. Nonr-1224(30), NR-062-234 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR March 1966

This report was also a dissertation submitted by -the first author in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1966.

ABSTRACT An experimental investigation is described in. which emphasis is given to revealing the structure of turbulence near the wall in a boundary layer. Measurements made include space-time correlations between the fluctuating wall pressure and the span-wise velocity component w, and between the various velocity components. The velocity correlations include measurements of the space-time correlation of the streamwise component of the fluctuating wall shear stress. Experiments have been conducted in a thick (5 in.) turbulent boundary layer with zero pressure gradient which is produced by natural transition on a smooth surface. Sufficient data have been obtained to allow us to propose a qualitative model. for the structure of turbulence near the wall. The proposed model outlines the sequence of events that result in the.production of intense pressure and velocity fluctuations by stretching of the vorticity after it is produced by viscous stresses within and near the edge of the viscous sublayer. The measurements are in qualitative agreement with the model. Here, qualitative agreement means that the size and shape of the contours of constant correlation and the sign of the measured correlations are in agreement with the proposed model for the turbulent structure. The proposed model as well as the result of measurements of the correlation between the fluctuating streamwise wall shear stress and the streamwise vrelocity component were found in favor of the concept that the t*rbulcence is generated in the wall. region. distending into the central parts of the boundary layer. Since it is well-known from measurements tLat eddies of small size dominate near the wall, this implies that smat.: eddies near the wall may become larger when they distend into the more central region. However, the physical mechanism of development from small to large eddies is not yet completely understood. ii

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v NOMENCLATURE viii Chapter I. INTRODUCTION 1 II. EXPERIMENTAL APPARATUS AND PROCEDURE 4 A. Wind Tunnel Facility 4 Bo Instrumentation 5 III. EXPERIMENTAL ENVIRONMENT 9 A. Extraneous Disturbances 9 B. The Nature of the Turbulent Boundary Layer Used in the Investigation 10 IV. THE MEASUREMENT OF MEAN WALL SHEAR STRESS USING A SINGLE HOT-WIRE AND THE CORRECTION FOR THE EFFECT OF WALL HEAT TRANSFER 13 A. Introduction 13 B. Calibration 14 Co The Measurement of Mean Wall Shear Stress 15 V. SPACE-TIME CORRELATIONS OF WALL SHEAR-WALL SHEAR, OF WALL SHEAR-VELOCITY, AND OF VARIOUS VELOCITY-VELOCITY COMPONENTS 17 A. Introduction 17 B. Measurements 18 C. Results 20 VI. SPACE-TIME CORRELATION OF PRESSURE-SPANWISE VELOCITY, Rpw 22 A. Introduction 22 B. Measurements 22 C. Results 23 VII. CONTOURS OF CONSTANT CORRELATION OF T7W T OF TwU AND OF pw 25 iii

TABLE OF CONTENTS (Concluded) Page A. Introduction 25 B. The Construction of the Contours 26 C. Results 27 VIII. DISCUSSION OF THE RESULTS OF MEASUREMENTS 29 A. The Propagation of Fluctuating Wall Shear (or Fluctuating Vorticity in xa-direction) Due to Convection and Diffusion 29 B. Phillips' Integral Condition and the Measurement of the Pressure-Velocity Correlation, Rpw 31 IX. THE STRUCTURE OF TURBULENCE NEAR THE WALL IN A TURBULENT BOUNDARY LAYER 34 A. Introduction 34 B. The Structure of Turbulence Near the Wall 35 C. Comparison with Experimental Results 38 1. Rvv measurements 38 2. Rwv measurements 39 5. Rpw constant contour diagrams 40 4. Ruv measurements 41 X. CONCLUSIONS 43 Appendix I. THE EXISTENCE OF INSTANTANEOUS LINEARITY IN THE VISCOUS SUBLAYER 46 II. ERRORS AND CORRECTIONS 48 III. CALCULATION OF THE MEAN WALL SHEAR STRESS 55 BIBLIOGRAPHY 62 INDEX TO THE POSITIONS OF PRESSURE-TRANSDUCER AND HOT-WIRES IN THE PRESENT CORRELATION MEASUREMENTS OF'uiuj AND pw 66 iv

LIST OF ILLUSTRATIONS Table Page I. PROPERTIES OF THE ACTUAL AND IDEAL TURBULENT BOUNDARY LAYER 11 Figure 1. Scale drawing of wind tunnel test section and massive vibration isolation mounting for the transducers. 73 2, Pressure transducer and hot-wire installation. Hotwire shown at closest spacing to plate, 0.05 in. 74 3. Hot-wire plug. 75 4. Mean velocity profiles in the turbulent boundary layer. Refer to Table I for other boundary layer parameters. 76 5. Hot-wire calibration curve for platinum u-wire. 77 6. Comparison of auto-correlations of velocity fluctuations, u, v, and w, near the wall. 78 7. Comparison of the space-time correlation of u-u near the wall. 79 8-9. Measured values of the space-time correlation of u-u very near the wall. 80-81 10-13. Measured values of the space-time correlation of u-u near the wall. 82-85 14-16. Measured values of the space-time correlation of v-v. 86-88 17-18. Measured values of the space-time correlation of w-w near the wall. 89-90 19. Measured values of the space-time correlation of w-w very near the wall. 91 v

LIST OF ILLUSTRATIONS (Continued) Figure Page 20-22. Measured values of the space-time correlation of u-w very near the wall. 92-94 23. Measured values of the space-time correlation of u-w near the wall..o 95 24~ Measured va;lues of the space-time correlation of w-v. 96 25-26 Measured values of the space-time correlation of w-v near the wall. 97-98 27, The measurement of Rwv near the wall, using different arrangements of x-type hot-wires (also see Fig. 26 for comparison). 99 28, Meas~J:ed values of the space-time correlation of w-v near the wall. 100 29, Prei.minrary consideration of Rpw measurements. 101 30. Measured values of the space-time correlation of fluctuating velocity component w with fluctuating wall pressure (x3/6* < 0). 102 31-38 Measured values of the space-time correlation of fluctuating velocity component w with fluctuating wall pressure (x3/6* > O). 103-110 39. Correlation contours of constant Ruu very near the wall. 111 40. Correlation contours of constant Ruu near the wall, 112 41-45. Correlation contours of constant Rpw in the x2-x3 plane. Origin of coordinate system at pressure transducer, 113-117 46. Three-dimensional diagram of contours of Rpw = const. (also see Figs. 41-45). 118 47~ The location of hot-wires for measurements of the displacement of eddies due to convection and turbulent diffu asion near the wall. 119 48. The behavior of eddies in a shear flows, 120 vi

LIST OF ILLUSTRATIONS (Concluded) Figure Page 49. The development of random vortex lines near the wall. 121 50. Random vortex flow in the x2-x3 plane near the wall. Points A and B are the locations of hot-wire measuring Rvv. 122 51. Random vortex flow in the x2-x3 plane near the wall. Points A, B, and C are the locations of hot-wire measuring Rwv. 123 52. Random vortex line near the wall. Points A, B. and C are the locations of hot-wire measuring Rwv. 124 53 Structure of a random vortex line near the wall and the explanation of measurements of contours of constant Rpw at different x2-x3 planes (also see Fig. 46). 125 54. Explanation of the separation of positive contours of constant Rpw in the present measurement (also see Figs. 44-46). 126 vii

NOMENCLATURE aw hot-wire overheating ratio e:f fluctuating voltage across hot-wire; e = Irw f frequency fw mean wall shear stress I current through hot-wire k coefficient of thermal conductivity Nu Nusselt number; Nu = iXk(Tw-Te) p f luctuating wall pressure rate of heat loss from hot-wire; Q = I2Rw Reynolds number based on distance from virtual origin of turbulent boundary layer Re Reynolds number based on momentum thickness Ud (ReY)w Reynolds number based on hot-wire diameter d; (Rey)w - Rpuj normalized wall pressure-velocity correlations; see, e.g., Eqn. (VI-1) ~Ru-u. normalized velocity-velocity correlations; see Eqn. (V-l) RUjuj R TwTw normalized wall shear-wall shear correlations; see Eqn. (V-2) RTuWU normalized wall shear-velocity correlations; see Eqn. (V-3) Re resistance of hot-wire at ambient temperature, Te resistance of hot-wire at O~C Rw resistance of heated wire at temperature, Tw ~rw ffluctuating resistance of heated wire viii

NOMENCLATURE (Continued) Te or 8e ambient temperature of hot-wire Tw or 8w temperature of heated wire t time U velocity in boundary layer in stream direction U0o free stream velocity UT wall friction velocity Uc convection speed of fluctuating velocity field which is correlated with fluctuating wall pressure or fluctuating velocity at a fixed point; Uc is determined by aRpw(xl,T)/aT = 0 or aRuiuj(Xi,T)/aT = 0 respectively. u,v,w fluctuating velocities in x, y, z directions Ui or uj fluctuating velocities in x, y, z directions (i or j = 1,2,3) xIx2,x3 spatial separations of pressure and velocity transducers in x, y, z directions? t I xl,x2,x3 spatial separations of two velocity measuring points in x, y, z directions x distance parallel to wall, increasing in 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 temperature coefficient of resistivity o boundary layer thickness b* boundary layer displacement thickness viscosity ix

NOMENCLATURE (Concluded) v kinematic viscosity fluctuating vorticity component in x3 -direction in viscous sub layer P density Pa, density of air T time delay X' time delay equal to transducer spacing divided by convection speed Uc j1 fluctuating shear stress Tw fluctuating wall shear stress W circular frequency; c = 2gf time average x

CHAPTER I INTRODUCTION Historically, the background for new investigations of the structure of turbulent shear flow are the extensive investigations of the turbulent velocity field made first by Townsend(2) (1951) and later by Schubauer and Klebanoff(20) (1951), Laufer(15) (1953), Klebanoff(9) (1954), etc. Most of our knowledge about the structure of turbulence in a shear flow rests on their measurements of the spatial correlations between turbulent velocities and the power spectra of turbulent velocity components. In recent years, quite a number of new investigations were reported, among which the more representative are Favre, Gaviglio, and Dumas(5) (1958), Grant(6) (1958), and Willmarth and Wooldridge(26'27) (1962). In this period a new technique using a tape recorder for the measurement of spatial correlations with variable time delay was developed by Favre, et al., so that one is able to study the correlation between the fluctuations of wall pressures and/or velocity components at one point and those of earlier or later history at the same or another point. This is the so called space-time correlation, which makes it possible to investigate the evolution of turbulent eddies. In the measurement of space-time correlations between the fluctuating wall pressure and the fluctuating velocities reported by Willmarth and Wooldridge,(27) the interesting result that they pointed out is that near the wall in the planes - = 0.20 and 0.10 the constant cor1'

2 relation contours of pv show a curious swept-back behavior (see Ref. 27, Fig. 57). In the beginning of the present work we made a few measurements of Rvv with hot-wires at various distances from the wall and found that near the wall at'2 = 0.2 there was a change of sign of Rvv from positive to negative (see Fig. 15). Grant(6) in his study of large eddies in the boundary layer measured the spatial correlations of each of the three components of the fluctuating velocity; however, he did not report any measurements of Rvv for 2 < 0.29. With these results it appeared to us that in the wall region there is a definite structure that is worth understanding; and therefore our measurements were mostly performed near the wall. The measurements described in the present work include space-time correlations between the fluctuating wall pressure and the span-wise velocity component w, and between the various velocity components. The velocity correlations include measurements of the space-time correlation of the streamwise component of the fluctuating wall shear stress. With these measurements we are able to propose a qualitative model for the structure of turbulence near the wall. The measurements of the correlation between the fluctuating wall pressure and the velocity components by Willmarth and Wooldridge(27) include pu and pv. In the present work we have continued the program and measured the correlation of pw and constructed its constant correlation contours. Interest was found in these contours of constant pw, which gave support to the proposed model of the structure of turbulence

near the wall. We shall discuss the details of this in Chapter IX and describe the measurements in Chapters VI and VII. A rough comparison of the present pw measurements with Kraichnan's(12) wall pressure theory was also made, and given in Chapter VIII. In Chapter IV, we have described the measurement of the mean wall shear stress with a single hot-wire using Wills'(28) experimental method to correct for the effect of wall heat transfer. We have used hot-wires to measure the correlation between the fluctuating streamwise velocity very near the wall. We have also studied the existence of the instantaneous linear velocity profile in the sublayer (see Appendix I). With the instantaneous linearity of the velocity profile near the wall confirmed, we are able to interpret our correlation measurements of uu performed in the sublayer region as the correlation of the fluctuating streamwise wall shear stress or fluctuating spanwise vorticity component, and to deduce the correlation distribution and the propagation in the wall region. These results are included in Chapter V, VII, and VIII. The present investigation was carried out in a 5-in. thick turbulent boundary layer at a nominal free stream velocity of 206 fps. The ratios of pressure transducer diameter and hot-wire length to boundary layer thickness were approximately 1:80 and 1:100, respectively, allowing a study of the detailed structure of the fluctuations in the layer. Corrections and errors of the present measurements are described in Appendix II.

CHAPTER II EXPERIMENTAL APPARATUS AND PROCEDURE A. WIND TUNNEL FACILITY The experiments were carried out in the fully developed turbulent boundary layer on the floor of the 5 by 7 ft low speed wind tunnel facility at the Aeronautical Engineering Laboratories, The University of Michigan. The wind-tunnel test section is 25 ft long and is indoors. The settling chamber, fan, and steel ducting that recirculates the air are out of doorso The total distance around the wind-tunnel circuit is 332 fto The contraction ratio of the nozzle is 15:1l A schematic diagram showing the general. layout is given in Fig. lo Natural transition took place in the contraction section (approximately 24 ft ahead of the measuring point) 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 ino in diameter and mounted on a heavy pedestal., which was inserted flush with the floor. The 0.0625 in, gap which was allowed between the plate and the windtunnel floor was sealed on the outside of the tunnel with a strip of rubber, The mounting pedestal was vibration-isolated from the floor by means of rubber shock pads~ Holes were drilled in the plate to

5 accept the pressure crystal assembly and the hot-wire plug. The hole for the former is 0.3125 in. in diameter while that for the latter is 1 in. in diameter. B. INSTRUMENTATION The fluctuating wall pressure was measured with a pair of 0.060 in. diameter lead-zirconate disks mounted back to back in a brass plug which was supported in the steel plate by a screw-controlled holder attached underneath the plate. The sensitive area of the pressure transducer which was newly made by Willmarth, is about 7.4 times smaller than that used in the former measurements by Willmarth and Wooldridge,(26 7) thus spatial attenuation caused by the finite size of pressure transducer is decreased and the resulting pressure measurement is more representative for the point to be measured. As shown in a schematic diagram of the transducer installation (see Fig. 2), rubber O-rings were used to prevent air leakage around the transducer body. The transducer was connected through a low-noise cable to a low-noise preamplifier having a cathode follower input with an input impedance of 1l2 x 108 ohms. The capacity of the lead zirconate disks connected in parallel was 60 micromicrofarads, allowing a low frequency response down to approximately 25 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 kcps.

6 The fluctuating velocities were measured with constant current hotwire equipment manufactured by Shapiro and Edwards. The frequency response of the uncompensated amplifier was flat from 1.3 cps to 320 kcps. Adjustable filters in the amplifier were used to cut out frequencies above and below the range needed in the experiments (1.3 cps < f < 80 kcps) to eliminate unnecessary noise. In the measurements of velocity-velocity correlations one additional amplifier made according to the design of Kovasznay(32) was used. Its frequency response without compensation was flat from 5 cps to 25 kcps when its filter adjusting switch was set at the widest band position (1 cps < f < 50 kcps). Both platinum and tungsten wires of 000015 in. or 0.0002 in. in diameter were used. Their lengths ranged from 0.03 to 0.05 in, and in any case a slenderness ratio greater than 150 was employed to take care of the effect of finite wire length.(13) The wires were attached to the supports by the usual method of copper plating and soldering. In case of velocity measurements very near the wall, silver-coated platinum wires were glued directly on the smooth surface of a plug 1 in. in diameter (see Fig. 3). Then the sensitive parts of the wires were obtained by the method of etching with dilute nitric acid and electrolysis. The plug was made of insulating material such as, bakelite and Plexiglas. The time constant of the wires was approximately from 0.35 to 0~50 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. As

7 reported by Willmarth and Wooldridge(27)the microscale of the turbulence near the wall was calculated from the measured spectra and found to be approximately twice the hot wire length (~ = 0.05 in.). Moreover, the upper frequency limit on the tape recorder described below limits the smallest eddy which can be studied to a scale of approximately the hot wire length. For the measurements of fluctuating velocity at a distance greater than 0.05 in. from the wall, a probe made of 0.19 in. brass tube and strengthened by thin sheet metal was used to support the hot-wire needles; see Fig, 2. Measurements very near the wall were obtained by fastening the hot wire probes directly on the wall. In case of the measurement of streamwise wall shear correlations and wall shear-cross flow correlations the wire itself was glued on the surface of a plug as described above to reach the points inside or near the edge of the viscous sublayer (bSL - 0.0043 b*). For the closest spacing to the wall, a microscope with a filar eye piece was used to measure the distance between the wire and its surface reflection. The signals from the pressure and velocity transducers were recorded on a three-channel Ampex Model FR-1100 magnetic tape recorder which utilized 0.5 in. wide tape travelling at 60 in. per sec. The recorder had a bandwidth extending from d-c to 10 kcps. In the present measurements special filters were used to replace the original 10 kcps filters in the record and reproduce units for each channel. Thus the limiting upper frequency for the measurements was extended to 20 kcps

8 so as to be able to measure small eddies of approximately the same size as the hot wire length. A special movable play-back head was used to replace the original recorder head to allow the introduction of time delay between a pair of signals. The smallest incremental time delay which could be obtained 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 inverter and then into the summing circuit. A provision for external filtering of the signal before it reached the thermocouple allowed the measurement of correlations in adjustable frequency bands; a Krohn-Hite Model 310-AB filter was used for this purpose (see Section III-B). The output from the thermocouple was measured on a sensitive research d-c millivoltmeter having a time constant of 3 sec.

CHAPTER III EXPERIMENTAL ENVIRONMENT A. EXTRANEOUS DISTURBANCES The accuracy of the measurements made in this report can be affected by the extraneous disturbances which 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. These disturbances were studied and reported formerly by Willmarth and Wooldridge; the following is quoted from their work (Refo 27, po 4-5): " A check on the effectiveness of the vibration isolation of the mounting showed that the spurious pressure signals caused by vibration amounted to less than 1/100 of the mean-square turbulent pressure fluctuationso "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 cycles per seco The wall pressure correlation measurements described in Reference 11 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 mean-square sound pressure in the free stream amounted to approximately 1/20 of the mean-square turbulent wall pressure fluctuationso The mean-square velocity fluctuations associated with the sound were less than 1/2 percent of the mean square turbulent fluctuations near the edge of the boundary layer. "The settling chamber of the wind tunnel contains four turbulence damping screenso The free-stream turbulence level in the test section increases with velocity and has been measured at 50, 1.00, and 150 fps. Extrapolation of the data to the 20C fps speed used in this experiment results in a value of /U~!U:.! x l0O3 for the turbulence level of the axial velocity component. The level of the transverse velocity component is approximately three times the level of the 9

10 axial component. "tLarge-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 caused by sunlight impinging on the outside of the tunnel produced density stratification near the walls which in turn produced vorticity when accelerated into the test section. Observations of streamers of smoke near 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 Taylor-Goertler boundary layer instability on the concave walls of the contraction and the density stratification." The wind tunnel condition has not been changed except that one of the four turbulence damping screens in the settling chamber of the wind tunnel had been damaged and removed sometime before the present measurements were made. We remeasured the free-stream turbulence level and found that 72j/Uo = 2.5 x 10-3 in axial flow direction. The level of the transverse velocity component is J:/U, = 3.2 x 10-3 Both were measured at an air speed of 200 fps, which was used to obtain a fully turbulent boundary layer for all measurements in the present work. B. THE NATURE OF THE TURBULENT BOUNDARY LAYER USED IN THE INVESTIGATION The mean velocity profile, velocity intensity profile, wall pressure spectrum and velocity spectra of the turbulent boundary layer used for the present investigation, were measured and reported by Willmarth and Wooldridge.(27) So, no such experiment has been repeated. The properties of the boundary layer profile measured by them are tabulated in Table I and reprinted in Fig. 4 together with the properties of Coles'

11 ideal boundary layer at the same value of Re, the Reynolds number based on momentum thickness. Figure 4 and Table I show that their measurements agree satisfactorily with the ideal case. TABLE I PROPERTIES OF THE ACTUAL AND IDEAL TURBULENT BOUNDARY LAYER ~T U. ~ ~ ~ ~ 6/ FIe * 8UT/Ue R Remarks ~F fps ft ft ft 7 Measured by 67 204 38,000 0.42 0.041 0.0315 1.30 0.0326 3. Measured by10 Willmarth and Wooldridge 38,000 1.30 0.0318 3.2x07 Coles' ideal boundary layer 45 203 43,000 0.42 0.041 0.0315 1.30 0.0325 3.85x107 Willmarth and Wooldridge 43,000 1.295 0.0515 4.0x107 Coles' ideal boundary layer In the measurements of wall pressure spectrum Willmarth and Wooldridge(26) found that the data were not repeatable for (I* < 0.13. The Uoo results varied with the amount of heat transfer to the tunnel by sunlight. Therefore, in the measurements of space-time correlation of pressurevelocity and velocity-velocity the Krohn-Hite filter has been used to reject all frequencies below - = 0.13. The upper limit on the freU00 quency, = 25, was provided by the tape recorder response. U00 At the closest spacing of the hot-wire to the wall, 0.002 in., for the streamwise wall shear-wall shear correlation measurements the convection speed of the disturbances is approximately 0.2 Up. Since the boundary layer thickness is 5 in. disturbances convected at this speed and at

12 wb* = 0l13 have a wave-length of 0.93 6. Hence, even with filtering and UOO very near the wall information can be obtained about eddies or disturbances whose scale is about the same as the boundary layer thickness. However, in correlation measurements the filter acts directly to eliminate the correlation in the frequency band below its cut-off point, and indirectly to boost the value of the normalized correlation coefficient at all values of time delay by reducing the magnitude of the denominator. The net effect with filter cut-off frequency at - 0.13, as studied by Willmarth and U00 Wooldridge(27) in the pressure-velocity correlation measurements, is not more than 5%. Their analysis also showed 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 or at large spacing between measuring points the correlation is mainly contributed from the low frequency (large scale) velocity fluctuations. The filter effect also increases with distance from the wall since lo~-, frequency fluctuations become increasingly important in the correlation measurement as this distance increases.

CHAPTER IV THE MEASUREMENT OF MEAN WALL SHEAR STRESS USING A SINGLE HOT-WIRE AND THE CORRECTION FOR THE EFFECT OF WALL HEAT TRANSFER A. INTRODUCTION The mean wall shear stress of the turbulent boundary layer used in the present investigation was measured by Willmarth and Wooldridge,(26,27) UT and reported in the form of a non-dimensional friction velocity T (see U00 Table I, Section III-B). The measurement was made by them with a Stanton tube using the calibration results reported by Gadd.(30) Other methods such as the "floating" element device developed by Dhawan(31) have been successfully used by different investigators: Hakkinen, Coles, and Korkegi (e.g., see Ref. 7). In the present work, however, the mean wall shear stress was measured by using a single hot-wire which was set up outside but very near the edge of viscous sublayer. The distance to the wall was determined with the aid of a microscope fitted with a filar eyepiece and was found to be 0.00439 5*. A piece of silver-coated platinum wire with a diameter of 0.0002 in. was glued directly on the flat surface of a plug as described in Section II-B. A sensitive length of 0.043 in. was obtained by etching off the silver coating in a very small jet of dilute nitric acid with an electric current of about 0.8 milliampere. Technically there was not much difficulty to place the hot-wire even closer to the wall to ensure X2UT that it was in the sublayer region, < 5; but when the wire was too 13

close to the wall the Reynolds number of the wire, (Rey)W, would be near unity and thus King's linear relation would not be valid (Ref. 13, p. 231). There were also difficulties in calibrating the hot-wire at a speed as low as 20 fps with the present equipments. Bo CALIBRATION At the beginning of measurements the hot-wire was calibrated and a curve was plotted with I Rw vs o Since, according to the usual form of King's formula for incompressible subsonic flow, we have Q = IRw - Te) (A + B XU) where A,B = constants However, the calibration curve obtained was not quite a straight line; it became curved in the region where. > lO(fps) After a study of the flow parameters involved in the present problem the above calibration 2 curve was replotted with I Rw vs N pU to include the effect of compressibility and then it turned out to be linear (see Fig. 5). This would be quite clear if one was guided in the study by a more general relation of the heat transfer of a thin wire with forced convection(13) (Prandtl No. = const.), Nu = a+b (ey where a, b = constants.

15 In the present measurement, a linear calibration curve was found very convenient to use, especially when correction was necessary in case that the ambient temperature and the density of air during calibration were different from those during measurement. C. THE MEASUREMENT OF MEAN WALL SHEAR STRESS Difficulties were experienced by many investigators when using a hot wire for velocity measurements close to a solid boundary. Besides setting up the wire at an adequate distance to the wall to ensure the validity of King's linear relation and measureing this distance accurately, one must also correct for the effect of wall heat transfer. Wills(28) (1962) made a comprehensive study of the latter problem and conducted a series of experiments. The results were plotted into curves which can be used to correct, at different heights, the errors introduced by the effect of wall heat transfer. Because of its simplicity and convenience Will's correction method was successfully employed in the present work to compute the mean wall shear stress. The result (see Appendix III'for, details) that we found was: f = 0.0912 lb/sq ft w where f = mean wall shear stress, w or the non-dimensional wall friction velocity,

16 T 1 U0 U0 / Pa 0.032, which agrees very well with the result by Willmarth and Wooldridge with Stanton tube, U = 0.0326 (see Table I); and by Coles' ideal boundary layer based on same Re, UT = 0o0318 (see Table I). UOO In calculation of the mean velocity, a correction for the error due to nonlinear effects of the turbulent velocity fluctuations was also studied. The correction found by our analytical method was 3% (see Appendix II-A), which was checked with the result by a graphical method.

CHAPTER V SPACE-TIME CORRELATIONS OF WALL SHEAR-WALL SHEAR, OF WALL SHEAR-VELOCITY, AND OF VARIOUS VELOCITY-VELOCITY COMPONENTS A. INTRODUCTION In recent years different investigators, either theoretical or experimental, have been interested in studying the turbulence near the wall in a boundary layer. The main purposes of their work may be summarized as: (1) To determine the structure of turbulence in the wall region; (2) To determine whether the so called "laminar sublayer" exists or not and to determine its structure; (3) To determine the mechanism which maintains the turbulence in a boundary layer. Many brilliant investigations have been carried out; however, to the author's knowledge, we are still far from reaching a general conclusion or a complete understanding, In the previous measurements by Willmarth and Wooldridge the contour diagrams of constant correlation pv in the planes which are parallel to the xl - X3 plane and near the wall (see Ref. 27, Fig~ 37, p. 52) showed a very interesting behavior. It was observed that in the plane x2 = o0.~20 a protruding part of the positive contour of R' started to ~~~~~~6* p ~~~~~pv appear in the negative region (xl < O). The protruding part became even sharper and extended further into the negative area when x2 = 0,10o. 17

18 Naturally, this phenomenon suggests that in the horizontal plane near the wall (2 < 0.20) the vertical component of fluctuating velocity changes its sign at certain distance along x3-axis and it seems that only the existence of a certain type of eddies may serve to explain it satisfactorily. All these previous investigations greatly stimulated the present work in an attempt to understand the turbulence in the wall region. A number of velocity-velocity correlation measurements in different directions and either at the edge or outside the viscous sublayer were carried out. The results revealed many interesting points, of which more discussions and a study related particularly to the structure of turbulence near the wall will be given in Chapter IX. B. MEASUREMENTS The fluctuating velocities in the turbulent boundary layer are stationary random functions of time and space. The velocity-velocity correlation coefficients which have been measured are defined by Ru(x2;x,xx1;T) - ui(xlx2,x3; t) Uj(xl+x',x2+x2,x3+x3;t + T) Ui (X,x2,x3; t) uj xlx+x, 1x2+x2,x3+x;3 t) In case both ui and uj in the direction of free stream (i = 1, j = 1) are measured in the linear viscous layer, then since Tw - u(,t) [The existence of instantaneous x2 linearity in the viscous sublayer will be discussed in Appendix I]

19 one can obtain U(Xl,x2,x3; t) u(xl+x,x2+x,x3+x3; t+T) RTWTW(X, 0,X3;T) = RU() [4 u(xx2 x3; tjl u(xi+x nx2+x2 x3+x3; ti2 X2 X2 2x U(Xl,x2,X3; t) ( U(Xl+uXX2+x,X3+X2; t+T) (V-2) u2(x1,x2,x3;t) / U(xl1,2+x, x2+xx3+x3;t) where x2 < 5SL (thickness of viscous sublayer) x2 + xz < 6SL similarly, we have U (Xl,x2,X3;t) U.(Xj+X,X2+X2,X3+X3;t+T) R,(XXX;3 (V-3) U (xl,x2,x3;t) Uj (X1+Xxl, x2+X2,Ix3+X3; t) where X2 < 5 Hence, one defines the wall shear-wall shear or the wall shear-velocity correlation coefficient in terms of a velocity-velocity correlation coefficient. The measurements of RTwTw RU, R RUU Rvv Rww, Ruw and RVw are shown in Figs. 6 through 28 as functions of the non-dimensional time Uoo(T -T' ) delay ( where T' is the time delay for maximum correlation of any particular curve. An additional vertical axis stands on each curve,

20 except those with xi = 0 (or 0), and shows the true origin 6* 6* U T U = 0. In each case the non-dimensional convection speed c can 6* U be computed by dividing the longitudinal spacing x_ by the correspondU T' ing non-dimensional time delay *. Since T' is the time delay for 6* maximum correlation, the convection speed determined in this way is de6Ru u.(XkT) fined by Riuj= 0. According to Wills(33)(1964), in measurements aT in a shear flow the convection speed, Uc, is a function of spacing 6 (between two measuring points) and time delay T; when Uc is defined by aR(6,T) cr aR(6,T) 0 a slight difference in magnitude was observed. aT a aRu However, for practical reasons the convection speed defined by Ui = was used in the present work. Co RESULTS The convection speed Uc in each case was computed by using the following relation, xl UooT UC -* 65* U) and was marked in the corresponding figures. The measured correlation curves of RT T are symmetrical about the w w vertical axis passing through the origin of the time delay axis. The symmetry was destroyed when one of the measuring points was raised from Y -= 0005 to 00204. When y/6* > 6SL' we actually measured RT u instead viu eion T pa l te relin e R was fn viscous re gion e The peak value of the correlation curve of RT U, was found

21 to increase slightly and shifted to the left side (T < 0) with the increasing y/b*. Such a destruction of symmetry and the shift of the peak of the correlation curves, which appear clearer in the contour diagrams of the constant correlation RT u (see Figs. 7, 40 and Section VII-C), will be explained in Section VIII-A. In the Rvv measurements at zero time delay a change of sign from positive to negative was observed when the two probes approached the wall as near as =2 - 0.16 (see Fig. 15). Grant(6) (1958) did not report any measurements of Rvv for -2 < 0.29. In his reported measurements near the wall, also 6* no negative Rvv has been observed. The change of sign was also found in the present measurements of Rwv near the wall at zero time delay when the two hot-wires which were set up at the same values of xl and x2 but different x3 were interchanged [e.g., X3 = + 0.223 for Rwv (see Fig. 26)]. Furthermore, when one studies the measurement of Rwv (see Fig. 28) one finds that Rwv changes from a negative to a positive value as Xi varies from + 1.52 to - 152. The change of sign in the above mentioned measurements and the unusual behavior of RPv show that there is a definite structure of turbulent eddies in the wall region. We shall study this structure in more detail in Chapter IX.

CHAPTER VI SPACE-TIME CORRELATION OF PRESSURE-SPANWISE VELOCITY, Rpw A. INTRODUCTION The pressure-velocity correlation measurements of Rpu ard Rpv reported by Willmarth and Wooldridge(27) were quite extensive and revealing. In the present work a number of measurements of Rpw were made for the following purposes: (1) To study how the turbulent velocity w is correlated to the wall pressure fluctuation; (2) To study the relation between pw and vw using Kraichnan's(12) theory for the wall pressure fluctuations (see Section VIII-B); and (3) To construct contours of constant R in order to study the pw turbulent structure near the wall. B. MEASUREMENTS The pressure-velocity cross-correlation coefficient, Rpw is defined by Rpw(X1X2x3;T) = p(x,y,z;t) w(x+xl,y+x2, z+x3;t+T) (VI-1) I p2(x,y,z;t) 1 w2(x+xl,y+x2,z+x3;t) The experimental measurements are shown in Figs. 30 through 38 as functions Uo(T - T ) of the non-dimensional time delay. The shift of the origin on the time delay axis, given by, is defined in the same way as 22

23 has been explained in Section V-B above. In consideration of the symmetry of Rpw about the xl-x2 plane, let us refer to Fig. 29 where the point of measurement of pressure fluctuations coincides with the origin and the points A and B are symmetrical with respect to the xl-x2 plane. The flow direction is into the plane of the figure along xl-axis which is not shown. Assume that a homogeneous turbulence exists in the plane which is parallel to the wall as is the case the boundary layer used in the present study. Then, if one defines the positive sign of velocity component w being along the positive direction of x3-axis, it will be quite obvious that +w at point A or -w at point B should give the same pressure response at point p merely because -of geometrical symmetry. One can write Rpw(xl,x2+x3) = -Rpw(Xl,X2,-x3) (VI-2) and infer that when X3 = 0 Rpw(xl,X2,O) = 0 (VI-3) from an argument of Rpw being unable to be either positive or negative. Consequently, in measuring Rpw, one can confine oneself to the quarter space between positive x2-axis and positive x3-axis and reduce the amount of work required. C. RESULTS The convection speed, Uc, of the disturbance as defined in Section

24 V-B was found equal to the local mean speed when spacing is not too large, or UC = U. Since the same result was also found in Rpu and Rpv measurements formerly (27) made by Willmarth and Wooldridge, one may conclude experimentally that all the three pressure-velocity correlations (i.e., Rpu, Rpv and Rpw) are convected with the local mean speed when spacing is not too large. Measurements of Rpw(see Figs. 50-31) showed that Rpw is an odd function of x3, which supports the discussion made in Section VI-B above. The measurements have been used also to construct contour diagrams of constant Rpw, which will be presented in Chapter VII.

CHAPTER VII CONTOURS OF CONSTANT CORRELATION OF TwTw, OF TwU AND OF pw A o INTRODUCTION Owing to the existence of the linearity (see Appendix I) in the viscous sublayer, the longitudinal fluctuating velocity u implies a fluctuating slope of the velocity profile in that region. Physically, a fluctuating slope of the velocity profile means a fluctuating local wall shear stress. Thus, we measured the streamwise wall shear-wall shear and streamwise wall shear-velocity correlations with hot-wires and plotted them as functions of nondimensional time delay. Since the fluctuating wall shear stress w -= x (x2 < bSL) X2 where x2 can be interpreted as the fluctuating vorticity component in x3-direction in viscous layer, these measurements can also be interpreted as the correlations of vorticity-vorticity (~-~) and of vorticity-velocity (5-u) where ~=fluctuating vorticity component in x3-direction in the viscous layer. With these results we constructed contour diagrams of constant RTwTW (or RSS) and of constant RTwu (or Rfu) in order to obtain a clear picture of the correlation field of the fluctuating streamwise wall shear stress or the fluctuating vorticity component in x3-direction in the viscous sublayer. The contours of constant correlations, pu and pv were measured and reported by Willmarth and Wooldridge.(27) In the present work the results of the measurement of correlation, pw have also been used to construct 25

26 similar contours of constant correlations. Thus, together with the former work, we have a complete set of constant correlation contours of pu, pv and pw which may help those working on the similar problems to get better understanding of how the three fluctuating velocity components in the semispace above the wall produce the pressure fluctuations at the wall. B. THE CONSTRUCTION OF THE CONTOURS A direct measurement of enough data points to allow the construction of the contours of constant correlations of TWTw, of TwU, and of pw would require tremendous tunnel work; 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 when the pressure transducer was located in the wake behind the hot wire probe. Hence, the spatial correlation contours were obtained from the measured time-delay correlations at zero longitudinal separation by using the convection transformation X1 UT Uc (VII-1) The correlation contours obtained by the use of this transformation are shown in Figs. 39 through 46 where in case of pw measurements the origin of the corrdinate system is located at the point where the wall pressure was measured, and in case of TWTw or TWU measurements the origin is located at the wall and directly beneath one of the two hot-wires.

27 C. RESULTS The contour diagram of constant RTwT (or R ) constructed above appeared to be a very narrow strip. It is some 126* in length and 26* in width. The maximum positive correlation was at the point where the fixed u-wire in the viscous layer was located during measurement (see Fig. 39). The contour diagram of RTu (or RIu) for x = 0.118 was similar in shape, but elongated along xIl axis (29V* in length and 1.86* in width) 6* and with the maximum positive value of RTwU located off the origin at x o.6 and X 0.13 (see Fig. 40). The angle of sweep of this 6* 6*IC maximum positive R was measured to be 12~ to the 5* -axis when projected Twu on the, - - 6* plane, and 110 to the -axis when projected on the xl plane. It is believed that this sweep represents the effect of convection and turbulent diffusion of the local wall shear fluctuations or the vorticity fluctuation in the x3-direction. To verify this point, a simple calculation has been made and will be found in Chapter VIII below. The contours of constant correlation Rpw which lie in vertical planes perpendicular to the flow at X1 _ 1, -0.5 and 0 are positive and are closed curves of constant Rpw. These positive contours have a tendency to move upward when F increases from negative to positive values. The positive contours are replaced by the negative contours near the wall when xl + 1, and finally break into two separate groups of positive contours 6F* with the negative contours near the wall growing even larger when xl - + 3 (see Figs. 41 through 46). An explanation of the variation of the 6*

28 R contours along X1 -axis has been attempted and will be included in Chapter IX. Chapter IX o

CHAPTER VIII DISCUSSION OF THE RESULTS OF MEASUREMENTS A. THE PROPAGATION OF FLUCTUATING WALL SHEAR (OR FLUCTUATING VORTICITY IN x3-DIRECTION) DUE TO CONVECTION AND DIFFUSION The contour diagram of constant correlation of TWTw (Fig. 39) and that of TwU (Fig. 40) show that there is a shift of the maximum positive correlation off the origin when the moving wire has larger distance from the wall than the fixed wire. In Fig. 40, the location of maximum positive correlation was observed to be: X_ x3 0.60 - 0.1 This deviation from the origin of the axes is believed to be caused by convection and turbulent diffusion, and thus shows how the fluctuating wall shear propagates from inside the sublayer region (X2UT < 5). AcV cording to the theory of turbulent diffusion (Ref. 29, p.48), at small time t diffusion proceeds proportionally with time, or x i - ui.t (VIII-la); and at large time t >> t*, where t* is the time for which the lagrangian correlation RL(t*) = i( i(t+t*) 0, diffusion is proportional to the s o7square root of time, or a xi 21L u t (VII-lb) 9L 1 29

3o0 where t* YL = | dT RL(T) Assume that very near the wall the turbulence is beginning to diffuse, hence we apply Eqn. (VIII-la) in the calculation below. Next we refer to Fig. 47 and suppose that a random disturbance first hits the ul-wire at xl = 0, x2 = 0.002 in, x3 = 0 (location of the fixed wire during measurement of RUu) and reaches a new point within a time period At xl = dl (unknown) x2 = 0.06 in (height of the 2nd wire; given) X3 = d3 (unknown). On this basis, we are able to equate the following: At d d d3 (VIII-2) (Uc)avg (\f_) )avg avg where d' = 0o06 - 0.002 o 0.058 in (difference of height between ul - and u2 - wire). The convection speed was found from measurements: (Uc)x2 = 0.002 in = 36 fps (Uc)x2 = 0.06 in = 110 fps hence (U) ) 73 fps; c avg

31 also from measurements (7)avg - 12.4 fps h) a - 14.4 fps. Substituting the given values into Eqn. (VIII-2), one finds dl = 0.34 in d3 = 0,068 in or d1/6* = 0.69 d3/6* = 0.14 (calculated) where 6* = displacement thickness of the B.L. = 0041 ft. This result can be compared with what we observed from the location of the maximum positive correlation of RTWu (see Fig. 40 and compare with Fig. 39): X1 - o.60 0.13 (observed) where xi, x3 are corresponding to dl, d3 respectively. As shown above agreement was found between the displacement determined by Ruu measurements and the displacement calculated on the assumption that the turbulent diffusion is at the initial stage, one naturally believes that the turbulence very near the wall is still very concentrated and is just beginning to diffuse. B. PHILLIPS' INTEGRAL CONDITION AND THE MEASUREMENT OF THE PRESSURE - VELOCITY CORRELATION, Rpw Phillips(l8) in his study of the aerodynamic surface sound in a turbu

32 lent boundary layer derived an integral condition which when applied to the present Rpw measurements reads 00 00 Rpw dx dx3 = 0 (VIII-3) -00 -00 where xl is the axis along stream direction and x3 the axis normal to the stream and parallel to the wall. Both the present measurement and our theoretical consideration of Rpw showed that it is an odd function of x3 (see Section VI-B and Figs. 30 through 31), therefore for Rpw one can even write down a more strict condition than Eqno (VIII-3) 00 Rpw dX3 (VIII-4) -00 In the report of measurements of the correlations pu and pv by Will(27) marth and Wooldridge, a qualitative comparison was made between experi(12) mental results and the approximate wall pressure theory of Kraichnan. For the present measurement of pw one can also infer a similar relation as follows~ V dr(r2) p(O,,) W(x;t,t - +P,) (r-x,-) rl dV(r) (VIII-5) r r(rl,r2,r,3) -- variable V - semi-space above the wall However, too much tunnel work is required to obtain enough information

about the correlation vw(r-x,r) for just making a qualitative comparison of the above equation, since our measurement showed that the vw was not conserved by convection. So we have only studied a special case: When T = 0 and x = x(O,x2,0), then we have pw = 0 (pw is an odd function of x3) at the left hand side of Eqn. (VIII-5). At the right hand side of Eqn. (VIII-5), 6U is always positive and our measurement of wv showed that 6r2 near the wall, e.g., X2 = = 0.102 (see Figs. 26 through 27) wv is an odd function of r3 and therefore the total volume integral at the right hand side of Eqn. (VIII-5) should vanish also. The further interest we have in studying the pressure-velocity correlation, Rpw lies in its relation with the turbulent structure near the wall which will be presented in detail in the next chapter.

CHAPTER IX THE STRUCTURE OF TURBULENCE NEAR THE WALL IN A TURBULENT BOUNDARY LAYER A. INTRODUCTION As discussed briefly in Chapter V above, the question, "What is the real structure of turbulence near the wall and what is the real mechanism which maintains the turbulence in a turbulent boundary layer?" has interested many investigators, both theoretical and experimental, for decades. To the author's knowledge no definite answer has yet been giveno This was not only theoretically due to the lack of a complete mathematics which is able to describe the whole history of the turbulence of different stages existing contemporarily in a turbulent boundary layer, but also experimentally due to the difficulty in techniques that it is almost impossible for us to extract any single turbulent eddy or disturbance to study its origin and development. However, it is believed that the statistical method, when it is properly employed, is still one of the best which can describe the average behavior of turbulence. In the present work the measurements of the correlations Rvv and Rwv showed that their'behavior in the wall region (X2 7 02) was completely different from that at greater distance from the wallo As mentioned in Section V-C, Grant(6) did not report any measurements of Rwv, or Rvv for X2 < 0O290 Also, no negative correlation has been observed in his reported 6* measurements of Rvv near the wallo The result of the present measurements 34

of RVv and Rwv together with the constant contour diagrams of Rpw and RT U stimulated us to propose a physical structure of the turbulence near the wall as presented in the following section. B. THE STRUCTURE OF TURBULENCE NEAR THE WALL In the problem of the transition from laminar to turbulent boundary layer either theoretical or experimental studies(e'g'' 2,8,10,11,22) showed that, in the course of development from two-dimensional TollmienSchlichting waves to the final stage when the turbulent spots are formed, a necessary intermediate step is the appearance of streamwise vortex components. According to Stuart,(22) it is this streamwise vortex component which produces the vertical convection of the span-wise vorticity component,, which is at the same time stretched along the x3-direction X2 so as to be intensified and eventually generate turbulence. We believe that such a s"treamwise vortex component, or together with the other two components a three-dimensional vortex line of a certain shape, also exists in the turbulent boundary layer and is an important part of the physical mechanisms which maintain the turbulence. To explain physically how a three-dimensional vortex line is formed and to find what shape it is we first refer to the work by Browand(5) (1965) who studied the behavior of small vortices in a shear flow. As shown in Fig. 48, Browand concluded that a small vortex of opposite circulation (positive) from the mean circulation (negative) always experiences a restoring force, + p(U2-U1)r (where r is the circulation of

the small vortex), when it is displaced either upward or downward; while a vortex with circulation in the same direction (negative) as the mean circulation (negative) always experiences a destabilizing force when displaced~ With this result in mind we now study the region near the wall in a turbulent boundary layer~ The viscous sublayer next to the wall is usually considered to be equivalent to a Couette flow in which all disturbances are highly damped,( 6) therefore an outer region near the edge of the sublayer, where the damping effect is much weaker but the shear flow is yet very strong9 would be more suitable for disturbances to grow and develop~ Suppose small disturbed waves, which are mainly two-dimensional, first appear and finally become concentrated to form discrete vortex lines around the edge of the sublayer and in the direction of x3-axis (see Fig~ 49a). As discussed at the beginning of this section the vortex line of opposite circulation from the mean circulation will remain one-dimensional along x3-axis9 while the vortex line with circulation in the same direction as the mean circulation will not remain one:-dimensional whenever there is a perturbed vertical motion imposed somewhere at a radnom position along this vortex lineo Assume that Browand? s two-dimensional result can be applied here locally, then the random disturbance may induce an upward or a downward local. motion at the latter vortex line and this local motion will continue in the region where'the shear flow is very strong. To illustrate these ideas, suppose there are disturbances of the vortex line as shown in Fig~ 49bo Once the vortex l.ine is bent vertically the stretching process in the

37 shear flow starts simultaneously, and finally the vortex line with circulation in the same direction as the mean circulation will take the form as shown in Fig. 49c. With the above physical strcuture in mind it seems that one can explain why an originally two-dimensional wave motion becomes three-dimensional in a strong shear flow and why the vortex flow in x2-x3 plane often occurs in pairs with streamwise vorticity components of opposite sign in a transition boundary layer as observed in experimental works (eogo, see Refso 6 and 10)o A similar physical structure which was predicted in a transition boundary layer by different authors is quite the same as above Hama(8) (1963) studied the behavior of a single vortex line in a shear flow and pointed out that besides the stretching due to the shear flow the plane of a curved vortex line rotates in a direction opposite to that of its circulation, which gave the same result as we discussed above. However, Hama did not discuss the case when the circulation of a vortex line has different sign from the circulation of the mean flow. Stuart(22) (1965) studied a problem pertaining to boundary layer transition in which longitudinal vortex pairs already exist in a boundary layer, and under his assumptions he linearized the equation of motion and solved the time dependent problem. He emphasized two things in his paper: the vertical convection and the span-wise stretching (due to aw ) caused by these 6x3 streamwise vorticeso Owing to the vertical convection the span-wise vorticity component, U2, travels upward and downward alternatively along the x3-direction and owing to the stretching it is intensified

38 in the course of its travelling. On this basis he finally proposed a structure of the vortex line similar to what we have presented above. Benney(I) (1964) also solved a problem mathematically in a boundary layer possessing constant shear, his result showed the existence of a vortex pair in the y,-z plane which agrees with the above discussion. When compared to the work by Stuart the structure discussed in the present work depends more upon the interaction between the vortex -line and the mean shear. Besides, we think that the vortex line picks up motion rin the plane perpendicular to the flow direction before its longitudinal vortex line component can be formed from stretching. In any case, Stuart's idea of using the convection and the stretching due to longi.tudinal vortices to explain the occurrence of intensified shear layers is valuable, since it is believed that such shear layers are closely related to the flow finally breaking into turbulence. (lo11l22) In the present work, the turbulence generating region near the wall but oautside the viscous layer is believed to contain a random distribution of vortex lines making an acute angle to the wall and extending downstream.. We shall compare this structure of turbulence near -the wall with the experimental results in the following sectiono CO COMPARISON WITH EXPERIMENTAL RESULTS 1. REv Measurements According to the discussion in previous section near the wall in the turbuile:at boundary layer there exists a region in which the vortex

39 lines as shown by Fig.(49c) are randomly distributed with respect to time and space. So, in the x2x3-plane one would see an average vortex flow as shown by Fig. 50. If we think that an average model exists there, then it is quite obvious that the double velocity correlation RvAvB(where vA = v measured at point A, etc.) is always negative in this region. Indeed, our measurements showed that when the two measuring hot-wire probes approached the wall region (x2 0.2) the sign of R was changed from 5* " ~~vv positive to negative (see Fig. 15). One also can refer to the report by Willmarth and Wooldridge (see Ref. 27, Fig. 37, p. 52) to find the corresponding drastic change in the Rpv constant contour diagrams as one approaches the wall at X2 - 0.5 and 0.2. 6* 2. Rwv Measurements Referring to Fig. 51, with w measured at point A and v measured at B one should find RwAvB to be negative, and similarly RwAvc to be positive; therefore R _ RW wiAvB AvCA when points B and C are in symmetrical position with respect to point A. The above result was found in agreement with our measurements (see Fig. 26). Next, referring to the average model of the vortex lines in Fig. 52 one finds RWBvA to be negative ana HwCvA to be positive, which was also verified by our measurements as shown in Fig. 28.

40 3. Rpw Constant Contour Diagrams If we set up an average model of our vortex lines as shown in Fig. 53a with one of the lower loops located at the point where the perturbation pressure, p, is measured then we can see that the distribution of the sign of velocity component w at different vertical planes which are parallel to the x2-x3 plane and located one after the other from upstream to downstream of the origin 0, will be as shown in Fig. 53bo Since the fluctuating pressure, p, recorded at the origin 0 which is directly underneath the lower loop of the vortex core should have a negative sign, one can easily find the distribution of the sign of Rpw at these vertical planes as shown by Fig. 53c. Again referring to Fig. 53a, if instead of point 0 we choose point A as the location where the perturbation pressure is measured, then since the pressure recorded there is positive and the sign of w in different vertical planes as shown by Fig. 53b is also reversed the distribution of the sign of Rpw as shown by Fig. 53c will still hold without any change~ The present experimental result of Rpw (see Fig. 46) was found to be in coincidence with the result as derived from the above discussion. Besides, the model proposed here can also predict that the pressurevelocity correlation, Rpw, is an odd function of x3 (see Fig. 53a), which agrees with the discussion in Section B, Chapter VI as well as with the experimental result presented in Figs. 30 through 31. In a closer examination of the contour diagrams of constant Rpw (see Figs. h4 through 46), one finds that at positive 1 the positive 6*

41 contours on the top gradually separated into two groups while the negative contours remained unseparated. To explain this, let us refer to Klebanoff's work (see Ref. 10, Fig. 19, p. 21) where the shape of an actual vortex flow in the X2-X3 plane during the nonlinear process of transition to turbulence was plotted and let us think that the contributions to Rp are mainly from the average of two cases: either with the pressure measuring point located at 0 or at A as shown in Fig. 54a. Then one can see that the two vortex flows (see Fig. 54b,c) when correlated with the pressure at point 0 or A, will agree with the constant contours of Rpw at positive x- as shown by our experimental result, i.e,, the positive constant contours on the top gradually separate into two groups while the negative constant contours remain unseparated. The scale OA in Fig. 53a was measured from direct observation of the constant contours of Rpw and found to be 36*. Referring to the same figure, the angle of inclination G between the plane of the sweeping vortex line of our average model and the wall was computed also by using the Rpw constant contour diagrams. From the displacement of the zero line between positive and negative Rpw contours at two different vertical planes where x- was equal to 1 and 3, we found this angle 0 to be ap6* proximately 35 degrees. 4. Ruv Measurements Schubauer and Klebanoff(20) (1951) measured the correlation of velocity components u, v at the same points in a turbulent boundary layer and found

it always negative. One may easily find that from the present proposed structure of turbulence the fluctuating flow field induced by the inclined vortex line as shown in Fig. 53a, has velocity components u, v which also give negative correlations.

CHAPTER X CONCLUSIONS The main results of this experimental work and theoretical study near the wall in a fully developed turbulent boundary layer can be summarized as follows: (1) Near the wall there was found a turbulence generating region which is filled with random vortex lines inclined at an acute angle to the wall when observed along the x3-direction. These vortex lines were stretched and intensified in a strong shear region near the wall, and owing to the streamwise component of these vortex lines there must be vertical convection and span-wise stretching (due to a- ) of the span6x3 wise vorticity component, -, to generate strong shear layers (which 6x2 are thought to be substantial elements for making turbulence) (l0 ll 22) It is believed that the proposed structure in the wall region is the real mechanism which serves to maintain the turbulence and to produce the wall pressure fluctuations in a turbulent boundary layer. The proposed structure was strongly supported by the present measurements of Rvv, Rwv and Rpw and it was identified to be quite the same as the structure in transition boundary layer recently studied by different (e.g. 1,,$22) investigators, except that in the present case in a turbulent boundary layer the appearance and the distribution of these inclined vortex lines are much more random.

44 (2) The diffusion of turbulence near the wall was found approximately proportional to time to Since this is true only at the initial stage of turbulent diffusion, the turbulence generated in the wall region can reasonably be thought of as fresh and new~ Both this result and the result presented in (1) above lead to the conclusion that in a turbulent boundary layer the turbulence is generated in the wall region and spreads to the outer region due to convection, diffusion and the interaction with shear flows Since it is well known from measurements that eddies of small size dominate near the wall, this implies that at least a portion of the small eddies near the wall may become larger when they distend into the more central parts of the boundary layer. However, the physical mechanism of development from small to large eddies is not yet completely understood, (3) The measurements show that the profile of the viscous sublayer next to the wall is fluctuating. However, the instantaneous linearity of a fluctuating sublayer profile was demonstrated either from a theoretical or from an experimental study (see Appendix I)o. On this basis, one can make streamwise wall. shear-wall shear (or x3-component of vorticity-vorticity) correlation measurement by using hot wires. (4) The zero to zero scale in X6 -direction of the measured Ruu contours near the edge of viscous layer (see Fig~ 39) can be interpreted as the average period of variation of u-component in x3-direction. This scale was measured to be some 26* which is comparable to the corresponding optical observation by Runstadler, Kline and Reynolds (1963) (see

45 Ref. 19, p. 262). (5) The mean wall shear stress can successfully be measured with a single hot wire set up at the linear region near the wall, using Wills' experimental method to correct for the effect of wall heat transfer. Correction of the mean-velocity measurement with a hot wire in the presence of velocity fluctuations in a turbulent boundary layer was studied. A correction term of first order to the mean square value of the fluctuating resistance of the hot wire was found analytically for constant current measuring equipment (see Appendix II-A). For the present problem the correction was computed to be 3% which agreed with the correction found by an alternative graphical method. In calibrating the hot-wire in the present measurement of wall shear stress in the wind tunnel, it was found that the calibration curve should be plotted with Nu vs. NRey)w 2 (not I vs. 4U) so that a linear curve can be obtained in order to make necessary correction easily when the ambient temperature and the air density during calibration are different from those during measurement. (6) In case that the turbulent velocity is not negligibly small when compared with its mean velocity, the calibration of the turbulent intensity measurement is not linear. The correction term of the u2 2 measurement was found approximately in terms of r4 and (rw) where r is the fluctuating hot resitance of the hot-wire (see Appendix w Il-B). For the present measurements the possible maximum error was estimated to be 6o5%.

APPENDIX I THE EXI STENCE OF INSTANTANEOUS LINEARITY IN THE VISCOUS SUBLAYER In the measurement of correlation of the fluctuating wall shear, an instantaneous linear viscous sublayer was assumed beneath the turbulent boundary layer. To verify this, we write the fluctuating part of shear stress in the viscous sublayer and neglect its variation with x2 (O < X2 < S.L. ), T [u -luv + _, (I-1) P ax2 The boundary conditions at the wall requires that u - v = w = O at x2 - O and the continuit- equation gives Kr2x] =0 o _X2 x2=o It follows from Eqn. (I-1) that Lai o - lw L %-~x2=0o p, F2U1 1 -uvuv] - F v u - v - v _x2 ax2 ax2 a x2= =0 similarly, 46

47 L3U] 1 a2 )ax2 =VX2 [uv-uv]X2-O but 4 1 a [uv-uv] (I-2) aX _ XZ-O 2aX23 X2-X 3au 62v au 2v. V 6x2 6x22 ax2 ax22 x2=0 As the second and third derivatives of the fluctuating velocity component, u, vanish at the wall, the variation of u will be closely linear for an appreciable range in x2 direction. The above argument was first used by (25) Townsend5) for the mean velocity profile near the wall in a turbulent channel flow. Experimentally, the measurements of -VUU, in turbulent boundary layer (9) (14) by Klebanoff and in turbulent channel flow by Laufer4) showed that such a linear fluctuating region does exist in the vicinity of the wall which covered the whole region of viscous layer. Since IJ7= c(xl,x3,t)'x2 implies u = c(x,,x3,t)'x2, their experimental results lend support to the existence of instantaneous linearity in the viscous layer as discussed above. In the present work no such experiment has been repeated, but our correlation measurement of Ru with zero time delay near the edge of viscous sublayer with one hot-wire varying its height gave approximately the same value, which also showed the agreement as predicted by the instantaneous linearity of the viscous layer (see Fig. 7).

APPENDIX II ERRORS AND CORRECTIONS A. CORRECTION OF THE MEAN VELOCITY MEASUREMIENT IN A TURBULENT BOUNDARY IAYER When we measured the mean velocity (see Chapter IV and Appendix III) a fixed mean overheating ratio aw was used. At constant current the fluctuating velocity causes the wire temperature and therefore the overheating ratio to fluctuate. However, the mean of the fluctuations of the overheating ratio does not correspond to the mean of the velocity fluctuation since the velocity does not vary linearly with the overheating ratio. During measurements we observed a mean overheating ratio, so the corresponding velocity found was in error. The overheating ratio, aw, has a linear relation with hot resistance, Rw Rw-Re aw = Re where Re = cold resistance (independent of velocity fluctuation). And since Rw shows up at both sides of King's formula, I2Rw = (Tw-Te)(A+B J) Rw"e ( ok { II- 1 ) Rw-Re we study the relation between velocity U and Rw. From Eqno (II-1) 48

49 B =IP- -A (' =Roa) Rw'Re or (I2a'-A) RW + ARe B (Rw-Re) let Rw = Rw + rw (where rw = fluctuating hot resistance) U - U + u (with v2 neglected) be substituted into Eqn. (II-2), and after squaring both sides of the equation we have [(I-cz-A)Rw+ARe ]2 + (I _ a iA) U+u B2(Rw-Re)2 [1 + _w 2 let C3(l+Clrw)2 ( +crw ) 2 where I2c-A I2'-A [ (I2' -A)Rw +ARe ] C2 = 1 (II-3) Rw-Re F (I2c'-A)Rw+ARe 1 2 L B ( w-Re) J then U+u = c3 (l+2clrw+c l2rw2) ( 1+2c2rw+c2arw2) T 4- W..r

5o C3 [ 1+2(cl-c2)r+(C l-c2) (c c-3C2)r -2c2(cl-c2)(cl-2c2)rw (II-4) +c22(c l-c2)( 5cl-llc2)r 4+... ] Taking average at both sides of Eqn. (II-4) and assuming u and rw to be waves symmetrical to their mean values, respectively, then U = c3[t1+[(c1-c2)(c1-3c2)]rw2 + 0(w4) +.., c3+c3[ (C1-c2)(C-3C2)]rw2 (II-5) After using the value of clc2, and c3 from Eqns. (II-3) and simplifying, Eqn. (II-5) becomes I 2 - r(I2t'-A)Rw+A.Re ( 1 2I2 ReI2'-A.) + -- I =.-__ )3 B ([I w — ) Bw-A ReW)3 3 (I2(Z, Re )e (~_Re)4 I rw (Rw-Re) J whe re c (I~ a'-A)Rw+ARe 2 U [ B'2w-Re Imean in error so, the correction for the mean velocity is A U = U - U mean in error 1 F 2I2X'Re(I-'-A) + 3(I Q'Re) 1 B L (-: )2 (RwRe)4 J (II-6) The value of B and the parameters in the bracket of Eqn (II-6) were given by measurements or calibration, and r,2 canl be obtained by measuring

51 e2 since e2 = I2'rw2 (I=const.) (II-7) Computation: In the present measurement of the mean wall shear stress (see Chapter IV and Appendix III), we have the following information from measurement s:,Umean in error = 39 fps rw2 = 0.0722 (p2) After using these figures and other given values in Eqn. (II-6), we find A U = l.16 fps therefore A U 1.16 error - - U 239+1'16 = 2.9% A graphical correction method tried by us gave the error A U B. THE ERROR OF THE a —' MEASUREMENT The error of the l' measurement is estimated as follows: From Eqn. (II-4) in Section A above U4-u - c3 [1+2(Cl-c2)rw+(c1c2) (ci13c2)r w2 (II-4) -2c2(C1iC2) (cl-2c2)w+.] where the coefficients ci, c2, and c3 are defined by Eqns. (II-3). With the aid of Eqrm (II-5), we have from Eqn. (II-4)

52 U - o3(Cl-C2)[2rw+(CI-3C2)(rw rw2)-2c2(c1-2C2)rW +...] (II-9) Equation (II-9) is squared at both sides and computed up to the 4th order of rw, then U2 c32( C-c2)2(4r2+(cl-3c2)2[rw4-2rw2rw2+(r2)2] +4(cl-3c2)rw(rw2_rw2)-8c2(cl-2c2)rw4 + a. oP Taking average of the above and assuming u and rw to be waves symmetrical to their mean values as we did in Section A, then u2 ~ c32( cl-c2)2 4rw2 + [ (c1-3c2)2_-82(0c-2c2) ]rw4 (II-10) -(cl-3c2)2 (rW2)2) Usually, in measuring small velocity fluctuations a linear relation was assumed between velocity fluctuations and electronic signals, i.e., u = se where s = sensitivity coefficient of the hot wire e = fluctuating voltage = Irw, hence 2 2 2 u = s e = (sI)2r2 (I = constant) Comparing this equation with Eqn. (II-10), we find the correction term approximate ly A U2 = C32(C -C1C2-(-c2c22cr -C 2) (r —)) (II-11)

53 To compute the possible maximum error in the present measurements, we use ( 0 U)max 0 35,9 U = 40 fps. Correspondingly, in our measurements we have r 2 = 0.168 (Q2), and if rw is assumed to follow a sine wave w variation, then'rW = 0042 (Q ) Using these values and other information from measurements in Eqn. (-), we find A 0.12 U2 Hence, erro r 5e7% C. OTHER ERRORS IN THE MEASUREMENT (1) From Meter Reading —The averages involved in computing the correlation were obtained by visual observation of the output mieter reading. As 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 about ~-5% of the maximum value. (2) From the Filtering Effect-See Section III-B. (3) From the Hot-Wire Alignment-In measuring v- or w-component of the fluctuating velocity an X-type hot-wire probe with a gap between the two cross wires, about 0.02 in~ was used. The two wires have different orientations when they changed side gave an effect to the measurement.

54 Such an effect was examined in the Rwv measurements (x2/5* = 0.102, X3/5* = +0.223), using hot-wire probes with opposite wire orientations (See Figs. 26-27). (4) From Using the Convection Hypothesis-When we constructed the contours of constant correlation of Tw-Tw, of Tw-u and of p-w the convection hypothesis was used (see Chapter VII). The amount of error introduced from this source can be determined at those values of xl/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. The maximum error in the correlation near the peak value of RTwTw or RTwu is approximately +20% of the peak value. In constructing the contours of Rpw in x2x3-plane at xl/b* = 0, actual measurements were made at every point, hence no error resulted from the convection hypothesis was noted. However, the two Rpw contours at negative values and two at positive values of xl/b* were constructed by using the convection hypothesis in order to make a qualitative study of the variation of the Rpw contour along xl/b* axis. Therefore, the maximum error in the correlation near the peak value of Rpw can be as high as +40% for these four contours.

APPENDIX III CALCULATION OF THE MEAN WALL SHEAR STRESS In Chapter IV the measurement of the wall shear stress was introduced and the result was reported. Here we give the details of calculation procedure as follows: A. INDIRECT APPLICATION OF WILLS'(28) CORRECTION FOR THE WALL HEAT TRANSFER From the present hot-wire measurements we have the following information: b = distance between the hot-wire and the wall = 0.00216 in. a = radius of the hot-wire - 0.0001 in. = length of the hot-wire 0.043 in. (0.00358 ft) Pa = density of air at test section 0.00207 slug-ft-3 = coefficient of viscosity -6 - 2 0.406 x 10 lb - sec - ft (for air at 113~F) k = coefficient of thermal conductivity 47 x 10-3 watt - ft-1 - ~R-' (for air at 113~F) I = electric current through hot-wire 53.34 x 1C0-3 ampere 55

56 = average resistance of heated wire = 8,78 ohms Re = cold resistance of hot wire = 5.85 ohms e e= ambient temperature = 3180K (5730R) = temperature coefficient of resistivity 0.00392 (~C)-1 or 0~00218 (~F)-z for Pt wire UO = free stream velocity = 206 fps First, we compute rate of heat loss = I2R-w = (533 4xlO )2x878 =2497X10-3 watt, then, from hot-wire calibration curve, we find w- = 0 CCo302 hence, one can calculate the flow velocity Oo ~0022 U = 0~00207 44K1 fps and the hot-wire Reynolds number IJ<(2a)pa (Rey)w = 44. l x0o00207 CO 406x10x12 3 76

57 or o. 45 45 (Rey)w = (3.76)0~45 = 1.82 Referring to Fig. 2 Wills' report (p. 392) for o0,45 b (Rey) = 1.82 - - 21.6 w a we have o-0* 17 ANu( ) = 0,116 Ge The above value was found by measuring the vertical distance between the curves b/a = 21.6 and b/a = 0o (when b/a + co the wall effect is absent). Physically, this vertical distance is equivalent to the amount of additional heat loss of the hot-wire due to the effect of heat transfer to the wall. -0.17 The value of Nu (Qw/~e) can either be measured from the same figure or can be calculated as follows: since one can write the relation between resistance and temperature of the wire, Re = Ro [l+a(Qe-Qo)] (III-1) similarly, Rw = Ro [l+o(Q-wo)] (III-2) subtract Eqn. (III-1) from Egqn. (III-2) and rearrange the result, we have Re Qw = ~e + aw (III-3) where aw = overheating ratio of hot -wire Rw-Re Re = 0.5 Re

From Eqn. (III-1), Ro was found to be 4.98 ohms, which together with other known values was substituted into Eqn. (III-3), hence 5.85xO.5 w = 318 585x5 0.003 92x4K. 98 = 4680K (or 843~R) Then, one can compute -0.17 -O.L1 Nu K (@wa) @ 24.97xO10-3 468 -0 17 47x10-3XO.x0 00358x(843-573) \318 1.64 and the percentage of correction of heat loss, (Qw \ n-0*17 A Nu Qe 0.116 707 or Nu =w )-707% A 2R 7.07W IaRw Therefore, after the correction is applied (IT2Rdfte = 24.97x10-3x(1-0o0707) -3 23.2x10 watt Referring to the hot-wire calibration curve again, we found JpT'= 0.262 and U = 33.2 fps. Thus, before correction U = 44.1 fps after correction U = 3352 fps

59 This result gave the correction of U for a laminar boundary layer. In case of a turbulent boundary layer Wills found empirically that a factor 0.5 should be applied to the correction for a corresponding laminar boundary layer (see Wills, p. 395), so 44.1+33.2 corrected 2 for turbulent B.Lo 38.7 fps. The error in the mean velocity measurement caused by velocity fluctuations was studied in Appendix II-A. The error was +3% for the present measurement, therefore the corrected mean velocity U = 38o7x1o03 = 39.8 fps Finally, the mean wall shear stress can be computed dU f = C - dx2 U since wire was set up near the 1 x,2 edge of viscous sublayer, x2=0.00216 in. 0.406x10o-6 x 39812 0.00216 = 0.09 lb/sq ft or the non-dimensional. wall friction velocity, UT 1 UOO Um 00 Pa 1 0 o.09 206 4 0.00207 = 0.052

60 B. DIRECT APPLICATION OF WILLS' CORRECTION METHOD As computed in Section A we have (Rey) = 1.82, and referring to Fig. 4 Wills' report (p. 394) for b/a = 21.6 we found the correction constant, Kw to be 0.2, hence o.45 0~45 (Rey)w corrected =(Re)w - Kw = 1.82 - 0.2 = 1.62 And after correction (Rey)w corr, = 2.92 (Rey) w C corr. 2a pa 2.92x0. 406xlO- x12 0 002xO. 00207 34.4 fps Similar to what was mentioned in Section A this result would only meet the case of laminar boundary layer. For a turbulent boundary layer a factor 0.5 should be applied, thus with U = 44.1 fps (without correction) and U = 34.4 fps (with correction for laminar boundary layer) we can find 44. 1+34.4 (U) corrected for 2 turbulent boundary layer = 39.3 fps

Since the correctioh of error due to turbulent velocity fluctuations was 3% as stated in Section A, U = 39.3x1.03 = 40.5 fps Finally, one can compute the mean wall shear stress _-X U f = I w X2 40o5x12 = 0o406x10-6 x 0,00216 00912 lb/sq ft and the non-dimensional wall friction velocity, UT - 1 AT U00 U Pa 1 0o0912 206 o000207 = 0.0322 This result agrees very well with the result found in Section A.

BIBLIOGRAPHY 1. Benney, D. J., "Finite Amplitude Effects in an Unstable Laminar Boundary Layer," Phys. Fluids, 7, 1964, p. 319. 2. Benney, D. Jo and Lin, C. C., "On the Secondary Motion Induced by Oscillations in a Shear Flow," Phys. Fluids, 3, 1960, p. 656. 3. Browand, F. K., "An Experimental Investigation of the Instability of an Incompressible, Separated Shear Layer," MIT Report ASRL TR92-4, 1965. 4. Einstein, H. A. and Li, H., "The Viscous Sublayer along a Smooth Boundary," Proc. Amer. Soc. Civ. Engrs., Paper No. 945, 1956. 5 Favre, A. J., Gaviglio, J. J. and Dumas, R,, "Space-Time Double Correlations and Spectra in a Turbulent Boundary Layer," J. Fluid Mech., 2, 1957, p. 313; J. Fluid Mech., 3, 1958, p. 344. 6. Grant, H. L.,, "The Large Eddies of Turbulent Motion," J. Fluid Mech., 4, 1958, p. 149. 7. Hakkinen, R. J., "Measurements of Turbulent Skin Friction on a Flat Plate at Supersonic Speeds," NACA TN 3486, 1955. 8. Hama, F. R., "Progressive Deformation of a Perturbed Line Vortex Filament," Phys. Fluids, 6, 1963, p. 526. 9. Klebanoff, P. S*, "Characteristics of Turbulence in a Boundary Layer with Zero Pressure Gradient," NACA. TN 3178, 1954. 10. Klebanoff, P. S., Tidstrom, K. D. and Sargent, L. M., "The ThreeDimensional Nature of Boundary-Layer Instability," J. Fluid Mech., 12, 1962, p. 1. 11. Kovasznay, L.S.G., Komoda, H. and Vasudeva, B. R., "Detailed Flow Field in Transition," Proc. 1962 Heat Trans. Fluid Mech. Inst., Stanford Univ. Press, 1962, p. 1. 12. Kraichnan, R. H., "Pressure Fluctuations in Turbulent Flow over a Flat Plate," J. Acoust. Soc. Amer., 28, No. 3, 1956, p. 378 13. Ladenburg, R. W., Physical Measurements in Gas Dynamics and Combustion, High Speed Aerodynamics and Jet Propulsion Series, IX, Princeton Univ. Press, 1954, Section F. 62

BIBLIOGRAPHY (Continued) 14. Laufer, J,, "Investigation of Turbulent Flow in a Two-Dimensional Channel1," NACA TN 2123, 1950. 15. Laufer, J., "The Structure of Turbulence in Fully Developed Pipe Flow," NACA TN 2954, 19535 16. Lin, Co C., The Theory of Hydrodynamic Stability, Cambridge Univ. Press, 1955, p. 11. 17. Lindgren, E Ro., Arkiv For Fysik, 12, 1, 1957o 18. Phillips, 0. Mo. "On the Aerodynamic Surface Sound from a Plane Turbulent Boundary Layer," Proc. Roy. Soc. (London), A., 234, 1956, p. 327~ 19. Runstadler, Po W., Kline, SO J. and Reynolds, W. C., "An Experimental Investigation of the Flow Structure of the Turbulent Boundary Layer," Ph.D Dissertation of Stanford Univo, also available as AFOSR TN 5241, 1963 0 20. Schubauer, G. B. and Klebanoff, Po S., "Investigations of the Separation of Turbulent Boundary Layers," NACA Report No. 1030, 1951. 21. Sternberg, J., "A Theory for the Viscous Sublayer of a Turbulent Flow," J. Fluid Mech., 13, 1962, po 16. 22, Stuart, J. T., "The Production of Intense Shear Layers by Vortex Stretching and Convection,," AGARD Specialists' meeting on "Recent Developments in Boundary Layer Research," Naples, Italy, May 1965. 23. Theodorsen, T,, "The Structure of Turbulence," 50 Jahre Granzschichtforschung (Edo H. Gortler & W. Tollmien), Braunschweig: Veiweg und Sohn, 1955, p. 55, 24. Townsend, A.o A., "The Structure of the Turbulent Boundary Layer," Proc. Camb, Phil. Soc., 47, 1951, p. 375. 25. Townsend, A. A., The Structure of Turbulent Shear Flow, Cambridge Univo Press, 1956, p. 218. 26. Willmarth, Wo Wo and Wooldridge, Co Eo, "Measurements of the Fluctuating Pressure at the Wall Beneath a Thick Turbulent Boundary Layer,!? JO Fluid Mecho, 14, 1962, p. 1870

64 BIBLIOGRAPHY (Concluded) 27. Willmarth, W. W. and Wooldridge, C. E., "MeasurIements of the Correlation between the Fluctuating Velocities and the Fluctuating Wall Pressure in a Thick Turbulent Boundary Layer," NATO, A.GARD Report 456, 1963; also Univ. of Mich., T:;-: Report 02920-2-T, 1962. 28. Wills, J.A.B., "The Co:L-rection of Hot-Wire Readings for Proximity to a Solid Boundary," Jo Fluid Mecho, 12, 1962, p. 388. 29. Hinze, J. 0., Turbulence, New York: McGraw-Hill Book Company, 1959, p. 48. 30. Gadd, G. E., "A Note on the Theory of the Stanton Tube," Brit. R, and M. 3147, 1960. 1, Dhawan, S., "Direct Measurements of Skin Friction," NACA Rep. 1121, 1953. 32. Kovasznay, L.S.G., "Development of Turbulence-Measuring Equipment," NA.CA. Report 1209, 1954, Fig. 21. 335 Wills, J.A..B., "On Convection Velocities in Turbulent Shear Flows," J. Fluid Mech,, 20, 1964, p. 417.

FIGURES Note: Figures 1, 2, and 4 are reprinted from AGARD Report 456, Figs. 1, 2, and 3 by Willmarth, W. W., and Wooldridge, C. E., 1963

66 INDEX TO THE POSITION OF PRESSURE-TRANSDUCER AND HOT-WIRES IN T]EE PRESENT CORRELATION MEASUREMENTS OF uijuj AND pw Note: Coordinate systems are defined as shown by the following figure: Mix' A B(xx,x2,) ) A (X1X21X3) 0Uc X3 tunnel floor where (1) the origin 0 is the point at which pressure-transducer measuring p is located; (2) the point A(xl,x2,x3) is the point at which the first hot-wire measuring u, v, or w is located; and (3) the point B(x,xixI) is the point at which the second hot-wire measuring u, v, or w is located.

Point A Point B Correlation Velocity x X3 Velocity XI X Figure ~~~~Pressure 6" 6" Figure Measured Pressure Component Component U V W U V W Ruu X 0 0.122 0 X 0 0 0 6a R w X 0 0.122 0 X 0 0 0 6b Rww X 0 0.122 0 X 0 0 0 6c Ruu X 0 0.004 0 X 0 0 0 6d " X 0 0.004 0 X 0 0.001 0.204 7a tf x 0 0.oo004 0 X 0 o.oo6 0.204 7b it x o0.0oo4 o x o 0.098 0.204 7 t" X 0 0.004 0 X 0 0.20 0.204 7d "i x 0 0.004 0 x 0 0 0.095 8a i" X 0 0.004 0 X 0 0 0.190 8b "i x 0 0.004 0 X 0 0 0.286 8c "I x 0 0.004 0 x 0 0 0.518 8d " i x 0 0.004 0 X 0 0 o.69o 8e " X 0 o.oo004 0 X 0 0 0.095 9a i" x 0 0.004 0 X 0.169 0 0.112 9b it X 0 0.004 0 X 0.51 0 0.095 9c t x 0 0.004 0 x 0 0.118 o.o64 10a 1" X 0 0.004 0 X 0 0.118 0.127 10b i" X 0 0.004 0 x 0.118 0.191 lOc "i x 0 0.004 0 x 0.118 0.254 10d i" X 0.004 0.118 0.381 lla " X 0 ooo0.004 0 X 0 0.118 0.508 llb " 0 0.004 0 X 0 0.118 0.635 11c x 0 o.004 0 x 0 0.118 0.889 lld i" X 0 0.004 0 x 0.118 0.127 12a i" X 0 0.004 0 X 1.016 0.118 0.127 12b

Point A Point B Point O F - Correlation Velocity Velocity x Figure Measured Pressure Component Component Pressure 5 U V Wg U, v W PRuu x o 0.004 0 X 2.032 0.118 0.127 12c X o 0.004 0 x 4.o64 0.118 0.127 12d " X O 0.004 0 X 0 0.118 o.5O8 13a "t X 0 0.004 0 X 2.032 O.118 o.508 13b x O 0.004 0 X 4.o64 O.118 0.508 13c Rwv X 0 1.89 0 X 0 0.191 0 14a TI X 0 1.75 0 X o 0.337 0 14b t"tX 0 0.254 0 X 0 o.406 0 14c I" X 0 6.14 0 X 0 0 0.35 15a TI X 0 2.07 o X O 0 0.35 15b "I X 0 1.14 0 X 0 0 0.16 15c "t X 0 0.16 0 X 0 0 0.35 15d TX O o.o64 0 X 0 0 0.16 15e o TI X 0 6.14 0 X -0.35 0 0.35 16a o I X 0 2.03 0 X -0.35 0 0. 35 16b " X 0 1.14 0 X -0.16 0 0.16 16c T x 0 0.16 0 X -0.35 0 0.35 16d "T X 0 o.o64 0 X -0.16 0 0.16 16e Rww X 0 1.04 0 X 0 0 0.203 17a I" X 0 0.122 0 X 0 0 0.203 17b "I X 0 0.122 0 X 0 0 0.508 17c " X 1.67 0 1.67 0 X 0 0.224 0 18a X 0 0.426 0 X 0 0.224 0 18b X 0 0.122 0 X 0 0.224 0 18c X 0 0.004 0 x 0 0 0.284 l9a

Point 0 Point A Point B Correlation Velocity Velocity Figure ~, 2 Xlu3 ty x Figure Measured Component Component Pressure vwu v wCmoe U V W U1 V W Rww X 0 0.004 0 X 0 0 0.568 19b X 0 0.004 0 x o 0 0.852 19c X 0 0.004 0 x 0 0o155 0 l9d Ruw X 0 0.004 0 X 0 0 -0.284 20a X 0 0.004 0 X 0 0 -0.568 20b it x 0.00o4 0 X 0 0 -0.852 20c " X 0 0.004 0 X 0 0 -1.136 20d " X 0 0.004 0 X 0 0 -0.284 21a " X 0 0.004 0 X 0 0 0.284 21b "I X 0 0.004 0 X 0 0 -0.852 22a t" X 0 0.004 0 X 0 0 0.852 22b X 0 0.127 0 X 2.03 -0.123 -0.284 23a X 0 0.127 0 X 0 -0.123 -0.284 23b o kO X 0 0.127 0 X -2.03 -0.123 -0.284 23c i X 0 0,127 0 X -4.06 -0.123 -0.284 23d X 0 0.127 0 X -6.og09 -0.123 -0.284 23e Rwv X 0 0.508 o.508 X 0 0 -0.223 24a,. X O o.508 o.5o8 x 0 0 0.223 24b,."X 0 0.508 0.508 X 0 0 0.446 24c ii" X o.508 o. 5o8 x 0 0.254 -0.223 24d i"ix o.o8 0.o50 8 X 0 0.254 0.223 24e x 0 o.o8 o. 5 0 8 x 0 0,254 0 24f x 0 o.508 0.508 X 0 -0.286 0.254 24g " X 0 0.004 0 X 0 0.155 0,284 25a oX o 0 004 0 x 0 0.155 -0.284 25b

Point A Point B Point 0. Correlation Velocity x2 3 Velocity Figr Pressure Component Component U V W lU V W Rwv X 0 0.004 0 X 0 o.186 0 25c X 0 0.102 o0.508 X 0 0041 -0.233 26a X 0 0.102 0.508 X 0 0o041 0.223 26b X 0 0.102 o. o8 X 0 0.041.0.382 26c X O 0.102 0.508 X 0 0.041 0.223 2'7a X 0 0.102.508 0 0 0041 -0.223 27b X 0 0,102 o.508 X 0 0.041 -0.223 27c X 0 0.102 0 X 1.52 0.041 0.222 28a X 0 0.102 0 X 0.71 0.41 0.222 28b X 0 0.102 0 X 0 0.041 0.,222 28c X 0 0.102 0 X -0.71 0.041 0.222 28d X 0 0.102 0 X -1.52 0.041 0.222 28e RPW X x o 0.127 -0.508 30a X X 0 o.50o8 -o.508 30b X X 0 2.032 -1.016 30c'? X X 0 2.032 -2.032 30d x x 0 0 0.127 0.508 31a X X 0 0,508 0o.508 31b X X 0 1.016 0,508 31c X X 0 1.524 0.508 31d X X 0 2.032 0.508 31e x x 0 3.o48 o.508 31f X X 0 0.127 1.016 32a X X 0 o.508 lo016 32b X X 0 1.016 1,016 32c

Point 0 Point A Point B Point 0 Correlation Velocity Velocity xI x Figure Correlatio Velocixy1 x2 V 3 e C m o t Measured Component Comonent Pressure Component U V W u V W Rpw X X 0 1.524 1.o016 32d X X 0 2.032 1.o016 32e it x 0 3.o48 1.016 32f I X X 0 0.127 1.524 33a x x 0 o. 508 1.524 33b X X 0 1.016 1.524 33c X X 0 1.524 1.524 33d Ix x 0 2.032 1.524 33e X x 0 3-.048 1.524 33f Ix x 0 0.127 2.032 34a "tx X 0 0.508 2.032 34b "iX X 0 1.016 2.032 34c " x X 0 1.524 2.052 34d "X X 0 2.032 2.032 34e t x x o 3.o048 2.032 34f I X X 0 1.524 2.540 35a iX X X 0 2.032 2.540 35b I? X X 0 2.540 2.540 35c ii X X 0 0.127 0.508 36a X X -0.51 0.127 o.508 36b " X X -1.02 0.127 0.508 36c X X -1.53 0.127 0.508 36d X X -2.54 0.127 0.508 36e X X 2.032 0.127 0.508 36f " x x 0 0.508 0.508 37a

Point A Point B Point O Correlation elocity x elocity x x Figure Measured Pressure Component Component 2 Pressure 6* V* *moe u V W u v w X X -0.51 0.5o8 o.eo8 37b X X -1.02 o. 5o8 o. 5o8 37c x x -.53 0.508 o0.58 37d " x x 2.032 o.508 o.508 37e 1" X X 2.032 0.127 o.o508 38a " x x 2.032 o.508 o.508 38b x x 2.032 1.524 o. 508 38c x x 2.032 1.524 2.032 38d X X 2.032 3.048 2.32 38e...

_ 0 5 10 15 2025 30 40ft. i' I I' I'I Airflow 15 ft. Fig. 1. Scale drawing of wind tunnel test section and massive vibration isolation mounting for the transducers.

4; CHO.RD AIRFOIL PHANTOM VIEW OF PRDBE HOLDER USED I RUBBER 0-RiN6 / I I I L 1 j /,-_' t I I f-F~-~F~/-t,~~~~PRO HODER CLAY FARI, IN i~ HOT WIRE PROBE t I I CLAN AIR SEAL PRESSURE TRANSDUCER STEEL PLATE Fig. 2. Pressure transducer and hot-wire installation. Hot-wire shown at closest spacing to plate, 0.05 inch.

75.D1in At —.0 2 in J w-wires k- r.28in-4 co "- ~Top view, u-wires 0. 00015 dia 25 i -1 in /a X2 0. 032 in 0. 015 in Bakelite (or Plexiglas) LX3 II CEI.002/ 1 in Wr II Tunnel floor Wire l I Front view,.U Liu U UU l lPlug for u-wires Leads Note: w-wires are arranged in the same horizontal plane owing to the large variation of velocity with x2 near the wall. This arrangement gives poorer spatial resolution in the x3 direction than the x-type wires but allows measurements near the wall. Fig. 3. Hot-wire plug.

OCD O bO 30 25 Ur 1i5 10 |/- Coles' Ideal Turbulent Boundary Layer o Re8e38,000 Um =204 ft/sec T=67~F 5 \ 0 o Re8= 43,000 Uco=203 ft/sec T=45~ F 0 L i _ I _i t iii i _ I I 1 I I I I 10 100 1000 10000 Ury Fig. 4. Mean velocity profiles in the turbulent boundary layer. Refer to Table I for other boundary layer parameters.

77 30 - I2R x 10 w (watts) R' e=1 ~R~R — /0 e R e 20 / / 0 —-O Original curve awTe = ReF ~~~~~~~~/ ~~Re =4. 50Q. - _ Alter correction for new:/ = o cold resistance aw = 0. 5 / Te = 113~F / R' = 5.85 Q 0 0.2 0. 4 1 0.6 pU ([ slug/ft2 sec]2) Fig. 5. Hot-wire calibration curve for platinum u-wire.

~(R, R)x2 Ruu(Rvv Rww) X2 6* O —-O RUU 0.122 (a). R 0.122 (b) 1. 0 R 0.122 (c) X-X R orR 0.004 (d) uu T T 0( ww 0. 5 x 10 20 30 U T Fig. 6. Comparison of auto-correlations of velocity fluctuations, u, v and w near the wall.

RUU (or R u) UU. TU W X2 0.4 ul-wire u2-wire a b c d 1 2 Sc(2 u2x X x z-1UI = / x* 0 0 0 0 jj t2 /0. - 0.004 6I= 0.005.0102.102 204 x X z - 30 ~ - 20 lo io 20,. ~ o o o d xX Tunnel front section view. Air zthe flow enters the paper and takes. positive direction along x -axis. -oo6* -0 20!0 20 30 b -o0. 2 d X —x Fig. 7. Comparison of the space-time correlation of u-u near the wall.

8o R UU(or R ) U T W ~~x ~~~~U ~0. 4 = 5. 3 x1 x2 x5 6=0 6*=o. 004 0.2 6 *=0 - 2*= 0 X3 UT' (a) = 0 0 = 0. = 09 52 6 Uoo('-T') - 40 - 20 0 20 40 6* (b) =0 =0 =0 =0. 190 (c) 0 _ =, = = 0 =0. 286 (d) = 0 =0 = 0 = 0. 518 (e) 0 =0 = 0 = 0. 690 Fig. 8. Measured values of the space-time correlation of u-u very near the wall.

R (or R ) UU T T i W W 0.4 x2 453 = 5. 3 X1 X2 2 -i*= Oe2 = 0. 004.2 * = = 6* 6* 6* x3 / I'x (a) 3 0 = 0. 0952 4-40 -o 20 020 40~ =0.169 =0 (b) = 0. 83 = 0. 112 =0.51 =0 = 2. 44 =0.0952 Fig. 9. Measured values of the space-time correlation of u-u very near the wall.

82 R uu(or R u) UU T U 2 = 5. 3 156 X x2 x2 2 = 0 = 0. 004 = O. 118 x3 Uco3 = 0 0 = 0. 0635 -=00.2 (a) U o(r -') - 40 <- 20 0( ~ 2X 40 6r = 0 = 0. 118 = 0 = 0. 127 (b) = 0 = 0. 118 = O = 0. 191 (c) =0 = 0.118 = = 0. 2 54 (d) Fig. 10. Measured values of the space-time correlation of u-u near the wall.

x2U x Ur 5. 3 R (or R ) 156 iV ~UU TWU () - O- 0 iX8 2 =0. 004 0.-2 =0 - =0. 118 6* 6* 6* x3 U oT' (a) = =0= 0. 381 406* 6* U (T-r') -40 -0 0 20 40 6* 0 =0.118 (b) 0 = 0. 508 0o =0 = 0. 118 (C) =0 = 0. 635 =0 = 0. 118 (d) 0 = 0. 889 I o 0 01 o% n,, n n a J a a a L n 1 n O e, I Fig. 11. Measured values of the space-time correlation of u-u near the wall.

84 R (or R U) uu Tu w -0. 6 XU 0.4 XU 2 753 T 2 =56 (a) - 5. = 156 v v I1 X xi I(Xl 2 x3 -40 -20 0 U 40r 5' —-, P.0 = O. 016 =0118 (b) 1. 74 =0.127 = 2.032 = 0.118 - 4. 064 = 0. 118 (d) = 6.-96 = 0. 127 Fig. 12. Measured va=lues of the space-time correlation of u-u near the wall.

= 5. 3 Uu (or R ) =156 x1 x2 l* = -, 2* =ok 004 0. 2 xl,-=0 = 0. 118 r = 2. 032 =0. 118 (b) =4= 3. 54 = 0. 508 c> - -4.064 =0.118 (c) 7.08 =0. 508 Fig. 13. Measured values of the space-time correlation of u-u near the wall.

R VV 0. 6 X x0 X XI 2 X1 X2-~~ 0. 4 "1 2 1 6*0 61.5 =8 0 60.191 -3 0. =0 =0.2 6 66 * =0 = 1.75 =0 =0.337 =0- =0 =0 (b) =0 =0. 254 =0 =0. 406 (C) =0 =0 Fig. 14. Measured values of the space-time correlation of v-v.

R VV 0. 6 0. 4 (a) =:6.14 - 0. 36 = 0.35 ga 6* 6* -d I 10 U (T-') -20 0 20 -=0 -=0 =-0.35 =0 x2 (b) * -=2.03 =- 0.42 =0.35 x x3 1 3 0 -=o -:=- 0.16 =0 x2 (c),*=1. 14 = — 0.21 =0.16 x1 x2 x3 = - 0.35 =0 (d) 6 — = -: 0-:O.1: - O. 57 = 0. 35 xl x2 X3 xg= - 0.16 = 0 (e) 6*=0 * =0.064 * = - 0.29 = 0..16 Fig. 15. Measured values of the space-time correlation of v-v.

88 R vv 0. 6 0. 4 X1 x3 (a) 5- = 6.14 0 0 0. 435 10 = 0 x2 (b) = 2. 07 1 =0 = 0. 35 OF= 0.35 20 (c) *=1.14 0 =0.16 (d) 0 0.16 0 x1 x2 X3 0 =O (e) -=0 =0.064 0 =00.16 Fig 6 Measured values of the =space-time =0. 16 Fig. 16. Measured values of the space-time correlation of v-v.

R WW 0. 6 x x Mo. 4 x =0 - 1.04 0 0 =0 =012\=0 =0 0b ==0 0. 203 0. 2 (a) - 20 20 U - 40 4 40 co = 0 = 0. 122 =0 = U = 0 1 I b = 0 =0.203 (b) =0 = 0. 122 =0 =0 (c~) = ~0 /- = 0 = 0. 508 Fig. 17. Measured values of the space-time correlation of w-w near the wall.

90 WW 0. 6 0.4 xI x2 x 1 2x xoo0 6*=1.-67 =0 - = 0. 224 x3 U T7' x 6* 0 0. 2 6* = 0 = 0 -20 -40 0 206 40 6* =0 =0.426 =0 =0.224 (b) =0 =0 =0 -0 =0.122 =0 =0. 224 (c) =0 =0 =0 Fig. 18. Measured values of the space-time correlation of w-w near the wall.

WW ww x x2 =xO 6*= 0. 004 0. 6 = U "r "I x3 x0 3 a= 00 = 0. 284 ( 6* 6* 6* U (-) ~~ I I ~ 1' i * Ioo a -40 -20 0 20 40 6* =0 =0 = 0 = 0. 568 =0 =0 (c) = 0 = 0. 852 I I I I I Rww WW 0. 2 0.1 =0 =0.155 =0 =0 20 -40 -20 0 Fig. 19. Measured values of the space-time correlation of w-w very near the wall.

X1 x2 Ruw xi Xi - 4 0 -0.004 0 0 x -X3~~0. I UO T-' x3 x_ 3 (a) - 40 - 20 =0 = -0. 568 (b) =0 =0 (c) = 0 =- 0.852 =0 =0 (d) =0 =- 1.136 Fig. 20. Measured values of the space-time correlation of u-w very near the wall.

X1 X1 x2 R x2 o=0 = 0. 004 RW 5 =0 0 =00 x 0.1 U T' x 3 = 0 00 0 = 0. 284 5* 6'* - 6* - ~~~0. 1~ l — _ =0 =0 ~~(b})~~ I ~ \ = ~ = + 0.284 Fig. 21. Measured values of the space-time correlation of u-w very near the wall.

R =0 = 0.:004: 2 = 0 0.1 x3 U T" (a) x (a) 6*= 0 |* = ~ * — 0.852 - 40 - 20 0 20 40 6* =0 =0 = 0 =+ 0. 852 (b) Fig. 22. Measured values of the space-time correlation of u-w very near the wall.

95 ~xlx2~ R~uw xl x -* g0 -- = O. 127. 1 * 2.03 -0.123 X3 U 00 (a) 6T = 0 t6* = 3. 21 X =-0.284 U- o0 0 6 —- 20 -0. 1 = M=0 =- 0. 123 = 0 = - 0. 284 = 2. 03 = - 0. 123 - 3. 21 =-0. 284 (C) - 4.06 = - 0.123 =- 6. 42 =- 0.284 (d) =- 6. 09 = - 0. 123 =- 9. 63 =- 0. 284 (e) Fig. 235 Measured values of the space-time correlation of u-w near the wall.

96 R WV Xl x2 xi xI aI = 0 2 = 0. 508 0.1 Il 0 2 o (6) * S6* 6* (a) 63* = 0. 508 J5 t A* =- 0 223.I I 12o a I uoo(' I') -40 -20 20 40 6* =0 =0 (b) | \ = =+ 0.223 =0 =0 (c) = =+ 0. 446 = 0 =+ 0.254 (d) =0 0= - 0. 223 =0 =+ 0.254 (e) = 0 =+ 0. 223 =0 = + 0.254 (f) =0 =0 =0 0 = - 0. 286 (g) =0 =+ 0. 254 Fig. 24. Measured values of the space-time correlation of w-v.

R WV X1 X2 X 01 1 = 0,2* = 0. 004 -. I 1 = =. 15 x3 U T'* 3 0 00 0 3 = 0. 284 (a) 6* 0 0- ~ - 20 = 0 =0.155 (b) = 0 =- 0.284 =0 =0.186 (c) I 60 =o =0 Fig. 25. Measured values of the space-time correlation of w-v near the wall.

R wv xW x. = 0 - 2= 0.102 0. 1 = 0 = 0. 041 6- = 0.1 6* 6* x3 U 7' = 0. 508 = 0 - =-0.223 (a) 5*=86 6* 6' - 40 - 20 40 6 = - = 0. 041 (b) = 0 =+ 0. 223 kO = 0 =0. 041 (c) = 0 =+ 0.382 Probe Used in Measurements Curve w-wire v-wire Probe A Probe B a A A b A A c A A Fig. 26. Measured values of the space-time correlation of w-v near the wall.

WV x1 x2 xi x =0 2 = 0. 102 0. 1 = 0 = 0. 041 ( 6 * 6 0 * =O. 02 X[ Uoo' (a) = 0. 508 0 = + 0. 223 (a) 6 (T- )6* U ) oo - 40 -20 0 0 6* = 0 = 0. 041 = 0 = - 0.223 (b) = 0 =0.041 (c)= 0 =- 0.223 Probe Used in Measurements Probe A Curve w-wire v-wire U a B b A B U Probe B e R B X Fig. 27. The measurement of Rwv near the wall, using different arrangements of x-type hot-wires (also see Fig. 26 for comparison).

100 R wv x1 x2. x' 0. 1 6r I - = 0.571 = 0. 041 x3 U - x0b 0 a = 2. 62 0. 222 - 40 20 o 20 40 = 0. 71 = 0. 041 (b) = 1. 21 = 0. 222 = 0 = 0. 041 (C) = 0 =0. 222 =- 0.71 =0.041 (d) - 1 20 = 0. 222 =- 1. 52 = 0. 041 (e) = - 2. 62 = 0. 222 l l! c =]' l- I I Fig. 28. Measured values of the space-time correlation of w-v near the wall.

101 x2!Us~ XXl U __X_3 X2 Hot-wires measuring w-velocity fluctuations B(o, x2 -x3) - A(o,x2,+x3) x3 Tunnel floor Pressure transducer, p, located at origin, o. Fig. 29. Preliminary consideration of Rp. measurements.

102 R 1 x2 -* =~ 0. 127 U x3 0.1 =0 0. =-.508 (a) 20 -20 ___ _~0__ 40 20 40 6* =0 0. 508 (b) = 0 = - 0. 508 = 0 = 2. 032 (c) = 0 = — 1. 016 =0 =2. 032 (d) =0 =- 2.032 Fig. 30. Measured values of the space-time correlation of fluctuating velocity component w with fluctuating wall pressure (X3/b* < 0).

103 R pw X1 X2 0 — = 0.127 0.1 U r' X3 (a) e =0 =0. 508 _I - L I u00(7- r') -40 -20 0 20 40 6* = 0 = 0. 508 (b) = 0 = 0. 508 = 0 = 1. 016 (C) = 0 = 0. 508 I' e ~!,,. ~ -- I [ i 0 =0 = 1. 524 (d) = 0 = 0. 508 = 0 = 2. 032 ~(e) = = 0. 508 =0 =3. 048 (: = =0 0. 508 Fig. 31. Measured values of the space-time correlation of fluctuating velocity component w with fluctuating wall pressure (X3/B5 > 0).

1o4 R pw x1 x2 $=~0 = 0. 127 U " x3 = 3 =0 -1. 016 (a) 6* 6 =._ uoo( -I0') 4T- 20 0 2 40 6* =0 =0. 508 (b) = 0 = 1. 016 =0 = 1.016 (c) =0 =1. 016 = 0 = 1. 524 (d) = 0 = 1.016 = 0 = 2. 032 (e) =0 = 1. 016 = 0 = 3. 048 (f) =0 =1.016 Fig. 32. Measured values of the space-time correlation of fluctuating velocity component w with fluctuating wall pressure (X3/8* > 0).

105 R pw 6= 0 6= 0. 127 0. 1 U 7" X (a) = 0 = 1. 524 6* 6* UI U- (T) =0 =0.508 (b) =0 = 1.524 =0 =-1. 016 (c) = 0 = 1. 524 =0 =01. 524 (d) = 0 = 1. 524 = 0 =2.032 (e) = 0 = 1. 524 = 0 = 3. 048 (f) =0 = 1. 524 lw_ Fig. 33. Measured values of the space-time correlation of f luctuating velocity component w with fluctuating wall pressure (X3/6* > 0).

106 R x x2 2.032 * = 0. 127 0. 1 U' x3 = 3. 40 * 0. 508 u (T-rT') - 0 20 = 2. 032 = 0. 508 (b) = 2.99 =0. 508 = 2. 032 =1.524 (c) = 2. 66 =0. 508 = 2. 032 =1.524 (d) = 2. 66 = 2.032 =2.032 = 3.048 (e) = 2. 40 = 2. 032 Fig. 34. Measured values of the space-time correlation of fluctuating velocity component w with fluctuating wall pressure (x3/6* > 0).

R pw X1 x2 o0 6- = 1. 524 0.1 U T' x 00 3 (a) = =o 2. 540 (a) 6* (T — 020 0'`p- 20 40 6* = 0 = 2. 032 0 =2. 540 O (b) o =0 ==2.540 = 0 = 2. 540 (c) Fig. 35. Measured values of the space-time correlation of fluctuating velocity component w with fluctuating wall pressure (X3/6* > 0).

1o8 R 6 ~ 0 6 2 = 0. 127 0.1 (a) * = - =0. 508 Uo(T-') - 40 - 20* 20 40 = - 0. 51 = 0. 127 (b) = - 0. 81 = 0. 508 =-1.02 =0.127 (c): - 1. 61 = 0. 508 - 1.53 = 0. 127 (d) = - 2. 42 = 0. 508 - 2.54 = 0. 127 (e) = - 4. 03 = 0. 508 =2. 032 =0. 127 (f) = 3. 22 = 0. 508 Fig. 36. Measured values of the space-time correlation of fluctuating velocity component w with fluctuating wall pressure (X3/8* > 0).

109 pw -0 0. 508 — 1.0 62=0.508 0.1 U r' x3 =0 ='0. 508 (a) * = -. 51 = 0. 508 (b) - - 0. 75 = 0. 508 =- 1. 02 = 0. 508 (c) =- 1.50 = 0. 508 = - 1. 53 = 0. 508 (d) = - 2.25 = 0. 508 = 2. 032 = 0. 508 (e) =2. 99 = 0. 508 Fig. 37. Measured values of the space-time correlation of fluctuating velocity component w with fluctuating wall pressure (x3/6* > 0).

110 R pw x1 x2 2.032 = 0. 127 0. 1 U (' x' _ _ 3 (a)'3.40 -=0. 508 - 20 0 20 6* = 2. 032 = 0. 508 (b) =2.99 =0. 508 =2. 032 = 1. 524 (c) = 2.66 = 0. 508 =2.032 =1.524 (d) = 2. 66 = 2. 032 =2.032 =3.048 (e) = 2.40 = 2.032 Fig. 38. Measured values of the space-time correlation of fluctuating velocity component w with fluctuating wall pressure (x3/6* > 0).

xl 8 x1 x2 x3 6 —=0 6-=O. 004 6=O 6 0 -0.01 -0.03 -0.04 4 -0.05 -0.07.01 -0.09 -o0. -0. 125 I.u -0.01 0.044 *0.095 0.17 0.27 0-0.41 3 0 -0.03 -0.04 -2 -0.07 0 o0.09 0.01 -6 0 1 U 00 -8 Fig. 59. Correlation contours of constant RUu very near the wall.

112 x1 6* -20 15 f 0 6/n* = 6* = 0. 004 6* =0 -.04 07; 1 x2 -.2/ 0. 118 -0.I 10 -0.15 -0.117 - 0.i -0.17 -0.17 -0.1 0 -025-0.20.30.40 0.5 0.55.05 3 0.40 6* 0.30.20 -0.1 -0.13 i -02 _.i3 -.047 -OIo 1o oo -15 0 \ty0..o 0 -20 Fig. 40. Correlation contours of constant Ruu near the wall.

-aonpsu'riq axnssaad qji maIsgwss as-euipaooo jo pUTSlao GuWed sx-ax aLq; uT'dw g;usqsuoo jo sa-noc uo UO;laiooD 01t *g *9 stlo ~AW O'I 0'0+= * 990'0+ 910'+ 0 > IO'0+,9 O'I - = *-9 a.'pa j *g Ix

114 10. 5 x1 B. L. edge = -0. 5 10.0 X2 +0. 015 +. 025 3 0 - Rpw = +0. 0 +. 035 +0. 04.. 05 1.0' 0 1.0 2.0 x3 Fig. 42. Correlation contours of constant Rpw in the x2-x3 plane. Origin of coordinate system at pressure transducer.

115 10. 5 x1 B. L. edge 1 0 10. 0 x2 R = +0. 025 3. o 0 7 +. 055 R =+0. 065 2. /+0. 085 1 0.0 0 1.0 2.Ox3 x3 Fig. 43. Correlation contours of constant Rpw in the x2-x3 plane. Origin of coordinate system at pressure transducer.

10. 5 x1 B.L. edge 10.0 2 3 0.02 0.03 0.04 0.02 6* O. 00.05 3. 0 0. 04 0.05 2. 0 0. 025 -0. 040 -0.01 0 2. x3 Fig 44. Correlation contours of constant Rpw in the x2-x3 pLane. Origin of coordinate system at pressure transducer.

L17 10. 5 x B.L. edge 6 3 10.0 2 X.1 0.1.0 0.015 0. 01. 0 0.01 0.0015 3.0 0.005 2.{0 -0.05.0 l>/ -0. -006 1. O -0. 003 1. 0 0 1.0 2.0 x3 Fig. 45. Correlation contours of constant Rpw in the x2-x3 plane. Origin of coordinate system at pressure transducer.

10 U La t Lto-c 6* 0082oo 0 -4 + 2 3 5 4 3 3Lkr' + + + +.02 4 -3 "-+ 2 +0.035 0 13 I2P 0 O04-5 +0e0e Fig 4-0.05 2. b, + 0.01 00O 3 22 - +00.0 1 2 3 +.04 5 Fig. 46. Three-dimensional diagram of contours of Rpw = Const. (also see Figs. 41-45).

119 3 Hot-wires U2 x3 X2 3 Hot-wires u2 d2 l -+ \ 1 Iz. o0 in 0. 002 in 0 x i Tunnel floor Fig. 47. The location of hot-wires for measurements of the displacement of eddies due to convection and turbulent diffusion near the wall.

120 x2 Mean circulation Mean velocity profile Unstable eddies Stable eddies U2 A —U 1 Fig. 48. The behaior of eddies in a shear flowx 0 Fig. 48. The behavior of eddies in a shear flow.

121 x2 U 00 Perturbed vortex lines (a) xl SL x3 O2 _ U U|00/ Vortex line, bent vertically (b) TX' ///////- X1 SL x`4x3 Vortex line, remains one dimensional U Vortex line, after |~ ~ adzbending and stretching Ij-g<) //////-X1 (c) SL x3 Fig. 49. The development of random vortex lines near the wall.

122 X2 Vortex flow, B. L. edge average model x uFig. U0 RhnII7IIorteIIIIowi h xx3 Fig. 50. Random vortex flow in the x2-x3 plane near the wall. Points A and B are the locations of hot-wire measuring Rw.

123 X2 Vortex flow, average model B.L. edge x1 Q/ BQIC UccX a /A _D x3 Fig. 51. Random vortex flow in the x2-x3 plane near the wall. Points A, B, and C are the locations of hot-wire measuring Rwv.

X2 Vortex line, -B. L. edge / average model B,\B''///C __ 3 X1 Fig. 52. Random vortex line near the wall. Points A, B, and C are the locations of hot-wire measuring Rwv.

125 x2 Vortex line, average model (a) U x Pressure measured at 0 x3 x2 Vortex line average model (b) w x pressure, p < 0 X2 pw (c) x3 pressure, p Fig. 53. Structure of a random vortex line near the wall and the explanation of measurements of contours of constant Rpw at different X>-X3 planes (also see Fig. 46).

126 x2 x2 u H H 00 1 1 Ic)(a).0e * X1 /Al X A -- L3` X3 /A 1 B'~'"~ xB x3 Pressure measured Pressure measured at o; p < 0. X2 X2 U 00 /-) (b) x3B x~ X3 w-component of this vortex flow X2 / is correlated with positive pressure 2I ~ /4 / 8 And w-component of this vortex flow is correlated with negative pressure - /I (c) o',A' Fig. 54. Explanation of the separation of positive contours of constant Rpw in the present measurement (also see Figs. 44-46).

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Unclassified Security Classification DOCUMENT CONTROL DATA R&D (Security claaaification of title, body of abetract and indexing annotation must be entered when the overall report is claesiled) 1. ORIGINATING ACTIVITY (Corporate author) a. REPORT SECURITY C LASSIFICATION The University of Michigan Unclassified Department of Aerospace Engineering, Aerodynamics 2b GROUP Ann Arbor, Michigan Laboratory 3. REPORT TITLE AN EXPERIMENTAL STUDY OF THE STRUCTURE OF TURBULENCE NEAR THE WALL THROUGH CORRELATION MEASUREMENTS IN A THICK TURBULENT BOUNDARY LAYER 4. DESCRIPTIVE NOTES (Typo of report and inClulve dates) Technical Report 5. AUTHOR(S) (Laost name. firet name, initial) Tu, Bo-Jang Willmarth, William W. 6. REPO RT DATE 7. TOTAL NO. OF PAGEI.b. NO. OF-REFS March 1966 126 33 5a. CONTRACT OR GRANT NO. 94a. ORIGINATOR'S REPORT NUMBIR(S) Nonr-1224(30) 02920-3-T b. PROJECT NO. NR-062- 234 c. S. OTH ER REPORT NO(S) (Any other numbera that may be assinted is..port d. 1 0. A V A IL ABILITY/LIMITAtION NOTICES Qualified requesters may obtain copies from DDC. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Department of the Navy Office of Naval Research Washington, D. C. 13. ABSTRACT An experimental investigation is described in which emphasis is given to revealing the structure of turbulence near the wall in a boundary layer. Measurements made include space-time correlations between the fluctuating wall pressure and the span-wise velocity component w, and between the various velocity components The velocity correlations include measurements of the space-time correlation of the streamwise component of the fluctuating wall shear stress. Experiments have been conducted in a thick (5 in.) turbulent boundary layer with zero pressure gradient which is produced by natural transition on a smooth surface. Sufficient data have been obtained to allow us to propose a qualitative model for the structure of turbulence near the wall. The proposed model outlines the sequence of events that result in the production of intense pressure and velocity fluctuations by stretching of the vorticity after it is produced by viscous stresses within and near the edge of the viscous sublayer. The measurements are in qualitative agreement with the model. Here, qualitative agreement means that the size and shape of the contours of constant correlation and the sign of the measured correlations are in agreement with the proposed model for the turbulent structure. D oD 1JAN64 1473 rUnclassified Security Classification

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