THE UN I V E R S I T Y OF MI C HI GAN COLLEGE OF ENGINEERING Department of Aerospace Engineering Interim Technical Report AN EXPERIMENTAL INVESTIGATION OF THE UNSTEADY FLOWS ABOUT SPHERES AND DISKS Frederick W. Roos William W. Willmarth ORA Project 01954 DEPARTMENT OF THE ARMY U. S. ARMY RESEARCH OFFICE —DURHAM GRANT NO. DA-ARO-D-31-124-G711, PROJECT NO. 5590-E DURHAM, NORTH CAROLINA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR December 1968 Distribution of this document is unlimited.

This report was also a dissertation submitted by the first author in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1968.

TABLE OF CONTENTS Page LIST OF FIGURES v NOME NCLATURE viii I. INTRODUCTION 1 Ao Interest in Sphere and Disk Aerodynamics 3 B. The Research Program 7 II. EXISTING KNOWLEDGE OF SPHERE AND DISK AERODYNAMICS 8 A. Introduction 8 Bo Aerodynamics of a Sphere 9 1. Flow Regimes 9 2. Steady Drag 16 3. Unsteady Behavior During Quasi-Steady Motion 25 4. Accelerated Motion 31 C. Aerodynamics of a Disk 35 o1 Flow Regimes 36 2. Steady Drag 40 3. Unsteady Behavior During Quasi-Steady Motion 43 IIIo EXPERIMENTAL APPARATUS 47 Ao Introduction 47 B. Design Considerations 47 C. Mechanical Equipment 53 1o Towing Channel 53 2. Air Bearings 55 3. Carriage and Strut 59 4. Motor and Towing Cable 61 D. Models and Strain Gage Transducers 64 1o Sphere and Disk 64 2. Disk Balance 65 3. Sphere Balance 71 40 Accelerometer 75 E. Electrical Equipment 77 o. Amplifier Circuits 77 2. External Units 81 F. Flow Visualization Equipment 82 1iii1

Page IV. EXPERIMENTAL RESULTS 87 A. Experimental Technique 87 1. Fluid Measurements 87 2. Balance Calibration and Circuit Adjustments 88 3. Operation of the Experiment 89 4. Data Reduction 91 B. Results of Sphere Experiments 91 1o Sphere Drag 91 2. Transverse Force on a Sphere 99 C. Results of Disk Experiments 109 1. Disk Drag 109 2o Moment on a Disk 111 Vo CONCLUSIONS 117 REFERENCES 120 iv

LIST OF FIGURES Page Figure 1. Sphere streamlines for Red - 0. 129 Figure 2. Sphere streamlines for Red = 5. 129 Figure 3. Sphere streamlines for Red 100. 129 Figure 4. Stages in loop shedding process for a sphere. Wake is shown in cross-section; dotted line indicates position of vortex axis. 130 Figure 5. Alternate loop shedding by a sphere at Red 350. 131 Figure 6. Effect of boundary layer transition on flow separation from a sphere. 131 Figure 7. Sphere C - Red relationship according to existing fata. 132 Figure 8. Disk CD - Red relationship according to existing data. 133 Figure 9. Towing channel installation. 134 Figure 10. Photograph of towing channel facility. 135 Figure 11. Photograph of carriage assembly with disk mounted on balance. 136 Figure 12. Cross-section of V-groove air bearing. 137 Figure 13. Photograph of sphere balance (before application of protective coating to drag transducer). 138 Figure 14. Photograph of disk balance mounted on sting and strut. 139 Figure 15. Drawing of disk balance. 140 vy

Page Figure 16, Diagram of circuit used with disk balance. 141 Figure 17. Drawing of sphere balance. 142 Figure 18. Drawing of accelerometer. 143 Figure 19. Flow visualization equipment. 143 Figure 20. Typical sphere balance calibration data. 144 Figure 21. Sphere drag coefficient data from this experiment. 145 Figure 22, Typical sphere drag record (Red = 532). 146 Figure 23. Data record of sphere drag during acceleration from rest, 147 Figure 24. Sign conventions for forces on a sphere. 148 Figure 25. Transverse force coefficient for a sphere. 149 Figure 26. Frequency of oscillation of transverse force on sphere. 150 Figure 27T Data record of sphere lift and side force showing low-frequency oscillation at Red = 399. 1 51 Figure 28. Sphere data record showing transition of fluctuation from list to side force (Red = 433). 152 Figure 29. Photographs of steady flow about a sphere. 153 Figure 30 Vortex loop wake behind a sphere at Red = 295. 154 Figure 31. Sphere shedding vortex loops at Red = 377. 155 Figure 32. Data record and flow photographs of sphere showing regular side force oscillations at Red= 3329 156 vi

Page Figure 33. Sphere data record showing irregular fluctuations in list and side force at Red = 715. 157 Figure 34. Model relating unsteady transverse force to sphere wake fluctuations. 157 Figure 35. Wake behind a side-supported sphere at Red = 304. 158 Figure 36. Disk drag coefficient data from this experiment. 159 Figure 37. Typical disk drag data record (Red = 530). 160 Figure 38. Sign conventions for drag and moments on a disk. 161 Figure 39. Moment coefficient for a disk. 162 Figure 40. Frequency of oscillation of disk moment. 163 Figure 41. Photograph of steady flow about a sting-mounted disk at Red 157. 164 Figure 42. Data record and wake photograph for disk showing regular oscillations in yaw at Red = 318. 165 Figure 43. Data record and flow photograph for disk showing combined pitch and yaw oscillation at Red = 278. 166 Figure 44. Data record and flow photograph for disk showing initial vortex shedding at Red = 305. 167 Figure 45. Model relating unsteady disk moment to vortex shedding. 168 vii

NOMENCLATURE a maximum value of unsteady acceleration, ft/sec2 max a acceleration, ft/sec2 A acceleration modulus, A = ad/U2 CD drag force coefficient, CD = D/(7/8)pU d CFpp peak-to-peak coefficient of total transverse force, CFpp Fppp/(lT/8)pU2d2 CL lift force coefficient, CL = L/(7r/8)pU2d2 CL mean value of CL C 2 root-mean-square value of CL CMIpp peak-to-peak coefficient of total moment, CMpp= Mpp/(7r/8)pU2d3 C M coefficient of disk moment about fixed axis in, wind M tunnel tests, CI = M'/(P-/8)pU2d3 \ C' t2 root-mean-square value of C? d sphere or disk diameter, ft D drag force, lb f frequency of force or moment fluctuation, Hz f* dimensionless frequency, f* = fd/U F total transverse force, lb FPP peak-to-peak magnitude of unsteady total transverse force, lb I* dimensionless moment of inertia of disk I* = npd-/64pd viii

k characteristic dimension of arbitrary flow, ft L lift force, lb m mass of sphere or disk, sl m apparent mass, sl M total moment on disk, ft-lb M' moment on disk about fixed axis in wind tunnel tests, ft-lb n oscillation frequency of freely falling disk, Hz P pitching moment, ft-lb R force resisting acceleration, lb Re Reynolds number for arbitrary flow, Re = Utlv Red Reynolds number based on sphere or disk diameter, Red =Ud/v Redmin minimum Reynolds number for wake fluctuations S side force, lb t time, sec T fluid temperature, 0C U steady relative velocity between fluid and sphere or disk, ft/sec x distance along track, ft Y yawing moment, ft-lb F unsteady circulation about sphere A wavelength of periodic sphere or disk motion, ft ix

tL -.fluid viscosity, sl./ft-sec kinematic viscosity of fluid, ft2/sec p fluid density, sl/ft3 uniform density of freely falling disk, sl/ft3 disk thickness, ft

ABSTRACT Several recent aerodynamic problems, e.g., the use of freely-rising balloons to measure atmospheric wind profiles and the unsteady cross-axis loads acting on tall launch vehicles, have prompted renewed interest in unsteady separated flows over bluff bodies. The quantitative features of three-dimensional bluff body flow fields are quite well known, however, very little quantitative data exists describing the unsteady loads and nothing is known concerning the relationship of the flow field geometries to the unsteady loads. An experimental program was undertaken to provide some of this information. A large part of the research effort was spent in the development of a towing channel facility equipped with an air bearing supported model carriage suitable for making measurements in glycerol-water mixtures of the unsteady drag and pitching moment on a circular disk and the unsteady drag and side force on a sphere. The measurements were made with strain gage balances over a wide Reynolds number range (5 < Red < 120,000). The hydrogen bubble flow visualization method was adapted to these glycerol-water mixtures to permit correlation of wake configurations with the unsteady loads. Steady drag coefficients for the sphere and disk agreed with existing data except for the disk in the range 100 < Red < 1000, where the experimental CD was as much as 50% lower than previous results. Drag force unsteadiness during steady motion was always < 5% for the sphere and < 3} for the disk, Sphere drag measurements under constant acceleration from rest proved the apparent mass concept to be valid at high Red until the sphere had travelled approximately one diameter. Nondimensional frequencies of disk pitch and yaw were in the range 0.100.14 for 100 < Red < 500. Although previous data indicates that vortex shedding normally begins at Red - 100, the sting support in these tests stabilized the wake to the extent that shedding did not occur for Red < 160o Peak-topeak moment coefficients were found to be of the order of.007 over this Red range. It was learned that reliable unsteady force data cannot be obtained with a sting supported sphere because the delicate vortex shedding process is seriously disturbed by the sting, however, based on the data gathered, an order of magnitude estimate of the peak-to-peak unsteady force coefficient is 0.020.10 for 270 < Red < 560. Flow visualization studies revealed that the unsteady transverse force on a sphere can be related to the concept that unsteady bound vorticity exists about the body in connection with the vortex-shedding process while the unxi

steady moment on a disk results from the upstream flow in the wake being deflected when vortices are shed. xii

I. INTRODUC TION Bodies as geometrically simple as the sphere and the circular disk can exhibit exceedingly complex aerodynamic characteristics. Anyone who has observed the tumbling fall of a coin through water or the swerving flight of a tennis ball can attest to this fact Thle unsteady motion of such objects results from the fact that they are aerodynamically bluff. bodies. The characteristic feature of bluff body flows is the separation of the flow from the surface of the body. A shear layer exists between the main flow past the body and the "dead fluid" in the space behind the body. Instability of this shear layer produces fluctuations in the pressure and shear stress distributions over the surface of the body, The resultantunsteady forces on the body cause it to move erratically. The situation is further complicated by the fact that the erratic motion of thie body alters the flow field; helnce a complex interaction (of body and fluid dynamics occursS The existence of this unrsteady behavior has long been known~ However, the mathematics of the problem are so difficult that prediction oi the unsteady motions is impossibleo In fac't, the much simpler problem

2 of steady separated flow about a body has yet to be solved analytically. Consequently, insight into the nature of bluff body flows must be obtained through experimentation. This paper describes an experimental investigation of the unsteady aerodynamics of spheres and disks which was conducted in the Gas Dynamics Laboratories of the University of Michigan. The remainder of Section I contains a discussion of the past and current interest in sphere and disk flows. An outline of the research program concludes this Section. A study of the existing literature on sphere and disk aerodynamics revealed that the general features of sphere and disk flows are quite well understood, but very little quantitative work has been done on the unsteady flow problems. The available information on sphere and disk flows that is pertinent to the present study is presented in Section II. Section III consists of a detailed description of the equipment used in obtaining the experimental measurements. The factors influencing the design of the experiment are also noted. The experimental results obtained in this investigation are presented in Section IV. Significant features of these results are discussed; where possible, the measurements are compared with existing published data. Finally, the conclusions that can be drawn from the results of this research are given in Section V.

3 Ao INTEREST IN SPHERE AND DISK AERODYNAMICS Interest in the behavior of a sphere or disk moving relative to a fluid goes back many years. In fact, the first recorded sphere drag measurements were published by Sir Isaac Newton, who determined the times of fall of spheres descending in columns of water and later timed the fall of various spheres dropped from the dome of Sto Paul's Cathedral in London[ 1, pp. 353-366]. Since then, a multitude of studies of the aerodynamics of spheres and disks has been undertaken. Many of these have been noted by Torobin and Gauvin[ 2-7] in their comprehensive review of the sphere problem. The first such investigations were prompted primarily by curiosity regarding the nature of fluids and their behavior in the presence of solid bodies. Nevertheless, there were applications for the results of such studies. For example, artillery officers were interested in the behavior of spheres moving through air in order to predict the ballistics of their cannon balls. Also, the rate of fall of a small sphere provided a convenient measurement of the viscosity of a fluid once the limits of validity of Stokes'[ 8] sphere drag relation, D = 3r dpAU had been established. Academic interest in the fundamental characteristics of body-fluid interactions is as keen as ever; moreover engineering situations

4 abound today in which the nature of the flow of a fluid past a sphere or disk is significant. In fact, in. the past ten years or so, several important new sphere and disk flow problems have arisen. One of these problems, that of rocket motor performance losses resulting from the entrainment of solid and liquid particles in the exhaust gases, was reviewed recently by Hoglund[ 9] These entrained particles are composed chiefly of metal oxides and, therefore, are much more dense than the surrounding gas. As a result, they lag behind the exhaust gas as the gas-particle mixture accelerates through the rocket nozzle. The resultant relative flow of the gas past the particle-s contains a momentum-loss mechanism in that the drag of the particles absorbs momentum from the gas' this mechanistm can often produce losses of rocket motor specific impulse greater than 10%. The significance of this problem is that the metallic oxide particles result from the burning of metals which are incorporated into rocket fuels for their potential. ability to increase specific i.mpulse. As Ho:gland noted, attempts to calculate these losses have been severely limited by a lack of knowledge of particle drag in the presence of accele ration and high-intensity turbulence effects. Another important sphere-flow problem exists in the meteorological use of freely rising balloons to measure the vertical profile of the horizontal component of the atmospheric wind. Although spherical

5 balloons have been employed in this manner for many years, Scoggins [10] points out that most of the data so obtained are averaged over at least 600 meters altitude. However, with the advent of large ballistic missiles and launch vehicles which must rise vertically through the atmosphere, much more detailed measurements of wind profiles are needed. A system to provide these measurements was devised wherein a precision radar was used to track a superpressure constant volume Mylar sphere rising through the atmosphere at a rate of about 25 feet per second. Careful analysis of the high-resolution profiles obtained with this system revealed small scale features that were not truly representative of atmospheric winds. Further tests led Scoggins to conclude that "spherical balloons do not rise vertically in a calm atmosphere." The small-scale horizontal balloon motions were found to be caused by the aerodynamics of the balloon itself. These balloons typically operated in the critical range of Reynolds numbers, 50, 000 to 800, 000[ 11];- the significance of this point will be made apparent below. Becuase of the discovery of these aerodynamically-induced balloon oscillations, the past few years have seen considerable effort put into the study of unsteady sphere aerodynamics[ 11 15] and attempts to minimize the spurious balloon motions. [10, 11, 13, 14, 16] Still another need for knowledge of the behavior of spherical particles in a fluid exists in many areas within the chemical industry.

In the first article of their series, Torobin and Gauvinl 2] note many processes wherein relative motion between particles and a fluid plays an important part: evaporation, drying, crop-spraying, and fluidizedbed reactions, to n.ame but a few. Although the flow field about a disk has not received the great attention given to the sphere flow field, nevertheless, there do exist some situations where such flows are important. A common example is the ubiquitous post-mounted traffic sign, which can oscillate violently, even to destruction, in a strong wind. This oscillation is due simply to the vort:ex-shedding which occurs in the wake of the sign, Willmarth, Hawk, Galloway, and Roos[ 17] recently investigated this phenomenon, Another reason for studying the aerodynamics of a circular disk is that the disk flow field provides a valuable comparison. for the sphere flow, particularly in the case where the respective bodies are rigidly fixedo Several features are common to both bodies, notably axial symmetry and the occurrence of a large region of separated flow. However, one significant difference between the two existso The flow separation line is free to shift about on the sphere, since the surface curvature is uniform~ On the other hand, the disk geometry is such that separation always occurs at the rimo

7 Bo THE RESEARCH PROGRAM The principal goals set for this research program were the quantitative analysis of the unsteady forces and moments acting on a translating sphere or circular disk, and the elucidation of the manner in which the fluctuation of the separated flow produces these loads. To eliminate the coupling between body dynamics and flow unsteadiness, the sphere and disk were constrained to move in a straight line, without rotation. Two categories of unsteady flows were investigated, In one, the velocity of the body relative to the fluid was held constant while the unsteady forces superimposed on the steady drag were studied, In the other, the velocity of the body was varied, i. eo, the motion involved acceleration effects. The research effort was focused on the range of Red from 100 to 10, 000. This range was selected because it encompasses a wide variety of wake types, from the steady laminar wake to the fully turbulent wake. The delimitation of the research was completed with the decision that the influence of such variables as compressibility, free stream turbulence level, and surface roughness would not be studied.

II. EXISTING KNOWLEDGE OF SPHERE AND DISK AERODYNAMICS A, INTRODUCTION The characteristics of the flow of an unbounded, viscous, incompressible, turbulence free fluid about a smooth, rigid sphere or disk are summarized in this Section. With this set of restrictions, the flow field depends primarily on the Reynolds number Redo Consequently the discussion in this chapter will focus on the variation with Red of the characteristic features of sphere and disk flows. The influence of other parameters, eo g., acceleration, will also be discussed. Most of the experimental results noted here contain one of two flaws, Wind tunnel and towing channel tests suffer from flow interference produced by the structure used to support the sphere or disk, On the other hand, experiments with freely rising or falling bodies often involve relatively short trajectories, which preclude the study of long wavelength fluctuations and, in some cases, even prevent full development of the flow field. In addition, the coupling between body and wake motions during free flight tests does not permit the study of flow instability alone. This is particularly important in the case of the disk, as will be noted below.

9 Hopefully, the advent of magnetic wind tunnel model suspensions will, eventually lead to interference-free studies of the flow about a rigidly held sphere. The same does not appear to be possible with a disk, as the disk geometry precludes incorporation of a magnetic dipole aligned with the disk axis. Since such a dipole is a necessary part of a magnetic suspension system, there seems to be no way to magnetically support a disk without significantly altering its shape. One further point should be mentioned. Because the experimental results included in the following discussion have been obtained with myriad different test setups and conditions, the Red ranges specified and even some of the details of the various flow phenomena may not be completely accurate. B, AERODYNAMICS OF A SPHERE 1, Flow Regimes As Red is varied, the flow about a sphere passes through a wide variety of configurations, In the following outline, an attempt has been made to piece together a coherent picture of the development of the sphere flow field with increasing Red from the results of many experimental investigations. ao 0 < Red <0~ 1o In this regime, the flow in the vicinity of a sphere is completely dominated by viscous effects. Streamlines near

10 the sphere are essentially symmetrical fore and aft of the sphere (Fig. 1). However, this symmetry can not exist far from the sphere, As Lamb[ 18, pp. 608-614] shows, the ratio of inertial forces to viscous forces in the sphere flow field is of the order of Ur/v, where r is the distance from the center of the sphere. Thus, no matter how great the ratio of v to U may be, inertial forces will become dominant at very large r. b. 0. 1 < Red < 24: The flow in this range of Red is still dominated by viscous effects, but fluid inertia exerts an increasing influence. The streamline pattern near the sphere is no longer symmetrical fore and aft. As indicated in Fig. 2, comparison of streamlines equal distances upstream and downstream of the sphere discloses that the point of maximum transverse deflection of a streamline shifts downstream from the equatorial plane of the sphere, the magr.itude of the shift increasing for streamlines farther from the stagnation streams line. The flow pattern in Fig. 2, it should be noted, is taken from the numerical solution by Jenson[ 19] for this flow regimeo His results, although obtained for a confined flow*, are qualitatively similar ito observed flows with boundaries far removed from the sphere. tJenson assumed that the flow was contained in a cylindrical stream tube of diameter equal to six sphere diameters.

11 co 24 <Red 130- At Re d24, a closed region of separated flow appears at the rear stagnation point[ 20]o Primarily because of the difficulty in observing the onset of separation, considerable disagreement exists as to the value of Red at which this occurs. However, the value noted above is thought to be very close. As is pointed out in most textbooks on fluid mechanics, Schlichting[ 21, ppo 27-29] for example, the separation is caused by the inability of the fluid adjacent to the surface to work against an adverse pressure gradient all the way to the rear stagnation point owing to loss of momentum through surface friction, As Red increases in this range, the separation circle moves upstream and an annular vortex forms within the separation bubble; this is indicated in Fig. 3o d. 130 <Red < 210~ As Red increases above 130, the separation bubble continues to grow, but now the downstream end of the bubble pulsates faintly with a very low frequency[ 20], This pulsation beecomes more conspicuous as Red increases, but the laminar wake downstream of the wake stagnation point remains stableo eo 210 < Red <270 For Red in this range, Magarvey and Bishop[ 22] noted a double laminar wake indicated by two parallel dye threads combined with an asymmetrical separation bubble, Because they were studying wakes produced by a liquid drop falling through an immiscible field liquid, it is not certain that this phenomenon occurs

12 in the case of a rigid sphere. However, it seems reasonable to expect that the same flow would occur for a solid sphere. This expetiation was verified in this laboratory by Hawkj 23], who obser'ved a twin wake behind a dyed steel sphere falling in water at Red; 250~ According to Magarvey and MacLatchy[ 24], this twit,-wake situation arises when the rate of vorticity production in the sphere boundary layer can not quite be balanced sufficiently by diffusion and 4coI.vection into the single laminar wake. Hence the separation bubble; grews with accumulated vorticity until it becomes unstable. When this happens, the vortex ring within the bubble becomes distorted arnd eventually a segment of it becomes pulled out behind the sphere by the flowing fluid (see sequence sketched in Fig. 4). Once this occurs, the pair of stretched vortex filaments linking the discharged loop with the- por - tiorn of the ring remaining in the separation bubble provide a mechai - ism for- discharging additional vorticity irnt the wake. Apparently ttiis mechanism carries away sufficient vorticity for the coriigiiratiior Uil remain stable until Red approaches 270. It would be expect:ed:a-t tthe asymmetry associated with this wake arrangement should priods( e - a transverse force or the sphere; Magarvey and Bishop" 22 did, iri fa4i observe that the sphere followed an inclined path, indieanirng the a-triom of a transverse force. fo 270 < Red < 290~ Once Red reaches approximat.ely 270, ihe twin vosrtex mechanism is no longer capable of discharging enough

13 vorticity to maintain equilibrium and waves begin to appear in the wake [ 22]. These undulations amount to slight additional discharges of vorticityo go 290 / Red i 700. The undulations which occur for Red just beg. 290< Red e e, low 290 become discrete vortex loops in this Red range. These loops are shed from opposite sides of the separation bubble with a definite periodicity[ 22, 25 ]. As Red increases in this range, the discharge of vortex loops becomes more random. The flow is sketched in Fig. 5. ho 700 <Re < 10, 000 (est. ): With Red above 700, the vortex loops, which have become quite random in size and orientation, are discharged with increasing frequency[ 22]. Above Red 1500, the shed vortex loops tend to cluster in clumps or "vortex balls" as Mtbller [26] termed them. Measurements by Cometta[ 27] in this Red range also seem to show these vortex clumps. Hot wire investigations of sphere wakes in this range of Red by Kendall[ 28 ] and wind tunnel flow observations by Winny[ 29] appear to indicate that an overall helical pattern exists in the wake, possibly produced by a progression about the wake axis of the location of the clustering vortex balls. Photographs by Mbller[ 26] and Schmiedel[ 30] also seem to show a helical wake configuration. This helical wake, if it does exist, may well occur far into the next Red range.

14 In the course of his hot wire experiments, Kendall[ 28 j noted that the free shear layer of the separated flow near the sphere surface oscillated with, a frequency apparently unrelated to the frequen.cy of thk. wake helix. In the range 3000 < Red < 10, 000, he found that the shear layer frequency scaled with U, whereas the frequency in the helical wake was 0. 18U/d (for Red > 400). Winny[ 29 ] also reported hot wire measurements, noting a per iodic - ity in the sphere wake at an average frequency of 0. 24U/d for Red 47, 000. In addition, he measured unsteady surface pressure at various points around the sphere. At Red = 130, 000, he found that pressure impulses occurred near the separation point with two frequencies. The overall frequency of these pulses was 0. 91U/d, but on the average, every fifth pulse was much larger, giving a frequency of approximately 0. 20U/d for the large impulses. The above indicates that there are two types of instability in the sphere wake. One of these leads to the overall spiraling motionr of the wake. The other is a higher-frequency local rolling up of the separated shear layer into vortices. In this connection, a most inriteresting discussion of similar processes occurring i.n the wake of a cylin..der was recently given by Morkovin[ 31 ]. i. 10, 000' Red < 200, 000~ In this range the vortex loop shedding becomes so rapid that it is nearly a continuous process and the sphere

15 wake takes on a turbulent appearance. The separation circle continues to move upstream on the sphere as Red increases, approaching a point about 800 from the front stagnation point[ 32 ]. Fluctuation of the wake exists throughout this regime, corresponding to the shedding of vorticity clumps. jo 200, 000 < Red < 400, 000: This is the Red range commonly termed the "critical range"~ At Red: 200, 000, the sphere boundary layer begins to undergo natural, transition from laminar to turbulent flow before separation. When this occurs, the enhanced mixing of the turbulent boundary layer adds momentum to the fluid layer adjacent to the sphere surface and thereby permits it to proceed farther against the adverse pressure gradient than could the laminar layer. This delay of separation significantly reduces pressure drag and is responsible for the well-known dip in the sphere CD versus Red curve. In fact, over a portion of this Red range not only CD but the drag itself decreases with increasing Redo In this critical range of Re the transition process is neither uniform nor steady, with the result that large changes can occur in the position of separation, Since the separation can shift up- and downstream at any given point around the sphere, significant random fluctuations of the wake occur. Because the boundary layer transition process is highly sensitive to sphere surface condition and steadiness of the free

16 stream, the actual range of Red for transition varies widely in different situations. For comparison, flow configurations before and after transition are sketched in Fig. 6.. Red > 400, 000~ When Red has reached 400, 000, the transition process is well established and has begun to move forward in the sphere boundary layer. The consequent thickening of the turbulent portion of the layer causes the separation circle to begin shifting upstream again, gradually increasing the pressure drag as Red increases. Because boundary layer transition is a result of flow instability, it is highly sensitive to many factors, e. go, surface imperfections and unsteadiness of the pressure field about the sphere. As a result, the flow separation line and the wake at supercritical Red continue to fluctuate strongly, especially at very low frequencies[33 ]j These fluctuations are particularly pronounced with freely moving spheres such as balloons, which generally have surface irregularities, owing to the ease with which the low-mass balloon. can rotate and move transversely in response to unsteady forces. 2. Steady Drag Many investigators have used a wide variety of techniques to measure the aerodynamic drag on a sphere and its dependence on Red,

17 surface roughness, and free stream turbulence. Consequently, a great deal of data on sphere drag has accumulated. In spite of all this effort, however, the relationship between sphere CD and Red is not very accurately known, as can be seen in Fig~ 7, which is a plot of selected representative sphere drag data for steady sphere motion. The scatter in this CD versus Red data is believed to arise from five principal causes. Perhaps the most significant of these is the difficulty in accurately determining the kinematic viscosity v and temperature T of a fluid, This is especially important in much of the sphere drag data because many of the experiments were performed early in this century, when viscosity and temperature measurement techniques were not nearly as precise as those used today. Because most of the low Red data was obtained with very small spheres, problems involved in measuring and weighing these small particles are believed to be partly responsible for the variation in the CD versus Red data at low Redo Another problem involved in many of the low Red experiments was the inability to accurately account for the effect produced by the walls of the fluid container. Several correction factors had been proposed:*, but no means existed for determining a priori which, if any, *A brief discussion of these correction factors is given in reference 34~

18 was appropriate. The dilemma arose, of course, from the fact that all the corrections were based on approximate solutions of the flow equations. The recent experimental work of Maxworthy[ 35 J has shed some important light on this problem; his results indicate that., as one might expect, none of the theoretical wall-correction factors is correct. A fourth source of scatter in the drag data is the unsteadiness in the sphere flow field for Red above a few hundred. Since almost all the available data for Red below approximately 100, 000 was obtained from tests of freely falling or rising spheres, this data suffers from the fact that the spheres were able to deviate from a straight vertical path when. the wake fluctuated. The path fluctuations tend to depend on the ratio of sphere density to fluid density and, as noted above, are fairly random. The shape of the CD - Red curve in the critical range of Red has been studied by many investigators. Because of the sensitivity of the sphere flow field to many factors in this transition. regime, as desc:ribed above, there have resulted almost as many different drag coefficiernt curves as there have been experimentso In the remainder of this section, the characteristics of'the CD - Red curve will be discussed. In addition, brief mention will. be

19 made of the various experiments which provided the data given in Fig. 7. a. 0 <Red <0, For Red below 0 1, the CD - Red relationship is in very close agreement with the prediction of Stokes'Drag Law, which is, in coefficient form, 24 D Re o Stokes obtained his result by omitting the inertia terms from the Navier-Stokes equation and solving the resultant linear differential equationl 8]. As was noted in the above discussion of the sphere flow field, Stokes' calculation is incorrect far from the sphere. However, the close agreement of his expression for sphere drag with the experimental results indicates that the flow far from the sphere has negligible influence on the drag in this Red regime. The flow near the sphere is dominated by pressure and viscous forces- 2/3 of the drag calculated by Stokes is caused by skin friction, the remaining 1/3 of the sphere drag resulting from the distribution of pressure about the sphereo The data plotted in this range comes from experiments by Arnold r 36]. who dropped rose metal spheres in a cylindrical tube filled with colza oil, and MJl.ler4 26], who used steel spheres in glycerol. The

20 results of Mb11ler are believed to be more reliable, primarily because of the smaller sphere diameter to tube diameter ratio in his experiments. bo 0. 1 Red <1: As Red increases above 0 1, CD becomes increasingly greater than the value predicted by Stokes? Law. This is, of course, caused by the fact that the contribution of pressure to the sphere drag is increasing because of the rising influence of fluid inertia, which Stokes omitted in his calculation. This behavior is also indicated by the streamline asymmetry mentioned above and shown in Fig. 2. Further, the skin friction itself becomes greater than Stokes' value owing to the increasing tendency, as Red increases, of the vorticity to be concentrated in a thin boundary layer. Drag data in this range agrees satisfactorily with the theoretical expression obtained by Oseen[ 37], which is expressed ru-nodinlh...siosnally as 24 C 2-4 +4. 5 D Red OseenVs result was calculated from an approximate form of tihe Navier-Stokes equation in which fluid inertia was partlially aeccoutted for through linearized inertia terms. The solution of Oseen-s equation takes the form of a series; Oseen computed the first Two terms9 as indicated in the above CD expression. Goldstein4 38 later

21 calculated further terms in the series, but agreement between experimental and theoretical results was not materially improvedo Experimental data for 0. 1 < Red < 1 was obtained by Arnold[ 36], Mtdller[ 26], Liebster[ 39], Schmiedel[ 30], and Maxworthy[ 40]o The recent experiment by Maxworthy, very carefully performed using sapphire spheres falling in a temperature-controlled glycerol-water mixture, has yielded the most accurate low Red drag data obtained to date. Liebster tested steel spheres falling in glycerol, sugar solutions, and water, while Schmiedel studied the fall of steel and aluminum spheres in glycerol-water mixtureso co 1 < Red < 2000~ With increasing Red in this range, the drag on a sphere undergoes a gradual transition from domination by viscous effects to pressure drag resulting from flow separation. An interesting feature of the CD - Red curve is the absence of any indication of the separation bubble formation. In other words, there is no noticeable change in the curve in the vicinity of Red 24, where separation commences. Evidently the pressure distribution of the outer flow is not perceptibly altered by the separation bubble, which is very thin when it first formso Furthermore, very little skin friction is lost because the sphere surface from which the flow has separated is nearly perpendicular to the mean flow, and consequently friction on this surface contributes essentially nothing to the drag force even

22 when there is no separation Apparently the pressure drag produced by separation is only. slightly greater than the loss of skin friction drag as the separation bubble grows, resulting in a smooth CD - Red curve of gradually increasing (though negative) slope as Red increases. Drag coefficient data for 1 - Red < 2000 was provided by many sources. In addition to the tests of Arnold, Mtller, Liebster, Schmiedel, and Maxworthy, which have already been cited, experiments in this range were performed by Allen[ 41], Hawk[ 23], and Wieselberger[ 42]. Allen measured the rate of fall of amber spheres in water, while Wieselberger determined the drag on spheres mounted in a wind tunnel. The only experiments with buoyant spheres at Red below 1000 were performed by Hawk, who studied the free rise of hollow aluminum spheres in glycerol-water mixtures. The discrepancy between his data and that of others in this range is believed to result from the low ratio of sphere density to fluid density, and also may involve inaccurate determination of fluid properties. d. 2000 1 Red < 200, 000: When Red has reached approximately 1000, the separation circle has advanced nearly to the equator of the sphere[ 20]. For Re above this value, skin friction drag is much smaller than the drag created by the low pressure in the separation

23 region, so that the drag is almost entirely pressure drag. Although the separation circle continues to move upstream gradually as Red increases, the net pressure force on the sphere is quite insensitive to such shifting (as the surface near the sphere equator is nearly parallel to the drag direction and, therefore~ CD remains nearly constant, increasing very slightly as Red is increased up to the critical range. Because CD is almost constant, the drag force increases 2 as Red; for this reason the range 2000 <Red < 200, 000 is known as the quadratic drag range. Experiments covering this range of Red have been reported by Wieselberger[ 42], Shafrir[ 43], Lunnon[ 44], Allen[ 45], and Shakespear[ 46]. Shafrir, in his recent work, observed spheres of various materials falling in water, as Lunnon had done earlier. Allen's tests were performed with steel spheres dropped in water. Shakespear performed his experiments by dropping weighted celluloid spheres through air in a tower. eo 200, 00 </ Red~ Available sphere drag data for Red > ____________ d 2009 000 has been generated primarily by wind tunnel experiments, although a number of other techniques have been employed. The four sets of critical range data plotted in Fig. 7 plainly show the extremely wide variety of CD - Red behavior existing in this regime.

24 Millikan and Klein[ 47] obtained their data with a sphere mounted on an airplane. They actually obtained 291 data points; only the spread of these points is indicated on the plot. This curve is the best representation known of the CD - Red relationship for a smooth, rigidly mounted sphere in a turbulence-free wind stream. For comparison, Flachsbart's[ 48] wind tunnel results are also shown in Fig. 7o The situation is the same as for the Millikan-Klein data (smooth, fixed sphere) except for the low but finite turbulence level in the wind tunnel stream. The only noteworthy difference between the two sets of data is the shift to lower Red produced by the turbulence. The other two sets of data in this regime resulted from freeflight tests. The Bacon and Reid[ 49] data was obtained by dropping wax, rubber, and wooden. spheres from an airplane into still air and tracking their fall with a recording theodolite. Radar tracking of freely rising balloons provided the data given by Scoggins[ 11]J In both these cases the spheres were free to rotate, alter direction, and change speed. These effects were, of course, much more pronounced in the case of the balloons, due to their much lower de:nsity The influence of the removal of rigid constraint is plainly evidenlt in both sets of data and clearly increases with i.ncreasing ability of the sphere to respond to wake fluctuations~

25 3. Unsteady Behavior During Quasi-Steady Motion Although studies of the unsteady behavior of freely-moving spheres have only recently been undertaken, the fact has long been known that a freely rising or falling sphere does not always follow a precisely vertical path at uniform speed. For example, Newton[ 1, p. 364] noted that light weight spheres "did not always fall directly down, but sometimes fluttered a little in the air, and waved to and fro. " In 1920 Schmidt[ 50] reported observing periodic changes in the speed and direction of a wax sphere dropped in water as vortex loops were released in the wake. Bacon and Reid[ 49] mentioned unusual irregularity in CD measurements performed with balloons rising in very still air, Liebster's[ 39] measurements of falling spheres showed that lateral movements occurred simultaneously with periodic changes in the vertical velocity. And in 1928 Lunnon[ 44] mentioned "the slightly curved and spiral path which is always followed by a sphere falling through liquid. " More recently, Jones, Peterson, Schaefer, and Schulte[ 51], in the course of measuring upper atmosphere density with a freely falling accelerometer-equipped sphere, discovered that under certain conditions excessive scatter appeared in their data, produced by lateral accelerations of the sphere. In his recent thesis, Sivier[ 52] reported that he was unable to obtain drag data for smooth spheres with his

26 magnetic suspension equipped wind tunnel for Red between 300 and 550. The reason for this, he noted, was excessive transverse motion of the sphere. The greatest impetus to the study of unsteady motions of freeflight spheres has come from the realization that spherical meteorological balloons do not accurately measure wind profiles. The need for high resolution wind profiles has been great enough that several research programs have been initiated to a&.alyze the unsteady behavior of spheres. Most noteworthy of these studies is the work reported by MacCready and Jex[ 13]. They present an enlightening discussion of the sphere flow problem in general, and, in particular, they give a very informative fluid dynamical analysis of the behavior of freely moving spheres in the vicinuty of the critical Red regime. For example, they point out that a risinrg or falling sphere anrinot have the same CD - Red rel.ationship in. She critical regime as an identical sphere rigidly held in a wind Itunrel. This is so because the portion of the CD - Red curve with slope less I-ha:r -2 is a regio:r of irsstabflhty for a:< ascei idirsg or descending sphere. Anopther sign.ificarnlt point brought out is the importance of the relative mass ( ratio of sphere mass to displaced fluid mass) in determinirg "he unsteady motion of a sphere.

27 Data presented by MacCready and Jex, gleaned from balloon ascent tests and experiments with rising and sinking balls in a swimming pool, covers the range 10, 000 < Red <1, 000, 000o In general, this data shows that average CD values (based on average vertical velocity) are greater for low relative mass spheres at any Red. Also, a very rough correlation is shown between a "motion magnitude" factor (essentially the ratio of lateral excursion amplitude to the excursion wavelength) and relative mass, the motion magnitude factor being generally greater for lower relative mass spheres. The balloon data corroborates their prediction regarding "forbidden" segments of the CD - Red curve by showing jumps from supercritical to subcritical Red along CD-Red lines with slopes of precisely-2. * Commenting on the general features of the motions observed in the swimming pool tests, MacCready and Jex point out that coupling between lateral and rotational spheres motions often occurred. They summarize by stating that lateral motions at subcritical Red were regular spirals or zigzags, with a wavelength on the order of twelve diameters, while motions at supercritical Red appeared to be "wandering spirals " *It should be noted that most light-gas filled balloons with diam eters of a few feet will, in the course of rising several miles through the atmosphere, pass through the critical range of Red, going from high Red to low Red

28 Mac Cready and Williamson[ 14], in a continuation of the work just described, showed that most of the differences between CD - Red curves for freely moving spheres of large and small relative mass were eliminated when CD was based on true velocity rather than average vertical velocity. In fact, CD data obtained this way all fell below the "standard" sphere drag curve (corresponding to Fig. 7 in this paper). However, MacCready and Williamson used the buoyancy forceassuming it to be equal to the average drag force, in. calculating their values of CD. Since the true value of CD must be computed from the instantaneous velocity and drag in an unsteady situation, data such as that presented by MacCready and Williamson must be evaluated with some caution. MacCready and Williamson also showed that- the average wavelength of lateral, motion. for 100, 000 Red 1, 0009 000 was approximately eleven sphere diameters for low relative mass spheres. A very extensive set of tests and data analyses was performed by Rogers and Cami.t4 15], who used a high precision pulse Doppler radar to track rising helium-filled superpressure Mylar balloons i.n order to obtain power spectra of the random unsteady motions. They determined that at supercritical Red appreclable spectral content exists in the wavelength range from 12 to 44 diameters (for two meter

29 diameter balloons) with a peak in the spectrum consistently occurring at a non-dimensional wavelength (X/d) of approximately 24, and that these fluctuations arise primarily from horizontal random velocity perturbations having a dimensionless root-mean-square value of about 0. 37. They note a considerable variation in spectral shapes and caution that one spectral shape may not be sufficient for adequately describing smooth sphere behavior~ Finally, they mention that in the supercritical regime balloons appear to rise with essentially constant vertical speed and fluctuating horizontal velocity components, while the velocity fluctuations in the subcritical regime seem to be of equal intensity in all directions. Unsteady sphere motion data has been obtained by Shafrir[ 43] in the range 1000 < Red < 50, 000. These tests were all conducted by photographing stroboscopically illuminated spheres of relative mass from 1. 15 to 8& 47 as they fell through water. No fundamental relationships were obtained, but several features of the data deserve mentiono Non-dimensional amplitudes of lateral motion were found to increase with Red; they also tended to increase with relative mass, contrary to the prediction of MacCready and Jex[ 13]. Non-dimensional wavelengths of the unsteady lateral motions increased from a minimum of 25 for spheres with a relative mass of 1o 15 to as high as about 80

30 for the large relative mass spheres~ Dimensionless wavelengths displayed a slight tendency to decrease with Red. Finally, the motion magnitude factors, while exhibiting considerable scatter, showed a decrease with relative mass and displayed an increase with Red up to Red 10, 000 followed by a decrease for higher Redo Preukschat[ 53] also studied freely moving spheres in water. On the basis of free fall tests over the range 500 Red < 42, 000, he concluded that no oscillations exist for spheres with relative mass greater than unity. This contradicts Shafrir's results, but the more refined technique used by the latter probably produced more accurate results. Oscillations were evident in Preukschat;s experiments with buoyant spheres for, Red between 800 and 9400 and relative masses from from 0. 526 to 0. 875. He determined that the unsteady motion was two dimensional, sometimes becoming helical after a few cycles. Dimensionless wavelengths of the transverse motion increased with -relative mass from approximately 12 to 26. Amplitude data was too scattered to reveal any dependence on relative mass or Red~ Onre other significant result:reported by Preukschat was that CD values based on local velocity (rather than average verti:cal speed) were independent of the extent of the lateral motion.s

31 Fluctuating lift forces on a sphere at supercritical Red were measured recently by Willmarth and Enlow[ 33]. They found that neither the power spectrum nor the root-mean-square value of the lift coefficient CL varied with Red in the range 484, 000 <Red < 2 1, 667, 000. The value of CL was approximately constant at 0. 062. No dominant frequencies appeared in the spectra of the fluctuating lift; the greatest spectral densities occurred at the lowest frequencies measured. During the experiment, a large-scale low frequency fluctuation was observed in the lift force; this fluctuation appeared to be almost a square wave with nearly constant amplitude but random period, The amplitude of this low frequency fluctuation was at least as large as the amplitude of the higher frequency fluctuations superimposed on ito However, the cause of the phenomenon was not determined because the wake behavior could not be studied (flow visualization equipment was not available)[ 54]. 4, Accelerated Motion Relative acceleration between a sphere and a fluid occurs in many situations and has therefore been a matter of considerable interest to fluid dynamicists. Examples of situations in which such accelerations are involved are the atomization of fuel sprays and the currently very important problem discussed above, of particle entrainment in rocket

32 motor exhaust gases. In their review of particle-fluid interaction problems, Torobin arid Gauvinl 4] discuss a number of experimental inrvestigations of sphere acceleration. Although they point out that knowledge of accelerated sphere motion is cruder and much less extensive than in the case of steady motion, several conclusions can be drawn on the basis of existing data. One important observation is that the'apparent mass"' concept is totally inapplicable beyond the initial stage of relative motion between the sphere and fluid. The apparent mass concept arises from potential flow theory, since it can be shown that the resistance to acceleration of a body in an inviscid fluid of infinite externt is simply R (m + m ) a, where R is the resisting force, a is the acceie.ration, m is the mass of the b: dy, antd m is the apparent mass, which depends only ton the a body geomdetry and the fluid density. For a sphere, m is just half a the mass of the displaced fluid[ 18, p. 124.o Early efforts at assessing the effect of acceleration o:r. particle drag in a vischrus fluid involved an attempt to include tbhe infi:ue'A-ne of fluid inertia by straightforward application Of the apparent mass concept. As Goldstei4n 55] shows, however, the flow about a body is

33 irrotational only at the instant of initiation of the motion. The vorticity generated at the body surface immediately begins diffusing into the fluid and is convected into a wake. If the motion is such that large Red is reached very rapidly, the vorticity will remain confined to a thin boundary layer adjacent to the body surface and a thin laminar wake, and the flow about the body will closely approximate potential flow until separation occurso Schlichting[ 21, ppo 222-223] presents flow field photographs and pressure measurements illustrating this behavior for a circular cylinder. In such cases, it is reasonable to expect that the apparent mass concept will be approximately valid until boundary layer separation becomes significant. More will be said on this matter in a later Section. Torobin and Gauvin also concluded that sphere acceleration phenomena are correlated by the acceleration modulus A = ad/U, citing the sphere results of Bugliarello[ 56] and the worls of several other authors who studied disks and cylinders for support of their statement. Although Lunnon[ 57] states that his results, which were obtained in an extensive series of measurements on steel spheres dropped in mine shafts, definitely do not scale with A, his data falls in the Red range from 10, 000 to nearly critical values and may very well have suffered from critical Re effects owing to sphere roughness and/or air currents in the mine d

34 s ha so Since the apparent mass concept has clearly failed to account foir the changes in drag produced by acceleration, it has become normal practice to present results of acceleration tests in terms of functional relationships among Red, A, and CD, with CD defined in the traditional manner. Existing data indicates that CD increases with A, for A> 0 and Red constant, and that for fixed positive A, the difference between CD and CD(A=O) diminishes as Red is increased. For negative acceleration, A< 0, the situation is somewhat unclear. Ingebo[58] found a unique CD - Redrelationship for clouds of solid spheres and evaporating liquid droplets at values of A = 0(-10-5). His values of CD faired into the standard sphere drag curve at Red 6 but fell increasingly below the CD(A=0O) curve as Red increased to 4000. In a more recent experiment which was free from the interactions that rmay have occurred in Ingebo's clouds of particles, Selberg and Nicholls 59J used a shock tube operating at low Mach nnumbers to accelerate solid spheres (deceleration relative to the gas) and cons cluded that the low values of A (-A 10-4 to 10 ) had negligible effect on CD in the range 200 S Red - 1500. The latter data is believed to be correct, because the experimental arranrgement used by Ingebo appears to have produced large flow

35 disturbances. However, the matter could certainly bear further investigation, particularly in light of its relevance to the particle entrainment problem. Another point regarding sphere deceleration (A 0) deserves discussiono It has been suggested by Lunnon[ 57] and by Torobin and Gauvin[ 4] that CD(A0O) should be greater than the steady state value regardless of the sign of Ao This view can not be correct for all - AO Consider, for example, a sphere moving through a still fluid at constant velocity and with a fully-developed wake, The fluid entrained in the wake moves, relative to the undisturbed fluid, in the direction of sphere motion, but with a lower velocity. If the sphere is brought rapidly to rest, corresponding to large, negative A, the momentum of the wake flow will cause it to wash over the rear surface of the sphere, exerting a force on the sphere in the upstream direction (i. e., a negative drag force)o Relating this argument to the case of a sphere entrained in a fluid stream, it is clear that for large -A the CD can be significantly lower than CD (A=0) owing to the sphere being accelerated into its own wakeo C. AERODYNAMICS OF A DISK The characteristic features of the flow of an unbounded fluid past a circular disk nominally oriented with its face perpendicular to the

36 stream will be outlined in this section. The fluid is considered to be viscous, incompressible, and free of turbulence; the disk is thin and sharp-edged. As in the case of a sphere, these limitations imply that the nature of the flow about a disk is primarily a function of Red. 1. Flow Regimes The flow field about a circular disk does not assume as wide a variety of configurations as does the sphere flow field. Principally, this is a result of the separation line being fixed at the disk rim. Because of this, the well-known rapid decrease in CD that occurs with separation line shift on spheres and cylinders does not occur with the disk. For this reason, and also because the disk shape does not appear as frequently as the sphere in engineering applications, the disk flow has been less extensively studied than the sphere flowo Nevertheless, it is believed that the changes in the flow past a disk that occur as Re is varied are well understood in general, although several ded tails are still lacking. The following is an outline of the development of the disk flow field as Red is increased. The discussion is primarily concerned with the flow past a fixed disk, but effects that occur when the disk is free to oscillate and translate are also noted.

37 a. 0 K Red < Go 1 The flow field about a disk in this regime of Red is controlled by a balance between pressure and viscous forceso This does not imply, however, that the forces acting on a disk are viscouso In fact, skin friction can not contribute to the net force acting on a disk oriented with its axis aligned with the flow direction, so the force on the disk is entirely due to the pressure distribution about the disk. When Red < 0o 1, the near-field streamlines of the flow past a disk exhibit complete symmetry about the plane of the disko b. 0o 1 < Red c 1o As Red increases above 0O 1, fore-and-aft asymmetry begins to appear in the streamlines, indicating that inertial effects near the disk are growing to a significant level. co 1 < Red 100OO Separation of the flow from the rear face of the disk occurs at Red 1. Experimental evidence on this point is very sketchy~ Stanton and Marshalli 60] injected dye through the rim of a disk supported by a sting in a water tunnel and, by observing the flow of the dye, estimated that formation of the separation bubble with enclosed vortex ring occurred at Red: 4. 5. They point out, however, that the difficulties in obtaining uniform flow at this low Red and in observing the details of the dye flow near the disk make their estimate of the separation Red rather uncertain. In addition, they mention elsewhere in their paper that the velocity profile in their test channel was

38 nearly parabolic, throwing considerable doubt on all their results, The value Red - 1 suggested above for the onset of flow separation was estimated from the results of Michael'sL 61] computer calculation of disk flow fields for Red of 2, 8, and 30, although some features of this work raise questions. His streamline plot for Red = 2 shows separation occurring on a circle of diameter about half that of the disk, the maximum thickness of the separation bubble being only about 5% of the disk diameter. However, there is some doubt in this author's mind that flow separation from a thin disk would begin at the rear stagnation point rather than at the disk edge, where the flow must execute an abrupt reversal in direction, This doubt is not diminished by the fact that Michael's solutions seem to indicate a slow asymptotic approach of the separation circle toward the disk edge. His streamlines for Red - 30, while showing a separation bubble whose length is nearly equal to the disk diameter, indicate that the diameter of the separation circle is still about 10% smaller than the disk diameter. In. apparent contrast with this is a photograph by Schmiedel[ 30] showing clearly that the flow is separating at the rim of disk at Red'- 48. 6. Experiments to answer this question have yet to be made. As Red increases from 1 to 100, the separation bubble grows until it is more than 1 1/2 disk diameters long at Red 100. Vorticity tends

39 to concentrate in a boundary layer on the disk face and in the separated sheet springing from the edge of the disk and is convected and diffused into the surrounding fluid through a steady laminar wake. do 100 <Red <250 (est.): According to Simmons and Dewey[ 62], the separation bubble of a disk held perpendicular to the flow becomes unstable at Red 100 and the process of shedding vortex loops commences. In this Red regime, the loops all appear to be shed from the same location on the rear face of the disk, analogous to the early stages of sphere vortex-loop shedding described above. Evidence of this behavior is given in wind tunnel flow photographs by Simmons and Dewey and by Stanton and Marshall's[ 60] water tunnel flows. The flow field is very different in the case of a freely falling disk. The ability of such a disk to pitch about a diameter in response to pressure variations produced by asymmetry in the separation bubble allows a complex coupling between disk orientation and wake generation to ensue. The resulting oscillatory character of the motion of the diskplus-wake system was extensively investigated by Willmarth, Hawk, and Harvey[ 63] and will be discussed below. e. 250 < Red The details of the remainder of the Red range for fixed disks are not clear, but a few important features can be noted. At Red;m 250, the vortex loop shedding begins to alternate,

40 that is, successive loops are shed from diametrically opposed sites on the rear face of the disk. For 250 < Red < 350 (roughly), the loop shedding is essentially confined to a plane. As Red increases above 350, loops tend to be shed from random positions about the disk circumference. The frequency of loop shedding increases with Red and the appearance of discrete vortex loops gradually disappears. The disappearance of discrete vortex loops in the disk wake may be explained by the rolling up, because of instability, of the separation layer into small-scale vortex filaments which are carried directly into the wake by the main flow. As Red increases the shear layer instability increases and increasingly smaller vortex filaments are cast into the wake which thereby gradually acquires a truly turbulent character. The wake may be considered fully turbulent when Red > 10, 000. Despite the changes occurring in the disk wake, the geometry of the separated shear layer near the disk remains essentially constant for Red > 1000 (approximately). 2. Steady Drag Disk drag data is not nearly as abundant as that for spheres. However, the existing data is consistent enough so that the CD - Red relationship for disks may be considered as definitely established. It should be noted, though, that one segment of this data, between

41 Red 100 and Red 1000, will receive critical comment in a later d d Section. All available CD data for disks has been plotted in Fig. 8. In the following, the features of the CD - Red curve will be noted along with brief discussion of the experiments from which the data of Fig. 8 was obtained. a. 0 < Red <1: When Red < 1, CD is very adaquately represented by the relation 20. 37 C + 37 242, D Re which was obtained by Oseen[ 64] in a manner similar to his calculation for sphere drag. Again, the solution of Oseen's equation is an infinite series, only the first two terms of which were calculated. The first term reproduces the earlier result of Oberbeck[ 18, ppo 604-605], whose computation paralleled Stokes' sphere drag calculation. The experimental CD data for this Red range was obtained by Schmiedel[ 30], with aluminum, gold, silver, and steel disks which were dropped in mixtures of glycerol and water. b. 1 < Red < 100~ Drag data for this range has been taken from the results of Schmiedel and of Willmarth et. alo [ 63], who also dropped disks made of various materials in glycerol-water solutions. Fluid inertia plays an increasingly important part in determining the flow in this range of Red, causing the CD curve to deviate increasingly from

42 the Red behavior characteristic of purely viscous flows. CD values calculated by Michael[ 61] in the course of his computer work are also plotted and indicate excellent agreement with the experimental data. c. 100 <Red <250~ For these Red, the experimental CD values display a great deal of scatter. This is explained by the fact that the data in this range was all taken from the free fall tests by Schmiedel and Willmarth et. al. in which the disks executed pitching motions and lateral excursions. The CD values here were based on average vertical velocityo However, as Willmarth et. al. pointed out, the concept of a simple drag coefficient loses meaning when this unsteady disk motion occurs because of the complicated set of unsteady forces that actually act on the disk. d. 250 < Re < 1000: The data in the next range, 250 <Re K d d 1000, was obtained during the wind tunnel experiments performed by Simmons and Dewey[ 62]. This author believes the apparent agreement between these CD values and the previously-discussed data (Red <2 50) to be accidental. These disks were mounted in the wind tunnel. in such a way that they were free to oscillate. Had they been rigidly mounted, it is probable that the resulting CD values would have been considerably lower, close to unity. It might also be noted that Simmons and Dewey estimate their accuracy at ~ 20%0for the

43 lowest Red. e, Red > 1000: Wind tunnel disk drag measurements by Wieselberger[ 42], Knight[ 65], Prandtl[ 66], Shoemaker[ 66], and Simmons and Dewey[ 62] produced the CD data for Red > 1000. The apparent constancy of CD in this range results from the fixed geometry of the separated layer, as mentioned above. 3, Unsteady Behavior During Quasi-Steady Motion The fact that freely falling disks oscillate and translate laterally when vortex loop shedding occurs has already been mentioned. The interaction of the disk dynamics and the wake fluctuations is in some respects a more complex phenomenon than the sphere-wake interaction, Basically, this results from the ability of pressure to rotate a disk about a diameter, the variation of disk CD with orientation, and the ability of the disk to generate lift when its axis is not coincident with the direction of motion. A detailed study of the unsteady motions of freely falling disks has been made by Willmarth et. al.[ 63] for Red up to approximately 10, 000. They have been able to map out the boundary between stable (non-oscillating) and unstable (steadily oscillating) disk fall situations in terms of Red and I*, the latter being the ratio of the moment of inertia of the disk about a diameter to the moment of inertia of a fluid

44 sphere of the same diameter. The boundary occurred at Red - 100 -4 -3 except for I* in the range 10 to 4 x 10, where it curved into high-3 er values of Red, peaking at Red~ 170 for I* = 10. At the larger I* values, where the disks tend to behave more like fixed disks owing to the greater inertia, the onset of oscillation at Re k 100 correlates with the initiation of vortex loop shedding observed in wind tunnel tests. The increase in Red on the stability boundary with decreasing I* was thought to be caused by changes in.the phase relationship between the disk pitching motion and the vorticity generation and shedding processes. The unsteady disk motions occurring for Red > 100f were found to fit into three general categories. Disks with large I* (> 10 ) tumbled; that is, after executing two or three pitching oscillations with growing amplitude, the disks would overturn and fall away in a random manner. Small amplitude lateral motions and large amplitude pitching oscilla-2 tions characterized the behavior of disks with I* slightly below 10; decreasing I* brought a decrease in pitch amplitude and an increase in the amplitude of the lateral excursions. Near the stability boundary (specifically, for Red between 100 and 1000) the dimensionless frequency of disk oscillation nd jU where n is the cyclical frequency, was a function of both I* and Red, but for

45 Red > 1000 the dependence on Red vanished and nd/U became a linear function of I* given roughly by nd- 0. 62 - 470 I*. U Lateral motion amplitudes were not recorded. Willmarth, Hawk, Galloway, and Roos[ 17] reported a series of experiments involving disks mounted in a wind tunnel. Two basic cases were studied: a rigidly held disk and disks fixed in place but free to rotate on a diametrical axis perpendicular to the wind stream. A large quantity of data, much of it statistical, was generated by the experiments. Just a few of the more pertinent results will be discussed here. The pivoted disks, which were tested in the Red range from 68, 000 to 636, 000, all were found to oscillate continuously with randomly varying amplitude and phase. Dimensionless frequencies of these oscilla-0. 44 tions scaled with (I*) ~ independently of Red In no case did the oscillations tend to diverge nor did auto-rotation occur. The rigidly held disk was equipped with a moment balance designed to measure the moment acting on the disk about a diametrical axis normal to the free stream. Root-mean-square values of the coefficient of unsteady aerodynamic moment in the range 10 < Red < 10 d

46 displayed no Red dependence. The constant value ofl CM, where MT C - M 7 2d3' M D being the randomly-fluctuating aerodynamic moment, was 0. 0047. Dimensionless power spectra of the fluctuating aerodynamic moment were also obtained for several Red between 187, 000 and 1, 050, 000. These, too, demonstrated a lack of dependence on Red, scaling completely with dimensionless frequency. This implies that there exists no significant variation in the character of the unsteady wake, which is responsible for the fluctuating moment, in the Red range from 10 to

III. EXPERIMENTAL APPARATUS A. INTRODUCTION This Section deals with the design, construction, and operation of the equipment used in the experimental study of sphere and disk flows. The factors considered in the overall design of the experimental apparatus are explained first. This is followed by a discussion of the details of the mechanical and electrical equipment and the manner in which this equipment was used to obtain the experimental measurements. B. DESIGN CONSIDERATIONS The first design question to be settled involved the selection of a working fluid for the experiment. Since rather low Reynolds numbers were to be encountered, the objective was to obtain relatively large, and therefore easily measureable, forces at these low Red To accomplish this it was necessary to operate with a fluid having a large gNv. To see this, consider the expression for sphere or disk drag~ D CD pU d o 47

48 Now Re v p d v P so that 2 v 2 v 2 U Re 2 d 2 d d pd and therefore Dz-~R 2 D 8v Red CDO Since the drag coefficient CD is a unique function of Red, it is clear that at a fixed Reynolds number the magnitude of D depends only on the product Av of the fluid. While v of air is comparable to v of a number of liquids, e. g., ethylene glycol, the density of air is much less than the density of any liquid. Consequently, jv of air is considerably smaller than Mv of most liquids and therefore wind tunnel testing (in air) was eliminated from consideration, at least for low Red Because all usable liquids possess a specific gravity within about 50% of unity, the problem was reduced to one of selecting a suitable liquid with as large a ai as possible. Several liquids were considered, notably ethylene glycol, propylene glycol, glycerol, and various oils. Glycerol was finally chosed for its stability, safety (i.t is non-toxic and non-flammable ), clarity (to permit use of flow visualization) and complete

49 miscibility with water~ This last characteristic was desirable because it permitted adjustment of p, by dilution with appropriate quantities of distilled water. * It was necessary to have this capacity because the strain gage transducers which were used in this experiment had a useful force-sensing range covering a factor of no more than 1000o Referring to the above expression for D, it is seen that at a fixed gi (and v) this limitation on the magnitude of D permitted investigation of a Red range over a factor of about 35 or so, assuming an approximately constant CD. Hence, coverage of a broad range of Red required either several strain gage transducers or adjustment of p, The next matter to be decided was how the relative motion between test body and fluid was to be achieved, i. eo, whether to move the fluid past the body or the body through the fluid. A number of practical considerations supported the choice of the latter method. From a mechanical standpoint, the design and construction of a suitable fluid tunnel would have been much more complicated than was the establishment of a towing apparatus. The power necessary to move *An additional advantage of this fluid system was the availability of tabulated density and viscosity data, e. go, reference 67. Using these tables, p and v could be determined simply by measurement of fluid temperature (assuming a mixture of known proportions)~

50 a large quantity of high viscosity liquid through a tunnel would have been enormous compared with that required to tow the test body through the fluid. Also, the large quantity of energy dissipated in the moving viscous fluid would have produced time variation of fluid properties due to temperature increase unless a heat exchanger were built into the tunnel. Another difficulty with a liquid tunnel would have been the rapid growth of wall. boundary layers at very low Re. In addition to reducing the cross-section dimensions of the uniform flow within the test section, the growing boundary layer would have accelerated the uniform core flow unless the test section were designed to compensate for this growth. However, it would have been impossible to achieve such compensation. over the wide range of Re to be studied in this experiment since boundary layer growth rate depends strongly on Re. An. example of this sort of difficulty is provided by the test section velocity profiles shown by Shair, Grove, Petersen, and Acrivos!s 68] for their blow Re oil tunnel. These profiles indicate that the flow is neither uniform across nor uniform along the test section. Finally, although it was a secondary consideration, the great difficulty of performin.g unsteady motion studies in a fluid tunnel influenced the decision.

51 One significant disadvantage inherent in the technique selected is the rather brief permissible duration of a test run imposed by the finite length of the fluid channelo In the present situation, the test body is limited to a translation of about 120 body diameters. This inhibits the study of unsteady flow phenomena occurring at low Red, where wake instability is slow to develop. Once the decision to move the sphere or disk through a stagnant liquid had been made, a suitable tracking system was necessary to permit smooth one-dimensional motion. Instead of a conventional wheels on rails setup, a linear air bearing system was selected to provide the tracking. Although more expensive than the former, it was hoped that the air bearing system would minimize jiggling and vibration of the translating carriage through elimination of all solid to solid contact, Also, a commercially available V-groove air bearing*. was-uti'lized in. the system, so the expense of designing and fabricating such a unit was not encountered, although designing and building a simple secondary bearing was necessary. The track system consisted of a 32 foot long V-groove air bearing and an equally long 2 inch square tube, parallel to the V-groove EThe Search Linear Air Trough, manufactured by the Search Corporation and distributed by the Macalaster Scientific Corporation:

52 unit and 9 inches to one side of it (Figs. 9 and 10). The carriage supported on these bearings was a triangular plate attached via pivots to three gliders, two of which rode in the V bearing. The V unit fixed the location of the carriage vertically and laterally and prevented pitching and yawing of the carriage. However, owing to the long moment arm of the model support strut, the V bearing was incapable of effectively counteracting roll. The incorporation of the square-tube bearing into the system permitted cancellation of rolling moments. The sixth degree of freedom of the carriage, translation along the track, was controlled by a motor and cable-loop arrangement. The next design problem was the selection of a means of measuring the forces and moments acting on the test body. External, internal, and sting balance systems were studied; strain gages, piezoelectric crystals. and differential transformers were considered for use as transducing elements. Two important factors eliminated an external (i. e., outside the fluid) balance from consideration. First3 an external balance would have had a much lower natural frequency, and therefore much poorer frequency response, than the other balance types owing to the necessarily greater mass supported by the sensitive elements. The other factor was the very low sensible moments due to the long moment arms between the test body and the transducers in an external balance.

53 For the best mixture of simplicity, sensitivity, and good frequency response characteristics, a combination of internal and sting mounted transducers was selected. Since static measurements were required, piezoelectric devices could not be used. Strain gage units were chosen for the sensing elements because of the relative bulk and poor sensitivity of differential transformers. Visualization of the flow past the sphere or disk was chosen over other means of studying the relationship between the flow field and the unsteady wake. Since the expense of the glycerol prevented repeated renewal of. the working fluid, the visualization method had to be noncontaminating. This ruled out the use of dyes or dispersed particles such as fine aluminum powder. The hydrogen-bubble technique was finally settled upon. In this method a fine platinum wire moving through the fluid with the test model is used as the cathode in the electrolysis of water. The very small hydrogen bubbles which are produced along the wire are swept off to act as tracers in the flow. C. MECHANICAL EQUIPMENT 1o Towing Channel The towing channel (Figs. 9 and 10) in which all the experiments were performed is 31 feet 8 inches long, 24 inches wide, and 26 inches

54 high and is normally filled to a depth of 24 inches. The charnel itself, seemingly a very simple piece of equipment, actually called for a great deal of time and effort. The original channel installation was made of formed galvanized steel sheets riveted and soldered together and supported externally by a framework of slotted steel angle. The initial soldering of the channel seams was rinot successful, with the result that the channel leaked badly. This leakage was worsened by flexing of the steel panels as the channel was filled, which tended to crack the seams open at weak points. Considerable energy was expended in various attempts to eliminate the leaks in this metal, channel, includilng repeated soldering, caulking, and taping of the seams. Finally, a liner of fiberglas cloth impregnated with epoxy resin was installed. However, even this last effort failed to stop leakage over a period of time due to failure of the epoxy-to-metal bond combined with leakage through pinholes left in the epoxy-fiberglas matrix. Therefore, the channel. was torn out and scrapped, except for the steel supporting frame. The second and final channel was started with the steel frame retained when the sheet metal channel was torn out. Side and end panels of 1/2 inch plywood were bolted to this frame, flush-head bolts being used to keep the inner surface smooth. The building floor itself formed the bottom of the channel; the corners between the floor and the side

55 and end panels were filleted to approximately a two inch radius with concrete. Using this structure as a permanent form, two laminations of fiberglas cloth thoroughly impregnated with polyester resin were built up on the inner surface. Finally, a surface coat of resin was applied. Black pigment was dispersed in this batch of resin to provide a dark background surface for the flow visualization studies. This installation has proved to be very successful: in nearly a year of use, no leakage whatever has occurred. Two 1000 gallon collapsible rubber tanks* are utilized for storage of the glycerol-water solutions. Fluid is transfered to and from the channel by an electric motor driven pump**, When ordinary tap water is used in the channel, filling is accomplished via the building water lines; the channel is emptied in such cases by simply siphoning the water into a nearby floor drain. 2. Air Bearings As was noted above, the test model and carriage structure was supported on two parallel linear air bearings, one of these being a Vgroove type and the other a simple square tube (Figs. 10 and 11). *Uo SO Rubber Company Sealdrum 1000. **MP Pumps, Inco, model R 15303B.

56 The V-groove trough consisted of three Search Air Troughs, two of which were 12 feet in length while the third was 8 feet long. * A single 32 foot unit would have been better in many respects, but the Air Troughs were only available in lengths of 12 feet or less. Consequ.ently joining mechanisms, one of which is visible in Fig. 11, had to be made; these allowed adjustments to be made to provide precise alignment of the units. Each Air Trough consisted of three aluminum extrusions welded into a rigid unit (Fig. 12). Rigidity was provided by a 4 inch deep I beam. Atop this beam was mounted a special extrusion which includes the 900 V, each surface of which was backed up by a hollow chamber through which the air was supplied. Midway up each side of the V a series of small holes had been drilled and a narrow longitudinal milled slot connected all the orifices in the face of each half of the V-groove. The third extrusion in the structure was a simple 1/8 inch thick right angle which also had a string of holes drilled in each side of the V with the same spacing as the holes in the other tV extr-uSioin The positioning of this piece was such that when it was welded into place the holes intersected the longitudinal slots in the first V mid-way *These Troughs are essentially production versions of the laboratory demonstration unit devised and described by Neher and Leighton [ 69]. Search part numbers are 11001 (8 foot unit) and 11003 (12 foot units)o

57 between the orifices in that extrusion. The purpose of this complex structure was to provide long, narrow air passages with concomitant large pressure drop to minimize the mass flow of air at a given operating pressure without having to resort to using extremely small diameter orifices, which are very difficult to drill. Air was fed into both ends of each manifold at 15-20 psig. At these pressures the flowrate was estimated to be 60-80 scfmo Air flow through the orifices in the center portion of the 32 foot Trough was not observably different from that at the ends of the unit based on a rough check of glider-floating ability, The two gliders used with the V bearing were 7 inch long aluminum blocks, cut down to minimize weight, each with a deep vertical hole in the center into which the pivot support for the appropriate corner of the carriage was fitted. These gliders can be seen in Fig. 11. It was found necessary to place the pivot point of the load attachment as deep in the glider as possible, If the pivot were too high, io eo, above the center of pressure of the glider, instability occurred and, when floated, the glider oscillated in pitch and yaw, producing a noticeable'"chatter" as the glider ends came into contact with the air bearing surfaces, and an equally noticeable increase in friction between the moving glider and the bearing.

58 The square tube outrigger bearing consisted of two 16 foot lengths of 2 inch square, 1/8 inch wall extruded aluminum tubing joined end to end with a fitting that maintained alignment and i.ncluded a pressure seal. The top surface of the unit was carefully filed to eliminate surface irregularities. Two longitudinal rows of 0. 018 inch diameter (#80 drill) holes were drilled in this surface. These rows were 3/4 inch apart, equidistant from the centerline of the tube surface. The holes in each row were spaced at 1 inch intervals and the rows were staggered by 1/2 inch. Air was fed into this unit at the mid-poirnt of each 16 foot section; normal operating pressure was 5-10 psig, requiring a flowrate of about 25-50 scfm. The glider for the corner of the carriage supported by the outrigger tube was a 4 inch long, 1 3/4 inch wide piece of 1/2 inch thick aluminum plate. A vertical hole was provided in the center for the carriage support pivot. Support for both air bearings was provided by cantilever beams attached to the building wall at 6 foot intervals. Each bearing was mounted on each support by a pair of adjustable stud bolts which permit vertical and rotational adjustment of the bearing. Because of the flexibility of the square-tube unit, it was found necessary to support this bearing at more points along its lengthk To accomplish this, Unistrut

59 steel beams were installed between the cantilever supports and parallel to the bearings. Additional pairs of studs were installed between these beams and the outrigger bearing so that that unit is now supported every 2 feet. The air to operate both bearings was drawn from the 100 psi building air line and was passed through a large capacity porousstone filter* and thence through a pressure regulator**, one of which was used for each bearing. One inch copper tubing then carried the air to the various bearing inlet fittings. 3. Carriage and Strut The strut-sting-model assembly was attached to an aluminum carriage (Fig, 11) which was supported through ball and socket pivots by three gliders, two V-type and the other flat-bottomed. Each pivot included a screw thread vertical adjustment for leveling of the carriage. Attached to the carriage was a hairpin shaped spring which engaged a pin on the flat-bottom glider and maintained the alignment of that glider with the outrigger bearing. The base (actually the top) of the carriage was a triangular aluminum plate, with the center removed to reduce weight. One of the supporting pivots was attached to each corner of the carriage base. Fas*Ro Po Adams Co. Type TR Poro-Stone Air Filter **Schrader noo 3468XG Air Pressure Regulator

60 tened to the outside edge of the base, i. e., the edge away from the outrigger bearing, was a vertical frame to which the strut was affixed. Finally, a stiffener plate lying in a plane perpendicular both to the carriage base and the vertical frame was mounted to these pieces to increase the rigidity of the structure. A horizontal arm extended under the I beam of the V-groove air bearing from the lower end of the vertical frame. To the end of this arm was attached the towing cable. This attachment point was selected as a compromise between eliminating the couple produced by drag on the strut and model and eliminating the couple caused by acceleration of the moving mass. The upper end of the model-support strut was mounted to the carriage assembly with two screws. This strut, the leading edge of which had been finished to an approximately parabolic profile, was originally made of Plexiglas to minimize weight. However, the Plexiglas strut was discovered to be far too flexible, so an aluminum strut was installed. The lower end of the strut was a cylindrical aluminum housing, halved lengthwise, for the model-support sting. Mounted atop the carriage base plate was an aluminum box which contained several solid-state amplifiers and circuitry for these, and some of the wiring for the strain gage transducer circuits, This box,

61 referred to below as the amplifier box, also contained a linear accelerometer positioned to sense acceleration along the track system. The equipment carried in the amplifier box will all be described more fully in Part E of this Section. 4, Motor and Towing Cable The carriage is propelled along the track by a continuous loop of stainless steel cable which passes around pulleys at each end of the track. One of these pulleys is driven, through a rubber coupling, by a 1/3 hp. electric motor equipped with a transmission which provides continuously variable output shaft speed in both rotations. * The setup is presently such that the maximum attainable carriage speed is approximately 6 feet per second. The locations of the motor and cable are indicated in Fig. 9. Considerable attention has been given to the problem of isolating the carriage system from the Zero-Max electric motor vibration. The base plate to which the motor and the cable drive pulley were attached was mounted to the building wall and floor through a very rigid structure, The pulley housing was mounted directly on the base plate, but the motor was mounted on vibration-damping motor mounts and, as t Zero-Max model JK3-M3o

62 mentioned above, was connected to the pulley through a rubber coupling. In spite of these precautions, however, considerable vibration, produced by rotational and translational oscillations of the cable drive pulley, was transmitted along the towing cable, Originall.y, a loop of Mylar tape was used for towing in the hope that the vibrations noted above would damp out within the tape. However, the low elastic modulus of the Mylar allowed the tape to act as a weak: spring, permitting low-frequency oscillations of the carriage. Also, the propensity of the tape to develop an electrostatic charge through friction caused it to cling to parts of the track system. This tended to produce slippage and jerkiness in the towing system. Several means of connecting the towing cable to the carriage system in such a way that transmission of vibration from the cable to the carriage would be eliminated were. tried' Though not completely successful., the best: method devised was a very soft rubber spring between. the cable and the carriage. This eliminates much of the vibration, but introduced several other undesirable effects The most serious of these was oscillation of the carriage relative to the cable.' for constant-speed cable motion, any variation in the drag acting on the carriage system(including bearing friction and

63 fluid drag on model and strut) changed the tension in the rubber spring and, therefore, changed its lengtho So, in effect, a lowfrequency vibration, or at least the possibility of one, was substituted for one of high frequency. Two techniques have been used for determining the carriage speed~ The method originally employed utilized a precision 3600 potentiometer driven through a reduction mechanism by the idler pulley in the towing cable loopo The wiper of this potentiometer was connected to one of the two channels of a strip chart recorder and the slope of the resulting x vso t trace gave U. When both channels of the recorder were needed for other data, a microswitch riding on the rim of the drive pulley was wired to one of the event markers (a non-proportional pen) on the recorder. A depression was cut into the pulley rim so that the microswitch would make and break contact once for each revolution of the drive pulley. In the course of this experiment it became evident that some of the mechanical unsteadiness originally attributed to towing cable vibrat ion was actually being produced by non-uniformity of the rotation speed of the drive motor output shaft. This was discovered during the analysis of some test data and was confirmed by a strobe light check of the shaft speed.

64 A geared down, electronically controlled d~ cG. motor* was installed i.n place of the original drive motor. Strobe light tests showed that the shaft speed of. this motor was constant within 1%, With this motor i.:n. use, steadiress of the drive system increased and it became possible to con.rect. t.he towing cable directly to the carriage, thereby eliminating the oscillation permitted by the rubber spring~ Data records from drag tests with the revised drive system displayed a marked decrease in the magnitude of drag force fluctuations. Flow visualization results were also improved, and a puzzling pulsation of the disk wake disappeared with the change of drive motors. Do MODELS AND STRAIN GAGE TRANSDUCERS o. Sphere and Disk The sphere and circular disk (Figs, 13 and 14, respectively) used i.n. the present. tests were both machined from Plexiglas~ Each was 3 inches in. diameter; the disk was 3/8 inch thick, giving a thickn.ess to diametser ratio of 1/ 8. Each. body was made in, front, and rear halves; the rear part, with a circular h.ole to clear the sting support, attached with. screws to the internal strain, gage balance, while the *Gerald TK Heller Co Model 6T60 Variable Speed Motorn.

65 front half-body was slip-fitted into position over a lip on the rear piece, The sphere was checked for sphericity; the results showed that diameter variation was less than 1%. The disk was actually a very short segment of a right circular cylindero The intersections of the front and rear disk faces with the cylindrical side surface were sharp edges. Earlier tests at low and high Reynolds numbers[ 63, 17] indicated that, as long as the disk axis remained nearly parallel to the free stream, there was no apparent difference in aerodynamic properties between a true plane disk and one of small, but finite, thickness. 2. Disk Balance The d. c. -powered strain gage balance used with the disk was originally intended to be used with the sphere also. However, early test results pointed out the need for a separate strain gage balance for the sphere. Such a balance was designed and built, the original balance then being used exclusively with the disk. The original balance will be referred to below as the disk balance. The disk balance consisted of two units, made of aluminum, one of which was to be the drag-sensing transducer and the other a combination pitching and yawing moment transducer (see Figs 14 and 15).

66 The moment: transducer consisted of a solid central disk: and an outer ring which were connected by thin, flat, radial aluminum strips positioned at 900 intervals around the disk circle. These strips were 0. 050 inch wide, 0O 015 inch thick (dimension perpendicular to plane of disk:), and were securely clamped into the central disk and outer ring, the exposed length being 0o 250 inch, The Plexiglas disk mounted to the outer ring, while the sting support connected to the central disk., Consequently, all loads acting on the disk were transmitted through the aluminum strips, which acted as bending flexureso A pair of strain gages was cemented to each flexure, one each on front and back, Considerable attention was paid to positioning the front and back gages as nearly identically as possible. Two types of strain gages were used, two beams being gaged with each type. One type was a U-shaped unbacked semiconductor (piezoresist:ive) strain gage;,* the other, a conve:nt:ional metal foil gage with epoxy film backing.o:'- The for mer has the advantage of possessing a much larger gage factor (140 as opposed to 3 for the foil gage) but. also has several shorteomings when compared with:Kulite type VFP-500-090, 500 ohms resistance. *' Mic ro-Measurements gage nC.O MD-DY-050AH-350, with resistance of 350 ohmso

67 foil gageso * Chief among these shortcomings is the comparative difficulty of installing semiconductor gages. Because each semiconductor gage is a single, thin, brittle crystal filament of silicon, these gages are extremely fragile. The gages must be handled very carefully during cementing and can not be clamped into place while the cement cures. A result of this is a variation from gage to gage in the condition of the cement bond, e. g., the thickness of cement between gage and metal surface might vary. This variation, in turn, causes a variation in the strain response and temperature sensitivity among the gages, A second problem is that, because of the small size and singlecrystal nature of the semiconductor gage, properties of the gage itself can not be controlled as closely during manufacture as can those of conventional foil strain gageso Consequently, gage-to gage variations in gage properties are greater with semiconductor gages. Another disadvantage of the semiconductor gage is that there exists no simple means of trimming the resistances of four such gages, after they have been mounted, to balance the Wheatstone *Definitions and significance of various strain gage characteristics are given in Section 2 of reference 70. Properties of semiconductor strain gages are extensively discussed by Dorsey[ 71].

68 bridge circuit into which they have been wired. * The resistance of a foil gage, on the other hand, can readily be increased simply by rubbing gently on the foil grid with powdered pumice. A fourth, and quite severe, problem with semiconductor gages appeared in the course of this work. It is appropriate to note it here even though it was not recognized at the time the choice of gage types was being made. This problem was "creeping" of the semiconductor gages when a load was applied to the gaged specimen. By this it is meant that, when the surface to which the gage was cemented was strained quickly from zero to a constant non-zero strain level, the semiconductor strain gage initially followed the surface strain but immediately began to relax back to its unstrained length. This phenomenon occurred regardless of whether the surface strain was tensile or compressive. It is not known whether this effect was a plastic flowing of the cement itself or a slippage at the interface between the strain gage and the cement. Several factors are felt to be possible causes of this problem, which did not occur with the foil gages: the necessity of using a *The theory of the Wheatstone bridge circuit is discussed in Perry and Lissner[ 70], Chapter 4. Some basic concepts relating to the use of strain gages in transducers are put forth in Chapter 11 of that work.

69 different cement with semiconductor gages;* the very smooth surface of the semiconductor crystal (making a good cement bond difficult to achieve); larger elastic modulus of the semiconductor crystal, compared to that of metal foil; and the considerably smaller wetted surface area of the semiconductor gage. Because of the combination of advantages and disadvantages, both types of gages were used in order to directly compare the two in operation, The result of this comparison has been that, in the present application, the sensitivity and performance of the foil gages were more than adequate. Consequently, when the sphere balance was designed and built, foil strain gages were used exclusively. Initially, the eight gages on the disk moment transducer were wired into two circuits, the four gages on each diametral pair of flexures being connected together in a four-active-arm Wheatstone bridge in such a way that this bridge circuit would become unbalanced when a couple was applied about an axis perpendicular to both the disk axis and the diameter along which the flexures lay, but would remain balanced when a pure axial (drag) load acted on the disk. This wiring, "A solvent-thinned epoxy compound, Bean Adhesive RP-43, was used with the foil gages. This cement was not compatible with semiconductor gages because the impervious nature of the gage material would not readily permit the evaporation solvent to escape, so Bean RTC, a room-temperature curing epoxy cement free of solvent, was used to bond these gages.

however, had t:o be altered when it becanmwe mvli."-.:?t. that perfonarnce of the originally-planned drag transducer is w;cona pie y w n~3adequate. The failure of the original drag u:niit K.dyolveed several factors, chiefly poor design and the use of semiconldu ct2:r strain gages. The drag transducer, it might be noted, haci bci;e-. designed specifically for semiconductor gages and could not be uodified to accept toil gages, The first alteration of the mnaoment tra~f'sdauei strain gage circuits to permit measurement of drag invol vE'd simply rewiring of the yawing moment gages into a bridge circuit seisi'tive to drag i nstead of yawing moment. This actually required ottly the transposition of two connections on a strip of solder terminalso The desirability of measuring both moments simultaneously with d'rag remrained, however, so another modification was introduced., In this second alteration, the foutr, gages in the yawing moment circuit were separated into two haltf bridges, a sitgle pair of precision. fixed resistors serving to complete, k*r' Whciat;jij e b t-idges (See Fig. 13 for circuit diagram), Each active,a'i b1i idge co'-sisted of the t:wo gages mounted on one of the yawinlg.:n n fleit fl.extutres. Using operatiornal amplifier circuits to be descriibed betlOw, the osutputs from the two bridges were added to produce a signal pr:-ioportional to drag and subtracted to yield a yawing momen.t sig'tali The gages in these circuits were foil gages; the pitching morniert h balan~::e which utilized the

71 semiconductor gages, was not involved in any of the modifications. Lead-out wires for the strain gage circuits were carried through the hollow brass sting support; at the rear of the sting they entered a rubber tube which carried them up and out of the fluid along the rear edge of the strut. The entire installation was then carefully waterproofed with a flexible epoxy compound. * The natural frequency of the disk balance as finally used was 96. 7 Hz. 3. Sphere Balance As was mentioned above, it was originally intended that the disk balance, measuring pitching and yawing moments and drag force, would be used in conjunction with the sphere as well as the disk. However, several factors combined to indicate that a separate sphere balance would be worth while, First, whereas the moment acting on a disk is produced by pressure forces, when the wake is separated, and consequently can be quite large (approaching roughly the order of magnitude of the product of drag force and disk radius), the moment on a sphere in incompressible flow must necessarily arise entirely from azimuthal nonuniformity *Bean Gagekote Number 5, a two-component material that cures at room temperature~

72 of viscous stresses at the sphere surface and is therefore much smaller in comparison, This means that accurate measurement of tile moment on a sphere would be considerably more difficult than in the case of the circular disk. Second, high Reynolds number tests of a twro foot diameter Styrofoam sphere in the University of Michigan 5 foot by 7 foot Low Speed Wind Tunnel showed a very large correlation between yawing moment and side force and also that the side force coefficient, which is pressure produced, is much larger than the yawing moment coefficientf 33]. Thus it appeared that, on a sphere, measurement of lift and side force would yield just as much information as measurement of pitching and yawing moments. Also, because of the greater magnitude of the side force coefficient, side force measurement could be accomplished with a larger ratio of signal to noise. Finally, the elimination of the necessity of fitting the balance within a thin disk, coupled with the knowledge gained in the construction and operation of the disk balance, meant that a significantly more accurate and reliable drag force transducer could be devised. The sphere balance (Figs. 13 and 17) was made up of two units, a drag force transducer and the supporting sting. The lift and sideforce transducers were built into the sting.

73 The drag transducer was a simple one-piece aluminum parallelogram flexure in the form of two parallel rectangular bars connected at their ends by thin strips. One bar was connected to the sting while the other was attached through an aluminum yoke to the Plexiglas sphere. Therefore, the forces acting on the sphere were transmitted to the supporting structure through the thin strip flexures. Four foil strain gages* were bonded to these flexures and wired together in a Wheatstone bridge in such a way that the bridge output voltage was dependent only on the drag force. In operation, the resonant frequency of the drag transducer was 31 Hz. The lift and side force transducers operate on the cantilever beam principle. The bending moment produced in a simple cantilever by a transverse force at the free end increases linearly with distance along the beam away from the free end. On the other hand, a pure couple applied to the free end of the beam gives rise to a bending moment that is constant along the beam. Thus, if the difference between bending moments at two stations along the cantilever is measured, it is proportional only to the transverse force at the end of the beam, whether or not a pure couple is also present. The bending, moment on the sting was measured at two stations by Meicro- Measurements type MA-13-045AL- 350.

74 mounting a pair of strain gages, * one on each side of the sting, at each station. These four gages were wired into a differential bridge, the output of which was linearly proportional to the transverse force acting at the center of the sphere. Two such units, oriented at right angles to one another, permitted measurement of lift and side force. The sting was a hollow, circular, brass cylinder, the open center being necessary for the lead-out wires from the strain gage circuits. The cross-section of the sting at the strain gage stations was reduced to a square for two reasons. First, the reduction in cross-section area at the strain sites increased the strain level of the strain gages, yielding greater sensitivity. At the same time the rest of the sting was kept stiff, which made the natural frequency of the sting high enough to allow good frequency response of the unit. Second, provision of flat surfaces on which to cement the gages considerably facilitated the accurate positioning and firm bonding of the gages. When the sphere unit (model, transducers, sting, and strut) was assembled and submerged, the natural frequencies of the lift and side force transducers were 18 Hz and 8. 5 Hz, respectively. The lower natural frequency of the side force transducer resulted from coupling *Micro -Measurements gage number MA- 13-0 62AQ- 350.

75 of sting bending with torsional deflection of the supporting strut. The lead-out wires from the drag, lift, and side force transducers were carried through the sting and up out of the fluid through a rubber tube fastened to the trailing edge of the strut. As in the case of the disk balance, the sphere balance unit was completely waterproofed with the epoxy compound. 4. Accelerometer The accelerometer mounted in, the amplifier box on the carriage was originally constructed solely as a device on which to develop techniques for handling and installing semiconductor strain gages and with which to study some of their characteristics. When completed, the unit showed an appropriate combination of sensitivity and frequency response to be useful in the experiment, so it was incorporated into the equipment L However, the creeping effect associated with the semicornductor gages (and noted above in the discussion of the disk balance) was observed during the course of the experiment. Shortly thereafter, the accelerometer was damaged, one of the semiconductor gages being destroyed- This necessitated regaging of the unit, which was accomplished with foil gages to eliminate the creeping effect. Measurement of acceleration was useful for two different reasons in this experi:sment First,. it provided a measure of the

76 unsteadiness in the carriage propulsion system. The other use was cancellation of that component of the drag balance output which was due to acceleration of the sphere or disk mass when the carriage motion was unsteadyo Whenever the carriage was accelerated, the sphere (or disk) and the fluid within it resisted the acceleration with a force equal to the total mass times acceleration. Because it acted along the line of motion, this force was sensed by the drag balance. Consequently, the output voltage from the drag balance was not proportional to drag under these conditions. This situation was corrected by using analog circuits to subtract the proper portion of the accelerometer signal (equal to the inertial force) from the drag balance signal to produce true unsteady drag. A strain-gaged aluminum cantilever beam, with a lead mass on its free end, was the active element of the accelerometer. Four foil strain gages* were cemented to the beam, two on either side. These gages were wired together in the usual Wheatstone bridge circuit. Figure 18 shows a sketch of the accelerometer, which has a natural frequency of 238 Hz and a sensitivity of approximately lOPv/ft/sec The accelerometer was mounted so that the beam hung vertically. When accelerated, the suspended mass exerted a transverse inertial *Micro.-Measurements type EA-09-050AH-120.

77 force on the end of the beam, thereby bending it and inducing strains of equal magnitude but opposite sign in the strain gages on opposite sides of the beam. No attempt was made to damp the motion of the accelerometer beam, so a sharp peak existed in the accelerometer frequency response curve at the natural frequency of the beam. However, the influence of this resonance peak was minimized by passing the accelerometer output through a notch filter (described below). E. ELECTRICAL EQUIPMENT 1. Amplifier Circuits Since the output levels of the several strain gage circuits used in this work were on the order of a few millivolts or less, amplification was necessary to bring these voltages up to the magnitude required for recording or display on an oscilloscope. For this reason, and also because several operational amplifiers were required for compensation purposes (e. g., removing inertial forces from the drag transducer signal, as referred to above) and for correlation and filtering of signals, it was decided to utilize inexpensive solid-state differential operational amplifiers, with appropriate attendant circuits, in

78 the experimental setup. In addition to their very low cost, * these amplifiers possess the attributes of low noise ( 2 Iv rms referred to input), great flexibility, and small size. ** As presently used, the basic solid-state amplifiers are wired into three types of circuits. The various circuit types are all labeled in the overall circuit diagram of Fig. 16. Four amplifiers operate as simple negative-feedback floating-input d. c. gain amplifiers, the gains of which are set by appropriate combinations of input and feedback resistors. These units amplify the output signals from the strain gage transducers. Several other amplifiers, also operating with negative feedback, are used as unity gain adders. By proper adjustment of the level of several inputs with potentiometers, these amplifiers achieve compensation of the sort noted above. With a single input signal, one amplifier operates as a sign inverter, necessary in the process of summing and differencing two signals mentioned above in the description of the disk balance. *At the time of purchase (mid-1966), the Philbrick PP85AU amplifiers selected cost $17 per unit. ~*Five amplifiers and necessary circuitry were mounted in the 2x3x8 1/2 inch amplifier box.

79 One other amplifier is used in an active notch filter circuit, the design and characteristics of which are fully described elsewhere[72]. The very sharp notch of this filter is set at the resonant frequency of the accelerometer so that, when the acceleration signal is passed through the filter, the resonance component of the accelerometer output is greatly attenuated. This is necessary because of the combined effects of the large resonance peak in the accelerometer response and the fact that this resonance is readily excited by the mechanical vibrations that remain in the carriage towing system. The experimental results are not significantly influenced by this selective filtering due to the relatively high frequency involved. The filter frequency is 238 Hz, while the sphere or disk vortex shedding frequency, the dominant fluidproduced periodicity, is no greater than 5 Hz, almost two orders of magnitude lower. The d. c. gain amplifier and notch filter circuits are all mounted in the amplifier box on the carriage. Several factors were involved in the decision to locate them there. Because of the great length of

80 wiring between the strain gage transducer circuits and the external signal-handling equipment (approximately 80 feet via the overhead instrumentation cable and back to floor level), considerable pickup of hum and noise by these lead wires was expected. By placing the gain amplifiers very near the signal sources, the effect of these disturbances on the signals of interest was minimized. Another factor involved the construction of the carriage-strut-model assembly, which was such that the carriage was out of balance due to the center of gravity lying outside the V-groove air bearing. To remedy this condition, some additional mass was required near the outrigger bearing corner of the carriage to shift the center of gravity to a position between the bearings. Finally, little penalty was involved in locating the amplifiers on the carriage because they weigh very little and also because the necessary wiring was already present, used by the strain gage circuits. Power for the amplifiers is provided by +15 vdc and -15 vdc lines from an external source through the overhead instrumentation cable. These voltages are also stepped down with dropping resistors to +3 and -3 vdc to power the strain gage circuits. Outputs from the strain gage transducers are fed directly to the inputs of the gain amplifiers; the amplified signals pass through the overhead instrumentation cable to the external equipment.

81 The operational amplifiers (summers.and:sign.inverters) are mounted.in. a box similar to the one on the carriage but are not carried with the moving system. The wiring of these amplifiers is readily changed to provide the specific operations required by the test being performed. These amplifiers use the same ~ 15 vdc power as the carriage mounted units. 2. -External Units Three pieces of electronic equipment in addition to those mentioned were generally used in conjunction with the force and moment measurements. These were a two-channel strip chart pen recorder, * a dual-beam oscilloscope, ** and a regulated direct current power supply. *** The power supply was actually two identical but independent units in the same chassis, each of which had a floating output. Both outputs were adjusted to 15 ~ 0o 01 volts. The outputs were wired together to provide +15 v and -15vw, relative to a common ground, to power the amplifier and strain gage circuits. *Moseley Autograf Model. 7100B Strip Chart Recorder. *Tektronix Type 502A Dual-Beam Oscilloscope. #**Trygon Dual Lab Supply DL40-1.

82 F. FLOW VISUALIZATION EQUIPMENT The hydrogen bubble flow visualization technique has come into extensive use in hydrodynamic studies over the past several years. It is commonly used for observation of water flows, although Criminale and Nowell[ 73] recently reported using the technique in aqueous mixtures of up to 40 % glycerol. * The many uses and refinements of the method are covered in detail elsewhere[ 74, 75]; only the application will be considered here. A sketch of the flow visualization equipment is included as Fig. 19. Because distilled water and glycerol, the two constituents of the working fluid, are very poor electrolytes, the addition of sodium chloride was necessary to permit an adequate rate of hydrogen bubble generation at reasonable voltages, i. e., under 100 vdc. The basic glycerol-water mixture produced no noticeable bubbling even with 200 volts potential difference between anode and cathode. Only a small concentration of salt was required, typically 0. 05 -0. 10% by weight of solution. *To the author's knowledge, the experiment reported here marks the first application of the hydrogen bubble technique to high-concentration (>50%) glycerol mixtures.

83 Originally, the platinum bubble-generating wire was stretched between the tips of two insulated brass rods which were fastened to the ends of a plastic strut*. This strut was bolted at its center to the model-support sting in such a way that the wire was positioned ahead of the test body and lay in the horizontal plane which included the sting centerline. A second wire was later added in the wake of the test body. This wire was actually two wires, the outer ends of which were soldered to the two brass rods supporting the original wire. The inner ends of these wires were attached to a semicircular ring which was, in turn, soldered to a third brass rod running parallel to the sting and fastened to the plastic strut. The semicircular ring was included to permit this second wire to lie in the appropriate plane without interferring with the operation of the sting balance used with the sphere. Early flow visualization tests with the original bubble generator demonstrated the existence of a serious flow interference problem. Examination of sphere and disk flow photographs indicated that the plastic strut holding the wires had a strong stabilizing effect on the *Bubble-generating wires were either completely bare or insulated in segments to produce streaklines. Commonly used "kinked" wires, made as described by Clutter et al [ 74] by rolling the wire through a pair of gears, did not work successfully in the highly viscous fluid.

84 wake because it was close enough to the sphere or disk to penetrate the tail of the separation bubble. To circumvent this wake interference, a 1/8 in. diameter brass rod bent into the shape of an inverted "U" was employed to position the bubble-generating wire. The cross bar of this wire holder, which was above the fluid surface (see Fig. 19), was fastened to a plastic arm on the carriage. The wire was held in tension at the appropriate depth by the insulated vertical legs of the U-shaped holder. A wire in the overhead instrumentation cable connected the bubble generator to the grounded negative terminal of a direct current power supply. * The positive terminal of the power source was connected to an aluminum strip attached to one wall of the towing channel below the fluid surface. This strip thus acted as the anode in the electrolysis process. The sheet of hydrogen bubbles which streamed from the platinum wire as it moved through the fluid was illuminated by a horizontal plane of light over a 6 ft long section of the towing channel. Photospot lamps were mounted in aluminum housings on both sides of the towing channel. Light from the lamps in each housing passed through a pair of collimating slits (and a 450 mirror in one housing); the light *Oregon Model A3A Variable Voltage Regulated Power Supply.

85 sheets so produced shone straight down into the fluid, parallel to the channel walls. These vertical light beams were reflected from mirrors attached to the channel walls at 450 angles, thereby producing a horizontal light sheet which illuminated the bubbles and the sphere or disk. Illumination from both sides was necessary to avoid loss of detail in shadow regions. Optimum results were obtained when the collimating slits in the lamp housings were adjusted to produce a light sheet that was approximately 1/2 in. thick at the center of the channel. This thickness was required in order that the intensity of illumination be great enough to produce worthwhile photographic results. A thinner sheet would have been more desirable in that it would have reduced the time required for the hydrogen bubbles to rise out of the light sheet. With the 1/2 ino thick sheet, this time was of the order of one hour. Since it was not usually possible to proceed at the leisurely rate of one test run per hour, many of the flow visualization tests suffered somewhat from background scatter caused by bubbles from previous test runs remaining in the light sheet. (This will be evident in some of the results presented in Section IV.) A Polaroid camera was attached to a frame fastened to the towing channel. The camera position is indicated in Fig. 19; the viewing direction was made as close to vertical as possible without allowing

86 the air bearing to obscure part of the bubble field. One of the event markers in the strip chart recorder was wired to the flash triggering switch in the camera shutter so that flow field photographs could be related to the recorded forces and moments.

IV. EXPER IMENTAL RESULTS A. EXPERIMENTAL TECHNIQUE 1. Fluid Measurements Evaluation of: the fluid properties needed to reduce the experimental data to coefficient form was accomplished in two ways, the choice of method depending on the type of working fluid in use. Whern the channel fluid was water or a simple glycerol water mixture, values of g and p were obtained by interpolation from the tables in reference 67 after T of the fluid had been measured to within 0. 10C with a mercury thermometer. In the case of glycerol water mixtures, the glycerol concentration in each such mixture was determined by measurement of the fluid specific gravity with a pycnometer. and an-.. analytical balance. * Introduction of sodium chloride into the glycerol water mixtures meant that the tables of reference 67 could not be used for,L and p, despite the fact that the viscosity of a mixture appeared to be unaffected by the salt addition, because specific gravity measurement was no longer sufficient to determine the glycerol content of the mixture once the salt was added, Consequently, when a glycerol-water-salt mixture was used, *Hydrometers could nost be used for this measurement because of the high fluid viscosityo 87

88 v and p of the mixture were carefully calibrated as functions of T, using a Cannon-Fenske Routine Viscometer and a pycnometer, respectively, for the measurements. These calibrations were then used in conjunction with measurement of T of the fluid to determine v and p for each run. It should be noted that T of the fluid was always measured along the test model path in the channel, as it was found that T could differ by as much as 1 0C between this path and the channel walls or fluid surface. Although the viscosity gradients resulting from this variation of T did not significantly affect the test results, accurate reduction of the experimental data required that v (and therefore T) of the fluid passing over the test model be known as precisely as possibleo 2I Balance Calibration and Circuit Adjustments The sphere and disk balance systems were calibrated by simply hanging various brass weights at appropriate places on a balance and recording the system output on the strip chart recorder. Typical calibrations of the sphere balance are given in Fig. 20; clearly evident is the high degree of linearity of the transducer outputso Not apparent in the calibration plots, but confirmed in the course of calibration tests, are the lack of interactions between transducers and the ability of the balance to accurately resolve an arbitrary force into its

89 lift, side force, and drag components. Transducer interactions were not absent in the disk balance and, therefore, several operational amplifiers were required to compensate for these interactions. The calibration and adjustment procedure for this balance became a trial and error cycle of applying weights, reading transducer outputs on the recorder, and adjusting the various potentiometers in the circuit (Fig. 16) to correct for the interactions that occurred. Although it was not possible to comnpletely eradicate transducer interactions in this manner, the reduction achieved in degree of interaction was sufficient to permit reasonably accurate measurement of P and Y (to within 10 %). To eliminate the force due to test body inertia from the drag measurements, the following technique was employed. The test body was filled with the working fluid and mournt-ed on the carriage in its proper orientation, but in. such a manner that the body was not submerged. The carriage was then accelerated up and down the track with the appropr:iate potentiometer being adjusted until no drag force was indicated on the recorder. 3. Operation of the Experiment Once the calibrations and fluid property measurements had been performed, operation of the experiment was straightforward. After

90 allowing at least 1/2 hour for the electronic equipment and strain gage transducers to warm up to thermal equilibrium, the remaining equipment was activated (air bearings, chart recorder, voltage on bubble generator, and flow visualization lights). The actual test run was controlled entirely from the drive motor, which was manually operated. This was done to avoid the possibility of accidentally damaging the strain gage balances. When the flow visualization equipment was being used, test runs were made at night, since normal daytime light in the building provided far too much background illumination in the channel. This background light was intense enough that, even on a cloudy day, the light reflected from the hydrogen bubbles was indistinct to the eye and did not register at all on film. Even at night and with high-speed Polaroid film, * an exposure duration of 1/30 second at the maximum camera aperture (f/4. 7) was required to produce a worthwhile photograph. This meant that run speeds had to be restricted to approximately one foot per second or less, because at higher speeds the photographic image became too blurred to be meaningful. Since the camera shutter was tripped manually, the experiment generally required two operators when flow visualization studies *Polaroid Polascope, with an equivalent ASA speed of 10, 0000

91 were made, the second operator handling the camera and also acting as observer at the flow visualization section of the channel. Upon completion of a test run, the carriage was backed up to the starting position and a period of 10-15 minutes was allowed to elapse before another test run was begun. This was done to permit the fluid disturbances to die out. 4. Data Reduction Quantitative data for a test run consisted entirely of the traces on the strip chart record. Frequencies and magnitudes of forces and moments were measured on the strip chart and converted to the appropriate dimensions through use of the calibration data. Velocity was measured, either from the slope of the x-t trace if the position-indicating potentiometer was used,or from the period of the event marker pips if the microswitch on the cable drive pulley was in use. The values of U,,9 and p were used thento reduce the data to dimensionless coefficient form. Bo RESULTS OF SPHERE EXPERIMENTS 1o Sphere Drag a. Steady Drag. Steady sphere drag was measured through most of the range 5 K Red 120, 000. This data, reduced to

92 coefficient form (CD versus Red), is plotted in Fig. 21. The solid curve in Fig. 21 is a mean curve drawn through the data of Fig. 7. The general overall agreement between the data from these tests and the previously existing data serves both to corroborate the existing data and to indicate the accuracy of the system used for the tests reported here. The greatest data scatter occurs at high subcritical Red (5500 to 120, 000), where it also appears that the CD values are consistently higher than the curve. Two comments regarding the data in this range are pertinent. First, the actual spread of the CD - Red data through which the curve of Fig. 21 was drawn is worth noting. This data spread is evident in Fig 7. When the data of Fig. 21 is compared directly with that of Fig. 7, most of the points fall within the band of data points in Fig. 7. The second point to be noted is the fact that almost all of the data in this Red range was obtained in what amounted to a shakedown test of the experimental apparatuso A great deal of mechanical unsteadiness existed in the system at that time, resulting in drag force records that were difficult to evaluate with accuracy, particularly at the lower Red values in this range. The experience gained in this early testing led to considerable refinement of the experimental equipment.

93 Tests in this Red range with the refined apparatus would yield results with much less scatter. However, such tests have not been performed because attention was focused on lower Red ( < 1000), chiefly because the flow unsteadiness at lower Red is more regular. Some scatter also exists in the rest of the data (Red < 2000), although no consistent deviation from the curve is evident here. An interesting feature of this data is the greater consistency of points for 400 < Red < 2000~ Data in this range was obtained solely with the balance designed specifically for the sphere, whereas the rest of the data in Fig. 21 came from tests with the sphere mounted on the disk balance. It became very evident in the course of these experiments that the disk balance was not as capable of producing repeatable drag results as was the sphere balance. To some extent this is attributable to the interactions among the tranducers in the disk balance. Another factor was the nonlinearity of the disk drag transducer at drag loads greater than about 0o 45 pound~* b. Unsteady Drag. The principal result of the attempt to study the unsteady component of drag during steady sphere motion is that it is very small0 In all cases the unsteadiness in the measured drag was no more than about 5% of the total drag, regardless of whether or not vortex shedding occurred. This, of course, does not include sWall effects were not studied directly however the close agreement of the drag results with existing data led to the conclusion that such effects were negligible0

94 the early stages of wake buildup, where large drag fluctuations sometimes existed. The magnitude of the first two or three fluctuations depended greatly on the acceleration imparted to the sphere at the start of a run. In general, more nearly impulsive starts produced greater fluctuations. Once the sphere wake had built up to its quasi-steady configuration (usually after two or three cycles of vortex shedding), the drag unsteadiness remained small and did not demonstrate a clear relationship to the vortex shedding process. Figure 22 is a copy of the drag and side force recorded during a typical test run at Red = 532. * Despite the large (though irregular) fluctuations in side force, which indicate vigorous unsteadiness in the sphere wake, very little related unsteadiness appears in the drag record. In fact, the variation in drag is small enough that it may be due entirely to mechanical unsteadiness in the experimental apparatus. Other run records have been obtained which display even less drag fluctuation (at similar Red) than exists in Fig. 22. A noteworthy feature of the data record in Fig. 22 is the gross difference in the nature of the development of unsteadiness in drag and side forceo The drag force had built up to its maximum value and *It should be pointed out that high frequency oscillations attributable to mechanical unsteadiness were smoothed out in the process of tracing all data records presented in this paper.

95 passed through three cycles of oscillation before the side force built up into its first (and largest) significant peak. This feature, a characteristic of all data records in which drag was combined with either side force or lift, indicates that the first few cycles of vorticity shedding in the sphere wake are nearly axially symmetrical and are apparently unrelated to the lateral wake oscillations which develop later in the process of vortex loop shedding~ c. Drag During Acceleration. For the purpose of studying acceleration effects, a number of test runs were made wherein the carriage was disconnected from the towing cable and attached instead to a long rubber band, the other end of which was fastened to the track system. The carriage was then drawn back along the track, held in place against the tension in the rubber band until the channel fluid became still, and then released, with drag and acceleration being recorded during the ensuing run. A data record from one such test run is given in Fig, 23I The run which produced the record of Fig~ 23 was made in water so that high Red was reached: the maximum value of Red, corresponding to the instant when the acceleration trace passed through zero, was 66, 000o Velocities at several instants were determined by graphical integration of the acceleration record. Red values were computed from these velocities and used to obtain appropriate CD values from Figo 7,

96 these in turn being used to calculate the D values making up the dotted curve in Fig~ 23. As may be seen, the measured drag is closely approximated by the "steady-state" drag curve except at three points. By far the most interesting of the three points of difference is the one at startup of the run, where a non-zero flat spot occurs in the measured drag. The constant value of D existing during this initial portion of the run, where the acceleration is also essentially constant, is within 3% of the D value computed according to the apparent mass concept discussed in Part B. 4o of Section II. Nearly identical results were obtained from several other test runs in which a varied from max 3. 0 to 6. 4 ft/sec2; calculations of sphere displacement revealed that the sphere travelled a distance approximately equivalent to one sphere diameter before the wake had grown large enough to destroy the potential flow about the sphere. An important comparison can be drawn between the above result and theoretical calculations of the flow about an accelerated sphere. Although no such calculations have been made for the case of constant acceleration, Chow[ 76] recently studied the flow past an impulsively started sphere~ Using the method of matched asymptotic expansions in a boundary layer type of analysis, he determined that, at high Red, flow separation would appear at the rear stagnation point on an impulsively started sphere after the sphere had travelled 0. 21 of a diameter.

97 A more correct comparison can be made by considering the corresponding results for a circular cylinder: calculations[ 21, po 214 and p. 221] show that separation on a cylinder occurs after a travel of 0. 26d for constant acceleration from rest compared to 0. 16d for an impulsive starto If it is assumed that the result for constant acceleration of a sphere bears the same relationship to the impulsivestart result as exists in the case of the cylinder, Chow's result scales up to 0. 34d for the distance covered by a constantly accelerating sphere before separation appearso Since the sphere travels id before the apparent mass concept becomes invalid, it is evident that considerable growth of the separation bubble must occur before the pressure distribution about the sphere becomes significantly different from the potential flow pressure distribution~ Similar acceleration tests were attempted in a glycerol-water mixture which reduced the maximum value of Red to about 300, but the low values of Red existing during the early stages of these test runs meant that the vorticity about the sphere diffused so rapidly that no period of potential flow could be observed, Also, incorrect drag compensation precluded the type of drag comparison shown in Fig. 23. The other two points at which the measured and calculated D values in Figt 23 differ significaently are probably related to the shedding of vortices in the sphere wakeO This would correspond to the

98 measurements by Schmidt[50] of the velocity of a sphere dropped from rest in a vertical column of fluid. He found that fluctuations in sphere velocity during acceleration coincided with the shedding of vortex loops. A similar phenomenon occurred during startup of most of the normal sphere drag test runs- the drag force measured during a rapidlystarted drag test would exceed, or "overshoot", the steady value. Some drag overshoot is evident in the low Red drag record of Fig. 22, but the effect was much more pronounced in high Red runs, where the overshoot was as much as 30% above the steady drag. Corresponding phenomena for impulsively - started circular cylinders and flat plates were recently studied by Sarpkaya[77]. Flow visualization enabled him to relate the drag overshoot to the growth and shedding of the initial vortex. What probably happens is that, when the sphere is put into motion very rapidly at high Red, vorticity is slow to diffuse and, therefore, accumulates rapidly in the growing separation bubble behind the sphere. Although this bubble of trapped vorticity soon reaches unstable proportions, the growth of the bubble is so rapid that the bubble becomes much larger than its steady-state size before instability succeeds in causing a vortex loop to be cast into the sphere wake. During this period the pressure drag on the sphere rises above its steady value

99 both because the separated flow area is temporarily larger than it should be and because the enlarged wake bubble forces the outer flow aside, reducing the base pressure below its eventual steady value. 2. Transverse Force on a Sphere Transverse forces (lift and side force) acting on a sphere constrained to rectilinear motion were studied at values of Red up to 2000. In general, the results of these tests serve only to provide order of magnitude estimates of forces and frequencies. The principal factor behind the lack of consistency in these results was flow interference produced by the structure employed to support the sphere. The extent of the flow interference effects will become evident in the discus - sion of the test data below. The conventions for lift (L) and side force (S) used in this discussion are indicated in Fig. 24. a. Magnitude of the Force. Magnitudes of the transverse forces measured in these tests are shown in Fig. 25. Although normal practice is to use the root-mean-square coefficient form to present data of this nature, these results are given in peak-to-peak coefficient form (CFpp) for two reasons. First, the test runs were of such short duration that meaningful time averages of the transverse force could not have been obtained in most cases. Consequently,the strip chart

100 records were analyzed to provide this data~ Then, because the data records for L and S were generally quite irregular, it was not possible to directly reduce the peak-to-peak measurements to root-mean-square valueso Only occasionally did anything approximating a sine wave appear in. the L and S data recordso It should be noted that the coefficient values plotted in Fig. 25 are for the total transverse force rather than for L or S aloneo Willmarth and Enlow[ 33], as noted in Section II, found that CL2 - 062 at supercritical values of Red. Comparing this number with the data in Fig. 259 it is apparent that the nondimensional transverse forces are weaker by a factor of three or four at these low Red. Where it could be determined, the dimensionless frequency of the transverse force was calculated; these values are noted in Fig. 25. The data appears to indicate that the stronger force fluctuations are associated with, lower frequencies~ bo Frequency of the Unsteady Force~ Many test runs were made in which L and S were not both recorded. Although values of CFpp could not be determined from such runs, it often was possible to measure the frequency of L or S aloneo As a consequence, frequency data is more abundant than is the force datao All dimensionless frequency (f*) data obtained with the sphere is plotted against Red in Fig~ 26W For comparison, the results of Kendall's[28] hot wire

101 experiments are shown, as are the results of the falling-sphere tests performed by Magarvey and Bishop[ 22, 25]. As was the case with the CFpp data, this frequency data, taken as a whole, makes little sense. In addition, the data subset that seems most consistent, that obtained with the side sting, can not be considered as truly representative for reasons to be discussed below. Much of the scatter in the data can be explained by two features evident in the original data records. One of these features was the fact that whenever a very low frequency (f* <. 05) appeared in a data record, a higher frequency fluctuation was superimposed on it, although the higher frequency could not always be determined because of irregularity in the record. Figure 27 is a reproduction of a data re cord in which this low-frequency phenomenon appeared. Because the wave-length of these low-frequency oscillations was a large fraction of the total run length, it was not possible to investigate this phenomenon in any detail. The second feature noted in a number of data records at Red above 500 was a breakdown of regular L and S oscillations (with f* in the vicinity of O0 15) into irregular fluctuations at roughly half the original frequency. The.end. of the L record in Fig. 28 illustrates this. Although the evidence was not conclusive, this phenomenon was taken to indicate a transition from strictly planar vortex loop shedding into a

102 helical mode of sheddingo In this connection it is worth recalling the work of Preukschat[ 53] noted in Section II, in which he noted that the unsteady motion of a freely rising sphere was invariably two-dimensional at first, but often evolved into a helical motion after a few cycles Because of the effect of these two phenomena on the frequency data, it is justifiable to consider the group of data points above fd/U = 12 in Fig~ 26 as being most reliableo If this is done, it is apparent that the data defines a smooth curve joining the high and low Red data given by Kendall[ 28]0 c. Flow Visualization Results-Sting Interference. The minimum Red for flow unsteadiness to occur, sometimes termed the lower critical Red, varied with the sting orientation. Early tests with the short sting in its normal position aft of the sphere indicated a value of 290 for Red min This unexpectedly high value, combined with the fact that the unsteady transverse forces were quite irregular when they did occur, was somewhat of a puzzle at first (recalling the discussion of sphere flows in Section II), Since it was already known at this point that the Plexiglas bar originally used to hold the hydrogen bubble wire had profoundly stabilized the sphere wake through its interference with the tail of the

103 separation bubble, * the possibility existed that the supporting strut was stabilizing the wake and preventing the regular shedding of vortex loops. As can be seen in the flow visualization photograph of steady flow (Red = 269) in Fig. 29a, the leading edge of the strut was indeed in close proximity to the tail of the separation bubble~ To test this hypothesis, then, a 15 ino extension was added to the sting~ However, repetition of the short-sting tests produced nearly identical resultso The correct solution to the puzzle was discovered after one of these tests when the strip chart recorder was inadvertently left on while the carriage was being backed up to its starting positiono The resulting data record showed a very regular oscillating transverse force~ The obvious conclusion was that the sting itself, an inherent part of the experimental apparatus, was interferring with the sphere wake flow. The existence of significant sting interference was proved in a series of test runs in which the sphere was removed from the strutsting support, weighted, and suspended from the carriage by two threads~ Flow visualization tests with this setup revealed vortex loop shedding with a high degree of regularity~ In addition, Redmin in the absence of sting interference was found to be approximately 215. *With the Plexiglas bar in place, one test run at Red = 500 produced no noticeable unsteady transverse force'

104 Photographic results from several of the flow visualization tests with the suspended sphere are presented as Fig. 29b, 30, and 31. Figure 29b is a photograph from a test run at Red = 192. * The flow was steady and, except for the difference in length of the separation bubble, appears very similar to the flow in Fig~ 29a. Figure 30 is a sequence of three flow photographs, taken at the same position along the towing channel, illustrating the rolling up that occurred in the sphere wake when vortex loops were being shed in the L (lift) mode (iL e. the loops were being alternately shed toward and away from the camera, which was looking down into the channel). The Red for this test run was 295. Visual observation during this (Fig. 30) and other runs with the suspended sphere at low Red disclosed the fact that most of the wake activity stems from the tail of the separation bubble, rather than from the vicinity of the sphere itself~ The tail of the separation bubble appeared to pulsate longitudinally. Whether the bubble was actually stretching and shrinking or simply oscillating laterally into and out of the light beam is not knowno It is interesting to recall that Taneda [20] also observed this pulsation, but as he, too, used a sheet of light *The random bright streaks appearing in this and several other flow visualization photographs are produced by hydrogen bubbles remaining in the illuminated region from previous test runs.

105 to illuminate a plane through the wake, the question is not resolved. It was very evident, however, that at low Red the vortex loops that were carried into the sphere wake were drawn out from the tail of the separation bubble~ This explains why the presence of the sting had such a profound effect on the wake unsteadiness By tying down the tail of the wake bubble through the no-slip condition at the sting surface, the sting prevented oscillation of the tail of the wake bubble and thereby hindered the process of vortex loop shedding. The separation bubble had to become more unstable than it would have been in the absence of the sting to overcome the damping effect of the shear stresses at the sting surface, and consequently Redmin was raised above its normal value. Vortex loop shedding in the S mode is clearly in evidence in Fig. 31, a flow visualization photograph taken during a run at Red = 377. Here the lateral oscillations of the wake were somewhat more pronounced than if the sphere had beers rigidly held because the sphere was free to sway from side to side in response to the transverse forces produced by the vortex shedding~ In contrast to the very evident vortex loop shedding of Fig. 30 and 31 are the flow photographs in Fig. 32, taken during a run with the short sting at Red = 332~ Also included in Fig~ 32 is a portion of the data record from this run, showing a high degree of regularity compared

106 with most of the test datao The first photograph, taken at the point indicated on the data record, is nearly identical with the steady flow photograph of Figo 29ao Closer inspection reveals a slight curvature of the wake behind the strut and an apparent thickening on the lower side of the separation bubble~ The second photograph is of the same section of the towing channel at a later time and shows the distortion of the wake by the shedding process. In spite of the magnitude of the unsteady side force (close to 20% of the magnitude of D), the wake appears relatively calm and displays no evidence of discrete vortices. This absence of clear, distinct vortices was characteristic of all the flow visualization tests with the sting-supported sphereo One other data record is included here as Fig. 330 This run, at an Red of 715, is presented to illustrate the high degree of irregularity normally demonstrated by the transverse forces at higher values of Redo Clearly, it was impossible to obtain reliable frequency and force level data from such data records as thiso Because vortex loop shedding was never clearly evident in the tests of the sting-mounted sphere, firm conclusions could not be drawn regarding the relationship between the unsteadiness in the wake and the unsteady forces. However, it was possible to obtain some insight into this relationship by studying the data from the few flow visualization test runs in which the fluctuations in S were relatively regular~

107 Considering Fig. 32, for example, it can be seen that two events occurred simultaneously when the sphere was about five diameters back from its position in the first photograph: the sphere wake was deflected slightly toward the top of the photograph, and S passed through a negative peak, corresponding to a force acting toward the bottom of the picture. This suggests that whenever the separation bubble behind a sphere shifts laterally away from the wake axis, a transverse force acts on the sphere in the opposite direction. Although the evidence is not very strong, it also appears that the magnitude of the transverse force depends directly on the extent of the lateral deflection of the separation bubble. This relationship between wake fluctuations and the transverse force can be explained quite simply. When the separation bubble moves to one side of the wake axis, as in Fig~ 34, it retards the fluid flowing past that side of the sphere and accelerates the flow on the other side, producing what can be thought of as circulation (I) about the sphere. This circulation moves around the sphere away from that side of the sphere toward which the separation bubble has shifted. The interaction of r with the mean flow produces the transverse force. Evidently the magnitude of r, and therefore the magnitude of the transverse force, is proportional to the deflection of the separation bubble

108 do Side Sting Results. Once it had been definitely established that the presence of a sting exerted a profound influence on the sphere wake, other means of supporting the sphere were sought in an effort to obtain meaningful data on the unsteady flow. The tests with the sphere suspended on threads have already been noted. However, as it was not possible to obtain unsteady force data with the suspended sphere, several test runs were made with the sting-mounted sphere held perpendicular to the direction of motion. That is, the sting was removed from the strut and attached directly to the carriage so that it hung down vertically. These tests yielded very regular oscillating L data; magnitudes and frequencies of the L oscillations are indicated in Fig. 25 and 26 respectively~ In spite of the regularity of the L oscillations and the fact that Red in for this configuration was only 220, flow visualization tests failed to reveal any clear indication of vortex loop shedding. Two photographs from one such test at Red = 304 are shown in Fig. 35; the only distinct feature observed in this and the other tests was a strong upward flow in the wake, indicated by the bright-edged dark streak in the first photograph~ (The dark streak resulted from the upflow carrying the hydrogen bubbles out of the light beam ) This upflow was evidence that the side stirg produced a serious disturbance in the flow about the sphere. The extent of this disturbance became apparent

109 with the calculation of average lift coefficients (CL) for several of the side sting test runs. For 270 < Red < 550, values of CL ranged from -. 18 to -. 14. The upflow in the sphere wake suggested the existence of flow circulation about the sphere; the negative values of CL, meaning that the lift force acted downward, confirmed this idea. C. RESULTS OF DISK EXPERIMENTS 1o Disk Drag a. Steady Drag. Coefficients of steady drag for a circular disk were measured through most of the range 5 < Red < 70, 000. The results of these measurements are given in Fig. 36, where they are compared with a curve that represents the data of Fig. 8. The most obvious feature of Fig. 36 is the disagreement between the new data and existing data for 100 < Red < 1000. This lack of agreement stems from the fact that all previous data in this Red range was obtained with disks that were free to oscillate, whereas the disk used in the tests reported here was rigidly constrained to one-dimensional motion. The cause of the increase in CD due to oscillation of a disk is not understood, but may lie in the ability of the disk to generate lift when it executes translational oscillations. Another interesting feature of the data in Fig. 36 is the rather abrupt flattening out of the curve indicated by the data at Red slightly

110 below 2000 This phenomenon cannot as yet be explained, but the fact that Redmin = 160 for the one-degree-of-freedom disk is probably involvedo The primary factor in the general scatter of the data is the rather poor repeatability of the disk drag transducer; this was pointed out above in the discussion of sphere drag results. As was the case with the sphere, the disk drag results for 9000 < Red < 70, 000 were obtained early in the course of the research. Considerable mechanical unsteadiness existed in the experimental apparatus at the time, and this showed up in the drag measured during these early tests. The resultant uncertainty in the determination of D is responsible for the values of CD being a bit high~ The CD data at low Red also appears high in comparison with the curve in Fig~ 360 However, a look at Fig. 8 reveals that the existing data in the range 0 < Red < 50 is not extensive, and so it is believed that the data presented here is essentially correct. bo. Unsteady Drag~ The drag data record shown in Fig. 37 is typical of the disk tests. Except for the unsteadiness during the wake development period, very little fluctuation exists in D in spite of the vortex shedding indicated by the oscillations in yawing moment (Y). Analysis of many disk drag data records showed that, during the steady state portion of a test run, the unsteadiness in D never exceeded 3% of D.

111 2. Moment on a Disk The moment acting about an axis in the plane of a circular disk was extensively studied in this experiment. Although the data derived from many test runs exhibited considerable scatter, enough consistency appeared that the results are considered to be reliable. In contrast with the sphere results, the uncertainty in the disk data did not seem to be caused by sting interference, but was a result of the fact that in a large number of the tests a steady state was not achieved. This situation will be discussed in the presentation below of specific resultso Sign conventions for pitching moment (P) and yawing moment (Y) are shown in Fig. 38. a. Magnitude of the Moment. Coefficients (CMpp) of the peakto-peak value of the total moment measured at various Red are presented in Fig. 39. The reasons given above for expressing the sphere transverse force results in peak-to-peak form also apply to:CM. A rather large spread in CMpp values is apparent. However, the two points connected by an arrow give a clue to the cause of this data spread. The two values of CMpp were both taken from the same data record, the high value from early in the test run and the low value from near the end of the run. This decrease in the magnitude of the unsteady moment appeared in almost every run in which regular oscillations

112 occurred, although the variation in magnitude was not generally as severe as these values indicate. The clustering of data points at CMpp z. 007 in Fig. 39 suggests that this value is approximately correct for the steady state CMpp. In comparison, Willmarth, Hawk, Galloway, and Roos[ 17] reported 2,5 6 thatCM2 = 0047 for a disk at Red between 10 and 106 Taking into consideration the fact that their moment was measured about a fixed axis and the difference between ro mo so and peak-to-peak values, it is seen that the dimensionless moment at high Red is larger by roughly a factor of three than it is for Red < 500. Data in Fig. 39 does not extend above Red = 500 simply because the moment records have become too irregular to permit a justifiable estimation of the moment magnitude. bo Frequency of the Unsteady Moment. The frequency of moment oscillations was determined from a number of moment data records. In nondimensional form, this data is plotted in Fig. 40. At several values of Red a pair of data points is connected with an arrowo This serves to illustrate the fact that the frequency, as well as the magnitude, of the unsteady moment decreases during the course of a test run. The individual points on the plot each represent an average value of f* over that portion of a run during which the moment oscillated regularly. Although most of the data is quite consistent, the trend

113 indicated by the arrows raises the question of what the steady-state value of f* should be. The answer to this question has yet to be determined; however, it seems likely that the correct value of f* is approximately o 10 for Red < 500~ The reduction in density of data points for Red > 320 attests to the fact that the moment fluctuations become increasingly irregular as Red increases above 320~ Above Red - 500 the moment records are so irregular that f cannot be determined. Data records for Red < 320 exhibit very regular oscillations. Shown on Fig. 40 is Redmin for the disk; its value is 160. This is considerably higher than the accepted value of 100 (noted in Section II). The cause of the discrepancy is not definitely known, but it is probably a combination of sting and strut interference effects. An attempt was; made to assess the role of sting interference alone by doubling the sting diameter. Redmin increased only slightly to 175. co Flow Visualization Resultso Flow visualization tests with the disk provided much insight into the processes of vortex formation and shedding in the disk wake and into the relationship between these processes and the unsteady moments on the disk. The first disk flow visualization photograph, that in Fig. 41, clearly shows the ring vortex within the separation bubble behind the disk in

114 steady flow at Red = 157. The close proximity of the tail of the separation bubble to the strut leading edge suggests that some strut interference may indeed have existed. Figure 42 presents flow visualization results from a test run at Red = 318 which featured regular vortex shedding in the Y mode (to eo alternate loop shedding in the horizontal plane). The portion of the wake shown in the photograph was about 12 disk diameters aft of the disk~ In contrast with the sphere results, vortices are clearly seen here. Apparent in the Y data record of Fig~ 42 is the decrease in magnitude and frequency of Y during the course of the run. Also evident is a characteristic feature of the disk moment records, the very large moment fluctuation marking the shedding of the first vortex loop of the run. This initial fluctuation was always much greater than the succeeding oscillations~ Vortex loops cast into the disk wake are very strong~ That this is so can be seen in two features of the photograph in Fig. 42. One of these features is the considerable difference in the extent to which two successive vortices (center bottom and upper right), shed slightly more than one second apart, have rolled up. The other important feature is the pronounced lateral motion of the vortices~

115 Regular vortex loop shedding from the disk is demonstrated in the photograph in Fig. 43~ The vortices are not as distinct here as in Fig~ 42 because, as the moment record shows, the plane of the loop shedding was nmt horizontal. The fact that the amplitude of the P oscillations fell off rapidly while the Y amplitude remained constant indicates that the loop-shedding plane rotated toward the horizontal as the run progressed~ It should be noted that the photograph in Fig~ 43 is somewhat misleading in that it appears to show two vortex loops shed in succession on the same side of the wake axis. This is not the case, of course: there is a vortex loop on the other side of the wake axis, but it is obscured by the strutL Figure 44 is included here because it illustrates the initial shedding from an almost-impulsively started disk. The intensity of the initial vortex ring generated by the startup of the disk is much greater than that of succeeding vortices. This is not obvious in the photograph, although it is suggested by the rapidity with which the first vortex loop is moving away from the disk axis~ The great intensity of the initial vortex is explained by the following argument. Once the disk is set into motion and flow separation occurs, the ring vortex within the separation bubble must grow to a certain size before instability can set in to start the loop-shedding

116 process, Even after the vortex has reached unstable proportions, however, the instability takes some time to develop. If the growth rate of the vortex is very high, as it is for a rapidly accelerated disk, the intensity of the vortex increases greatly during the time taken for the instability to develop, Since the flow instability is already present when the succeeding vortices are generated, they do not have the extra growth, time and consequently are less intense than the initial vortex, The relationship between the vortex loop shedding process and the unsteady moment can be determined by considering the photograph and the Y record of Fig, 44 togethero The first vortex loop to be shed came from the part of the disk nearest the bottom of the photograph; the initial fluctuation in Y was positive, corresponding to a Y vector coming out of the photograph, (by the right-hand rule). This relationship between the moment and vortex shedding agrees completely with, the disk flow field model proposed by Willmarth, Hawk, Galloway, and Roos[ 17]j The key to their model is the idea that the moment is produced by t;he upstream flow in the center of the wake, A sketch of the model is shown in Fig~ 45; it is self-explana-!tory~

V. CONCLUSIONS The research program that is the subject of this paper had two principal objectives. The first of these was the establishment of an experimental facility suitable for the study of sphere and disk flows over a wide range of Reynolds numbers; the second objective was the utilization of this facility in an experimental study of the unsteady flows about spheres and disks and the relationships between the flows and the forces acting on these bodies. Both objectives were met: a towing channel was developed and used in conjunction with strain gage equipped models and flow visualization to shed considerable light on the nature of unsteady sphere and disk flows. Specific conclusions that may be drawn from the results of this research program are listed below. 1o The hydrogen bubble flow visualization technique was successfully adapted for use in glycerol-water mixtures with glycerol concentrations as high as 91%. 2. Sphere drag results for 5 < Red < 120, 000 agreed closely with previously existing data. 3. Results for the drag on a circular disk constrained to rectilinear motion reproduced existing data except for 117

118 100< Red< 1000o In this range the measured drag coefficient for the non-oscillating disk was as much as 50% smaller than that indicated by existing data, which was obtained with oscillating diskso 4. The drag on a sphere or disk moving rectilinearly at constant speed is not significantly affected by vortex shedding in the wake; maximum drag unsteadiness was less than 5% of the total drag on the sphere and less than 3% of the total disk drago 5. Drag on a sphere accelerated from rest to a constant velocity exceeds the steady state drag by as much as 30% at high Red, until vortex shedding becomes establishedo 6. The potential flow apparent mass concept is valid for the first diameter of motion of a sphere undergoing constant acceleration such that Red 30, 000 when the sphere has moved one diameter. 70 Asymmetrical unsteadiness in the wake of a sphere normally commences at Redmind 215; wake stabilization caused by a sting support delayed this unsteadiness until Red had reached 290. 8. Wake stabilization caused by a sting raised Redmin to 160 from the value 100 characteristic of unsupported diskso

119 9. When a sphere or disk was accelerated from rest to a constant velocity, the first vortex loops to be shed were more intense than those shed later; correspondingly, the first fluctuations in unsteady force or moment were greater than later fluctuations. 10. The vortex shedding process in the wake of a sphere was materially impaired by the presence of a sting support; such impairment was not evident in the disk wake~ lio Peak-to-peak values of the total transverse force coefficient for a sphere moving at constant velocity were in the range o 02 - o 10 for 300 < Red < 600; the dimensionless frequency of the unsteady force in this Red range increased with Red from.13 to o16. However, further study is required to verify and extend these results in the absence of support interference~ 12. The peak-to-peak value of the moment coefficient for a circular disk moving at constant velocity in the direction of its axis was approximately o007 for 160 < Red < 500; the dimensionless frequency of the moment was approximately constant at f* =- o10 for these Redo

120 13o The unsteady transverse forces on a sphere result from circulation about the sphere which is caused by lateral fluctuations of the separation bubble. 14o Unsteady moments on a disk are produced when the direction of the reverse flow in the disk wake is shifted during the shedding of a vortex loopo

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127 67. Anon0 Physical Properties of Glycerine and Its Solutions. Booklet compiled by the Soap and Detergent Association (available from the Dow Chemical Coo ). 68, Shair, Fo Ho, Grove, A. So, Petersen, E. E., and Acrivos, A. "The Effect of Confining Walls on the Stability of the Steady Wake Behind a Circular Cylinder," J. Fluid Mech., Volo 17, part 4, po 546 (December, 1963)o 69o Neher, Ho V., and Leighton, R. Bo "Linear Air Trough," Am. J. Physo, Volo 31, p. 255 (April, 1963)o 70. Perry, Co C., and Lissner, H. Ro The Strain Gage Primer (McGraw-Hill Book Coo, Inc., New York, 1962), 2nd edo 71. Dorsey, J, Semiconductor Strain Gage Handbook (BaldwinLima-Hamilton Corpo, Waltham, Masso, 1965). 72, Cowles, L. G, "The Parallel-T Resistance-Capacitance Network, Proco IRE, Vol. 40, No. 12, po 1712 (December, 1952). 73. Criminale, W. 00, Jr., and Nowell, R. W. "An Extended Use of the Hydrogen Bubble Flow Visualization Method, " AIAA J., Vol. 3, Noo 6, p. 1203 (June, 1965), 74. Clutters D. W,, Smith, Ao M. 0O, and Brazier, Jo Go "Techniques of Flow Visualization Using Water as the Working Medium, " Douglas Aircraft Division Repto Noo ES 29075 (April 15, 1959)o 75. Schraub, F. A., Kline, S. J., Henry, Jo, and Runstadler, P. Wo'"Use of Hydrogen Bubbles for Quantitative Determination of Time Dependent Velocity Fields in Low Speed Water Flows, " Repto MD-10 of the Thermosciences Divo, Stanford University Depto of Mechanical Engineering (February, 1964)o

128 76. Chow, R. "On the Initial Development of Incompressible Flow Past an Impulsively Started Sphere, " Grumman Research Department Report Re-317 (January, 1968). 770 Sarpkaya, T. "Separated Flow About Lifting Bodies and Impulsive Flow About Cylinders, " AIAA J., Vol. 4, No. 3, p. 414 (March, 1966).

129 Figure 1. Sphere streamlines for Red 0. Figure 2. Sphere streamlines for Red = 5. Figure 3. Sphere streamlines for Red" 100.

~~~~~~~a3~~~~~~~~~~ 7,00- ~ ~ Figure 4. Stages in loop shedding process for a sphere. Wake is shown in cross-section; dotted line indicates position of vortex axis.

Figure 5. Alternate loop shedding by a sphere at Red 350. a. Laminar separation. b. Turbulent separation. Figure 6. Effect of boundary layer transition on flow sep. ration from a sphere.

10-' 0 ALLEN o ARNOLD * BACON & REID o FLACHSBART * HAK v LIEBSTER V LUNNON O MAX WORTHY _ 7' MILLIKAN 8 KLEIN _ MOLLER A SCHMIEDEL CD 0O SCOGGINS ~D' SHAFRIR,2* SHAKESPEAR 0 WIESELBERGER 10' 1' |a Ro OSEEN STOKES v V1:00 V v I. + III t~~a$ Se I0- 1' I 0~ 10' 10Z Red 0o 10'4 lO 10' Figure 7. Sphere CD- Red relationship according to existing data.

CD[ o KNIGHT 102 o MCHAEL a PRANDTL o SCHEDEL o S4OEMAKER O SIMMONS & DEWEY v WIESELBERGER A WILLMARTI, HIAWK,/. st.ady I0' o & HARVEY ol - ciatlon OSEEN OBERBECK t i 100 1 I i,I I I J I I II I, I I i I II I I -Ii I, 11 10-1 100 10' 0 RPed 103 I10 IO' Figure 8. Disk CD- Red relationship according to existing data.

SQUARE TUBE C.ARRIA6E BEARING MOT V 6ROOVE BEARING / ADJUSTMENT / OVERHEAD INSTRUMENTATION 7E NLARGED CROSS SECTION / 7U /ENLARGEP END VIEW / / 0 40 g AlR IDLER PULLEY & CARRIAGE EQUIPMENT FLOW VIUALEZATION POT. DRIVE & MODEL DRIVE MOTOR Figure 9. Towing channel installation.

::i::':::::lj~~~~~~~~~~~~~~~~~~~~:':c::::j:::;:~~~~~~~~~~~~~~~~~~~~~::i::::::ii~~~~~~~~~~~~~~~~~~:'ll::i:::i:::::i-::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:i;:::~:1:-::::::I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~~i! ~i~]~~111~~ ~ ~ ~~~~~~~:''::::~:::::::i::i:::'::i:::/ il::i:: I::::::-9::~::'::iii,:::'i j:::ii-:ii~i:~:::- I::: i:::::' ki.........~~~~~~~~~~~~~~~~~~~~~~~~~:~~:: ~Pl:'i:::::Xi[:: ~I::ii ~~!~~~ ~.....................~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i:::::::::i::::~ i~ i.... i~ Figure 10. Photograph of towing channel facility.~~~~~~-i::

~~~~~";ac " c:~~~~~~~~~~~~~~~~~~~~~~~~~.. o""9~~~~~~~~~~~~~~~~8s~~~~~~~~~e~~~~~~~~~~~ "I~~~~~~~~~~~~_"~~~~~~~~~~-''': ~ ~ ~ ~ ~ ~,xxx-,,'-~,,,~~~ ~',,~"~...... ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,,'x,'' —"' x x.4." —,~~~,.~~~~~ i~~~~~~~~~~~~~~~~~:~~~~~~~~~~~~~~~~~~~~~~~~~~:~~~~~~~,.. ~..,~,~~...,,. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~:':'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'.::~~~~~~~~~~~~~~~~~~~~~-::~ ~~~~~::i::i-~::,::::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:!~:i;(i?.i' ~ ~~~~~:: - "::?::';:::....., i i!~~~~~~~~:-: }?Z~; Figure 11. Photograph of carriage with disk mounted on balance.~~::::::~8:i-~

137 -- GLIDER /'e.\?, EXTERNAL HOLE INTERNAL HOLE AIR CHAMBER Figure 12. Cross-section of V-groove air bearing.

Figure 13. Photgraph of spher balance (befor application o protetivecoatng t drg rasdce)

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PITCH TRANSDUCER (SEMICONDUCTOR GAGES) r-i Figure 15 Drawing of disk balance. \F 1 Figure 15. Drawing of disk balance.

+15v 0 POWE R TO ANMoPL IIF ERS j- GAIN AMPLIFIERS jt111r_,_ NOTCH FILTER iM2r - - - - - - 15K K w 8 25K 110 Io1 KLSI S Figure 16. Diagram oitb Uaw ~~ I ~s~c SK S 1M 5K r~~~~~~10KADE 350 10 100K tK I I I I I K 100 W 0K 1'25 3% 350 ~~~~~~~~~~~~~~~~~~100K( 500 ~ ~ ~ I 10 IOK i, I 1 ~~~~~l100K 20K 50dL\1~~~ 500 )5~K 5K POWER & rPIM CONNECIIN TO AMPLIFIE IRS NOT SiON STRAM GCAGE BRIDGE'S Figure 16. Diagram of circuit used with disk balance.

STRAIN GAGES I I l \ DRAG LIFT & SIDE FORCE \ TRANSDUCER / TRANSDUCERS ~~~~~~Fiu\ / Figure 17. Drawing of sphere balance.

143 —.50- 1 -'.50.50 "-~ STRAIN.0-16 GAGES.016 Figure 18. Drawing of accelerometer. CAMERA LAMP HOUSINGS-'/ WIRE HOLDER. UBBLE WIRE MIRRORS Figure 19. Flow visualization equipment.

4 I0 mv/in range 6 3 500 mv/in ronge deflkeAion on chart deflec+ion (o on chrt 4 * L (in) 3 slope'.1 B73 16/in -.094 ".03 -.02 ".0I.01.02.03.04 L,-S (ib) slope.00833 Ib/in -2 0.3 0 /.2 A:6 S 1.0 1.2 D (lb) 4 a. Drag transducer. b. Lift and side force transducers. Figure 20. Typical sphere balance calibration.

10' CD 100 gpi~~~~~~~~~~~~~~~~~~ 100 IC' 102 Re102 Figure 21. Sphere drag coefficient data from this experiment.

.0W S (Ib) D (Ib) ~~~~~~~0~ ~~~~~~~~~~~~~~~~~~~~ 02I I I I I I I l l I 0 a 4 6 0 I 14 a16 b Figure 22. Typical sphere drag data record (Red = 532).

147 3 a (ft/seca) 2 0.21 D (lb) /\ x measured calculated i from steady CD O0 - 0 I 3 4 t (sec) Figure 23. Data record of sphere drag during acceleration from rest.

148 L Figure 24. Sign conventions for forces on a sphere.

.12,.10 e o0.030.07?.08 0 SHORT STING C o LONG STING 0 PP * SIDE STING.06 0.00.04.070 fd e e.s, U..140.02 0.133 I I I, I, I 0 100 200 300 400 500 600 700 Red Figure 25. Transverse force coefficient for a sphere.

150 *20.50 0 ap 1.10 0 0 0.08 fd o 0. U o 0....L.....06 * L o, * UDST- LUPT 0 - KIALL N LCAL (NU Wm)a.o04 LO a - 145Y & sesOP - LOO (PLLV6S UoPn).02,'. I I -l'.02.IO.0 10 50 0 400 SOO GO0 00- _ o Figure 26. Frequency of oscillation of transverse force on sphere.

151 S (lb) o +.005 -.005 cONST. - 01! - bSPEED Ut d 10 20o 30 40 50 60 70 o Figure 27. Data record of sphere lift and side force showing low-frequency oscillation. Long sting; Red= 399.

152.040 S (lb).oZO -.040 L (Ib).O0Ut II d- "m 40 60 80 100 120 Figure 28. Sphere data record showing transition of fluctuation from lift to side force (Red = 433).

153 a. Sphere on sting (Red = 269). b. Sphere suspended on threads (Red = 192). Figure 29. Photograph of steady flow about a sphere.

154 a. Sphere shedding a vortex loop. b. Same segment of wake after 16 sec. c. Same segment of wake after 35 sec. Figure 30. Vortex loop wake behind a sphere at Red = 295.

,~"~"-.. Figure 31. Sphere shedding vortex loops at Red = 377.

156 Flow about sphere. Same position 15 sec. later. first photo +d 5 040 ) o lIb) o~~~~~~~~~~I ed IO 0 30 4( SO 60 7' 80 Figure 32. Data record and flow photographs of sphere. regular side force oscillations at Red I 332.

157.005 (lb) 0 -.005 L (lb) o -.0054 o 20 40 60 80o 1o Figure 33. Sphere' data record showing irregular fluctuations in lift and side force at Red = 715. wake axis circulatory eflec1ion of flow deflec~ion of ~f ~~~~~low.separation bubble U x direcion F (in U x r direction) Figure 34. Model relating unsteady transverse force to sphere wake fluctuation.

158 a. Wake immediately behind sphere. b. Same segment of wake 15 sec. later.

CD 100 100 10' O 03 104 105 Red Figure 36. Disk drag coefficient data from this experiment.

.003 (in-lb).002.001.08.0D 6 o (Ib).0o4.02 CONST. SPEED 0 4 8 t (sec) 12 16 2024 Figure 37. Typical disk drag data record (Red = 530).

161 Y U Figure 38. Sign conventions for drag and moments on a disk.

. I ~~.016 CMPP'0DS ~0 0 0 0 0 0.004 0 100 20D 300 400 SO0 600 Red Figure 39. Moment coefficient for a disk.

cl, 2 a -mN N s t m: S ___ 01___ e~~~ I a II I. o ~~~ )j~~~~~~~( C C -- c~:0 0~~~~~~ CD 0 i o - r. o 0 8] J CD P1 ~ U1~ O9

...i i 5M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0;9~~~0 _Ii~~iiiici~i:i:-ii~l::;!!:! H!? Al_ Figure 41. Photograph of steady flow about a sting-mounted disk at Red 157.

165 (in-lb) o —--- +.0010(in- b) 0 —.0010 - W.CONST. dU ~ -, 0 50 75 100 Figure 42. Data record and wake photograph for disk showing regular oscillations in yaw at Red = 318.

166 pp'to Y (in-lb) 0 -.0005-5 -0010, Figure 43. Data record and flow photograph for disk showing combined oscillation in p-itch and yaw at Red = 278. +.0005 / (in.lb) cnA A /

167 photo 7000 +000. I +.0010 (in-lb) +0005Co,ST. S- I5t _ LlO t _ 0 20 40 60 Figure 44. Data record and flow photograph for disk showing initial vortex shedding at Red = 305.

STAGNATION POINT OF REVERSE FLOW SHIFTS OFF CENTER, PRODUCING MOMENT VORTEX LOOP SEPARATION SHEET BEGINS BEING SHED ROLLING UP TO FORM A NEW VORTEX U M FLOW AHEAD OF DISK Is NEGLIGIBLY ALTERED BY SHEDDING Figure 45. Model relating unsteady disk moment to vortex shedding.

Unclassified Seciuti CtsW~r'tkat"n DOCUMENT CONtROL DATA. R & D.ieel rlt. el.*.ftl.r.a#n ti tI. *, hnty oIf h*errt and indetfnd enointlot metl h enleted we the oerall re tit In clrs#lfled) I. g9RIGNA TING Aftiviy( (VopForaftf Autlhor) r28. REPORT SECURITY CLASSIFICATION Department of Aerospace Engineering Unclassified The University of Michigan 2b. GROUP Ann Arbor. Michig.an 3. R-APORT tiTLIL An Experimental Investigation of the Unsteady Flows About Spheres and Disks 4. DESC RIPtVE MO.tteS (Tpe of tepott and inclteive date.) Interim Technical Report S. AU t"Ho S) (Pitat name, middle Initial, laat name) Frederick W. Roos William W. Willmarth 6. REPORT DATO 7sa. tOTAL NO. OF PAGES lb. NO. OF REFS December, 1968 77 *6. Italy," CPGRANt NO. 98. ORIGINAtOR'S REPORT NUMBER(IS DA-AR0-D-31-3124-G711l b. PROJECT NO. 5590-E C. lib. OTHER REPORT NO($) (Any other number# thea may be aelsgned this report) 10. DOISTRIUTION *STATEMENT Distribution of this document is unlimited. 11. SUPPLEMEINTARY NOTtES 12. SPONSORING MILITARY ACtlVITY Department of the Army U. S. Army Research Office —Durham Durham, North Carolina 13. ABSTRACT The various features of unsteady three-dimensional bluff body flow fields have been studied in a towing channel filled with glycerol-water mixtures and equipped with an air bearing supported model carriage. Steady drag coefficients for the sphere and disk agreed with existing data except for the disk in the range 100 < Red < 1000 where CD is as much as 50% lower than previous results. Peak-to-peak moment coefficients developed ona disk are of the order of 0.007 for 100 < Red < 500 at nondimensional frequencies of 0.10 < fd/U < 0.14. The sphere wake was seriously disturbed by the sting model support, however, the unsteady side force coefficient is of the order of 0.02 to 0.10 for 270 < Red < 560. Flow visualization studies using hydrogen bubbles were used to relate the unsteady forces to the shedding of vorticies in the wakeDD NO 1473 Unclassified Security Classification

Unclassified Security Classification KIEV WORDY S | ~LINK A | LINK a LINK' C KEY WORDS ROLE WT.ROLE WT- ROL' -, WT Unsteady flow 7 3 Sphere flow 7 3 Disk flow 7 3 Separated flow 7 3 Uncu lass ilfied.. Stcurity Cluslitic~tirrn

L305 0 69 5 5550