THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Aerospace Engineering Gas Dynamics Laboratories Technical Report WALL PRESSURE FLUCTUATIONS BENEATH AN AXIALLY SYMMETRIC TURBULENT BOUNDARY LAYER ON A CYLINDER Chi-Sheng Yang William W. Willmarth ORA Project 02149 under contract with DEPARTMENT OF THE NAVY OFFICE OF NAVAL RESEARCH CONTRACT NO. N00014-67-A-0181-0015 WASHINGTON, D. C. administered through OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1969 This document has been approved for public release and sale; its distribution is unlimited.

ABSTRACT Measurements of the turbulent pressure field on the outer surface of a three inch diameter cylinder aligned with the flow were made at a point approximately 24 feet downstream of the origin of the turbulent boundary layer in an air stream of 145 ft/sec. The boundary layer thickness was 2. 78 inches and the Reynolds number based on momentum thickness 4 was 2. 62 x 10 The wall-pressure measurements were made with pressure transducers constructed from 0. 06 inch diameter lead-zinconate-titnate disks mounted flush with the wall. The measurements included root-meansquare, power spectrum and correlations of the wall pressure and were compared with the existing experimental results for the turbulent wall pressure filed beneath a plane boundary layer. The root-mean-square wall pressure was 2, 42 times the wall shear stress, somewhat lower than that measured beneath a plane boundary layer. The general shape of thenormalized power spectrum was similar to the spectrum associated with a plane boundary layer. However, at high frequencies (ow6*/Uo > 10) the spectrum contains approximately twice the energy density measured in the plane boundary layer. The streamwise convection speed deduced from longitudinal spacetime correlation measurements was almost identical to that obtained in the plane boundary layer. The rate of decay of the maxima of the space iii

time correlation of the pressure produced by the convected eddies was twice as fast as in a plane boundary layer. The longitudinal and transverse scale of the pressure correlation were approximately equal, in a plane boundary layer the transverse scale is larger than longitudinal scale, and were one-half or less than the longitudinal scale in the plane boundary layer. It is concluded that the effect of the transverse curvature of the wall is an overall reduction in size of pressure-producing eddies. The reduction in transverse scale of the larger eddies is greater than that of the smaller eddies. The smaller eddies then decay more rapidly and produce greater spectral densities at high frequencies owing to the unchanged convection speed. iv

TABLE OF CONTENTS page ABSTRACT iii LIST OF ILLUSTRATIONS vii NOMENCLATURE ix' I. INTRODUCTION 1 II. WIND TUNNEL AND CYLINDRICAL MODEL 5 A. Wind Tunnel Facility 5 B. Cylindrical Model 6 III. INSTRUMENTS AND EXPERIMENTAL METHODS 9 A. Instruments and Methods for Measuring Mean Properties of the Flow 9 B. Instruments and Methods for Measuring Turbulent Pressure Field 10 1. Pressure Transducers 10 2. Electronic Equipment 13 IV. MEAN FLOW FIELD 15 A. Equations of Motion 15 B. Similarity Laws 17 C. Experimental Results for Mean Flow Field 24 1. Pressure Gradient 24 2. Skin Friction 25 3. Velocity Profiles 25 4. Summary of Mean Properties of the Boundary Layer 27 V. EXPERIMENTAL RESULTS 29 A. Longitudinal Space-Time Correlation of the Wall Pressure 30 B. Longitudinal, Lateral and Oblique Spatial Correlation of the Wall Pressure 34 C. Root-Mean-Square Wall Pressure 37 D. Power Spectrum of the Wall Pressure 39 E. Correction for Finite Transducer Size 40 F. Integral Spatial and Temporal Scales 42 v

page VI. SUMMARY AND DISCUSSION 44 APPENDICES A. Correction of the Wall Pressure for Effects of Sound in the Free Stream 48 B. Convection Speed Measured in Oblique Directions 50 C. Correlation of the Wall Pressure in Narrow Frequency Bands 51 REFERENCES 57 vi

LIST OF ILLUSTRATIONS Table Page 1. Summary of Some Results of Skin Friction Measurements 26 2. Properties of the Axially Symmetric Turbulent Boundary Layer 28 Measured in the Present Investigation. 3. Comparison of Root-Mean-Square Values of the Wall Pressure Fluctuations 38 4. Integral Spatial and Temporal Scale in Plane and Axially Symmetric Turbulent Boundary Layer 43 Figure 1. Schematic diagram of 2 in. diameter steel tubing installed in the wind tunnel as the backbone of the cylindrical model. 60 2. The test station and the rear support of the cylindrical model. 61 3. The symmetry of the velocity around the cylinder at y = 0. 65 in. 62 4. Velocity-survey apparatus. 63 5. Mouth of the pitot tube. 63 6. Pressure transducers mounted in the semi-cylinder lead shell. 64 7. The arrangement of the pressure transducers. 64 8. Pressure transducers assembly. 65 9. Circuit diagram for pressure transducers and cathode follower. 66 10. Pressure gradient along the cylinder. 67 11. Mean velocity profiles in the axially symmetric turbulent boundary layer (Law of the Wall). 68 12. Skin friction coefficient. 69 vii

Figure Page 13. Longitudinal space-time correlation of wall pressure. 70 14. Time delay for Rp maximum at constant x1. 71 PP 15. Local convection speed of the pressure-producing turbulent eddies. 72 16. Decay of wall pressure correlation in a reference frame moving at local convection speed. 73 17. Longitudinal wall pressure correlation in a broad frequency band. 74 18. Transverse wall pressure correlation in a broad frequency band. 75 19. Spatial correlations of the wall pressure along a line at 45~ to the flow direction. 76 20. Spatial correlation of the wall pressure, 77 21. Contours of constant wall pressure correlation. 78 22. Effect of transverse curvature on wall pressure correlations. 79 23. Measured wall pressure spectra. 80 24. Corrected wall pressure spectra. 81 25. Corrected wall pressure spectra. 82 26. Comparison of the mean velocity profiles. 83 27. Time delay for R maximum at constant xl for varied x3. 84 28. Local convection speed of pressure-producing turbulent eddies in various frequency bands. 85 29. Asymptotic convection speed as a function of frequency. 86 30. Amplitude of narrow-band longitudinal space-time correlation of the wall pressure. 87 31. Amplitude of narrow-band transverse space-time correlation of the wall pressureo 88 32. Amplitude of narrow-band space-time correlation of wall pressure along a line at 45 to the flow direction (Note x3 = x1)lo 89 viii

NOMENCLATURE a E (k1,k3, w) f kl k3 p P p q00 Qp(Xl, x3, T) Qp(XX31T,I) R R a R, B0.(xx3 ) Rpp(Xl,X3, T) Spp(Xl, 3x, r = y+ a U U00 U U C U V V radius of circular cylinder spectrum of the wall pressure frequency wave number mean pressure fluctuating wall pressure free stream dynamic pressure double pressure correlation double pressure correlation in a narrow frequency band which has central frequency at w radius of the pressure transducer Reynolds number U a/v 0o Reynolds number U O /v Reynolds number U006*'/v correlation coefficient of wall pressure correlation coefficient of wall pressure in a narrow frequency band which has central frequency at o radius distance from axis of the cylinder mean velocity in the boundary layer in the stream direction free-stream velocity wall friction velocity convection speed of the pressure-producing turbulent:eddies: fluctuating velocity in x direction mean velocity in the boundary layer in y direction fluctuating velocity in y direction ix

x y z X1, x2, x3 r 6 6* 0 A1 A3 A P T T C W () distance parallel to wall, increasing in the stream direction distance normal to wall, increasing away from wall distance parallel to wall and perpendicular to stream direction, forming a right-hand Cartesian coordinate system with x and y spatial separation of pressure transducers in x, z directions temporal cross-spectral density of the wall pressure boundary layer thickness boundary layer displacement thickness boundary layer momentum thickness integral scale of Rp in x1 direction integral scale of Rp in x3 direction integral time scale of Rp kinematic viscosity density time delay convection time of pressure-producing turbulent eddies wall shear stress meridian angle power spectrum stream function circular frequency time average x

I. INTRODUCTION Knowledge of the pressure fluctuations beneath turbulent boundary layers is desired for numerous problems in fluid mechanics. The problems include: aerodynamic sound produced by turbulence in the boundary layer when the surface is rigid, vibration and sound radiation produced when the turbulent boundary layer is developed on a slightly flexible surface that is set in motion by the pressure fluctuations, and the new knowledge of turbulence structure in a boundary layer that can be obtained from wall pressure measurements Most of our knowledge of wall pressure fluctuations has been obtained from experimental measurements beneath the flat plate boundary layer. There are a great many papers reporting measurements of wall pressure fluctuations on a flat plate for various conditions (which include subsonic and supersonic free stream velocities with various free stream pressure gradients and for boundary layers developed on smooth and rough surfaces). At low subsonic speeds with zero free stream pressure gradient, the investigations of Bull (1963), Willmarth (1958a) and Willmarth and Woolridge (1962) are representative of the results of pressure measurements beneath flat plate boundary layers developed on smooth walls at high Reynolds numbers. In general, it has been observed that the wall pressure fluctuations are random without periodic components and have 1

2 power spectra roughly similar in shape to the spectrum of the turbulent velocity component normal to the wall. Using the method of spacetime correlation measurements, it was discovered, Willmarth (1958a), that the pressure fluctuations were convected with a speed of approximately 0. 8 UQo. More detailed investigations, Willmarth and Woolridge (1962) and Bull (1963) have shown that the convection velocity varies with streamwise spatial separation of the measuring stations and that for small spatial separation the convection velocity is low, 0. 56 U. The increase in convection velocity with streamwise separation of measuring points is attributed to the more rapid decay of the smaller pressure producing eddies which onthe average lie closer to the wall and are moving slower than those eddies which are larger and, therefore, move at higher speeds owing to the higher speed at a greater distance from the wall. The purpose of the present investigation of the pressure fluctuations beneath the boundary layer on the outside of a cylinder whose axis is aligned with the free stream is to determine the effect of transverse curvature on the wall pressure fluctuations. Ideally, the determination of transverse curvature effects should be made by comparing measurements made on a cylinder and on a flat plate with exactly the same Reynolds number, pressure gradient, Mach number and surface roughness. Measurements have been made in the boundary layer on a smooth flat

3 plate, Willmarth and Wooldridge (1962), at slightly higher Reynolds numbers, based on momentum thickness, R = 38,000, than could be obtained on the cylinder, Ro = 26, 200. We also will compare measurements on the cylinder with the measurements of Bull,(1963), obtained beneath a flat plate boundary layer at Reynolds number, R0 = 19;, 500. Our knowledge of turbulent boundary layers with transverse curvature is not as extensive as it is for the flat plate boundary layer and is restricted to measurements and similarity laws for mean quantities only. The work of Richmond (1957), Yu (1958), Yasuhara (1959), Reid and Wilson (1963), and Rao (1967) contain measurements and in some cases similarity laws for mean properties of turbulent boundary layers with varying amounts of transverse curvature. It;is often profitable in attempting to understand mean properties of the flow in a turbulent boundary layer to first consider the flow in a laminar boundary layer. It has been found by Glauert and Lighthill (1955) that the laminar boundary layer developed on a cylinder (when transverse curvature effects are large) has a much fuller profile than the Blasius boundary layer on a flat plate. In fact, the velocity near the wall is proportional to the logarithm of the distance from the cylinder axis and departs from that of the Blasius profile (in which u is linearly proportional to y near the wall) as the cylinder radius is reduced. The cause of this behavior (as was clearly explained by Glauert and Lighthill

4 (1955)) is that "the shearing force, on, a cylinder of unit length, is equal to the shear stress JL du/dy multiplied by the circumference 277 (a + y) of the cylinder, and this force must be independent of y in the region where the acceleration of the fluid is negligible, that is, near the solid boundary. One can expect that the presence of turbulence in a boundary layer with transverse curvature will increase the rate of momentum exchange (just as it does in a flat plate boundary layer) with the result that the velocity profile will be fuller, the skin friction increased, and the streamwise rate of growth of the boundary layer thickness decreased. The effect of transverse curvature on the structure of turbulence in the boundary layer and on the pressure fluctuations beneath it have not been studied. As a result of the present wall pressure measurements, we have been able to qualitatively explain some of the effects of transverse curvature on turbulence structure.

II. WIND TUNNEL AND CYLINDRICAL MODEL A. WIND TUNNEL FACILITY The experiments were carried out in the test section of the 5 x 7 foot low speed wind tunnel at the Gas Dynamics Laboratories, Department of Aerospace Engineering, The University of Michigan. The wind tunnel test section is 25 feet long and is indoors. The settling chamber, fan and steel ducting that recirculates the air are out of doors. The total distance around the wind tunnel circuit is 332 feet and the contraction ratio of the nozzle is 15:1. The sound field in the tunnel test section has been measured by Willmarth and Woolridge (1962). They stated: "The sound field in the test section was first measured with 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 Hz. The wall pressure correlation measurements on the floor of the test section which were made later showed a small peak at negative time delay which was caused by sound propagating upstream. From the measurements, it was finally determined that the sound energy amounted to approximately 1/20 of the energy in the turbulent wall pressure fluctuations. " In the present wall pressure correlation measurements the sound energy is approximately 1/50 of the energy in turbulent wall pressure 5

6 fluctuations. The reduction of sound level was accomplished by better sealing against air leaks at the diffuser entrance and by reducing structural vibration in the downstream region of the test section and diffuser entrance. The free-stream turbulence level measured by Tu and Willmarth (1966) at 200 ft/sec free stream speed was /U = 2. 50 x 103 in the flow direction. The configuration of the wind tunnel has not been changed since that time. B. CYLINDRICAL MODEL A 40 foot long, 3 inch diameter cylindrical model on which the boundary layer measurements were made was installed along the centerline of the wind tunnel. It consisted of a 2 inch steel tubing used as the backbone of the model and a 3 inch steel tubing used as the aerodynamic surface. The 3 inch tubing was located on the inner 2 inch tubing by means of adjustable set-screws. A 6 inch long ellipsoid of revolution and an 8 inch cone, both made of wood, were attached to the upstream and downstream end of the model respectively. The supports for the 2 inch tubing were so designed that they could furnish moments to reduce the mid-span deflection of the tubing (Fig. 1). Upstream support consisted of 5 streamlined aircraft wires. Two wires, T2, and one wire, T1, produced a counter-clock-wise moment. The other two wires at the bottom were dummy wires. The downstream support was a unistrut frame with two

7 unequal height columns (Fig. 2). When the 2 inch tubing was bolted on the columns, a clock-wise moment was produced. The deflection of the model in the test section of the tunnel was reduced to a minimum value by properly adjusting the forces T1 and F2. The wind velocity near the wire, T1, was low, approximately 0. 09 UO, therefore, the larger deflection at the front end of the model did not effect the symmetry of the boundary layer flow in the test section. The wind velocity near the wires, T2, was approximately 0. 15 U. Since the terminals of the wires near the surface of the model were 1/4 inch in diameter, they caused an asymmetry of the flow field around the cylinder in the test section. Four airfoil shaped filets, made from balsa wood, were used to reduce the wake from the junction of the support wires and cylindrical surface with satisfactory results as described below. A circumferential pitot tube array (Fig. 4, left), which consisted of 8 tubes 0. 65 inch above the surface of the model, was used to measure the circumferential velocity distribution at 16 feet and 24 feet from the entrance of the test section. Initially the velocity distribution was very asymmetric with low velocity regions at four circumferential positions in the wake of the four upstream support wires. The use of balsa wood filets (see above) reduced the wake from the upstream wire supports and careful alignment of the model by moving the downstream supporting

8 struts reduced the asymmetry of the boundary layer to an acceptable levelo The circumferential velocity distribution measured at the pressure transducer station (x = 24 ft) for three different free stream velocities are plotted on Figo 3. The largest velocity deviation was 0. 02 U. At its final position, the maximum vertical deflection of the model in the test section of the tunnel was 0. 39 inches, the corresponding slope was 0. 2% approximately. The maximum lateral deflection was 0. 25 inches, the corresponding slope was 0. 1% approximately.

III. INSTRUMENTS AND EXPERIMENTAL METHODS A. INSTRUMENTS AND METHODS FOR MEASURING MEAN PROPERTIES OF THE FLOW To measure. the velocity profile very near the surface of the cylinder a pitot tube was installed on a traversing device (Fig. 4, right). The pitot tube. was a 0. 042 inch diameter stainless hypodermic tubing with a flattened mouth. The dimensions of the mouth of the tube are shown in Fig. 5. It was used in the Reynolds number range 50 < R = hU/v < 250 where h wias the internal height of the mouth. The correction to the reading of the tube owing to viscous effects was negligible (see McMillan (1 954)). The maximum angle of attack of the pitot tube was 10 degrees. It was experimentally confirmed that the angle of attack did not affect the total pressure reading up to an angle of 15 degrees (see Alexander (1953)). The pitot tube was moved normal to the surface by the operator outside the test section, using a linkage and worm gear mechanism that was driven by a flexible shaft. The distance of the pitot tube from the surface was measured with a cathetometer focused on the pitot tube through the window of the tunnel. The static pressure was measured with a static tube in contact with the surface of the cylinder. This static tube was a long 0. 065 inch diameter hypodermic tubing with one end sealed. A 0. 035 inch diameter hole was drilled in the tube 3 1/2 inches from the sealed end. 9

10 The difference between pitot and static pressure was measured with a precision single-tube manometer * with a resolution of. 001 inch- using water as the indicating liquid. The velocity profiles quite far from the wall, y/6 > 0. 2, were measured with a pitot tube rake which consisted of 10 pitot tubes and 2 static tubes at different heighths from the wall (Fig. 4, middle). The wall shear stress was measured according to Preston's method (Preston (1954)) using a 0. 042 inch diameter pitot tube in contact with the surface of the cylinder. The calibration given by Smith and Walker (1958) was used for calculation of wall shear stress. B. INSTRUMENTS AND METHODS FOR MEASURING TURBULENT PRESSURE FIELD 1o Pressure Transducers At the test station, the lower half of the model was replaced by a 4 1/2 inch long, 3 inch 0. D. semi-cylindrical lead shell (Fig. 6). There were 13 transducer elements, which were 0. 06 inch diameter and 0. 020, inch: thick lead-zinconate-titnate (PZT-5) disks mounted permanently flush with the outer surface of the lead shell. The arrangement of the transducers which was selected to efficiently obtain the spatial correlation of the pressure field is shown on Fig. 7o * The Meriam Instrument Co., Micro-Manometer Model 34 FB2

11 Fig. 8-a shows the cross section of a single transducer plug mounted in the lead shell. A fine copper wire (0. 002 inch diameter) was attached to the front surface of the phenolic plug and brought out through the hole in it which was then filled with wax. The PZT-5 disk was glued on the plug with conducting cement. * This plug was then inserted into a 0. 063 inch diameter hole drilled through the lead shell. The position of the plug was so adjusted that the PZT-5 disk was flush with the lead surface. Having put the plug in position, the remainder of the hole was filled with vacuum sealing wax, a product of Central Scientific Company. On the front surface, the transducer was electrically connected to the lead shell with a thin coat of conducting silver paint. * The cross section of the triple transducer plug is shown on Fig. 8-b. The construction differed from the single transducer plug in that an additional brass sleeve was used and the method of ground connection was altered. The outer surface of the brass sleeve was curved to match the surface curvature of the lead shell. The electrical connection between the brass and the PZT-5 disks was provided by gluing a piece of aluminized mylar plastic sheet (0. 0005 inches thick) on the surface. The brass sleeve was in contact with lead. *SC 12 MicroCircuits Co., New Buffalo, Michigan.

12 Thirteen short Microdot coaxial cables with Microdot connectors at one end were glued on the inner surface of the lead shell with epoxy cement * The copper wires at the back of the transducer plugs were soldered to the central conductor of the cables. Two seven foot long Microdot coaxial cables with additional outer shielding to make a triaxial cable were used to conduct the transducer signals to the input of the cathode followers and preamplifiers. To reduce parasitic capacitance of the long cable, the coaxial cable shield was driven by a voltage proportional to the transducer signal and the additional outer shield outside the cable was used as ground (Fig. 9). In order to seal the air gap between the lead shell and the steel tubing, the lead shell was held on the model by a thin rubber cuff (0. 008 inch, thick). Four rubber O-rings were placed between the contact surfaces of the lead and the steel to prevent the transmission of vibration of the tunnel structure and the model to the lead shell and transducers. The capacitance of the pressure transducer was 32 pf. The use of a driven shield, see Figo 9, effectively reduced the capacity of the long cables leading to the preamplifiers and cathode followers to a low value. The frequency response of the transducers was assumed to be flat over the frequency range of interest, f < 50, 000 Hz. Willmarth (1958b) has shown *Epo-lux Steelcote Mfg. Co., No. 185A.

13 by shock tube calibration that larger transducers than the present transducers which were made in a: similar manner did have a flat frequency response up to at least 50, 000 Kz. The absolute calibration of the pressure transducers was carried out "in situ" with the lead shell installed on the cylindrical model. A previously calibrated transducer, * Willmarth and Wooldridge (1962), and the new pressure transducers were put inside a Helmholtz resonator which was made from a bottle whose bottom was cut to fit the cylinder. The joint was sealed with clay and the Helmholtz resonator was excited at 250 Hz by a carefully placed low speed air jet. The calibration was obtained by comparing the output of the two transducers inside the Helmholtz resonator. The sensitivity of the transducers was typically 1. 32 x 106 volts/dyne/cm2 2. Electronic Equipment The transducers were connected to a cathode follower with high input impedence of 1. 2 x 10 ohms (Fig. 9) followed by a low noise preamp5 lifier and amplifier system with a maximum gain of 10, see Willmarth and Wooldridge (1962). The band width of the amplifier was adjustable between 1 Hz and 160 KHz. The electrical signals from the pressure transducers were recorded on a three channel Ampex FR 1100 tape recorder which had a band width from DC to 20 KHz. The root-mean-square wall pressure was measured with a Ballantine model 320- true RMS meter. *We checked the 1962 calibration using the 1962 method (an enclosed volume variable over a small range with a piston) and found no appreciable change in the calibration over 6 years!

14 The spectrum of the fluctuating wall pressure was obtained by passing the signal through a Tektronix Type 115 spectrum analyzer plug-in unit which was driven by a Tektronix Type 544 oscilloscope. The output was recorded on a Hewlett Packard Model 2DR-2M x-y recorder. The correlations of the wall pressure were measured with a Princeton Applied Research Model 101 correlation function computer whose output was recorded with the above x-y recorder. Two Kron-Hite Model 310-AB variable band pass filters were used for correlation measurements in narrow frequency bands.

IV. MEAN FLOW FIELD A. EQUATIONS OF MOTION The momentum equations for axially symmetric mean motion in a turbulent flow written in cylindrical coordinates are au au 1 aP 1 a au a2u au2 I 1 ( u a-+ v - ar- + - r (r r+ -(r uv) (1) ax ar pax rar ar 2 ax r ar ax and 2 av av lap 1 a av a2v 1 a 2 a U ax+ v r=-par+ v-rar(r )+ 2-ra(rv ) uv (2) ar where U, V and P are mean; quantities and it has been assumed that the free stream is parallel to the x axis and the velocity V is the radial velocity normal to the x axis. Using the usual boundary layer approximations, the boundary layer equations are au aU laP v a 1 a -ax Var p ax r r r ar 3 and aP = 0 (4) ar with continuity equation au 1 (rV (5) ax r ar 15

16 The corresponding boundary layer conditions are: U=V =, uv=O at r a U=UO, V = O, uv=O at r =a+ 6 where a is the body radius and 6 is the boundary layer thickness. With the aid of continuity equation and the fact that dp/dx = - pUO dUO/dx, Eq. (3) can be integrated with respect to r from r = a to r = a + 6, Hence, the momentum integral equation for axially symmetric turbulent boundary layer is obtained: T 2 7 dU.w d 2 2 2 00 2a^ - ^ (^ n9 + ^ v^^a)2w(6) ~2a p =d [( 0 + a) I+ Uj(6*+ a)2 (6) where Tw- is the wall friction. The displacement thickness (6*) and the momentum thickness (0) for a fluid of constant density are defined by (6+a)2 (6*+ a)2 a2= (1 - udr2 (7) 2 oo a. and (6+a)2 (0+a)2 -a=J I1 - U -dr (8) -'2 l co Uoo a for UO = Constant, Eq. (6) becomes -w 2 0 dO -=-U (l+a-) (9) P co a dx

17 Hence the skin friction coefficient is _w O dO C 2(1+ (10) Cf 2 2(+ d(10) 2pU0o B. SIMILARITY LAWS The correct similarity laws for the mean flow in an axially symmetric boundary layer with zero pressure gradient have not yet been firmly established. If the amount of transverse curvature is not too large, the most logical approach might be to assume that the usual law of the wall and law of the wake are still valid. However, one must recognize immediately that there is an additional dimensional length parameter, the radius of curvature of the wall, a, in the problem. Thus, in the wall region dimensional considerations indicate a functional relationship U yU (aU11) U v w and in the wake region, U -U TU^ " ' 0!^? a) (12) Here, we have made all the usual assumptions about the mean flow in the wall and wake region (see Clauser (1956)) and have simply added the additional length, a, the transverse radius of curvature of the wall.

18 It is quite possible that the traditional division of mean flow properties into a wall region and wake region may not be valid when the transverse curvature is large, 6/a> lo One may visualize a very small radius of curvature of the wall in which the region occupied by fluid motions obeying the law of the wall (which is assumed to be of the form of Eqo (11)) is a very small fraction of the region occupied by the turbulent boundary layer flow. In other words, the boundary layer on a slender rod (a/6 - 0) is almost all a wake-like flow and the region near the wall (which, if it is called the wall region, must be independent of free stream conditions) would be a very small region containing the viscous sublayer. It is possible that in the limit a -0 the wall region contains only the viscous sublayer. In our present work we have been restricted to boundary layers in which 6/a ~ 2. For this case we will assume that the traditional (but modified, see Eqo (11) and (12)) division of the mean flow into a wall region and a wake region is valid. Inaddition wewill also adopttheprocedure of Richmond (1957), for the flow in the region near the wallo Essentially what Richmond (1957) did (under ColesI guidance) was to assume that there was a region near the wall where the mean flow was dominated by the wall. He then obtained a law of the wall for the axisymmetric boundary layer using Coles' streamline hypothesis.in that region. Coles' streamline hypothesis, Coles (1955), (which is true in the region near the wall of a two - dimensional turbulent boundary

19 layer) asserts that U/UT is constant on mean streamlines. Therefore, we can expect Richmond's procedure for the wall region of the axisymmetric boundary layer to be valid in the region near the wall where the turbulent flow is still essentially two-dimensional (or, in other words, when the ratio y/a is small). If y/a is not small, the similarity law obtained from the streamline hypothesis may be incorrect. We will give for reference (since Richmond's paper is not readily available) a brief description of his procedure for obtaining the similarity law. He assumes (the streamline hypothesis) that U/U7 is constant on the mean streamlines. Therefore, using the stream function, J, U =(Ct) (13) or inverting the expression C- H (14) The continuity equation is aU 1 arV au+ arV 0 (15) ax r ar and defines the stream function ar a; Ur =ax Vr =- (16) ax aXI

20 Thus, from Eq. (14) Iu= i 1 H U 1 aU U=-r. =-H'r iU ar r ar rc u U Br UT= UT(x) (17) so that UT = Urc r Integrating UT over the area between the wall and radius r (18) r a U7 rdr = T c r I HW T U ar a (19) U/UT 1 H(y) H'(y) y = G -U so that CU,(r2 2CU - a ) = GU (20) Inverting this expression, we obtain U = F.CUrT u? 2. (r - a2)] (21) for convenience, we can evaluate the constant C as follows. From Eq. (21) ay - = UT F C U( - a ) U7r a

21 or Tw 2 = U C a F'(O) AI 7 let F'(O) = 1 and then since pU2 = T, to obtain C = 1/av. Thus, 7 Richmond's law of the wall, which assumes the validity of the twodimensional streamline hypothesis near the wall beneath an axisymmetric turbulent boundary layer, is U IT 2 2 U F (r -2a U, '= F v... or U = F[ (1 Y) (22) using the new coordinate U (1+ -), one can write the two-dimensional v. 2a law of the wall in the form U 5.751 og UY (1+ ) + 5.10 (23) U 10 ' 2a where the function F is the usual (empirically verified) form for twodimensional flow. It appears that the above Eq. (23) provides an adequate representation of our results for 6/a ~ 2. There are indications from Richmond's work (1957), that the region of validity of Eq. (23) becomes a rather small region near the wall as 6/a becomes large.

22 There are a number of other investigations of the mean flow field in the region near the wall. In the work of Yu (1958) (which includes experimental results) a different method was developed (at Iowa Institute of Hydraulic Research under the guidance of Rouse and Landweber) for correlation of the mean flow in the wall region and the wake region. In Yu's formulation, the wall region contains the free stream velocity as an additional parameter (in addition to the parameters v, U, and a). We have chosen Richmond's method for data presentation because in the wall region one should not have free stream velocity as a parameter. Yasuhara (1959), has reported mean velocity measurements on a slender cylinder for cases in which 6/a 1. Yasuhara presented his results in the form advocated by Richmond (1957), with results quite similar to Richmond's and to our present measurements. Rao (1967) has reported a different form for the law of the wallo His form is derived on the basis that for a slender cylinder, 6/a > 1, the sublayer thickness is comparable to the radius of transverse curvatureo This reasoning suggests that U a U Ua r U - In (24) U a This form is certainly correct in the sublayer where as Rao points out rr = const = aT, However, we doubt that the sublayer thickness can ever be as lge as the adus of curvae of the Thus we believe be as large as the radius of curvature of the cylinder. Thus, we believe

23 that the correct conception of the sublayer is a region dominated by wall effects and that even when a becomes very small the sublayer thickness is always small compared to the radius of curvature of the surface, a. If this were not true, the turbulent eddying flow would wash the fluid in the sublayer (assumed to be of the order of a distance, a, from the wall), completely off the cylinder. The flow in a region occupying a crosssectional area of the order of the cylindrical cross-sectional area, therefore, cannot be termed a sublayer and Rao's formulation, Eq. (24) cannot be correct. Rao goes on to propose that throughout the boundary layer, the form of the velocity profile should be U F 7 Inrj (25) where F is the function obtained for the law of the wall in a two-dimensional flow. We have not used this form, Eq. (25), because it has the obvious property (since the function F is a logarithm in the wall region) of taking the logarithm of the logarithm of r/a. This certainly reduces scatter of data points for large r thereby requiring extremely accurate measurements to determine the validity of the formulation, Eq. (25). It would appear that the mean flow field in a turbulent boundary layer with transverse curvature is not well understood when 6/a is large. The law of the wall, Eq. (23), has only a small region of validity and an appropriate similarity law for the wake region has not been firmly established.

24 C. EXPERIMENTAL RESULTS FOR MEAN FLOW FIELD 1. Pressure Gradient The streamwise static pressure distribution along the surface of the cylinder at two different free stream velocities was measured as described in Section III, Ao The results were plotted in Fig. 10, showing a slight streamwise pressure increase. This was caused by the too rapid divergence of the side walls of the wind tunnel test section. (The original designers over estimated the correction necessary for boundary layer growth on the test section walls). The mean flow field in the flat plate (two-dimensional) boundary layer on the floor of the test section has been investigated by Willmarth and Wooldridge (1962). It was found that in the pressure gradient of Figo 10, the mean velocity profiles were (within the accuracy of the measurements) those generally accepted for an equilibrium two-dimensional boundary layer, see Coles (1954). In addition, the dimensionless shape factor, r, of Buri (see Schlichting (1968)) where ( '4 ( )1/4 (26)4 r=U dx ~V 2 dx V (26) -4 -2 was rF- -6 x 10 or approximately 1/100 of the value, r= -7 x 10, required for separation of a two-dimensional turbulent boundary layer. We can conclude that the slight positive (adverse) pressure gradient aC /ax = 3 x 10 3fto, will not cause the mean flow in the boundary layer o deviate ppreciy from the zero pressure gradiet case to deviate appreciably from the zero pressure gradient case.

25 2. Skin Friction The wall shear stress was measured using a Preston tube as described in Section III. A. The results are displayed in Fig. 13 (see also Tables 1 and 2) along with the measurements of Richmond (1957), Yasuhara (1959), and Yu (1958) for axially symmetric turbulent boundary layers with approximately the same ratio of 0/a as our experiments. The present measurements agree reasonably well with previous results at R 0 104 4 and extend the skin friction R 0 3. 5 x 10 for 0. 15 < 0/a < 0. 25. Note that the skin friction coefficient; is;:appreciably larger than in a flat plate turbulent boundary layer. This is qualitatively the same trend that one finds in a laminar boundary layer with transverse curvature. Because Yu's (1958) definition of 0 is different from ours, Eq. (8), when 0 is calulated according to Eq. (8), Yu's value of R is reduced by 0 approximately 15%. 3. Velocity Profiles The velocity profiles measured at the location of the pressure transducers are plotted in Fig. 11 in the form suggested for the law of the wall with transverse curvature, see Eq. (22). The shear velocity UT was obtained from skin friction measurements using the relation UT = (7w/P) /2 The results of the velocity profile measurements, Fig. 11, show that Richmond's modified law of the wall, Eq. (22), agrees fairly well with measurements of the present investigation for 6/a < 2.

Table 1 Summary of Some Results of Skin Friction Measurements X(ft) 8 8 7 8 9 10 3.28 U(fps) 30 60 90 154 154 154 110 R a 15250 30750 45060 40200 40200 40200 21800 eo(inM) 0. 253 0. 247 0. 20 0. 069 0. 094 0. 109 0. 051 O/a 0. 253 0. 247 0. 200 0. 138 0. 188 0. 210 0. 154 R0 3860 7600 9010 5540 7540 8750 3360 Cf 0. 0037 0. 00338 0. 00315 0. 00323 0. 00305 0. 00290 0. 00347 Source Yu (1958) Richmond (19 57) Yasuhara (1959) Yasuhara (1959) X = Distance from the nose of the cylindrical model.

27 4. Summary of Properties of the Boundary Layer In Table 2, the various properties of the turbulent boundary layers with transverse curvature that we have measured are summarized.

Table 2 Properties of the Axially Symmetric Turbulent Boundary Layer Measured in the Present Investigation x(ft) Uo(fps) R 0 (in.) 0/a U Cf Cf/Cf R 6/2a 6*/0 00 a T f f 0 24 200 134,000 0.382 0.254 6.96 0.00244 1.17 33,800 1.007 1.15 24 145 115,000 0.341 0.228 4.98 0.00232 1.08 26,200 0.926 1.23 co 24 100 70,200 0. 359 0. 238 3.83 0. 00292 1.30 16,800 1. 073 1.14 16 198 136,000 0. 218 0. 146 7.30 0. 00274 1.21 19,800 0. 607 1.14 16 101 69,800 0. 244 0.162 3.92 0.00301 1.21 11,300 0.80 1.21 CfI (flat plate) due to Coles (1954).

V. EXPERIMENTAL RESULTS The measurements of the wall pressure fluctuations beneath the axially symmetric boundary layer will be discussed in the light of our knowledge of wall pressure fluctuations beneath a plane two-dimensional boundary layer. The most striking property of the wall pressure fluctuations beneath a plane boundary layer is the now well known fact that the random pressure fluctuations are convected at speeds of the order of 0. 5 U, to 0. 85 U. The convection and decay of the pressure fluctuations were first measured by Willmarth (1958) using the technique of space-time correlation in which time delay is varied for a constant separation distance between pressure transducers. The use of spacetime correlation measurements in turbulent flow was pioneered by Favre (1952) (who measured the space-time correlation between streamwise velocity fluctuations). For the present experiment we have found that the space-time correlation of wall pressure fluctuations beneath the boundary layer with transverse curvature also shows convection and decay, but with the important difference that the rate of decay of pressure correlations is more rapid. This will be discussed in detail below along with other statistical measurements of the random pressure field which include the mean-square; power spectrum; lateral, oblique, and longitudinal correlations (with zero time delay); and narrow frequency band correlations 29

30 of the wall pressure fluctuations. We will compare our present measurements with the two-dimensional measurements as we proceed with our descriptiono A. LONGITUDINAL SPACE-TIME CORRELATIONS OF THE WALL PRESSURE The results of our longitudinal space-time correlation measurements-are shown in Fig. 13. The correlation curves (reading from top to bottom) were measured at increasing values of spatial separation, The peaks of the correlation occur at larger and larger time delay as the spatial separation increases (note that the displaced time origins are indicated by small vertical bars on Fig. 13o ) At large spatial separations a small peak appeared on the left branch of the curveo This is the upstream propagation of pressure fluctuations caused by sound produced in the wind-tunnel diffuser and fan. The mean-square sound pressure in the test section can be estimated from the average maximum value of the peak on the space-time correlation curve for T < 0. It is approximately 0. 02 p2. This value was used in estimating the true value of the spatial correlations of wall pressureo (See Appendix A,) Qualitatively similar space-time correlations have been measured beneath a plane boundary layer. The essential difference between the present measurements and those in a plane boundary layer is that with transverse curvature the pressure correlation decays more rapidly,

31 However, in the present investigation, the convection speed is identical (within the experimental error) with the convection speed in a plane boundary layer. Figure 14 shows the location, for various streamwise spatial separations and time delays, of the maxima of the pressure correlation along with the two-dimensional results from Willmarth and Wooldridge (1962) and Bull (1963). Figure 15:shows the convection speed for increasing values of spatial separation. The convection speed of Fig. 15 is obtained graphically from the slope of the locus of the peaks of pressure correlation of Fig. 14. The reader should note that this definition of convection speed is not by any means unambiguous since the correlation of the pressure decays in time and space. A perfectly definite convection speed is easy to define and to understand if one has a convected frozen random pattern convected at constant speed because the lines of constant correlation and the convection path in the xl, T plane will be parallel straight lines whose slope is the convection velocity. In the case of a decaying random field in which turbulence of various scales moving at different convection speeds is present, we may refer to Wills (1964) who has discussed the difficulties with the various definitions of convection speed and has proposed a convection speed definition based on a maximum integral time scale that is different from the definition we have used. In practice, for our experiments, it is an almost impossible procedure to use Wills' definition of convection speed since one must measure pressure correlations while

32 varying the spatial separation between the pressure transducers. * The surface curvature and the inaccessibility of the cylindrical model (in the center of the tunnel) make this impractical. Also, it is not really necessary to use a precise definition of convection velocity because we are looking for changes from the rather well known convection properties of the two-dimensional wall pressure fluctuation field. The definition of convection velocity used for the present work is precisely the same as that used in our previous plane boundary layer experiments. As we have already mentioned, the convection velocity is the same as that found beneath a plane boundary layer. The measurements show in each case an increase in convection velocity from 0. 56 U, to 0. 83 U, as the longitudinal spatial separation is increased. The increase in convection velocity is caused by the more rapid decay of the smaller pressure producing eddies near the wall. After the smaller eddies have decayed only larger eddies remain and, since they are larger, their effective centers are farther from the wall and they move at a faster speed owing to the higher mean velocities farther from the wallo The concept of an effective center of such a poorly defined entity as an eddy is admittedly vague. It is more accurate to speak of a convected distribution *Or use numerous closely space transducers.

33 of vorticity which if of large scale (therefore, extending into the higher speed regions of the boundary layer) must move at a higher average speed than a small scale distribution of vorticity near the wallo Although the convection speeds in the plane and axially symmetric boundary layer are the same there are important differences between the space-time correlation for the two cases. In the boundary layer with transverse curvature, the rate of decay of the maximum of the spacetime correlation is more rapid than it is in the plane boundary layer. Referrring to Fig. 13, we see that when x1/6* = 9. 33 the maxir,^:X.m pressure correlation is R = 0. 1. In the case of the plane boundary layer, W.7illmarth and Wooldridge (1962), R = 0. 1 occurs for x1/6 = 22, 6. The pp decay of the maximum (or peak) value of the pressure correlations for plane and axially symmetric boundary layers are shown in Fig. 16, where R pp(x1, 0, ') is the peak value of the pressure correlation. From the figure, it can be observed that in the boundary layer with transverse curvature, the decay of pressure correlations is faster than it is in a plane boundary layer. Note that the decay rate is especially rapid at small spatial separations. For large spatial separations the rate of decay (the slope of curves in Fig. 16) is approximately the same. We will discuss these matters in more detail after we have presented all the experimental results. Additional measurements of convection:speeds have been obtained from, spacetime correlation measurements with x3 3 0o These are discussed in Appendix B.

34 Bo LONGITUDINAL, LATERAL AND OBLIQUE SPATIAL CORRELATIONS OF THE WALL PRESSURE The spatial correlations of the wall pressure (with zero time delay) have been studied in considerable detail to determine the contours of constant pressure correlation on the surface of the cylinder. The array of thirteen pressure transducers, see Fig. 7, was designed so that a large number of pressure correlations for different spatial separations could be efficiently measured with a small number of pressure transducers. All of the wall pressure correlations were measured in the frequency band 0. 144 < < 28. 8,and have been corrected for effect of sound in the U 00 free stream. (See Appendix Ao ) The results of the spatial correlation measurements are displayed in Fig. 17, 18, 19, and 20. Consider Fig. 17 and 18 which show the longitudinal and lateral pressure correlation. It is clear that the pressure correlation decreases with x1 or x3 more rapidly than has been observed in the plane boundary layer. For the present case of a boundary layer with transverse curvature in which 6/a ^ 2, the lateral or longitudinal spacing at which any given positive value of the pressure correlation is attained is the order of half the spacing at which the same value of pressure correlation is attained in the plane boundary layer. Figures 19 and 20 show measurements of spatial correlation in oblique directions, along a 45~ line in Fig. 19 and for various oblique locations in Fig. 20. In all these measurements, using the array of

35 transducers of Fig. 7, we have assumed that the random wall pressure field is statistically homogeneous. Thus, we ignore the rather slow streamwise changes in the statistical structure of the wall pressure field. This means that the pressure field is. homogeneous longitudinally R (x.3 0) = Rpp (-x, x3, ) (27) pp X x3' O pp x ' and by axial symmetry laterally homogeneous R (x1,x3 0) = R (x -x3, 0) (28) pp (Xl ' 3 pp 1 3' Therefore, the number of wall pressure correlations measurements necessary is reduced. (Actually it is sufficient to measure correlations in only one quadrant of the x1, x3 surface.) The results of all the measurements of pressure correlation with zero time delay are summarized and compared with the plane boundary layer case in Fig. 21 and 22, respectively. Figure 21 shows that the contours of constant pressure correlation. are very nearly circular and that the correlation- has decayed with distance to 1/20 of the maximum value in a distance of approximately 1. 5 6*o (Note that in this experiment 6* = 0. 42 inches and 6 = 2. 8 inches.) In the lateral direction, a distance x3 = 1. 5 6* corresponds to only 23. 7 degrees of arc along the cylinder. Note that in Fig. 18, the lateral pressure correlation is negligibly small for x /6* > 6. 2 and does not appear to -3

36 oscillateo We do not expect that appreciable correlation of the wall pressure (relative to the value of R = 0. 05 at x3/6* = 1l 5) for points on opposite pp sides of the cylinder would be found. Note that we have filtered out all pressure fluctuations with a frequency less than 100 Hz and, since the convection speed is of the order of 0. 8 U,,the half wave length of the pressure fluctuations that have been rejected is already quite large, that is, greater than or equal to 1/2(0. 8Uo /100 Hz) - 7 inches (or 2. 56). Recall that the pressure correlation is essentially zero in a distance of the order of 36*'0 O 56. Figure 22 compares the contours of constant correlation for plane and axially symmetric boundary layerso The iso-correlation contours in the axially symmetric boundary layer are nearly circular as compared to the case of the plane boundary layer in which the larger: iso-correlation contours, for smaller values of Rpp, are elongated in the transverse, x3, direction. For large values of R p(small size iso-correlation contours) the shape pp of the iso-correlation contours is nearly circular for both the plane and axially symmetric boundary layer. From these results it is apparent that the effect of transverse curvature is to reduce the scale of all the random pressure producing eddies. by a factor of two or more for 6/a - 2. The reduction in transverse scale of the larger eddies is greater than the longitudinal scale reduction. But for the smaller eddies, the scale reduction is approximately the same in the longitudinal and transverse directions,

37 C. ROOT-MEAN-SQUARE WALL PRESSURE The root-mean-square (rms) value of the pressure fluctuations on the surface of a cylinder was measured in the frequency band 0. 063 < U < 57. 6. The measurements are compared with the rms wall-pressure U00 associated with plane boundary layer obtained by other investigators in Table 3. The present data has been corrected for free sstreiam sound. * According to Bull (1963), the ratio rms wall-pressure to free stream dynamic pressure is not dependent on Reynolds number. But the ratio of rms wall-pressure to skin friction increases slightly when Reynolds number increases. In the later case, the increase occurs because the skin friction coefficient decreases as Reynolds number increases. If we compare the ratiosf p 7/q or p/T for the plane and axially symmetric boundary layers, see Table 3, we find that there is not a large change in either ratio caused by transverse curvature. For instance, if we compare the values listed in Table 3, the uncorrected ratio of rms wall-pressure to dynamic pressure beneath an axially symmetric boundary layer increases 9%. from the value measured by Willmarth and Roos (1965) at slightly higher Reynolds number. Recalling that transverse curvature increases the mean skin friction, we note that transverse curvature decreases the ratio p2/w by approximately 10%. In any case, there is not a large effect of transverse curvature on the root-mean-square wall pressure for 6/a < 2. *See Appendix A.

Table 3 Comparison of Root-Mean-Square Values of the Wall Pressure Fluctuations Vp P/q p/T wa Frequency Range Remarks Axially Symmetric Boundary Layer Plane Boundary Layer 5. 6x103 5, 99x10-3 5.14x103 5, 64x10-3 5. 35x103 5. 45x10-3 5 4x 310 2. 26,2. 42 2. 54 2. 66 2. 48 2 58 2. 80 26, 200 26, 200 38,000 38,000 19,500 26,000 33, 800 0. 063 < w6*/UV < 57. 6 0. 063 < w6*/Uoo < 57. 6 0. 14 < w6*/Uoo < 28.8 0. 14 < w6*/U < 28. 8 0. 016 < w6*/U < 19. 9 00 0. 087 < w65*/Uoo< 11 0.14 < w6*/U < 17 Uncorrected Corrected Uncorrected Corrected Corrected Corrected Corrected Present data R/6*= 0.072 ~~ Willmarth andRoos (1965) R/5* = 0. 061 Bull (1963)

39 D. POWER SPECTRUM OF THE WALL PRESSURE The power spectrum of the wall pressure fluctuations beneath the axially symmetric turbulent boundary layer measured with the present pressure transducers (R/6* = 0. 072) at U0 = 145 ft/sec was obtained in the frequency range from 100 to 20, 000 Hz (0. 144 < U < 28. 8). The data were compared with the power spectrum in the plane boundary layer measured with a pressure transducer with R/6 * = 0. 061 by Willmarth and Roos (1965) and with a pressure transducer with R/6* = 0. 095 by Bull (1963), see Fig. 23. The reason for choosing 100 Hz as the lower frequency limit for our measurements was the possibility of free stream temperature and vorticity distrubances caused by heat input or cooling at the wall of the wind tunnel circuit. A description of our observations of this phenomenon are given in Willmarth and Wooldridge (1962). However, in the present investigation, we did not observe the severe pressure disturbances at low frequencies that were found at the wall, Willmarth and Wooldridge (1962). The reason may be that the effects of external heat input or cooling remain confined to a region near the wind tunnel wall. Generally speaking, the shape of the wall pressure spectra beneath a twodimensional boundary layer and beneath a boundary layer with transverse curvature do not appear greatly different on a log-log plot. However, on closer examination one finds that at high frequencies, 6*/U00 > 10, the wall pressure spectrum beneath the boundary layer with transverse

40 curvature contains approximately twice the energy density that was measured beneath a plane boundary layer and at low frequencies, co6 */U < 1, the 00 energy density of the pressure spectrum beneath the boundary layer with transverse curvature is 75% less than beneath a plane boundary layer. The normalized power spectra are shown on a linear scale in Fig. 25. It is quite clear on the linear scale that there is, indeed, greater spectral density at high frequencies. The data of Fig. 25 have been corrected for the effects of attenuation caused by the finite size of the pressure transducer. This is discussed below. E. CORRECTIONS FOR FINITE TRANSDUCER SIZE Experimental resolution of the structure of the turbulent pressure field is limited by the finite size of the pressure transducer. For a transducer of diameter 2R, the measurements of the spectrum of the pressure at a frequency of the order of U0/2R * or greater will be subject to a considerable error which becomes larger as the frequency increases. Therefore, it is desirable to use as small a diameter transducer as possible. In the present investigation, the diameter of the transducer was 0. 06 inches and the frequency U o/2R was 30 Hz. (Note that most of the energy in the spectra of Fig. 23 occurs at frequencies less than 20 KHz. ) Therefore, the present results should not be subject to a large error. *This is the frequency for which the transducer diameter equals the wave length and for a one-dimensional pressure field and a transducer there would be a complete cancellation of the pressure signal at this frequency.

41 Corrections to the power spectrum measured by a finite size transducer have been computed by Corcos (1963). His calculations assume that the pressure field can be represented as a function of the variables wxl/Uc and wx3/Uc. For this reason and others, related to his method 1C o C of calculation, his corrections are an approximation to the true corrections. His approximate corrections agree very well with our experimental results and calculations, Willmarth and Roos (1965), when wR/Uc < 1. We have used. Corcos' (1963) computations for the correction of our power spectrum beneath the boundary layer with transverse curvature because wR/Uc < 1 in the range of interest for our spectrum. Note that using Corcos' computations means that the wall pressure must be expressible using the similarity variables and functions used by Corcos (1963). To a reasonable approximation this is true and is discussed in Appendix C. The corrected power spectrum non-dimensionalized by rms wall-pressure is presented on a logarithmic scale in Fig. 24and on a linear scale in Fig. 25. For comparison, the corrected dimensionless power spectrum of the plane boundary layer wall pressure measured by other investigators are also presented in the figures. The corrected root-mean-square wall-pressure is listed in Table 3. Note that the correction to the root-mean-square wall-pressure is not more than 10%.

42 F. INTEGRAL SPATIAL AND TEMPORAL SCALES The integral scales of the wall pressure were computed according to the formulae 00 oo A1= IRpp(X1,0,0) dx1 (29) -00 00 A3 j IRpp(0,X30) dx3 (30) -00 00 A = J IRpp(00,,T) dr (31) -00 where A, A3, and A are respectively the longitudinal, lateral and temporal integral scale. We have used the absolute value of pressure correlation in our definitions to ensure that spuriously small integral scales are not produced by oscillations of the pressure correlation caused by the rejection of low frequencies when the signals were filtered. The various integral scales are collected for comparison with the case of a plane boundary layer in Table 4. It is apparent that in all cases the effect of lateral transverse curvature is to reduce the spatial and temporal integral scales. It is clear and not too surprising that transverse curvature reduces the lateral integral scale, A3, by a larger amount (a factor of 2. 5 or more) than the reduction of the longitudinal and temporal scales (a factor of the order of 1. 5).

Integral Spatial and Frequency Range T U /6* Aoo A1/6* A3/6* Remarks Table 4 Temporal Scale.-in Plane. and Axially Symmetric.Turbulent Boundary. Layer Axially Symmetric Plane Boundary Layer Boundary Layer 0.144< <28. 8 0. 14< < 13.6 0.075 < 7. 5 U <7.5 oO oO oO 2. 92 3.90 3.84 1.31 2.06 3.20 1.04 2. 51 6.74 CO Willmarth and Wooldridge (1962) Bull (1963)

44 VI. SUMMARY AND DISCUSSION From the results of our measurements presented in Section V and in Appendix C, we can make a qualitative assessment of the effect of transverse curvature on the structure of turbulence and on the wall pressure. First consider the result of Fig. 14 and 15 which show that the convection speed in the boundary layer with transverse curvature is almost the same as in a plane boundary layer. Next, consider the velocity profiles in the two boundary layers which are compared in Fig. 26 where the displacement thickness, 6 *, has been used as a length scale. We have used 6* exclusively for our length scale in presenting our results because it is a more definite quantity than 6 and has been successfully used in the case of the plane boundary layer. (As a matter of fact, our qualitative discussion would not be changed if 0 or 6 were used as the characteristic length. ) From Fig. 26, it is apparent that the velocity profile is much "fuller" when transverse curvature is present. Consider a turbulent pressure producing eddy near the wall of any given size (relative to 6*). It is clear that if an eddy of this given size is present in a boundary layer with transverse curvature, it will have a higher convection velocity than it would have in a plane boundary layer, (because at every point in the eddy the mean velocity would be larger). On the other hand, as mentioned above, the measured convection velocity in the two

45 boundary layers is nearly the same. The explanation for the unchanged convection velocity is that the pressure producing eddies in the boundary layer with transverse curvature must be smaller and therefore nearer the wall where the mean velocity is lower. The assertion that the pressure producing eddies are smaller (relative to 6*) in a boundary layer with transverse curvature is in agreement with all our other measurements. Consider the relative size of the contours of constant pressure correlation of Fig. 22. In a boundary layer with transverse curvature, the contours are smaller,by approximately a factor of two or more,than they are in a plane boundary layer. Figure 16 shows the decay of the peaks of longitudinal space-time correlation. The decay is much more rapid at small spatial separation in a boundary layer with transverse curvature* and this is caused by the presence of smaller eddies that are created by the transverse curvature. (Recall that Willmarth and Wooldridge (1962) have shown that an eddy of any given size decays after traveling a distance proportional to its size. ) Thus, if there are relatively more smaller *Note that in Fig. 16 for x1/6* > 6 the rate of decay of the maxima of the pressure correlation in the plane and axially symmetric boundary layer is the same. This indicates that after the smaller eddies have decayed, the eddy size distribution of larger eddies is similar.

46 eddies, they will decay in a shorter distance. Finally, the power spectrum of the wall pressure (see Figo 24) contains a greater energy density at high frequencies than ina.plane boundary layer owing to the unchanged convection speed of smaller eddies. Consider now the shape of the contours of constant correlation in a plane boundary layer for R < 0. 1. (See Fig. 22. ) The contours for pp - R < 0. 1 are larger transversally than in the stream direction. Howpp - ever, the smaller contours for larger R (R > 0. 3) are nearly circular. In a boundary layer with transverse curvature all the contours are smaller and nearly circular. We believe that there are two primary effects in a boundary layer with moderate transverse curvature that reduce the size of turbulent eddies. The first effect which causes a reduction in size of the eddies is the increased fullness of the velocity profile. * The turbulent eddies near the wall moving at any given convection speed are necessarily smaller because the mean velocity corresponding to that speed is reached at a point nearer the wall. In addition, there is a second effect in which the larger eddies suffer a greater reduction in transverse scale than small eddies because the wall is curved transversely. Thus, if one visualizes a large eddy adjacent to the curved wall, it is apparent that in the transverse direction at either side of * See the discussion of Section I about the effect of transverse curvature on the mean velocity profile.

47 the periphery of a large eddy the mean velocity is higher than it would be at the sides of the same eddy in a plane boundary layer. Thus, there is a streamwise shearing motion along the sides of large eddies, in a boundary layer with transverse curvature that is not present in a plane boundary layer. This shearing motion acts to reduce the transverse scale of large eddies. * Additional confirmation for the above selective effect of transverse curvature on the lateral scale of large pressure producing eddies is contained in Appendix C. There, the effect of transverse curvature on the decay of the narrow band correlation of the pressure in a lateral direction is to cause a more rapid decay for large eddies (in a low frequency band) than it is for small eddies (in a high frequency band). *If we consider the extreme case of large transverse curvature with 6/a -0, the transverse extent of the largest eddy is limited to a distance of the order of 26. For a plane boundary layer there is no readily apparent limiting transverse length.

APPENDIX A CORRECTION-OF WALL PRESSURE FOR THE EFFECT OF SOUND IN THE FREE STREAM The pressure measured at the wall is P = P+ Ps where p is the turbulent wall pressure and ps is the pressure produced by sound waves in the free stream. Define the true value of the wall pressure correlation as P1P2 (Rpp)t p The measured pressure correlation is ((P1 + P) (P2 + P) (Rpp) + Ps2/P2 (ppm (P + ps)2 + / 2 -2 ---=p= where assume p1 = p2 and pp = 0 Therefore (Rpp) = (1+ p2/p2) (Rpp)m 2/P2 (32) For Willmarth and Wooldridge's (1962) data p2/p2 = 0. 05 and gels 1962 da s 48

49 (Rpp)t= 1. 05(Rpp)m 0.05 (33) For the present data p 2/p2 = 0. 02 and ( (Rpp) = 1.002 (R34) Pp t pp M.

APPENDIX B CONVECTION SPEEDS MEASURED IN OBLIQUE DIRECTIONS From the results of oblique (X3 # 0) space-time correlation measurements, the locus of the peaks of pressure correlation were plotted in Fig. 27 as a function of longitudinal spatial separation for several obliquities (various values of x3/6 *). Figure 27 shows that the location of the peaks of the pressure correlation in the xl, T plane remains the same as it was for x3 = 0. Thus, the decay of the wall pressure in the transverse direction does not directly effect the convection speed in the longitudinal direction. 50

APPENDIX C CORRELATION OF THE WALL PRESSURE IN NARROW FREQUENCY BANDS Let p(x, z, t; w)be the signal obtained by passing the output of a pressure transducer at (x, z) through an ideal filter which has a narrow pass band centered at a frequency, w, whose width is A c. Then the correlation of pressure fluctuations measured by two pressure transducers a distance (xl, x3) apart at central frequency co for a band width A co is T Qp (1, X3, T;) = p(x,z,t;o) Pp(x+ x, z+ x3, t+ T; ) dt (35) 0 Corcos (1962) has shown that* Qpp(x1,x3, 7; c) = r(x1, x3, w) cos(w)T+ a) Ac (36) where r(xl, x3, c) = lr(xl, x3, w)leia is the temporal cross-spectral density of the wall pressure fluctuations and a is the argument of the complex quantity r. The cross-spectral density is related to the spectrum by o00 r(x1,x3,;) =-2 JJ E(k,k3,w) exp(klx + k3x3) idkldk3 (37) -00 *See Bull (1961) for a complete discussion. 51

52 The narrow band correlation coefficient is defined as p(X3,3' Ti,'w) R ' (x x (Oo;) (38) pp Corcos (1962) suggested that r could be represented by the expression r(xl, =x3 (o) | A U B - exp U- (39) where A(wxl/U ) and B(wx3/U ) are functions to be determined experiI C 3 co mentally with properties A(O) = B(O) = 1 and the convection velocity Uc = U (w) is function of the central frequency w. Then C C Cox 1 ox 3 ox1 Q (xx3,T;)- ()I A(U o B(T-U c-os( - Aw (40) C eO CC and Qp(O, O, O;c) = (w)I Aw (41) Therefore, the correlation coefficient in a narrow frequency band at a central frequency o is Cox Cox3 WX (x x1, 3 r;w) =A()U BU COos(wT- U) (43) pp 1 ' The narrow- ^ 7~Bal spc-time co s r '- (43) "he arr n \a spc /\ c6i C The narrow-band longitudinal space-time correlation is R pp( 0,, wc) = A U coslwT - ) (44) \ Uc \

53 Where the function A(wxl/Uc) is obtained when r = 7c = X/U 1ic' c I7 c wx1 A u c = IR ( 0 -,T;w) I= - IR (X, T;W) I pp 1 ' pp 1' ' c (45) The narrow-band transverse space-time correlation is Rp (O,x3, pp, 3 T;C) = B - Cos W T \ 0 (46) The function B(wx3/U ) is obtained with T = 0 00 wx 3 B u~ =R (0 x3 Oio) \ 0 "s/ (47) Relatively narrow-band correlations of the wall pressure were measured in the four frequency bands 0. 61 < < 0. 97 o 00 Central frequency 06* at = 0.79 U oO 1. 28< < 2. 76 o3. 3.87 < U< 6.25 oo it iti t 2.02 t tt 1 t 5.06 5. 21 < <15. 63 00 "t i i" 10.42

54 To carry out those measurements two Krohn-Hite model 310-AB variable band-pass filters were-nsed. The high and low pass settings of the two filters were carefully matched to give identical phase shift as a function of frequency so that they were identical (within + 3 ). From the measured space-time correlation curves, values of correlation amplitude and convection speed were obtained. Figure 28 shows the convection speed for increasing values of spatial separation in various frequency bands. The convection speed of Fig. 28 is obtained in the same way as that described in Section V. A. For an ideally narrow frequency band ( (Aco/o) << 1) the convection speed would remain constant when spatial separation increased. However, Fig. 28 shows an increase in convection speed as the longitudinal spatial separation increased. This is because we have a finite band width and the smaller eddies in that band width decay faster than the larger ones. (See Section V. A). The asymptotic values of convection speed at large spatial separation were plotted in Fig. 29 as a function of dimensionless frequency o6 */U. Comparing Fig. 29 with Willmarth and Wooldridge's (1962) data shows that the asymptotic convection speed (i. e., x1/6* - oo) of given size eddy in axially symmetric boundary layer is almost identical to that obtained in the plane boundary layer.

55 The amplitude function A(cwxl/Uc) of the narrow-band longitudinal correlation of Corcos' representation can be found by plotting the measured maximum amplitude of space-time correlation, io e., R (x, 0, t;), pp 1' c for various spatial separations and central frequencies (Fig. 30). Figure 30 shows that the amplitude of narrow-band longitudinal space-time correlation is only slightly less than in the plane boundary layer. This means the rate of decay of a given size eddy is about the same as in the plane boundary layer and that the rate of decay is proportional to the size of the eddy. The amplitude of transverse correlation in Corcos' representation, B(cox3/U.), was obtained by plotting the measured transverse correlations at zero time delay as a function of wx3/UO, (see Fig. 31). The measurements show that the amplitude of narrow band transverse correlation for (x3/ *) = 0o 191 is nearly the same as in plane boundary layer. But for (x3/6*) >; 0 722, the transverse correlations fall off very much more rapidly than they do in a plane boundary layer as cx3/Uo increases. This means that one effect of the transverse curvature of the wall is to cause an increase in the decay of eddies in transverse direction. This has been discussed in Section V. The amplitude of narrow-band space-time correlation measured along a line at an angle of 450 to the stream direction are plotted against cWxl/Uc in Fig. 32. The data are compared with computations from Eq. (43) C -

56 using the experimentally determined values of A(wxl/Uc) and B(wx3/Uo) of Fig. 30 and 31. That is, IR (x1 x "3 T;W.: '= IRpp(X 0, T;o) I xl Rpp(0 X3, T, ) I pp 1P 3'.1 pp\ " P pp' 3 ' (48) Wx1 WX3 =A B This relation appears to give a good approximation to the measured values. provided the value of B(wx3/U ) appropriate to the large or small spatial separation x3/6* is used. * This is apparently an effect caused by transverse curvature since a single function B(wx3/U ) suffices for a plane boundary layer. Actually, the formulation of Eq. (40) is only an approximation even for the plane boundary layer and does break down in that case also; see Bull (1963) who showed that for very low frequencies A(cxl/Uc) and B(wx3/Uo) were not unique. *Clearly a failure of the formulation of Eq. (40).

REFERENCES 1. Alexander, L. G., Baron, T, and Comings, E. W., (1953), "Transport of Momentum, Mass and Heat in Turbulent Jets," University of Illinois Engineering Experiment Station, Bulletin,Series No. 413. 2. Bull, M. K., (1961), "Space-Time Correlations of the Boundary Layer Pressure Field in Narrow Frequency Bands," University of Southampton Report A. A. S. U. No. 200. 3. Bull, M. K., Wilby, J. F. and Blackman, D. R., (1963), "Wall Pressure Fluctuations in Boundary Layer Flow and Response of Simple Structures to Random Pressure Fields, " University of Southampton Report A. A. S. U. No. 243; also Jourhal of Fluid Mechanics,Vol. 28, 1967, p.. 719. 4. Clauser, F. H., (1956), "The Turbulent Boundary Layer, " Advances in Applied Mechanics, Vol. 4, Academic Press, p. 1. 5. Coles, D., (1954), "The Problem of the Turbulent Boundary Layer," ZAMP, Vol. 5, p. 181. 6. Coles, D., (1955), "The Law of the Wall in Turbulent Shear Flow," 50 Jahre Grenzshichtforschung, (ed. by Gortler and Tollmien), F. Vieweg und Sohn, Braunschweig, p. 153. 7. Corcos, G. M., (1962), "Pressure Fluctuations in Shear Flows," University of California, Institute of Engineering Research, Report Series No. 183, Issue No. 2; also Journal of Fluid Mechanics, Vol. 16, 1964, p. 353. 8. Corcos, G. M., (1963), "Resolution of Pressure in Turbulence," Acoustical Society of America Journal, Vol. 35, p. 192. 9. Favre, A. J., Gaviglio, J. J. and Dumas, R., (1952), "Some Measurements of Time and Space Correlation in Wind Tunnel, " 8th International Congress on Theoretical and Applied Mechanics, Istanbul; also NACA Tech. Memo. 1370 (1955). 10. Glauert, M. B. and Lighthill, M. J., (1955), "The Axisymmetric Boundary Layer on a Long Thin Cylinder, " Royal Society of London, Proceedings, Series A, Vol. 230, p. 188. 57

58 11. McMillian, F. A., (1954), "Viscous Effects on Flattened Pitot Tubes at Low Speeds," Journal of the Royal Aeronautical Society, Vol. 58, p. 837. 12. Preston, J. H., (1954), "The Determination of Turbulent Skin Friction by Means of Pitot Tubes, "Journal of the Royal Aeronautical Society, Volo 58, p. 109. 13. Rao, G. N. V., (1967), "The Law of the Wall in a Thick Axisymmetric Trubulent Boundary Layer, Journal of Applied Mechanics, Transaction ofASME, Vol. 34, p. 237. 14. Reid, R. O. and Wilson, W., (1963), "Boundary Flow Along a Circular Cylinder, " Journal of Hydraulic Division, Proceeding of ASCE, Vol. 89, p. 21. 15. Richmond, R. L., (1957), "Experimental Investigation of Thick Axially Symmetric Boundary Layer on Cylinders at Subsonic and Hypersonic Speeds, " Guggenheim Aeronautical Laboratory, California Institute of Technology, Hypersonic Research Project Memo, No. 39o 16. Schlichting, H, (1968), Boundary Layer Theory, 6th ed., McGrawHill Book Company, Inc., p. 629. 17. Schloemer, H. H., (1966), "Effects of Pressure Gradients on Turbulent Boundary-Layer Wall-Pressure Fluctuations," U. S. Navy Underwater Sound Laboratory Report, No 747; also, Acoustical Society of America, Journal, Vol. 42, 1967, po 93. 18. Smith, D. W. and Walker, J. H., (1958), "Skin Friction Measurements in Incompressible Flow, " NACA Tech. Notes - 4231. 19. Tu, B. J. and Willmarth, W. W., (1966), "An Experimental Study of the Structure of Turbulence Near the Wall Through Correlation Measurements in a Thick Turbulent Boundary Layer," University of Michigan, Technical Report, 02920-3-T. 20. Willmarth, W.W., (1958a), "Space-Time Correlation of the Fluctuating Wall Pressure in a Turbulent Boundary Layer, " Journal of Aeronautical Sciences, Vol. 25, p. 335.

59 21. Willmarth, W.W., (1958b), "Small Barium Titanate Transducer for Aerodynamic or Acoustic Pressure Measurements," The Review of Scientific Instruments, Vol. 29, p. 218. 22. Willmarth, W. W. and Wooldridge, C. E., (1962), "Measurements of the Fluctuating Pressure at the Wall Beneath a Thick Turbulent Boundary Layer," Journal of Fluid Mechanics, Vol. 14, p. 187. 23. Willmarth, W. W. and Roos, F. W., (1965), "Resolution and Structure of the Wall Pressure Field Beneath a Turbulent Boundary Layer, " Journal of Fluid Mechanics, Vol. 22, p. 31. 24. Wills, J. A. B., (1964), "On Convection Velocities in Turbulent Shear Flows, " Journal of Fluid Mechanics, Vol. 20, p. 417. 25. Yasuhara, M., (1959), "Experiments.on Axisymmetric Boundary Layers Along a Long Cylinder in Incompressible Flow," Transactions of the Japan Society of Aerospace Sciences, Vol. 2, p. 72. 26. Yu, Y. S., (1958), "Effect of Transverse Curvature on TurbulentBoundary-Layer Characteristics, "Journal of Ship Research, Vol. 2, No. 3, p. 33.

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ T2 F- F2 r, I A\\ -- 3 D Tubing 2' D Tubing o 2 4 6ft Figure 1. Schematic diagram of 2 in. diameter steel tubing installed in the wind tunnel as the backbone of the cylindrical model.

II I I II ~~~~~~~~~~~~~II A-A Section Tunnel Floor 0 0.5 I ft I., a Figure 2. The test station and the rear support of the cylindrical model.

1.0 0.8 4 0.6 U0 o 104 ft/sec 0.4- D 138 ft/sec A 195 ft/sec 0.2 0 I I,- L,..... a. I- I I 0 7- 37/r/ 27r Figure 3. The symmetry of the velocity around the cylinder at y = 0. 65 in.

63 Figure 4. Velocity-survey apparatus..0 1 o f c~0 *I I- T L \ \ s \ \ \/+oofT K *o s':.NN...............:....:.............. Qf) Figure 5. Mouth of the pitot tube.

go 02 ----------- ~~ ~C go~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.............. t~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.............................. '-S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.......... C~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...................... -C to~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...................... to~~~~~~~~~~~~~~~~~~~~~~~~~~~..... N '-S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~............. '4 - 7 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~................. -a~ ~~ ~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~................................. o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~................... p N p~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.................................... b to~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.......................... p 0 04~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.................................. 1~~~~~~~~~~~~~~~~~~~................ go 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~....................... o 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.........I...... 5 '-4 ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~............................. o I to~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.............................. Ks~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..... j~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.......... 04~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.................1 CA~~~~~~~~~~~~~~~~~~~~~~.... -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........... '-S S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~....................................... o at-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~................... to -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...................... '"4~ ~ ~ ~ ~ ~ ~~~~~~~~~~~1......... p 'S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.................................... 5 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~......... 04 t -.~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~............. C4 -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~::........ '-4 -, ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.......... 0 04~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..... 1...... Ct> 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~............................ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a 4 ~~~ ~~~ ~~~ ~~~ ~~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~............... to~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~................... 0 ~ ~ ~~~~ ~~~~~~~~~~~~~~........................... p~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~......... 04~~~~~ ~~~~~~~~~~~~~~......................... to~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..... 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~............... $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........................... ' —a~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.........

a, Single Transducer Plug;:3 Wax. 3 Phenolic / E^~ Brass ~, —~i0 02 inch jx,. p ZT-5 Lead \\\\ - ///.I-i,,l~a, IIi I b, Triple Transducer Plug Figure 8. Pressure transducers assembly.

Microdot low noise. coaxial cable Pressure Transducer -I —.To preamplifier 0.47 /iF F igur e 9. Circuit diagram for pressure transducers and cathode follower.

0.12r 0.10 Cp 0.06 -0 — _ U= 99 ft/sec U =199ft/sec Pressure Location 4 Transducers 0.02 so wow ---O 400000 MP 8 0 1 -- a % sop %*ft am awww -0.02L 0 I I I I I I I 4 8 12 16 20 24 28 32ft X Figure 10. Pressure gradient along the cylinder.

30 25 A U U A 00 I - Coles' Ideal Turbulent Boundary Layer 0 0 UL=200 ft/sec e U= 145 ftAec 4 U= 100 ft/sec 10 100 1000 U - (I +/2a) v 10000 Figure 11. Mean velocity profiles in the axially symmetric turbulent boundary layer (Law of the Wall).

10-2 Cf 10-3 3x104 I I I I I I I I I I I I I I Q- - a0 w --- Blasius Flat Plate _ - -- Coles Flat Plate I I I I I I I I. a e. 0 CD mm 8/a * Present 0 J3 Richmond Yu 0.15- 0.25 0.14-0.21 0.20-0.25 A Yosuhara 0.154 I I I I I I I A I I I I I I I I I I I I a ~ ~ A I I - ---.. - a E 2 m m m 0 m m - 3x102 104 105 Figure 12. Skin friction coefficient.

I 70 1.0 _Rpp 0.8 0.6, 00 0 0 0.4** X~/8 ri.Lf,4 0 * 0.31 0.53 2 - ZO 0.01 0. 3 0.31 0.53 * 0L4 -.0 0 0 0 I 0- 1.14 1 74 - u ~ ' " 0 a. - --- l ---- 0 0 a r o 0.51 0.79 1.45 2 27 i ----..........,... _ I ' - i --- —-- 0 -_. I 0 a I I L --—.1 p I I I I* I I p *. -1 * -— o. 'o ~* S U* '* * o 0 0.o * * * *0. 10 4.8 6.74 0I iji ~a0 0 0 O00 0 0 0 0 0 0.~ ~ 0 00 00000 0000' 6.93 9.41 'U ~ 0.0 9.33 12.1 ' 0 * 0_ L.0 o 0 O 0 ~ ~ ~ ~ i., o a - d [ lo 0 0 0 - Figure 13. Longitudinal space-time correlation of wall pressure.

71 16 12 8 4 0 8 a — *.. o 0 WillmarthaWooldridge(1962) --- 0 Bull (1963) 0 4 8 X/8* 12 Figure 14. Time delay for Rp maximum at constant xl. PP

1.0 0.8 Uc U 0.6 0.4 0.2 0 0 _ 2 0 Willmarth a Wooldridge (1962) ~ --- 0 Bull (1962) -3 w0 0 1 2 3 4 5 6 7 8 x/s" 9 10 Figure 15. Local convection speed of the pressure-producing turbulent eddies.

1.0 0.8 * o. 9_ 0.6 0 X 0 0.2 0 N\ Nb. -- 0 Bull(1963) R=25400... ---0 Willmarth a Wooldridge _ (1962) R= 38000 _0 _ 4 f ~~~-__~~~s -3 CAD 0 I 2 3 5 6 7 8 9 10 XI/s Figure 16. Decay of wall pressure correlation in a reference frame moving at local convection speed.

1.0 0.8 0 a. 0 X Q. ir d a A 0.6 0.4 8/a M, R *-X- 2 0.14 26200 m -- 0 0.2 38000 -. ---- 0 0.3 1 000 * — 0 0.3 19500 --- 0 0.5 11000 a ---- 0 0.5 24300 ---- 0 0.5 33800 Willmarth a Wooldridge (1962) Bu11(1963) -. $P 0.2 0 -0.2 0 0 I 2 3 4 5 X, /8* 6 7 8 9 10 F igur e 1 7. Longitudinal wall pressure correlation in a broad frequency band.

0.o 0.8 II' ~- 0.6 - j x Of 0' 0.4 - 0.2 -0.2 10~ 8/a A. M, Re 0.14 26200 4m. 0 O 0.2 38000 Willmarth a Wooldridge Jnnnn, (1962) 4 -- 0 0.3 -. -- - 0 0.3 * IV 19500 11000 I \ - -e - -- O 0.5 Bull (1963) A -- O 0.5 24300 J -& — -- O 0.5 33800 A A._. qkf No aof I z_.C1 0,_ w-' A I 2 3 4 5 x f * Figure 18. Transverse wall pressure correlation in a broad frequency band.

76 L Uvl/ 0 I 2 3 4X103 1.0 x 0 X CL 13 D~ 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 L/S~ Figure 19. Spatial correlations of the wall pressure along a line at 45~ to the flow direction.

77 X, U /v 3 4X103 0 I 2 0.7 0.6 0.5 0. I0 X x Xz CO CL 0.4 x3/* -' 0 00 -.-o- 0.191 0. ---- 0.478 0 _-.- 0.722 0 -. — 1.095 (3JUr/V,211.529 >.80.21 0.3 0.2 0.1 0 0 1 2 3 4 5 X,/ Spatial correlation of the wall pressure. Figure 20.

78 2.0- I I.I O-1.0 -2.0 -2.0 -- I -1I.I --- - -2.0 -1.0 0 X,/SI 1.0 2.0 Figure 21. Contours of constant wall pressure correlation.

/ ~ \\ I / — 8ODXI -8 /a=0 8/a = 2 Bull (1963) ---- Willmorth SWooldridge (1962) Figure 22. Effect of transverse curvature on wall pressure correlations.

80 I I I I I I Il I I 1 I I Il 1. I 10 U. ~ (w~u) 10 -lo-3 I 0 0 8/a R/8 Rg*\ * 2 0.07 32100 -— 0 0.10 25400 Bul1(1963) 0 0.06 49400Willmartha Roes (__.i9I5) I31:1I I I I i l l I I I'l lull,,.,,,,,,,,.,,.,....... -I I0 10 8u Co I U 10' Figure 23. Measured wall pressure spectra.

RIe4oads ainssaid 11Mm papa)ao3 'T z 9.inSi~ zol 101 001,,01 I t-01 -01I ad d(m) ~ o01 18

82 I 0.5 0.4 0.3 0.3 0.2 0.I 0 UI(w) 2 — 0 Willmarth & Roos(1965) 20 30 5 10 o10' 87~ U 0 10 8U 20 30 0 F igur e 2 5. Corrected wall pressure spectra.

83 12 10 8 6 4 2 0o 0 0 * * 2 26200 o 0 38000 Willmarth a Wooldridge (1962) S 0 0 0 0 0 00 0 * 0.2 0.4 0.6 U/u0 0.8 1.0 Figure 26. Comparison of the mean velocity profiles.

84 12 I0 8 u0Tc ~ 6 - / x3/! / - 0 (See Fig. 14) 4 o 0.478 a 0.722 2 A 1.095 02 -. I I I-I I 0 2 4 6 8 10 X,/I Figure 27. Time delay for Rp PP maximum at constant xl for varied x3.

1.0 r 0.8 - - - m -mo ~~~ ~d.~ -.m m Uc u. 0.6 wo/Uo 0.79 sm 0.4 0.2 2.02 5.07 I 10.42 I I I I a I I a I m I A a I I I I I R mma I 2 3 4 5 6 7 8 9 10 XI/ Figure 28. Local convection speed eddies in various of pressure-producing turbulent frequency bands.

1.0 0.8 Uc U0 0.6 mm _V -_W 0.4 0.2 * /a * 2 o 0 Willmarth a Wooldridge (1962) 00 0M I I r% L. u 0 2 4 6 8 w8/U. 10 12 14 Figure 29. Asymptotic convection speed as a function of frequency.

1.0 0.8:D x 3 <1 0.6 0.4 --, - -- A - 0 I mmmm m m 8/a 2 2 2 2 0 0, B/U 0.79 2.02 5.06 10.42 Bull (1963) Willmarth a Wooldridge (1962) 00 -. 0.2 0 0 5 10 15 20 cuXI/Uc Figure 30. Amplitude of narrow-band longitudinal of the wall pressure. 25 30 35 space-time correlation

1.0 =0.19 8 rZ) X 3 %f.-0 0.8 0.6 A — 0 — =. —A- - ~ _,-,.V - >0.72 _ -A!It a a 2 2 2 2 0 UD 0.79 2.02 5.06 10.42 Willmarth 8 Wooldridge(1962) 0.4 0.2 0 l 00 00 CO 0% %V***. m_ i- -- m alIft ==-=& I I vIIV C ) I 2 3 4 5 6 7 I WXa/Ue Figure 31. Amplitude of.narrow-band transverse space-time correlation of the wall pressure. 3

~It __o jX o0.19 20.72 ' UQ 0.8 2. 0.79 0.6 V\ a *2.02 X 0.6 A 2 5.06 0.6 A, j, C 0.04 0 E_ E.(48) 0 0. 2,tin_,_ -q. 50 0 x wxI/Uc 2 cceX/Uc 6 7a8 Figure 32 AmPlitude of narro-band spacetime correlation of wall piuresur along a line at 450 to the flow direction (Note X3 =

DISTRIBUTION LIST FOR UNCLASSIFIED TECHNICAL REPORTS ISSUED UNDER CONTRACT NO. NO00014-67-A-0181-0015, TASK NO. 062-234 Technical Library Building 131 Aberdeen Proving Ground, Maryland 21005 Professor W.. W Willmarth Department of Aerospace Engineering University of Michigan Ann Arbor, Michigan 48108 Defense Documentation Center Cameron Station Alexandria, irginia 22314 (20) Professor Finn C. Michelsen Naval Architecture and Marine Engineering 445 West Engineering Bldgo University of Michigan Ann Arbor, Michigafh 48104 Technical Library Naval Ship Research and Development Center Annapolis Division Annapolis, Maryland 21402 Professor Bruce Johnson Engineering Department Naval Academy Annapolis, Maryland 21402 Library Naval Academy Annapolis, Maryland 21402 Professor W. P Graebel Department of Engineering Mechanics The University of Michigan College of Engineering Ann Arbor, Michigan 48104 AUFOSR (REM) 1400 Wilson Boulevard Arlington, Virginia Dr. J. Menkes Institute for Defense 400 Army-Navy Drive Arlington, Virginia 22204 Analyses 22204 Professor S. Corrsin Mechani cs. Department The Johns Hopkins University Baltimore, Maryland 20910 Professor 0 M. Phillips The Johns Hopkins University Baltimore, Maryland 20910 Professor Wo R. Debler Department of Engineering Mechanics University of Mechanics Ann.Arbor, Michigan 48108 Dr. Francis Ogilvie Department of Naval Architecture and Marine Engineering University of Michigan Ann Arbor, Michigan 48108 Professor S. D, Sharma Department of Naval Architecture and Marine Engineering University of Michigan Ann Arbor, Michigan 48108 Professor Lo So Ge Kovasznay The Johns Hopkins University Baltimore, Maryland 20910 Librarian Department- of Naval Architecture University of California Berkeley, California 94720 Professor P. Lieber Department of Mechanical Engineering University of California Institute of Engineering Research Berkeley, California 94720 Professor M. Holt Division of Aeronautical Sciences University of California Berkeley, California 94720

Professor Jo Vo Wehausen Department of Naval Architecture University of California Berkeley, California 94720 Professor G. Birkhoff Department of Mathematics Harvard University Cambridge, Massachusetts 02138 Professo r J R. Paulling Department of Naval Architecture University of California Berkeley, California 94720 Commanding Officer NROTC Naval Administrative Massachusetts Institute of Cambridge, Massachusetts Unit Technology 02139 Professor E. V. Laitone Department of Mechanical Engineering University of California Berkeley, California 94720 School of Applied Mathematics Indiana University Bloomington, Indiana 47401 Commander Boston Naval Shipyard Boston, Massachusetts 02129 Director Office of Naval Research Branch Office 495 Summer Street Boston, Massachusetts 02210 Professor M, S. Uberoi Department of Aeronautical Engineering University of Colorado Boulder, Colorado 80303 Naval Applied Science Laboratory Technical Library Bldg. 1 Code 222 Flushing and Washington Avenues Brooklyn, New York 11251 Professor Jo J. Foody Chairman, Engineering Department State University of New York Maritime College Bronx, New York 10465 Dr. Irving C. Statler, Head Applied Mechanics Department Cornell Aeronautical Laboratory, Inc. P. 0. Box 235 Buffalo, New York 14221 Dro Alfred Ritter Assistant Head, Applied Mechanics Dept. Cornell Aeronautical Laboratory, Inc. Buffalo, New York 14221 Professor N. Newman Department of Naval Architecture and Marine Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Professor A. H. Shapiro Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Professor C. C. Lin Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Professor E. W. Merrill Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Professor M. A. Abkowitz Department of Naval Architecture and Marino Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Professor G. H, Carrier Department of Engineering and Applied Physics Harvard University Cambridge, Massachusetts 02139 Professor E. Mollo-Christensen Room 54-1722 Massachusetts Institu+te of Technology Cambridge, Massachusetts 02139 Professor A... ppen Department of Civil Engineering Massachusetts In-stitulte of Technology Cambridge, Massachusetts 02139

C ommander Charleston Naval Shipyard Uo So Naval Base Charleston, South Carolina 29408 A Ro Kuhlthau, Director Research Laboratories for the Engineering Sciences Thorton Hall, University of Virginia Charlottesville, Virginia 22903 Director Office of Naval Research Branch Office 219 Dearborn Street Chicago, Illinois 60604 Library Naval Weapons Center China Lake,, California 93557 Libra.y MS 60-3 NASA Lewis Research Center 21.000 Brookpark Road Cleveland, Ohio 44135 Professor Jo M. Burgers Institute of Fluid Dynamics and Applied Mathematics University of Maryland College Park, Maryland 20742 Acquisition Director NASA Scientific & Technical Informat-ion Po O o Box 33 College Park, Maryland 207h0 Professor Pai. Institute for Fluid Dynamics and Applied Mathematics University of Maryland College Park, Maryland 207h0 Technical Library Naval Weapons Laboratory Dahlgren, Vi:rginia 22448 Computation & Analyses Laboratory Naval Wea pons Laboratory Dahlgren, Virginia 22U48 Professor C, S. Wells LTV Research Center Dallas, Texas 75222 Dr. Ro H. Kraichnan Dublin, New Hampshire Commanding Officer Army Research Office Box CM, Duke Station Durham, North Carolina o3444 27706 Professor A, Charnes The Technological Institute Northwestern University Evanston, Illinois 60201 Dr. Martim H. Bloom Polytechnic Institute of Brooklyn Graduate Center, Dept. of Aerospace Engineering and Applied Mechanics Farmingdale, New York 11735 Technical Documents Center Building 315 U. S. Army Mobility Equipment Research and Development Center Fort Belvoir, Virginia 22060 Professor J. E. Cermak College of Engineering Colorado State University Ft. Collins, Colorado 80521 Technical Library Webb Institute of Naval Glen Cove, Long Island, Professor E. V. Lewis Webb Institute of Naval Glen Cove, Long Island, Architecture New York 11542 Architecture New York 11542 Library MS.85 NASA, Langley Research Center Langley Station Hampton, Virginia 23365 Dr. B. N, Pridnmcre Brown Northrop Corporation NORAIR-Div Hawthorne, California 90250

Dr. J. Po Breslin Stevens Institute of Techno] Davidson Laboratory Hoboken, New Jersey 07030 Mr. D. Savitsky Stevens Institute of Techno] Davidson Laboratory Hoboken, New Jersey 07030 Logy Mr. Alfonso Al.edan L. Director Laboratorio Nacional De Hydraulics Antigui Cameno A. Ancon Casilla Jostal 682 Lima, Pera Logy Clommander Long Beach Naval S-hipyard Long Beach, California 90802 Mr. C. H., Henry Stevens Institute of Technology Davidson Laboratory Hoboken, New Jersey 07030 Professor John Laufer Department of Aerospace Engineering University Park Los Angeles, Califonlia 90007 Professor J. Fo Kennedy, Director Iowa Institute of Hydraulic Research State University of Iowa Iowa City, Iowa 52240 Professor L. Landweber Iowa Institute of Hydraulic Research State University of Iowa Iowa City, Iowa 52240 Professor Eo L. Resler Graduate School of Aeronautical Engineering Cornell University Ithaca, New York 14851 Professor John Miles % IoG.P.P. University of California, San Diego La Jolla, California 92038 Director Scripps Institution of Oceanography University of California La Jolla, California 92037 Dr. B. Sternlicht Mechanical Technology Incorporated 968 Albany-Shaker Road Latham, New York 12110 Mr. P. Eisenberg, President Hydronautics Pindell School Road Howard County Laurel, Maryland 20810 (2) Professor A. Ellis University of California, San Diego Department of Aerospace & Mech. Engrg. Sci. La Jolla, California 92037 Professor J. Ripkin St. Anthony Falls Hydraulic Lab. University of Minnesota Minneapolis, Minrmesota 5544 Professor J. M. Killen St. Anthony Falls Hydraulic Lab. University of Mirmesota Minneapolis, Minnesota 55414 Lorenz G. Straub Library St. Anthony Falls Hydraulic Lab. Mississippi River at 3rd Avenue SE. Minneapolis, Minnesota 55414 Dr. E. Silberman St. Anthony Falls Hydraulic Lab. University of Mimnesota Minneapolis, Minresota 554l4. Superintendent Naval Postgraduate School Library Code 0212 Monterey', California 93940 Professor A. B. Metzner University of Delaware Newark, New Jersey 19711 Technical Libtbrary USN Underwrater Weapons & Research & Engineering Station Newport, R ihode Island 02840 Technical lI-brary Underwater Sound Laboratory Fort Truibuill New London, Cornnecticut 06321

Professor J. J. Stoker Institute of Mathematical Sciences New York University 251 Mercer Street New York, New York 10003 Library Jet Propulsion Laboratory California Institute of Technology 4800 Oak Grave Avenue Pasadena, California 91109 Engineering Societies Library 345 East 47th Street New York, New York 10017 Office of Naval Research New York Area Office 207 W. 24th Street New York, New York 10011 Commanding Officer Office of Naval Research Branch Office Box 39 FPO New York, New York 09510 Professor M. S. Plesset Engineering Division California Institute of Technology Pasadena, California 91109 Professor H. Liepmann Department of Aeronautics California Institute of Technology Pasadena, California 91109 Technical Library Naval Undersea Warfare Center 3202 E. Foothill Boulevard Pasadena, California 91107 (25) Professor H. Elrod Department of Mechanical Engineering Columbia University New York, New York 10027 Dr. J. W. Hoyt Naval Undersea Warfare Center 3202 E. Foothill Boulevard Pasadena, California 91107 Society of Naval Architects and Marine Engineering 74 Trinity Place New York, New York 10006 Professor S. A. Piascek Department of Engineering Mechanics University of Notre Dame Notre Dame, Indiana 46556 United States Atomic Energy Commission Division of Technical Information Extension P. 0. Box 62 Oak Ridge, Tennessee 37830 Miss 0. M. Leach, Librarian National Research Council Aeronautical Library Montreal Road Ottawa 7, Canada Technical Library Naval Ship Research and Development Center Panaman City, Florida 32401 Professor T, Y. Wu Department of Engineering California Institute of Technology Pasadena, California 91109 Director Office of Naval Research Branch Office 1030 E. Green Street Pasadena, California 91101 Professor A. Acosta Department of Mechanical Engineering California Institute of Technology Pasadena, California 91109 Naval Ship Engineering Center Philadelphia Division Technical Library Philadelphia, Pennsylvania 19112 Technical Library (Code 249B) Philadelphia Naval Shipyard Philadelphia, Pennsylvania 19112

Professor R. C. Mac Camy Department of Mathematics Carnegie Institute of Technology Pittsburgh, Pennsylvania 15213 Library & Information Services General Dynamics-Convair P. 0. Box 1128 San Diego, California 92112 Dr. Paul Kaplan Oceanics, Inc. Plainview, Long Island, New York Commander (Code 246P) Pearl Harbor Naval Shipyard Box 400 FPO San Francisco, California 11803 96610 Technical Library Naval Missile Center Point Mugu, California 93441 Technical Library Naval Civil Engineering Lab. Port Hueneme, California 93041 Commander Portsmouth Naval Shipyard Portsmouth, New Hampshire 03801 Commander Norfolk Naval Shipyard Portsmouth, Virginia 23709 Professor F. E. Bisshopp Division of Engineering Brown University Providence, Rhode Island 02912 Dr. L. L. Higgins TRW Space Technology Labs, Inc. One Space Park Redondo Beach, California 90278 Ridstone Scientific Information Center Attn: Chief, Document Section Army Missile Command Redstone Arsenal, Alabama 35809 Dr. H. N. Abramson Southwest Research Institute 8500 Culebra Road San Antonio, Texas 78228 Editor Applied Mechanics Review Southwest Research Institute 8500 Culebra Road San Antonio, Texas 78206 Technical Library ( Code H245C-3) Hunters Point Division San Francisco Bay Naval Shipyard San Francisco, California 94135 Office of Naval Research San Francisco Area Office 1076 Mission Street San Francisco, California 94103 Dr. A. May Naval Ordnance Laboratory White Oak Silver Spring, Maryland 20910 Fenton Kennedy Document Library The Johns Hopkins University Applied Physics Laboratory 8621 Georgia Avenue Silver Spring, Maryland 20910 Librarian Naval Ordnance Laboratory White Oak Silver Spring, Maryland 20910 Dr. Bryne Perry Department of Civil Engineering Stanford University Stanford, California 94305 Professor Milton Van Dyke Department of Aeronautical Engineering Stanford University Stanford, California 94305 Professor E. Y. Hsu Department of Civil Engineering Stanford. University Stanford, California 94305 Librarian Naval Command Control Commumications Laboratory Center San Diego, California 92152 Dr. R. L. Street Department of Civil Engine ring Stanford University Stanford, California 94305

Professor So Eskinazi Department of Mechanical Engineering Syracuse University Syracuse, New York 13210 Professor Ro Pfeffer Florida State University Geophysical Fluid Dynamics Institute Tallahassee, Florida 32306 Professor Jo Foa Department of Aeronautical Engineering Rennsselaer Polytechnic Institute Troy, New York 12180 Professor Ro C. Di Prima Department of Matheaatics Rennsselaer Polytechnic Institute Troy, New York 12180 Dr. M. Sevik Ordnance Research Laboratory Pennsylvania State University University Park, Pennsylvania ] Professor J. Lumley Ordnance Research Laboratory Pennsylvania State University University Park, Pennsylvania ] Dr. J. M. Robertson Department of Theoretical and Applied Mechanics University of Illinois Urbana, Illinois 61803 Shipyard Technical Library Code 130L7 Building 746 San Francisco Bay Naval Shipyard Vallejo, California 94592 Code L42 Naval Ship Research and Development Center Washington, D. C 20007 Code 800 Naval Ship Research and Development Center Washington, D.C. 20007 Code 2027 U. S. Naval Research Laboratory Washington, D.C. 20390 L6801 L6801 Code 438 Chief of Naval Research Department of the Navy Washington, D.C. 20360 Code 513 Naval Ship Research and Development Center Washington, D. C. 20007 Science & Technology Division Library of Congress Washington, D.C. 20540 ORD 913 (Library) Naval Ordnance Systems Command Washington, D.C. 20360 Code 6420 Naval Ship Engineering Center Concept Dpe$ign Division Washington, D.C. 20360 Code 500 Naval Ship Research and Development Center Washington, D.C. 20007 Code 901 Naval Ship Research and Development Center Washington, D.C. 20007 Code 520 Naval Ship Research and Development Center Washington, D.C. 20007 Code 0341 Naval Ship Systems Command Department of the Navy Washington, D.C. 20360 Code 2052 (Technical Library) Naval Ship Systems Command Department of the Navy Washington, D.C. 20360 Mr. J. L. Schuler (Code 03412) Naval Ship Systems Command Department of the Navy Washington, D.C. 20360 (3) (6)

Dr. J. H. Huth (Code 031) Naval Ship Systems Command Departmnnt of the Navy Washington, D.C. 20360 Code 461 Chief of Naval Research Department of the Navy Washington, DCo 20360 Code 530 Naval Ship Research and Development Center Washington, D.C. 20360 Code 466 Chief of Naval Research Department of the Navy Washington,.D.C. 20360 Office of Research and Development Maritime Administration 441 G. Street, NW. Washington, D.C. 20235 Code 463 Chief of Naval Research Department of the Navy Washington, D.Co 20360 National Science Foundation Engineering Division 1800 G. Street, NW.. Washington, D.C. 20550 Dr. G.. Kulin National Bureau of Standards Washington, D.C. 20234 Department of the Army Coastal Engineering Re.search Center 5201 Little Falls Road, NW. Washington, D.C. 20011 Code 521 Naval Ship Research and Development Center Washington, D.C. 20007 Code 481 Chief of Naval Research Department of the Navy Washington, D.C. 20390 Code 421 Chief of Naval Research Department of the Navy Washington, D.C. 20360 Commander Naval Ordnance Systems Command Code ORD 035 Washington, D.C. 20360 Librarian Station 5-2 Coast Guard Headquarters 1300 E. Street, NW. Washington, D.C. 20226 Division of Ship Design Maritime Administration 441 G. Street, NW. Washington, D.C. 20235 HQ USAF (AFRSTD) Room.D 377 The Pentagon Washington, D.C. 20330 Commander Naval Ship Systems Command Code 6644C Washington, D.C. 20360 Code. 525 Naval Ship Research and Development Center Washington, D.C. 20007 Dr. A. Powell (Code 01) Naval Ship Research and Development Center Washingtnn, D.C. 20007 Director of Research Code. RR National Aeronautics &. Space Admin. 600 Independence Avenue, SW. Washington, D.C. 20546 Commander Naval Ordnance Systems Command Code 03 Washingtnn, D.C. 20360 Code ORD 05411 Naval Ordnance Systems Command Washington, D.C. 20360

AIR 5301 Naval Air Systems Command Department of the Navy Washington, D.C. 20360 AIR 604 Naval Air Systems Command Department of the Navy Washington, D.C. 20360 Dr. John Craven (PM 1100) Deep Submergence Systems Project Department of the Navy Washington, D.C. 20360 Mr. Ralph Lacey (Code 6114) Naval Ship Engineering Center Department of the Navy Washington, D.C. 20360 Dr. A. S. Iberall, President General Technical Services, Inc. 451 Penn Street Yeadon, Pennsylvania 19050 Dr. H. Cohen IBM Research Center P. 0. Box 218 Yorktown Heights, New York 10598 Code 522 Naval Ship Research and Development Center Washington, DoC. 20007 Commander Naval Oceanographic Office Washington, D.C. 20390 Chief of Research & Development Office of Chief of Staff Department of the Army The Pentagon, Washington, D.C. 20310 Code 6342A Naval Ship Systems Commmand Department of the Navy Washington, DoC. 20360 Code 468 Chief of Naval Research Department of the Navy Washington, D.C. 20360 Director U. S. Naval Research Laboratory Code 6170 Washington, D.C. 20390 Code 473 Chief of Naval Research Department of the Navy Washington, D.C. 20360 Code 6100 Naval Ship Engineering Center Department of the Navy Washington, D.C. 20360

Unclassified DOCUMENT CONTROL DATA R & D (er.lrity rlarlltfrallon hf tfitl, rbody f athnfrl t d nde fln annonltlon mnt be entered when fho ovrall report Is cise ed 1I. 9R~INAT(NQ ACTIVITY (CO9portel 2ast'r.) ae. REPORT SECURITY CLASSIFICATION The University of Michigan Unclassified Department of Aerospace Engineering 2b. GROUP Ann Arbor, Michigan Not applicable 3. rrPORT tiTLE Wall Pressure Fluctuations Beneath an Axially Symmetric Turbulent Boundary Layer on a Cylinder 4. DEscRIPT IE' NO.T'S (Typ of report and Incloluve da:tce) Technical Report - - -- -- - - i s. AU T "n S~) (ritrat name, middle Iflttsl, talt name) Chi-Sheng Yang and William W. Willmarth 0. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS August 1969 89 26 sa. CONTRAC t OR GRANT NO. oa. ORIGINATOR'S REPORT NUMBER(S) N 00014-67-A-0181-0015 02 -1 -b. PROJECT NO. 02149 —T c. 9b. OTHER REPORT NO(S) (Any other numbers that may be aassegned this report) d. 10. OISTRCIUTION STATEMENT This document has been approved for public release and sale; its distribution is unlimited. It. SUPPLEMEN4TARY NOTES 12. SPONSORING MILITARY ACTIVITY Office of Naval Research I I 13. ABSTRACT 1 1 Measurements of the turbulent pressure field on the outer surface of a 3 inch diameter cylinder were made at a point 24 feet downstream of the origin of the turbulent boundary layer. The root-mean square wall pressure was 2.42 times the wall shear stress. The normalized power spectrum at high frequencies (Ub*/Uo > 10) contained twice the energy density of the spectrum beneath a plane boundary layer. The convection speed was the same as that in a plane boundary layer but the eddy size was smaller by a factor of two. The smaller eddy size and unchanged convection speed account for the greater energy in the spectrum at high frequencies. DD I'. OV.1473 Unclassified Security Classification

Turbulent Pressure Turbulent boundary layer Wall pressure fluctuations Unclassified Security CluiMifictilon

UNIVERSITY OF MICHIGAN illl1111111111111 3 9015 03527 4821