'W" -‘~—*"—"- ' --—""-—‘ -~~—'-"""" ""“‘ “ ' """"~*-v<-v-—--'—'V-—-—=--m¢-—~‘ THE EFFECT OF HEAT TRANSFER ON THE LAMINAR BOUNDARY LAYER AND LAMINAR SEPARATION OF WATER FLDWING PAST A FLAT PLATE AND A SPHERE r Thesis for .the Degree of Ph. D. MICHIGAN STATE UNIVERSITY SURINDER KAPUR 19.72 LIBRARY Michigan State University IllllllllllllllllIIIIIIIIIIIIIIIUIllllIIIIIIIIIIHIIHI 3 1293 10385 9611 This is to certify that the thesis entitled THE EFFECT OF HEAT TRANSFER ON THE LAMINAR BOUNDARY LAYER AND LAMINAR SEPARATION OF WATER FLOWING PAST A FLAT PLATE AND A SPHERE presented by Surinder Kapur has been accepted towards fulfillment of the requirements for Eh I I2. degree in fiyechanical Engineering m MA Major professor Date Jill-121:3! 17L 1972 0-7839 w 37 av ‘ "DAB 8: SUNS. I BUUKQ BINDERY INCA LIBRARY BINDERS T . n nan-canny autumn III ABSTRACT THE EFFECT OF HEAT TRANSFER ON THE LAMINAR BOUNDARY LAYER AND LAMINAR SEPARATION OF WATER FLOWING PAST A FLAT PLATE AND A SPHERE BY Surinder Kapur The effect of heating a body, which is in a uniform flow of water, is investigated numerically and experi- mentally. Primary interest is in the effect of heat transfer on the laminar boundary layer separation. Since the variation of viscosity with temperature is large for water, the velocity and temperature fields interact. This necessitates the simultaneous solution of the momentum and energy equations. Numerical results for various flow con— figurations are presented. Experimental results are pre— sented for the flow of water past a three-inch sphere in a ten-inch square horizontal test section. Flow visuali— zation techniques, hydrogen bubble, and the shadowgraph methods, were used to locate the separation point experi— mentally. Numerical results indicate that heating does sub— stantially shift the separation point backward for the Surinder Kapur linearly retarded flow past a heated flat plate. The effect of heating on the position of the separation point, for flow past a sphere, is small. Experimental results tend to confirm the small influence of heating on the position of the separation point on the heated sphere. Numerical results also include the effect of heat- ing on the various boundary layer parameters such as displacement thickness, thermal boundary layer thickness, and wall shear. THE EFFECT OF HEAT TRANSFER ON THE LAMINAR BOUNDARY LAYER AND LAMINAR SEPARATION OF WATER FLOWING PAST A FLAT PLATE AND A SPHERE BY Surinder Kapur A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1972 A DEDICATION This thesis is dedicated to my mother, without whose encouragement and confidence this work would not have been completed. This is also dedicated to my wife, Rani, who has displayed great patience and tolerance during the last year of this work. Finally, I dedicate this thesis to my son, Sanjay. ii ACKNOWLEDGMENTS The author wishes to express his sincere thanks to his major professors Drs. Merle C. Potter and Mahlon C. Smith for their valuable guidance throughout the course of this study. Special thanks to Dr. Potter for his constructive comments during the preparation of this manuscript. The author wishes to thank Dr. Norman L. Hills and Dr. George E. Mase for their time spent while serving on the guidance committee. The author wishes to thank the Mathematics and Science Teaching Center at Michigan State University for providing a graduate teaching assistantship during the course of this study. iii LIST OF LIST OF Chapter 1. 2. 3. TABLE OF CONTENTS TABLES . o o o o I o o o o o F IGURES . o o o o g o o o o 0 INTRODUCTION . . . . . . . . . MATHEMATICAL FORMULATION . . . . . NUMERICAL SOLUTIONS . . . . . . . 3.1 Introduction. . . . . 3.2 Procedure for Solving the Boundary Layer Equations. . . . . . . 3.3 Method for Solving the Momentum Equation . . . . . Method for Solving the Energy Equation . . . . . . . . . Finite-Difference Representation of 5-Derivatives . . . . . Method of Integration. . . . Starting the Solution. . . . Boundary Layer Parameters . . Computer Program . . . . . wwww w w \Dmflm U1 ub EXPERIMENTAL APPARATUS AND PROCEDURE . 4.1 Introduction. . . . 4.2 Description of the Heated Sphere . . 4.3 Description of the Test Section Loop. 4.4 Flow Visualization Techniques . . . 4.5 Operating Procedures . . . . . . iv Page vi vii 14 24 24 29 30 35 39 40 47 48 52 53 53 53 55 56 57, Chapter 5. NUMERICAL AND EXPERIMENTAL RESULTS Introduction. . . Verification of the Numerical Technique. . . . . Flow Past an Unheated Sphere: Similar Flows on a Flat Plate. Adverse Pressure Gradient Flows Flow Over a Sphere Hydrogen Bubble Experiment . Effect of Heat Transfer in Linearly Retarded Flows on a Flat Plate. Effect of Heating a Sphere . Simple Potential Flow ("Above Critical") . Experimentally Determined Velocity Distribution ("Below Critical") . . Effect of Introducing the Buoyancy Force Term ("Below Critical") . . Experimental Results. Effect of Treating Viscosity Constant in Linearly Retarded Flow Conclusions . . . . 5.1 5.2 A. B. C. 5.3 5.4 5.5 A. B. C. D. 5.6 5.7 TABLES. . FIGURES . . BIBLIOGRAPHY. APPENDIX . . Page 59 59 60 60 63 64 67 68 70 70 71 72 74 76 77 80 87 152 156 Table 10. 11. 12. LIST OF TABLES Similarity Solutions (Ue - xm) . . Calculated Separation Point for the Case Ue/U0° = l - x . . . . . . . . Values of Y at the Point of Separation (Illingworth [10]) . . . . . . Summary of Principal Methods for Solving the Laminar Boundary Layer Equations Steps in the Procedure for Starting Integration at Wall . . . . . . Value of ¢$ for Similar Flows . Effect of Heating and Cooling a Flat Plate on Skin Friction and Heat Transfer Parameters . . . . . . . . . Calculated Values of ¢§ for Howarth's Retarded Flow Ue/Um = l — 5/8. . . Separation Point Calculated for the Flow Ue/Um = 1 - g o o o o o o o 0 Comparison of Values of ¢§ on a Sphere as Calculated by the Present Author and Smith and Clutter [37] . . . . . The Effect of Heating the Sphere on the Separation Point "Above Critical" Flow . The Effect of Heating the Sphere on the Separation Point "Below Critical" Flow . vi Page 80 80 80 81 82 83 83 84 85 85 86 86 Figure l. 10. 11. 12. 13. 14. LIST OF FIGURES Velocity Profile Near the Separation Point. Boundary Layer Profiles . . . . . . . . Drag Coefficient for a Sphere as a Function of Reynolds Number . . . . . . . . . Boundary Layer on a Body of Revolution-- Coordinate System . . . . . . . . . Nondimensional Fluid Properties for Water . . Notation for Velocity and Temperature Profiles in the Boundary Layer on a Body of Revolution. . . . . . . . . . . Notation for Finite Difference Representation. Brass Sphere with Heating Element and Location of Thermocouples. . . . . . . Photograph of the Sphere with the Support Red 0 O O O I O O O O O O O O 0 Schematic Diagram of Test Loop. . . . . . Velocity Profiles for Similar Flows U ~ Em. C o 0 Q o o o o o o o o e The Effect of Heating and Cooling on the Velocity Profiles on a Flat Plate . . . . The Effect of Heating and Cooling on the Temperature Profile on a Flat Plate . . . The Effect of Buoyancy Forces on the Velocity Profiles on a Heated Vertical Plate, Two Feet from Leading Edge. . . . . . . . vii Page 87 87 88 88 89 9O 9O 91 91 92 93 94 95 96 Figure 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. The Effect of Buoyancy Forces on the Temperature Profiles on a Heated Vertical Plate, Two Feet from Leading Edge Velocity Distribution Around a Sphere . . Pressure Distribution for Flow Over a Sphere . . . . . . . . . . . Velocity Gradient at the Wall for Two Cases of Potential Flow Around the Sphere . Photograph of Hydrogen Bubbles Visualized on Unheated Sphere . . . . . . . Graphical Procedure for Extrapolating Separation Point from Hydrogen Bubble Pictures . . . . . . . . . . Velocity Profiles for a Heated Flat Plate Ue/Uw = l - 5/8; E = O. I o I o o 0 Velocity Profiles for a Heated Flat Plate Ue/U0° = 1 - 5/8; E = .575. . . . . . Velocity Profiles for a Heated Flat Plate Ue/U0° = l — 6/8; E = .7863 . . . . The Effect of Heating and Cooling a Flat Plate on the Velocity Gradient at the Wall Ue/Uco = 1 - 5/8 . . . . . . . . . The Effect of Heat Transfer on Separation POinto Ue/Um = l '- g/B O o o o o 0 Temperature Profiles for a Heated Flat Plate Ue/Um = 1 _ 5/8’ g = 0. o o o o 0 Temperature Profiles for a Heated Flat Plate Ue/Um = 1 '- E:/8' g = .575. o o o o 0 Temperature Profiles for a Heated Flat Plate tIe/IJm = l " g/B, E, = .786. o o o o 0 Velocity Profiles for a Heated Flat Plate Ila/Um: 1 '— g, g = o o o o o o o 0 Velocity Profiles for a Heated Flat Plate Ue/U0° = 1 - E, E = .083 . . . . . . viii Page 97 98 98 99 100 100 101 102 103 104 105 106 107 108 109 110 Figure Page 31. Velocity Profiles for a Heated Flat Plate Ue/Um = 1 " g, g = .118 . o o o . o . ll]. 32. The Effect of Heating and Cooling a Flat Plate on the Velocity Gradient at the Wall Ue/U.° = l - E . . . . . . . . . . 112 33. The Effect of Heat Transfer on Separation POint. Ue/Um = l - g . g . . . . o . 113 34. Displacement Thickness Over Heated Flat Plate Ue/Uco = 1 - E . . . . . . . . . . 114 35. Momentum Thickness Over a Heated Flat Plate Ue/U0° = l - E . . . . . . . . . . 115 36. Temperature Profiles for a Heated Flat Plate Ue/Um = 1 - E! g = 0047 . o . . . . o 116 37. Temperature Profiles for a Heated Flat Plate Ue/Um = l — E] g = .083 o . . . . o o 117 38. Temperature Profiles for a Heated Flat Plate Ila/t]m = 1 - E, g = .118 . . o . . . . 118 39. The Effect of Heating and Cooling a Flat Plate on the Local Nusselt Number Ue/Um = l — g I o o o o o o o o . 119 40. Velocity Profiles for a Heated Sphere. "Above Critical" Flow, a = 0° . . . . . 120 41. Velocity Profiles for a Heated Sphere. "Above Critical" Flow, a = 47.5° . . . . 121 42. Velocity Profiles for a Heated Sphere. "Above Critical" Flow, a = 67.8° . . . . 122 43. The Effect of Heating and Cooling a Sphere on the Velocity Gradient at the Wall. "Above Critical" Flow. . . . . . . . 123 44. The Effect of Heating and Cooling a Sphere on the Local Skin Friction Parameter. "Above Critical" Flow. . . . . . . . 124 45. Temperature Profiles for a Heated Sphere. "Above Critical" Flow, a = 0° . . . . . 125 ix Figure 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. Temperature Profiles for a Heated Sphere. "Above Critical" Flow, a = 47.5° . . . Temperature Profiles for a Heated Sphere. "Above Critical" Flow, a = 67.8° . . . . The Effect of Heating and Cooling a Sphere on the Local Heat Transfer Parameters. "Above Critical" Flow. . . . . . . . Velocity Profiles for a Heated Sphere. "Below Critical” Flow, a = 40°. . . . . Velocity Profiles for a Heated Sphere. "Below Critical" Flow, a = 62.6° . . Velocity Profiles for a Heated Sphere. "Below Critical" Flow, a = 82°. . . . . Displacement Thickness Over a Heated Sphere. "Below Critical" Flow. . . . . . . . Momentum Thickness Over a Heated Sphere. "Below Critical" Flow. . . . . . . The Effect of Heating and Cooling a Sphere on the Velocity Gradient at the Wall. "Below Critical" Flow. . . . . . . . Temperature Profiles for a Heated Sphere. "Below Critical“ Flow, a = 40°. . . . . Temperature Profiles for a Heated Sphere. "Below Critical" Flow, a = 62.6° . . . . Temperature Profiles for a Heated Sphere. "Below Critical” Flow, a = 82°. . . . . The Effect of Heating and Cooling a Sphere on the Local Heat Transfer Parameter. "Below Critical" Flow. . . . . . . . Definition of Positive and Negative Buoyancy Forces. 0 O O O O O O O O I O O The Effect of Buoyancy Forces on the Velocity Profiles, When the Sphere is Cooled. ”Below Critical" Flow, a = 82°, AT = -80°F. Page 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 Figure 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. The Effect of Buoyancy Forces on the Velocity Gradient at the Wall, When the Sphere is Cooled. "Below Critical" Flow, AT = -80°F. The Effect of Buoyancy Forces on the Velocity Gradient at the Wall, When the Sphere is Heated. "Below Critical" Flow, AT = 80°F . Comparison of Local Nusselt Number with Experimental Results of Brown [27]. Twall = 80°F; Twater = 70°F, Uniformly Heated Sphere . . . . . . . . Comparison of Local Nusselt Number with Experimental Results of Brown [27]. Twall = 190°F, Twater = 180°F, Uniformly Heated Sphere . . . . . . . . . . Comparison of Heat Transfer Parameter with Experimental Results of Brown [27] . . . Shadowgraph Picture of the Heated Sphere AT = 20°F. . o . . . o . o o o o Shadowgraph Picture of the Heated Sphere AT = 40°F. . . . o .. o . o o a o Shadowgraph Picture of the Heated Sphere AT = 70°F. . O O Q 0 I I I O O O Shadowgraph Picture of the Heated Sphere AT = 100°F . . . . . . . . . . . The Effect of Treating Viscosity Constant and Variable on the Velocity Profile Ue/U0° = l - 6, E = .083, AT = 80°F . . . The Effect of Treating Viscosity Constant and Variable on the Temperature Profile Ue/U0° = 1 - E, E = .083, AT = 80°F . . The Effect of Greating Viscosity Constant and Variable on the Velocity Gradient at the Wall. Ue/U0° = l — 5, AT = 80°F . . . The Effect of Treating Viscosity Constant and Variable on the Local Heat Transfer Parameter. Ue/Um = l — 5, AT = 80°F. xi Page 141 142 143 144 145 146 146 147 147 148 149 150 151 CHAPTER 1 INTRODUCTION The concept of flow separation is as old as that of boundary layer theory. Ludwig Prandtl, the originator of the boundary layer theory was concerned about flow sepa- ration before he formulated his ideas on the boundary layer. As a young engineer Prandtl found that the com— puted pressure recovery could not be achieved in actual diffusors [1]. He spent considerable time, prior to his presentation of the boundary layer theory, attempting to understand flow separation and the pressure losses in the diffuser. In 1904 Prandtl [2] presented his new theo— retical concept of the boundary layer in a paper entitled "Fluid Motion with Very Small Friction." In this paper he discussed flows over objects for which the Reynolds number was large. For such flows he made the following obser— vations: 1. Frictional effects are confined to a very thin layer, called the boundary layer, near the surface of the object. 2. The flow external to the boundary layer can be considered frictionless. 3. The pressure variation from the mainstream is "impressed" upon the boundary layer, i.e., Bp/By = 0. Because flow separation is caused by viscous effects confined in the boundary layer, it is often expressed as "boundary layer separation." Prandtl [3] states clearly that the necessary condition for separation from the wall is the increasing pressure in the direction of flow, i.e., positive (or adverse) pressure gradient along the flow path. The statement holds for compressible flow as well as incompressible flow. Within the boundary layer, the effect of viscosity is such that the velocity parallel to the wall changes along the distance perpendicular to the surface, i.e., the velocity gradient Bu/ay exists (u is the streamwise velocity and y is the distance normal to the surface). Since the flow velocity at the wall is zero u increases and finally reaches Ue' the inviscid flow velocity at the outer edge of the boundary layer. The momentum of flow near the wall is small and the ability of the fluid to move forward against the pressure rise is also limited. Downstream, this small amount of momentum along the body surface is used up to overcome the pressure rise, and, finally, the fluid particles are brought to rest at the "separation point." The point, at which the velocity gradient 3u/3y at the wall is zero, is defined as the separation point. At a point downstream of separation, because of the existing adverse pressure gradient, reverse flow occurs as shown in Fig. l, and, owing to this reverse flow, the flow in the boundary layer is forced away from the wall. At the point of separation, the flows begin to leave the surface at a small angle, maintaining the adverse pressure gradient. It has been demonstrated [1] that a laminar boundary layer can support only a very small adverse pressure gradient without the occurrence of separation. In the case where the boundary layer is turbulent, the danger of separation is intrinsically reduced, compared with laminar flow, because the turbulent flow boundary layer contains much more momentum and hence is able to resist the adverse pressure gradient for a greater distance than the laminar flow boundary layer. Typical velocity profiles of laminar and turbulent boundary layers are shown in Fig. 2. It is this transition from laminar to turbulent boundary layer flow around the sphere which explains the abrupt change in the drag coefficient around a Reynolds Number of 3 x 105 in Fig. 3. It is well known that for S Reynolds Number less than 3 x 10 , the boundary layer flow around the sphere is laminar until 84° where it separates and that for Reynolds Number above 3 x 105, the boundary layer flow undergoes transition to turbulence and the flow does not separate until it reaches 110°. This sudden backward shift in the separation point reduces the size of the wake which accomplishes the decreased drag coefficient. Thus by changing the shape of the velocity profile in the boundary layer (from a laminar to a turbulent one, see Fig. 2) a delay in the separation point is achieved. There are in existence several methods which have been developed for the purpose of artificially controlling the behavior of boundary layers, i.e., to delay or elimi— nate separation. Of the most popular ones are: l. Blowing—~acceleration of the boundary layer by injecting fast moving fluid parallel to the wall. 2. Suction—-removal of slow moving layer of fluid in the boundary layer near the wall. 3. Reduction in Viscosity of Fluid Near the Wall—- by heating the wall for liquids and cooling the wall for gases. It is obvious that the first method would give the fluid particles in the boundary layer the momentum they! need to overcome the adverse pressure gradient, while in the second method the slow moving fluid in the boundary layer is removed which again increases the boundary layer momemtum thus delaying separation. It is the third method, that of controlling the fluid viscosity near the wall by heating the wall, that is the subject of the present investigation. Only liquids will be considered. There has been some interest in the last twenty years in the effect of heat transfer on laminar separation par- ticularly in flows involving gases. Work with liquids has been very limited. A number of general qualitative conclusions on laminar separation with heat transfer in gases [4], which have been established (more or less independently) by various investigators are: 1. Cooling the wall tends to lessen the direct* effect of the pressure gradient. 2. Cooling the wall tends to delay separation. 3. Cooling the wall tends to diminish the skin friction for a favorable pressure gradient but tends to increase it for an unfavorable pressure gradient. All of the above conclusions were reached by Morduchow and Galorion [5] on the basis of a Karman— Pohlhausen type of analysis with a fourth degree velocity profile and were subsequently confirmed by the use of higher degree profiles by Morduchow and Grape [6]. More— over, these conclusions have also been reached on the basis *By "direct" effect is meant here the influence of the gradient term prOportional to dUe/dx. The influence of a pressure gradient, however, appears "indirectly" also, mainly, through the dependence of Ue and Te on x (cf., e.g., Refs. 5 and 6). of similarity solutions by Cohen and Reshotko [7] and Li and Nagamatsu [8], through the use of the Illingworth- Stewartson transformation [9], and by the analysis of Illingworth [10], Luxton and Yonug [11] and Low [1]]. By far, most of the solutions on the laminar bound- ary layer with heat transfer in gases which have been considered are the similarity solutions [7, 13] in which Ue ” xm, and adverse pressure gradients are represented by negative values of m. In connection with the effect of heat transfer on separation, the results of the similar solutions of pri- mary interest are the value of m, as a function of wall temperature, required for a zero—skin-friction boundary layer. These are shown in Table 1. It is noted that, as the wall is cooled, a larger negative m, corresponding to a larger adverse pressure gradient, is required for separation.* This may be considered to illustrate the tendency of cooling to delay or prevent separation in gases. The case where the external velocity may be repre— sented by U 6 = .- {-1— l X 00 *A comparison between entries Tw/Tm = .2 and .25 slightly contradict this trend, but this seems to be due to use of Pr = l in Ref. 7 and Pr = .7 in Ref. 13. may almost be regarded as the prototype of an adverse pressure gradient. It has probably been the most fre— quently studied case of an adverse pressure gradient. A number of investigators: Illingworth [10]; Morduchow and Grape [6]; Gadd [l4]; Curle [15]; and Poots [16], have investigated the effect of heat transfer on the separation point for such a flow of gas on a flat plate. Table 2 shows their results and confirms the result that "cooling the wall tends to delay separation." In addition to the similarity solutions and the various solutions for the case of Ue/U0° = 1 — x, there have been a few other cases for which the effect of heat transfer on laminar separation has been calculated. Poots [16] considered the case Ue/Uoo = l - g-and calculated the case of zero heat transfer (Tw/Tco = l) and of a heated wall (TW/Tm = 2). The results showed, that for low Mach Number, an upstream movement of the separation point when the wall is heated. Morduchow and Grape [6] have con- sidered the case in which a stagnation flow is followed by an adverse pressure gradient and have calculated the adverse pressure requirement for "immediate" separation as a function of wall temperature. Gadd [14] gives corresponding results if the initial region is one of zero pressure gradient, instead of stagnation flow. Baxter and Flfigge-Lotz [17] have calculated somewhat similar cases, in which a zero pressure gradient is followed by either a step pressure gradient or a ramp pressure gradient and found in each case that separation would occur sooner with a hotter wall. Fannelop and Flfigge-Lotz [18] have calculated the boundary layer over a flat—plate leading edge section followed by a semi— infinite wavy wall and found that separation occurred earlier for a heated wall than for an adiabatic wall, whereas cooling considerably delayed separation. Illingworth [10] presented an approximate analytical solution to show the effect of uniformly heating and cool- ing a circular cylinder, uniformly in motion in a gas. The results are presented in Table 3 where the position of laminar separation is y and A is defined as h (x) A=l- w e h and He being specific static and specific total enthalphy respectively. From Table 3, it is evident that there would be no appreciable difference in the position of separation be— tween the two cases (1) the temperature of the cylinder 0°C (A = .0521) and (2) 100°C (A = -,2951). Chang [1] explains this result of Illingworth by noting "that with the cylinder the effect of heat transfer on separation is small, due to the fact that the flow over the first 90% of the unseparated boundary layer is accelerated; conse- quently the deceleration region is short. Furthermore, the skin friction has taken time (see Table 4, Ref. 10) to reach a considerably larger value at the minimum pres- sure than for the unheated case. Thickening subsequently takes place more rapidly with a heated wall than with an unheated cylinder, but not enough to cause much change in the position of separation." One should also note that viscosity for a gas is quite insensitive to small changes in temperature, an increase of 100°C produces only a 30% change in the viscosity. Because the viscosity in liquids decreases with increasing temperature, the influence of viscosity vari- ation on the velocity profile is Opposite to that for gases. Consequently, heating should delay separation and lessen the "direct" effect of pressure gradient, while cooling should do the reverse. Surprisingly few solutions for liquids with vari— able viscosity have been presented. Schuh [19], Hanna [20], and Seban [21] solved the flat plate case (simi— 1arity solution, m = 0) with an inverse power law for the viscosity—temperature relationship, holding other fluid properties constant. Schuh calculated by successive approximation two flows having high Prandtl Number; one representing cooling and the other heating. Hanna pre— sented an approximate solution achieved by integral methods in which polynomical profiles were used for the 10 velocity and temperature. Hanna presented results where the Prandtl Number ranged from 0.25 to 1000. Seban ex- tended Schuh's results for a wide range of wall Prandtl Number. The three works reveal the effect of heating and cooling the wall in that they show that the skin friction decreases when the wall is heated and increases when the wall is cooled. Poots and Raggett [22], using experimental values (in the range of 0—100°C) for the viscosity, conductivity, specific heat and density, solved two different laminar boundary layer configurations. First the case of the heated flat plate and secondly the heat transfer effect on an infinite rotating disk is analyzed. Their results confirm the effect of heating and cooling on the skin friction and also indicate that heating tends to increase the heat transfer rate. Poots and Raggett have also pre— sented an analytical expression for the local heat trans— fer at the wall. The only other theoretical work that appears in the literature where the effect of heat transfer in liquids is treated is that of Kaups and Smith [23]. They have extended the method of Smith and Clutter [24], which solves the boundary layer equations, for calculating the laminar boundary layer in liquids having variable fluid prOperties, including viscosity. Kaups and Smith have, 11 like the previous researchers, presented numerical results for the flat plate case. They also present results of a flow past a semi-infinite body of revolution and indicate that by heating the body, the flow is stabilized, i.e., heating eliminates the inflection point from the velocity profile. They state: "In adverse pressure gradients, a heated wall appears to delay separation." However, they present no results which indicate the relationship between heating and the delay of separation. It should be noted that in the cases where the temperature difference brings about a difference in the density of the fluid, it becomes necessary to consider buoyancy effects also. Even though the density differ- ences in liquids are relatively small, see Fig. 5, the buoyancy effects are often considerable. Of the five references just cited [19 through 23] where the investi— gators have studied the effect of heating a body in liquids, only Kaups and Smith [23] have considered the effect of buoyancy. They present the cases of a heated vertical plate for both positive and negative buoyancy forces. The negative buoyancy force appears to have an effect similar to that of an adverse pressure gradient, while the positive buoyancy force has an opposite effect. This implies that the buoyancy force in the direction of fluid motion (positive buoyancy) tends to delay separation. 12 While there has been experimental work done on flows around spheres in air to determine the effect of heating the sphere, there exist only three papers cited in litera— ture on the experimental results of flows around spheres in water; Kramers [25], Vliet and Leppert [26], and Brown [27]. Kramers measured the heat transfer from an induction heated steel sphere to water and to oil in forced con- vection. The Reynolds Number ranged from 0.4 to 2100. These experiments were conducted with small sphere to water temperature differences. Vliet and Leppert measured heat transfer coefficients from an induction heated copper sphere transfering heat to water in the Reynolds Number range of 103 to 6 x 104 with substantial temperature differences (up to 130°F measured experimentally). Whereas Kramers [25] and Vliet and Leppert [26] have made measurements on a heated sphere that has a constant temperature surface, Brown [27] has measured the heat transfer coefficients for a uniformly heated sphere. Brown has presented results indicating the relationship between Nusselt Number and Reynolds Number for flow around a uniformly heated sphere. From his data on heat transfer and shadowgraph pictures, Brown points out that, for laminar flow around a uniformly heated sphere the sepa- ration point occurs at 90° (note: as mentioned earlier, 13 the separation point for unheated sphere for laminar flow is 84°). Thus Brown's work shows a backward shift of the separation point. In the experimental papers just cited above [25, 26, 27] the heated spheres were suspended in vertical test sections with upward fluid motion, which means that the buoyancy forces were positive. In the present investigation a computer program is used to show numerically the effect of heating on the separation point and the various boundary layer parameters for various flow configurations. This is done by treating the viscosity as a temperature dependent variable. Buoyancy effects are included also. Flow of water about a heated sphere also is studied experimentally. CHAPTER 2 MATHEMATICAL FORMULATION Formulation of the incompressible laminar boundary layer problem with heat transfer in liquids is complicated by the strong dependence of viscosity on temperature. The equations necessary to describe such a flow are those of continuity, momentum, and energy. Also, the relations describing the dependence of viscosity and density of the fluid on temperature are needed. Axisymmetric, steady flow about a body of revolution will be considered. The simpler problem of plane two-dimensional flow is included in the equations by letting r, the body radius, be a con— stant.* The basic notation and coordinate system is shown in Fig. 4. U°° is the reference velocity and Ue(x) is the velocity of the main flow just outside the "velocity" boundary layer. T00 is the reference temperature and Te(x) is the temperature of the main flow field just outside the *This will be illustrated later. 14 15 "thermal" boundary layer.* gx is the gravitational acceleration in the free stream direction. In the curvilinear coordinate system x is the dis— tance along the surface of the body, measured from the forward stagnation point and the dimension y is measured perpendicular to x. The velocity components are u and v, u being parallel to x and positive when moving in the direction of increasing x. The velocity component v bears similar relation to the y-direction. The body radius rO is as shown and it may vary with x. The boundary layer equations for the axisymmetric case in the above coordinate system were first developed by Boltz [28] and Millikan [29]. They are listed as follows:** Continuity: 1 a a __ . Flé—x-(ru) + 37mm — o , (2.1) Momentum: A9532 . + r 3y 3y , (2.2) *In all the numerical examples to be presented in the subsequent chapters, Te(x) is considered constant and equal to T . **One can observe that if r = const the equations are identical to the boundary layer flow for plane 2 — D flow. 16 Energy: C,.§.2..§2_k32T.ur_a_T (23) p p 8x 3y 7 3y r y y ' ' In the above equations Cp and k are considered to be constant. Also the viscous dissipation and the com— pression effects are neglected. Eqs. (2.2) and (2.3) differ from the equations obtained for plane flow when the Prandtl boundary layer approximations are made in that they contain the transverse curvature terms (u/r) (Br/3y) (Bu/3y) in eq. (2.2) and (k/r) Br/By aT/ay in (2.3). The transverse curvature terms are important when the boundary layer thickness is of the same order of magni- tude as the radius of the contour of the body (i.e., 6 ~ r0), which would be the case for any long, slender body as indicated by Schlichting [30]. If 6 <+n)¢" (3.64) and energy equation to 6" = Prm[:§l + R](¢+n)6' (3.65) Hence, values of ¢ and 6 at previous stations are not re— quired. The E-dependent terms also disappear for similar type flows. Such flows occur when P and R and the wall boundary condition are constant for all 5. These include flow over a flat plate with constant pressure and wedge type flows (buoyancy forces not included). The procedure of solution described earlier requires that the momentum equation be solved first, but to do so requires the values of viscosity. To get started at the E = 0 station, a linear temperature profile that satis- fies the inner and outer boundary conditions for tempera— ture is assumed. The viscosity obtained from this temperature profile is then used to start the solution. After the first solution of the momentum equation at any 48 particular station is found, the viscosity obtained from the last solution of the energy equation is used. At 5 = O, the value of P, the pressure parameter is known. For the flat plate with zero pressure gradient, P = 0, while for a sphere, P = 1. At E = 0, the value of R, the radius parameter, for any conical body is zero, while for a flat plate R = 0 for any 5. 3.8 Boundary Layer Parameters Once the momentum and energy equations have been solved at a particular E-station, the conventional bound- ary layer parameters are calculated. 1. Displacement Thickness. A physically meaningful measure for the boundary layer thickness is the displacement thickness 61, which is the distance by which the external potential flow is displaced outwards as a consequence of the decrease in the velocity gradient in the boundary layer. = f (1 - 694dy (3.66) 0 e 61 Using the transformation equations (2.13), the definition of 61, equation (3.65) transformed into the (€,n) coordi- nates is 000x 0" u 61=/-fi—f (1"5")dy e 0 e 49 vmx / U0° w 6 = ——— U— f (l - f‘)dy (3.67) e o A dimensionless displacement thickness may be defined as in Smith and Clutter [24] 6* — / U—“L 61 (3 68) l “ v00 'R_ ' Substituting (3.67) into (3.68) * nm - fnzoo 61 = (3.69) /E/a U0° x 2. Momentum Thickness. The momentum thickness is defined as 5 ._. f B—(l - .13..)dy (3.70) 2 o Ue Ue In the (€,n) coordinates, 61 may be written as / vmgL / U0° w _ __ I - I 62 — U U f f (l f )dy m e o A dimensionless momentum thickness is defined [24] as * UmR 62 5 = __ (3.71) 2 ‘v TT" 50 or m f — I flzdn * n=00 0 (s = (3.72) 2 U /_2 A U x 3. Shear Stress at the Wall. The shear stress at the wall is defined as T=B_u w 11w 8y w or in (€,n) coordinates as / U _ e N Introducing the local skin friction coefficient defined [23] as: Tw of = L U 2 (3.74) zpme or C = 2 LEE " f / UeEL um w Define the local Reynolds Number Reg to be Re = fig: 5 V00 Thus 1 U cf(Re£)’5 =/% [Ufl]f; (3.74) L 00 51 A conventional shear parameter is defined [23] as which with (3.74) becomes u c * = ———£——— / L-[—EI f" (3.75) x um w 4. Heat Transfer Parameter. The heat transfer rate at the wall is given by or in (5,0) coordinates U qw = - k V e gQTm (3.76) W \) L (D Let the local Nusselt Number be defined as (for flat plates) qu kw(Tw - T ) Nu= x or with (3.76) and definition of Rex, I Nu /Re = ——E!—— (3.77) x x l - 9 w For the sphere, Nusselt Number is defined as qw D D kwfiw - Tm) Nu (3.78) 52 Since the external velocity field Ue is a function of E, it may be written as Ue = Umf(€) Thus (3.79) _ /—f(€) 95) NuD//R5fi — 2 5 [1 _ g 3.9 Computer Program W A computer program was written to solve the momentum and energy equations simultaneously by the method de- scribed in the proceeding sections. This program was run on the CDC 6500 and is reproduced in the Appendix. The results of the runs made are given in Chapter 5. CHAPTER 4 EXPERIMENTAL APPARATUS AND PROCEDURE 4.1 Introduction The experiments were conducted to provide a quali- tative estimate of the effect of heating a sphere on the separation point. A brass sphere was placed in a square horizontal test section of a loop designed to circulate water. The sphere was heated electrically by means of a heating element placed inside the hollow brass sphere. The temperature difference between the water upstream from the sphere and the wall of the sphere was measured with the use of thermocouples. Hydrogen bubbles generated off a fine wire placed on the surface of the sphere were used to visualize the flow around the unheated sphere, while the shadowgraph method was employed when the Sphere was heated. A description of the sphere, test 100p, and the flow visualization techniques follow. 4.2 Description of the Heated Sphere A sketch of the sphere with the heating element installed inside it, is shown in Fig. 8. The 3.00" 53 54 (1.004) external diameter hollow brass sphere consisted of two hemispheres joined together. The wall of the Sphere was 0.25 inches thick resulting in a constant temperature sphere. The heating element consisted of 21 feet of Nichrome wire with asbestos insulation. The heating wire was held in a plastic cage inside the sphere, as shown in Fig. 8. Four, 30-gage iron—constantine thermocouples were soft-soldered into the inside wall of the sphere at the position shown, i.e., No. l at 70°, No. 2 at 45°, No. 3 at 0°, No. 4 at 315° (all angles measured clockwise from the forward stagnation point). The thermocouples, when inbedded into the wall, were 1/16" from the outside sur- face. Since the temperature difference between the up- stream water and the wall of the Sphere is required, a thermocouple was placed upstream of the sphere and used as the reference junction. The heating element lead wires and the thermocouple wires passed through the Plexiglass support tube. A photo~ graph of the sphere with the support tube is shown in Fig. 9. The brass sphere was marked at 2 degree intervals between 80° and 100° (measured from the forward stagnation point). The sphere was coated with a fine coat of varnish, which insulated the brass Sphere from the fine wire which 55 generated the hydrogen bubbles. The hollow sphere was filled with water before it was placed into the test section. The water served to improve the heat transfer to the interior walls of the Sphere. 4.3 Description of the Test Section Loop A schematic diagram of the test loop, which consists of a variable flow rate pump, two tanks and a lO-inch square plexiglass test section is Shown in Fig. 10. Water was circulated by a centrifugal pump. The walls of the upstream and downstream tanks were lined with stainless steel sheets. The upstream tank contained three straighten— ing sections. The first of these contained glass marbles sandwiched between two stainless steel screens; the second, a fine mesh stainless steel screen; and the third, a honeycomb constructed by placing 3-inch long, 0.2 inches diameter plastic soda straws between two stainless steel screens. The honeycomb was placed at the inlet of the square test section, so as to insure uniform velocity profiles in the test section. No measurement was made of the low free stream fluctions level. However, hydrogen bubbles traces showed no observable fluctions. The piping used to transport water from the downstream tank to the upstream tank through the pump was made of P.V.C., a plastic material. 56 Care was taken to maintain the water as free from impurities as possible. Before any tests were made, the water in the test loop was passed through a water filter at low speeds for at least 12 hours. 4.4 Flow Visualization Techniques The two visualization techniques, the hydrogen bubble method and the shadowgraph method, used in the present work will not be discussed at any length because they have been used in the past by many researchers [40]. Mention will be made, however, of the problems encountered in using these techniques in the present Situation. The hydrogen bubble method consists of using a fine wire as one end of a d—c circuit to electrolyze the water. The tiny bubbles thus formed are visualized by means of an appropriate light source placed outside the test section. A .001—inch platinum wire was used to generate the hydro— gen bubbles. A Spectra—Physics Laser beam passed through a cylindrical lens was used as a light source. The plati— num wire was stretched across the top half of the meridian plane. Thus the hydrogen bubbles were generated next to the surface of the Sphere, in the boundary layer and the wake region. The Shadowgraph method is based on the phenomenon that light passing through a density gradient in a fluid is deflected. It measures the second derivative, there— fore allowing visualization of only those parts of the 57 flow where the density gradient change is sufficiently large. The shadowgraph system consists of a bright light source, a colliminating lens, and a viewing screen. Photographs were taken of the region near the separation point using the hydrogen bubble method and of the whole flow field around the sphere using the shadow- graph method. The laser beam used to visualize the hydrogen bubbles was not photographed directly; a video recording was made with a TV camera installed with a 130 mm lens. The video pictures projected on the TV screen were then photographed. The shadowgraph pictures were directly photographed. 4.5 Operating Procedures The test loop was filled with soft water and the brass sphere was installed in the test section. The flow rate was adjusted so that the pump circulated 94 gallons per minute through the test loop. This flow rate resulted in the flow rate of the water in the test section to be 0.3 ft/sec. The Reynolds Number, based on the diameter of the sphere, was thus 7500. This low Reynolds Number assured a laminar flow in the boundary layer prior to separation. Before the sphere was heated, the hydrogen bubbles were generated and photographs taken to insure the occur— rence of laminar separation (about 84°). The power to the heating element inside the sphere was then adjusted 58 to provide the desired temperature difference between the water upstream of the sphere and the wall of the sphere. The shadowgraph pictures were taken for various tempera— ture differences. Prior to making any tests, the sphere was heated to check the uniformity of the temperature of the Sphere wall. All the thermocouples (1 thru 4) were used to check the wall temperature of the sphere and it was found that the Sphere temperature was essentially constant. When the Sphere was heated to make tests two thermocouples (2 and 4) were used to determine the temperature difference between the water upstream and the wall of the Sphere. The sphere reached a constant temperature within three minutes after the power was turned on. CHAPTER 5 NUMERICAL AND EXPERIMENTAL RESULTS 5.1 Introduction In this chapter results will be presented to Show the effect of heating a body which is placed in a uniform stream of water. Numerical results will be presented for various flow cases of water flowing past a flat plate and a sphere. Experimental results, which are qualitative in nature, will be presented for the case of water flowing past a 3—inch sphere placed in a lO-inch horizontal square test section. Though the primary objective of this investigation is to show that heating does affect separation, numerical results will include the effect of heating on the boundary layer parameters. In Chapter 3, the numerical methods of solution, for the momentum and energy equations, were discussed. It Should be emphasized that the boundary layer equations (momentum and energy) are valid only if 6 << 1. Near the separation point this requirement is no longer valid and hence the boundary layer equations are not applicable. 59 60 If the boundary layer equations were extended up to the separation point, a mathematical singularity would exist at the separation point. This presents a serious problem to investigators interested in locating the separation point. One way to eliminate this mathematical singularity would be to solve the full Navier-Stokes equations instead of the boundary layer equations. Since the Navier—Stokes equations are very difficult to solve, what is usually done is that the boundary layer equations are solved to a point as near the separation point as possible and then the solution is extrapolated to the separation point. The usual procedure is to use the value of ¢§ along the sur- face of the wall and to extrapolate the separation point to be the point where ¢& = 0. (This is done by fitting a curve through the 6; data points.) 5.2 Verificatign of the Numerical Techanue The purpose of this section is to establish the accuracy of the numerical method of solution. It will be done by comparison with known numerical and exact solutions, and experimental data. A. Similar Flows on a Flat Plate The only boundary layer flows which can be solved exactly are similar flows for which Ue = Em, where con— stant coefficients are assumed. For such flows the momentum equation and the energy equations are uncoupled. 61 The momentum equation is independent of €, and since the flow is two-dimensional, the radius parameter R is zero; eq. (3.7) then reduces to 4'" = - 331-(¢+n)¢"+ 9((4' + 1)2 - 11 (5.1) In this equation P = m where Ue = Em. The flows included in this study are for m = 1.0, 0.33, 0, -.0476l9, and —0.090429. m = 1.0 is the stagnation point flow; m = 0 is the Blasius type or flat plate flow; and m = 0.090429 corresponds to the separation profile. Fig. 11 shows these five profiles and in Table 6 the values of ¢§ are compared with exact results. Kaups and Smith [24] present results for a flat plate (Ue = Em, m = 0) where the plate is heated. Heating causes a viscosity variation u(T) across the boundary layer coupling the momentum and energy equations; they are 7}”- (c4“) = - P—‘glwwm" + P[(¢>' + 1)2 - 1] 6" = Prw[2§l](¢+n)9' (5'2) where c is the variable viscosity parameter defined by eq. (2.9). The eqs. (5.2) are solved simultaneously. Figs. 12 and 13 show the velocity and temperature profiles respectively for the heated and the cooled flat plate. 62 It can be seen that the given temperature variation causes considerable deviation from the Blasius profile. As ex- pected, the inverse viscosity—temperature relationship for liquids makes the effect of heating and cooling opposite to that for gases. Also shown in Fig. 12 is the "seventh power" turbulent profile. In Chapter 1, it was pointed out that the reason why a turbulent boundary layer around a sphere separated "later" (110°) than a laminar boundary layer (84°) was that the turbulent boundary layer contained more momentum near the wall than the laminar boundary layer and thus could better overcome the adverse pressure gradient. The "seventh power" turbulent boundary layer profile in Fig. 12 is shown to indicate that with sufficient heating the laminar profile tends to look more like the turbulent profile (i.e., contains more momentum). Figs. 12 and 13 also give a check against the calculations of Kaups and Smith [24]. Table 7 shows the effect of heating and cooling on the skin friction and the heat transfer parameters; the results are compared with Kaups and Smith. Since buoyancy forces are to be included in the calculations for flow around a sphere, a check was made with Kaups and Smith's [24] example of flow past a verti- cal flat plate with Tw = 312°F, T0° = 40°F, and U00 = 6fps. Figs. 14 and 15 Show the velocity and temperature pro— files respectively at a distance of two feet from the leading edge. 63 It is concluded from the results presented for the Similar flows that the numerical method of solution being used in the present investigation is sufficiently accurate. Since in this investigation, attention is focused on the separation point, a check was made on decelerated, or adverse pressure gradient, flows. B. Adverse Pressure Gradient Flows The development of the laminar boundary layer in a linearly retarded velocity field C2 U_:=1-% (5.3) has been studied by Howarth [34], Von—Karman and Millikan [35], Hartree [36], and Smith and Clutter [37]. This fluid flow case was studied without the consideration of heat transfer effects. In this investigation, a method of solution which is identical to the method of the latter two [36, 37] is used. This particular type of flow, eq. (5.3), leads to separation. Table 8 compares values of 6; calculated by the present author and those by Hartree and Smith and Clutter. As was pointed out earlier, the boundary layer equations are not valid at the separation point, thus the separation point is extrapolated from the values of 6;. The present method extrapolated the separation point to lie at E = 0.96 for the flow given by eq. (5.3). 64 In addition, the retarded flow 1 - E (5.4) C'C‘. (‘D 8 has been studied by various authors [6, 34, 38]. Table 9 compares the separation point as calculated (extrapolated) by various workers and the present author. It is evident that the discrepancy is greater than for the retarded flow given by eq. (5.3). The reason for this seems to be that, because eq. (5.4) represents a very strong adverse pres— sure gradient, separation occurs very close to the leading edge. The boundary layer equations are valid in only a short region near the leading edge; hence the extrapolation procedure is assumed inaccurate. The flows discussed in parts A and B have been two— dimensional plane flows. For axially symmetric bodies, the radius parameter R = (E/ro)(dro/d€) has values other than zero. The boundary layer growth on a Sphere is pre— Sented next. C. Flow Over a Sphere Two cases for the potential flow around the sphere will be considered; in one case, the simple potential flow for an unseparated flow is employed, while in the other, use is made of a velocity distribution Ue/U0° which has been obtained experimentally [31] in the case when the flow separates from the forward portion of the 65 Sphere. The two velocity distributions are shown in Fig. 16. The pressure distribution around the sphere has been reproduced from [31] and is shown in Fig. 17. "Critical flow" is defined as the flow where the flow in the boundary layer "just" becomes turbulent at the point of separation. In Fig. 17 the "above critical" curve represents the pressure distribution around the sphere when the boundary layer becomes turbulent before it separates and the "below critical" curve represents the pressure distribution when the boundary layer remains laminar up to the separation point. It is obvious from Fig. 17 that the pressure distribution measured for the "above critical" flow is similar to that of the simple potential flow. Thus the solution obtained from the potential flow theory for flow around the sphere serves as an approximation for the "above critical" flow. The theoretical velocity distribution on the sur— face of a sphere as obtained from potential flow theory, is given by C e .IT- 00 NH» = % Sin a = sin 5. (5.5) The radius ro(€) is given by r = R1 Sln 6 where R1 is the radius of the Sphere. The boundary layer calculations for the flow around the Sphere, with no 66 heating, were carried out to determine the separation point. The value of 6; are tabulated in Table 10 and are compared with those calculated by Smith and Clutter [37]. The separation point, ¢& = 0, was extrapolated to be at 104°. The experimentally determined velocity distribution [31] for “below critical” flow around the sphere is given by C a: = 1.5 6- .3640263 + .2466865. (5.6) The boundary layer calculations using eq. (5.6) for the velocity distribution with no heating predicts the separation point to be at 87°. Tomotika [31] using eq. (5.6) calculated the separation point to be at 81° using a momentum integral technique. Page [41] measured the separation point to be around 85°. Fig. 18 Shows ¢$ plotted against a for the two types of flows around the sphere corresponding to eqs. (5.5) and (5.6). Referring back to Fig. 17, it is evident that the point of minimum pressure lies at 74° for the ''below critical" flow. It is because of this stronger adverse pressure gradient for the "below critical" flow case that the flow separates earlier (about 10° after minimum pres- sure point) than for the "above critical" flow case (about 20° after the minimum pressure point). 67 5.3 Flow Past an Unheated Sphere: Hydrqgenfiubee Experiment The purpose of this section is to present experi- mental results which show that the flow past the unheated Sphere is laminar. This is done by showing that separation exists around 85°. The hydrogen bubbles, generated by the platinum wire placed on the surface of the Sphere, made it possible to visualize the entire flow region around the sphere, i.e., the boundary layer and the wake region. Fig. 19 shows a photograph of the flow region between 80° and 100° (measured from the stagnation point). It is evident from the photograph that there is reverse flow beyond 88°, implying that the separation point must lie ahead of 88°. To define the separation point, the follow— ing procedure was employed. A tracing of the photograph was made, as is shown in Fig. 20 (solid lines). Then a dotted line was passed through the zero velocity point of the reverse flow "path" lines and extrapolated to a point on the sphere. This point was defined as the separation point. AS is evident from Fig. 20, the separation point lies near 84°. It must be emphasized that the major problem en— countered was the generation of large hydrogen bubbles. Researchers have experienced this problem in the past [41] but it was eSpecially a nuisance in these experi- ments because the large bubbles would stick to the surface of the sphere thus disturbing the flow. Conventional 68 techniques of brushing off the bubbles and Switching the polarity of the platinum wire were employed but without satisfactory results. 5.4 Effect of Heat Transfer in Linearly Retafded Flows on a Flat Plate Consider first the flow over a heated flat plate where the external velocity flow field is given by U 6 _ .. E. U "‘ l 8 o (507) Since the viscosity variation is important in heated liquid flows, the momentum and energy equations are coupled and thus solved simultaneously. The free stream fluid temperature is assumed to be 70°F and the flat plate is heated to temperatures of 100°F, 150°F, and 200°F. Also considered will be the case of the cooled flat plate, where the free stream temperature is 150°F and the plate is 70°F. Thus we have three heated cases with tempera— ture differences of AT = 30°, 80°, 130°, and one cooled case with AT = -80° (AT = Tw — Tm). For all the cases to be presented, the results will be given for the four temperature differences mentioned above. Figs. 21 through 23 show the velocity profiles at various E-stations along the flat plate for AT = 130°, 80°, 30°, 0°, and -80°. Fig. 24 Shows how ¢§ varies along the plate for various AT. The dotted line in 69 Fig. 24 represents the extrapolation to is, the separation point. Fig. 25 shows the separation point location for various temperature differences. Figs. 26 through 28 are plots of the temperature profiles at various E-stations. Next, the flow over a heated flat plate where the external velocity flow field is given by U 9 =1—6 (5.8) on “I was calculated. This external velocity flow field repre— sents a stronger adverse pressure gradient than the case represented by eq. (5.7). Figs. 29 through 31 Show the velocity profiles at various E-stations along the flat plate for different AT. Fig. 32 shows the variation of ¢§ vs. 6 and here again the dotted line Shows the extrapo— lated values of as. Fig. 33 shows the location of the separation point for various temperature differences, AT. Since heating tends to increase the E-component of the velocity vector (i.e., increased momentum in the boundary layer), it is obvious that as the plate is heated, there will be a corresponding decrease in the displacement thickness. Fig. 34 Shows this decrease in displacement thickness by comparing the curves of various ATS. Note that for the cooled flat plate, the displacement thickness is the greatest. Fig. 35 gives a similar plot for the momentum thickness. Figs. 36 through 38 show the 70 temperature profiles for various ATS. It may be inferred from these plots that as the plate is heated, the thermal boundary layer decreases with correspondingly higher gradients at the wall, thus increasing the heat transfer rate. Fig. 39 is a plot of the heat transfer parameter Nug/Reg, showing the increased heat transfer. 5.5 Effect of Heating a Sphere Numerical results will be presented for the two potential flows, eq. (5.5) and eq. (5.6), around the sphere (see Section 5.2). The effect of introducing the buoyancy force term will be discussed for the case of flow of water past a heated sphere in a vertical channel. Experimental results showing the effect of heating a 3—inch sphere in a lO—inch square horizontal test section will be presented. A. Simple Potential Flow ("Above Critical") The velocity profiles for the temperature differ- ences, AT = 130°, 80°, 30°, 0°, -80°, at various a locations along the surface of the sphere are shown in Figs. 40 through 42. Fig. 43 shows the variation of 6; vs. a for the temperature differences mentioned above. The dotted line shows the extrapolation to the separation point a Fig. 44 is a plot of the local skin friction S. coefficient along the surface of the sphere. Referring back to Fig. 17, it is evident that the pressure distri— bution represented by the "theoretical" curve is such 71 that the pressure gradient is favorable (negative) from the stagnation point (a = 0°) to a = 90° and that beyond a = 90°, the pressure gradient is unfavorable (positive). Fig. 44 shows that heating the sphere diminishes the skin friction before a = 90° and increases it beyond a = 90°. Figs. 45 through 47 are plots of the temperature profiles for various a-locations along the surface of the Sphere. Fig. 48 shows the variation of local heat transfer parameters, NuX//RE* for the temperature differences being discussed. Table 11 gives the extrapolated values of a for the various temperature differences. S B. Experimentally Determined Velocity Distribution (“Below Critical") The effect heating has on the velocity profiles in the boundary layer at various a-locations is shown in Figs. 49 through 51. It is evident that heating tends to decrease the velocity boundary layer thickness. Figs. 52 and 53 Show how the displacement thickness and momentum thickness is affected by heating and cooling the Sphere. Fig. 54 shows the effect of heat transfer on the velocity gradient ¢$ at the wall. The dotted lines indicate the extrapolation to the separation point as. Figs. 55 through 57 are plots of temperature profiles at various a-locations. The local heat transfer parameter Nux//RE* is plotted in Fig. 58. Table 12 gives the extrapolated values of a for the various temperature S differences. hwflflllrfli- ., ”1.3. 72 C. Effect of Introducin the Buo anc Force Term I Below Critical The buoyance force term in the momentum equation (3 .7) is retained to Show its effect. For the potential flxDW'around the sphere, use is made of eq. (5.6), which represents the "below critical" flow. Before the results showing the effect of the buoyancy force term are pre- Sen¢ed, a definition of positive and negative buoyancy force is given. The buoyancy force is considered positive when it is in the same direction as the free stream velocity and negative when in a direction opposite to the free stream velocity. Fig. 59 shows the positive and negative buoy- ancy force in a vertical channel that exists in the boundary layer when the Sphere is heated or cooled. The reason for presenting the flow in the vertical channel is that the flow remains axisymmetric and thus can be treated by the numerical solution. It is obvious that when the buoyancy force is posi— tive (Fig. 59a and 59d), it would delay separation, and that when the buoyancy force is negative (Fig. 59b and 59c), it would bring about separation earlier. Calculations were made, with the buoyancy force term included for the four cases listed below, where the sphere was either heated or cooled. 73 Case 1. T = 150°F; T0° = 70°F (heating) Fig. 59a Buoyancy Force Positive Case 2. T = 150°F; Tco = 70°F (heating) Fig. 59c Buoyancy Force Negative Case 3. T = 70°F; T0° = 150°F (cooling) Fig. 59d Buoyancy Force Positive Case 4. T = 70°F; Tco = 150°F (cooling) Fig. 59b Buoyancy Force Negative Fig. 60 shows the effect of the positive and nega— tive buoyancy forces on the velocity profile at an a— station on the Sphere when the Sphere was cooled (Case 3 and 4). Also shown in Fig. 60, for comparison, is the velocity profile when the buoyancy force was not included in the calculation. It is evident that the negative buoyancy force makes the velocity profile approach the separation profile. Figs. 61 and 62 are plots of ¢$ vs. a for the cases mentioned above. Both the cooled sphere (Fig. 61) and the heated sphere (Fig. 62) indicate a slight Shift in the separation point when the buoyancy forces are included. In Chapter 1, reference was made to an experimental study by Brown [27], where he studied the effect of heat— ing a Sphere in a vertical channel (as in Fig. 59a). The copper sphere was heated with a constant heat flux 74 su<:h that the temperature difference between the water upstream and the stagnation point of the sphere was 10°F. The two extreme cases for which Brown presents results area: (1) water temperature = 70°F, Sphere temperature = 80°TV and (2) water temperature = 180°F, sphere tempera— ture = 190°F. Brown plotted the experimentally deter- oumed local Nusselt number along the sphere for various Reynolds numbers. Also presented was a plot of a heat transfer parameter (NuD/Reg Pr'36) vs. a. Brown's experimental results are verified quali- tatively by the present numerical solution, with the buoyancy force term included. For the purpose of simu- lating the potential flow around the Sphere, use was made of eq. (5.6). The local Nusselt number NuD is plotted in Figs. 63 and 64, and compared with Brown's results. Fig. 65 compares the heat transfer parameter used by Brown, as calculated by the present analysis. The results Show that Brown's experimental data agree within 10% of the results calculated by the present work . D. Experimental Results Hydrpgen bubble method: The hydrogen bubble method was unsuccessful in visualizing the flow past the heated sphere. It was evident that the platinum wire generated large bubbles very rapidly and that these bubbles remained 75 on. the wire thus disturbing the flow field. Attempts to brtash off the bubbles fast enough and record only the tiny buldbles were unsuccessful. Repeatedly, a new platinum Idgre was used with the anticipation of generating uniform bubbles but all attempts failed. It was thought that the secondary currents produced by the heating element wire Othich had A.C. flowing through it) may be the cause of a large potential across the platinum wire, and reducing the input d.c. potential did not produce any different results. Thus the hydrogen bubble method was abandoned in favor of the shadowgraph method. Shadowgraph method: The 3-inch brass Sphere was heated in a lO—inch square horizontal test section. The temperature difference between the water upstream and the ‘wall of the sphere ranged from 20°F to 100°F. Shadow- graph pictures were taken for various temperature differ— enceS and these are presented in Figs. 66 through 69. Since Shadowgraph pictures Show dark lines where the derivatives of the density gradients are large, the wake region is visualized. It becomes difficult to extrapo— late the separation point, since the reverse flow patterns are not visualized. However, these pictures do merit some qualitative analysis. It is evident that as the sphere is heated to higher temperatures the flow about the sphere becomes 76 asymmetric, i.e., the wake region becomes asymmetric. This is probably due to the buoyancy forces. Also, note that the wake is apparently three—dimensional at higher temperatures. Even though the location of the separation point cannot be determined, it is evident that there is not a significant shift in the separation point, i.e., the separation streamline appears to lie in the same region. As is evident from the results presented, the shadowgraph method is not suited to determine the location of the separation point. 5.6 Effect of Treating Viscosity Constant in LInearly Retarded Flow The results presented in the preceding sections were calculated when the viscosity varied across the boundary layer. In the present section, calculations are made with the viscosity held constant across the boundary layer. The linearly retarded flow, eq. (5.4), past a heated flat plate is examined. Calculations are made using: (1) the value of viscosity at wall temperature u = u ; and (2) the value of viscosity at free stream w temperature, u = uw. These results will be compared with the results obtained previously, when the viscosity was allowed to vary, u = u(T). Calculations are made for 'the case when the temperature of the flat plate is 150°F and that of the free stream is 70°F (AT = 80°F) . 77 Figs. 70 and 71 compare the velocity and temperature profiles at a typical E—station, when the viscosity is considered constant and variable across the boundary layer. In Figs. 72 and 73, the effect of treating viscosity con- Stant on the velocity gradient at the wall and the heat transfer parameter Nux/Rex are shown. The plots show that the results obtained by treat— ing viscosity constant are substantially different than those obtained by treating viscosity variable. When viscosity is considered constant, the calculations do not Show any significant shift in the position of the separation point. 5.7 Conclusions Heating substantially increases the u—component of the velocity in the laminar boundary layer. With a AT of 180°F the u—component of the velocity in the boundary layer increases by 100% for the linearly retarded flow past a heated flat plate, while the u—component of the velocity increases by approximately 50% in the laminar boundary layer for the flow past a heated Sphere. Heating retards the laminar boundary layer sepa- ration. The shift in the separation point is more pro— nounced in the linearly retarded flow past heated flat plate (30%), than for the flow past the heated sphere* *The numerical calculations were made using the experimentally determined [31] pressure distribution from 78 (5%). Experimental results tend to confirm that heating a sphere has a small effect on the position of the sepa- ration point. Heating has the effect of decreasing the displace- ment thickness and the momentum thickness. The displace- ment thickness decreased by 22% and the momentum thickness by 6% for the linearly retarded flows past the heated flat plate with AT = 180°F. For the flow past the heated Sphere, the displacement thickness decreased by 30% and the momentum thickness by 20%. While the effect of heating and cooling is small on the velocity boundary layer thickness, cooling does sub— stantially increase the thermal boundary layer thickness. The effect of introducing the buoyancy terms is not significant and thus may be neglected for the flow of water past a heated sphere in a vertical channel. The effect of heating a body subjected to a uniform flow of water on the boundary layer parameters cannot be determined with high accuracy by treating viscosity con- stant. The Skin friction coefficient may be calculated, with up to a 10% error, using the value of viscosity at the wall temperature as constant, and the heat transfer an unheated sphere. This pressure distribution was not allowed to change during the calculations; but as the separation point Shifts the actual pressure distribution also changes. To properly account for this change an iteration scheme would be necessary which would un- doubtedly result in a greater Shift of the separation Point. 79 coefficient may be calculated, with up to a 15% error, using the value of viscosity at the free-stream tempera- ture as being constant. TABLE S TABLE 1. Similarity 8C) solutions (Ue = xm) Values of m for Tw/Ton au/ay = 0 Reference 2 -0.06 7, 13 1 -0.094 13 0.6 -0.109 7 0.5 ~0.ll78 13 0.25 —0.l351 13 0.2 -0.134 7 TABLE 2. Calculated separation point for the case Ue/Uco = 1 - x Value of x According to Several Authors Tw/TQ S Illingworth Morduchon Gadd Curle Poots [10] 8 Grape [6] [14] [15] [16] 2 0.067 0.073 0.072 0.071 0.075 1.295 0.093 0.106 0.8 0.128 0.135 0.6 0.152 0.16 0.5 0.168 0.195 0.3 0.19 0.195 TABLE 3. Values of y at the point of separation (Illingworth [10]) A Y -1 1.402 0 1.429 +1 1.550 81 .»mmn m»a um ma vocums 0:» been) mam moon» 0» unawaan ma uo>aou on o» acanoum on» »s:» »cdoa ma unoc0w»avsoo Hflgiv: %m. .auanoum >»wa«nd»u advances: oz .voz»oa »uaxu xcc mo ouso Iououa ucfi»us»a »non no: .aa»souo .noandfiua> »wowaueu .uooo aw 06H» ocwusaaoo pooavou on canon »:n »copcomoncw as added» .oocauouuao .annoua 0:» Gavan: .»AEMH ca mauoaxo .auanoua .a5wvua so: 034» mew»saaou Isoooo swasuu 5.x sown: o»«cAm can o .02 onus: «ammo cw noxaaIxusucson ass .auusowu m o» my now: >us aw :ofi»ssvo 85»coeos mo auauuosoz o .02 can» uo3oHo hansnoum waves: can aaasofi»ouooze Iuauod .»unxo >-00wuouoona sca»neu0uussu» «answm Ioou»uu= .n .auanoua suaaunau. gouau065s oz .poas .»uonu soon as: acofiumavo nacho on canons and» uca»saaou pcouon uOu boom .xowu»- s .cofi»aacu cad» oc«»sasou .noano =.»oonuuo>oc unauoom »aou aw mew»ameoo Iauausouu newans»no o» econ Iwun> ucoosoaupcfi own »aoa~msu mca>ao>ca quanoua no» wand .oEH» ucw»sasou canon soon as: xuoa »cowowuusacw 5.x nods: :« .cOw»osqo .oucau0umwo I»«=m »oc cofluosvo n.ouoouu moan o>ono m .02 as 038m »:n .»usxo >~H¢0d»0uoone uoEMOMucou» n.0000uu u»«c~m .o .EuHQOua >»aaanu»n Advances: aux :.»oosnuo>o. .uovuo vacuum ma sawuasvo ocw>ao>sw assanoua Lam cane n.oooouu can .H I s .o I a .ucOH ma .amann Iuasm »oc sawuuzvo u.oououu us one nowuavcson condemn 08w» usauaasou .Auusoau Iwuo> usoucomopca ouu »aoflamxm .voucufiov acousouo 20a: .:o«»oussu0u »so«cu>soo m o» my sown hoousood 3.x coax: cw .s0a»csvo .oocououuao mums: »uoum 06a» ocausaaou can »uoxu aaaaow»ouooca .»uaxo >H~aua»ouoona coaMOuncsu» n.oououu ouacam .m .Eodnoua >»«aana»n Hou IapOEdc no: .coocufiop >08 Innoos nod: «nun: u>wuuuoxu .»«6w~ cw >H»uoxo .cowusaou «Ea» osu»smeou .n:o«»ou EUHnOum waxed unconson has oau» new»amsoo .xosusooo IAHQEOU mcH»:dsou unwasno caucus coo xaaoofi»ouoone on» nmwans»mo o» 6:00 »a0aaaxm .>H»ooum umwuo> Eu»m>n was» .xouusuou 30H Law »mau soon was xuoz »co«u«uunncfi .mcofi»asvo xmuwco .oocwukuan ca unocxown» uuxca ausccsom >Hucuwo«uusu can oaasam »sn .»uaxo zaaoow»0uounh oco Ed»cmeoe vmfluHUOEca o»ac«m .v .aomou oau0>oa ca uOuum ucoouua pouvcsn .moanowus> »coocum .nc0w»anau»a«u a»«oo~o> Houo>om .»c«oa saw»ocma»u IopCA us» as >Haua» .uaqOuuwu no ouon» an macs uso»«s»u0u can» use: on .uomoo o~n¢uo>ou (comma cauadu >.x onus: nos. «cod»sn«u»uqu unannoun Iuoo Law xaaaso oosws»no »noe on» ad oanwunom coH»cavu Ed»coEos uo uofiuom auscavuo nausea »occno on can unawusHon »oaxm aoausuua ousmau a o» m unca»08u0umcuu» u:o«u¢> ouacwwcu .m .so«»ssuo cuxm .o>onn .o>ons Iuocxaam uo an .ncOA» auwunaaefim .m>ona as mean on» usond as use» on» »son< no 056a on» »=on< Inaon uonEwm no woman Hoooq .m .nnosx0an» Saco auoowncoo »H .uoaua unconson Edou»nms mo .uonsu sauce 0» o>«»wnconc« con»ox ouuo>os cw houuo usouuum .a»wawnd»u hound auscsson no «.0000 nu aosusooa ooa can» no»oouo .uonao usuanoua new >uo»uuuuw»sn quHnoum concauwpcoo Ham: maasuo>au »a0l 0:» ca sod»oavo asuum»sa Annmaucu Is: >Houwuso >0susou¢ :H .»:oaco>coo pad »usu auo> manwuuoa annuaoou ousmmu n E:»coeoe u.sweuwx co> es»cosoz .H saw»owuh cwxm nEu«o«»«uo ooms»:a>v¢ on» .» cw uuouum >n Emwa»mmumuwo pon»0: .oz canvass: as m0ausuu< h «x 3 occauasqo uo>sa Sunvsson unsweda on» ocqbaou Law apoc»oe HaQAOCwua mo aumEEsm .v mqm<9 82 % r e a: A Ar . 3m Imm.mv smsousu Am~.mv .. .. m NH Imm.mv smaousu io~.mv ca coHumHoaumuaH ca :4 (coaumaoamuuxm ucaomuv cav Ha Amm.mv nmsounu Amm.mc ~\c< = = cam OH Amo.mv nmsounp Amm.mc ~\c< = . ~\c Ham; 0 o ONHm OHDGOOOHA meow»msvm mm»m mHsEnom mo mmha c S“ mm»m Ham3 um coH»muum»cw mcfluumum How whopmooum 0:» ca mam»m .m mqmée 83 TABLE 6. Value of ¢$ for similar flows .____.._.___- m ¢$ Calculated ¢a Exact 1.0 1.232587 1.2325877 0.33 0.47413 0.4741 0.0 0.332057 0.3320573 -0.04 0.220325 0.220317 -0.0904285 0.00006 0.0 TABLE 7. Effect of heating and cooling a flat plate on skin friction and heat transfer parameters 1 C(RW N/R’fi Tw Tm f x u x °F °F Kaups & Present Kaups & Present Smith [24] Method Smith Method 312 40 0.3698 0.36985 1.0646 1.06462 130 40 0.5540 0.55401 0.8938 0.89387 40 312 0.8182 0.81823 0.2937 0.29375 84 TABLE 8. Calculated values of ¢§ flow Ue/U0° = 1 — E/8 for Howarth's retarded Present Smith & 5 Author Clutter Hartree 0.0 .33206 0.025 .325728 0.04795 0.32263 0.05 .321842 0.100 .312 .311979 0.1534 .30102 .301031 0.206 .28966 .290089 .3116 .26564 .265623 .417 .23962 .239712 .23972 .5226 .21108 .62808 .17922 .179232 .73356 .14254 .142554 .83904 .097337 .098627 .09773 .8865 .071046 .072033 .948 .02642 .026397 .0249 .956 .014267 .0114 .958 .009534 .0059 .9589 .006469 0—extrapo— lated .96 O—extrapo— O-extrapo- lated lated Computing Time 106 secs 70 mins 85 TABLE 9. Separation point calculated for the flow Ue/Um = l — 5 5 Sep. Reference .12 Present Author .122 6 .12 38 .12 34 .1198 39 TABLE 10. Comparison of values of 6% on a Sphere as calculated by the present author and Smith and Clutter [37] u a° ¢w Smith & Clutter Present Author [37] 0° 1.31189 1.31193 30° 1.26099 1.261 60° 1.08115 1.082 90° 0.64833 0.642 101.7° 0.2394 104° 0* 105.9° 0* *Value obtained by extrapolation. 86 TABLE 11. The effect of heating the Sphere on the separation point "above critical" flow AT(°F) d°* 130 107.9 80 106.6 30 105.2 0 104 -80 101.5 *Va1ues obtained by extrapolation. TABLE 12. The effect of heating the sphere on the separation point "below critical" flow AT(°F) d°* 130 91° 80 90° 30 88.5° 0 87° -30 85° *Values obtained by extrapolation. FIGURES edge of boundary layer "II: .c Separation poin Bu _ W-O y=0 Separated Region Figure 1. Velocity profile near the separation point. y Y U 1 Ue () e u ‘27 u :11 / #11 a) Laminar b) Turbulent Figure 2. Boundary layer profiles. (II-I..Ivlfint!un..vv n.fl.a.~a~ Drag Coefficient 100 10 88 Laminar separation d=84° Turbulent separation a=110° J L l l l l - O l 10 101 102 103 104 105 106 Reynolds Number Figure 3. Drag coefficient for a sphere as a function of Reynolds Number. y Te Y ‘33 'I‘ u r u x m// \ on r _ 4 ) _ \_ I x \ I / / Figure 4. Boundary layer on a body of revolution--coordinate system. Fluid Property Ratios 1.0' 0.4 89 / Figure 5. Nondimensional fluid properties for water. D/Dref = 491.69°R. :2: = 62.4 #/ff3. ref = 4.339 #/hr ft Pr = 13.66 ref /r LlflJref 1 1.2 1.4 T/Tref 90 ow" n 0 n 6 I V117117777 "Trrr‘_ -1 (ro)n-l (ro)n 5 n-2 E; n-l 5n E» n+1 Figure 6. Notation for velocity and temperature pro- files in the boundary layer on a body of revolution. n-3 n—2 n-1 n AS n ¢I gn—3 gn-2 -1 gn-l En Figure 7. Notation for finite difference representation. 91 ////F" Thermocouple locations Heating element Figure 8. Brass sphere with heating element and location of thermocouples. Figure 9. Photograph of the sphere with the support rod. 92 .mooa umo» mo smummwp oa»mEonom .oH ousmflm oQEoohmcon Gmmhum moanume mmoao mesa 93 10. 1.0 m -0.090429 -0.047619 0.0 0.33 1.0 MUCH)? Figure 11. Velocity profiles for similar flows. Ue~ gm 94 7.0 A T T W (D 6 3 __ A 40°F 312°F ° B Blasius Profile C 312°F 40°F D 1/7th Power Turbulent Profile 5-6 t' 5 } Kaups and Smith [23] 4.9 h— 0 4.2 ~ 3.5 ._ l 2.8 " I A I 2.1 —- B / 1.4 ._ I C / (7 r D ‘// / / -‘.—-—— / _——-—t " 1 1 1 0. .2 .4 .6 .8 1.0 u/Ue Figure 12. The effect of heating and cooling on the velocity profiles on a flat plate. 95 7.0 T T Pr w 00 (I) 6.3 _. A 40°F 312°F 1.13 B 312°F 40°F 11.49 “ x 5’6 r" 0 } Kaups & Smith [23] 4.9 b 4.2 H 3.5 — 2.8 - A 2.1 - A 1.4 - B 07 ”- 0. 1 1 1 1 0. .2 .4 .6 .8 1.0 G = T - Tw - T w w Figure 13. The effect of heating and cooling on the temperature profile on a flat plate. 96 6.0 F Body Force Negative .. c) ............ Positive T = 312°F; T = 40°F w (D 5.0 _ o x } Kaups & Smith [23] 4.0 3.0 2.0 1.0 0. u/Ue Figure 14. The effect of buoyancy forces on the velocity pro- files on a heated vertical plate, two feet from leading edge. 97 Body Force Negative ------ Positive 0 . 2' “' x } Kaups & Smith [23] / _ x’ 1. 2’ / /X / / /X / / / / / /X / / O l I l L O. O 2 0.4 O 6 O 8 G Figure 15. The effect of buoyancy forces on the temperature profiles on a heated vertical plate, two feet from leading edge. 1.5 1.0 98 #- / U //I -- potential m —_\\§>// //’ / U _ //, e / Uco expt l l l L L l l l l 0 10 20 3O 40 50 60 70 80 90 a degrees Figure 16. Velocity distribution around a sphere. ’1 Theoretical / _\\\‘7 I )— I I I a1 I Experimental ,’ ////’ \ (above critical) I I I \ \ , _ \\ I \ I \ , \ } I \ \ , '- \‘ / I, \\ / I \ I \ I \\ / Experimental \\\\\ I ) (below critical) \ I I .. \\ \/ [I -—.- a L l l I l 1 L L l 1 1 0° 30° 60° 90° 120° 150° 180° Figure 17. Pressure distribution for flow over a sphere. .oumnmm on» venous 30am Hmfiucouom mo mommo 03» Mom Ham3 on» »m »coflpmum >»Hooao> .mH ousmwm Ammoummpv o 99 NH .moa .om oem one cow .mv .om .vm .NH .0 u _ — _u — — q q _ — ..m~. Lm. (ms. 3 =9 1 o.H =Hmoauauo 3636m= m =Hmouuuuo «>064: "a 1mm.a m.H Figure 19. Photograph of hydrogen bubbles visualized on unheated sphere. Zero-velocity line Figure 20. Graphical procedure for extrapolating separation point from hydrogen bubble pictures. 101 10. 9. — AT(°F) A: -80° B: 0° 3. P- C: 30° D: 80° E: 130° 7. Figure 21. Velocity profiles for a heated flat plate. Ue/U0° = 1 - 5/8; E = 0 102 10. AT(°F) 9. _' A: -80° B: 0° C: 30° D: 80° 0 O 8. __ E. 130 7. _ 6. ~ 5. L 4. r 3. — 2. F B C\ D l. "‘ E O. 1, L 1 L O. 0.2 0.4 0.6 0.8 u/Ue Figure 22. Velocity profiles for a heated flat plate. Ue/U0° = l - 5/8; E = .575 103 10. AT(°F) 9' A: 0° B: 30° C: 80° D: 130° 8. '- 7. r— 6. '- 5. F 4. F- 3. *- A 2. ._ B D 1. ’ 0. 4 1 1 1 O. O 2 0.4 0.6 O 8 1.0 u/Ue Figure 23. Velocity profiles for a heated flat plate. Ue/Um,= l — E/B; E = .7863 m mxw . H a Box a .32. may on ucoflomuv wuHooHo> may no mumam umam m mcwaooo can mcflumwn mo uommmo use .vm musmam w mm.H ON.H mo.H 0. ms. 00. mv. om. ma. .0 104 . /, _./ 1, 4; q 111 q . q q I / I com com coma 4:10on €035 105 m\w t a u SD\mD .ucaom coflumummom co Hmmmcmnu ummz mo uowmmm one .mm ouomfim 3 m. A89- any .2 oOmH OOOH 00m 0 00ml OOOHI _ _ 1 . d .O _ _ mw o.a 11 m.H 106 3.0 AT(°F) _ A: -80° 2.7 B: 30° C: 80° D: 130° 2.4 —- 2.1 - 1.8 ~ 1.5 - A 1.2 - B C 0.9 r- \ D 0.6 *- O.3 - 0. 1 1 l 0. 0.2 0.4 G 0.6 0.8 1 Figure 26. Temperature profiles for a heated flat plate. Ue/U0° = l - 5/8, E = O 107 3.0 AT(°F) 2.7 P. A: _800 B: 30° C: 80° D: 130° 2.4 _' 2.1 h 1.8 ~ 1.5 b 1.2 — A 0.9 F B C D 0.6 _ 0.3 P 0' L l l l 0. 0.2 0.4 0 6 0 8 G Figure 27. Temperature profiles for a heated flat plate. Ue/Uoo = l - E/B, E = .575 108 AT(°F) A: 30° B: 80° C: 130° 28. Temperature profiles for a heated flat plate. Ue/U0° = l - E/B, E = .786 109 10. AT(°F) 9'0F- A: -80° B: 0° C: 30° D: 80° 8.0- E: 130° 7.0'- 6.0 ~ u/Ue Figure 29. Velocity profiles for a heated flat plate. Ue/U0° = l - E, E = O 110 lO. AT(°F) 9. h- A: ‘800 B: 0° C: 30° D: 80° E: 130° 8. - 7. ” 6. r S. T 4. '— A 3. — B C 2. — D E l. “ 0. L 1 l L 0. 0.2 O 4 0.6 0.8 u/Ue Figure 30. Velocity profiles for a heated flat plate. Ue/Ua,= l - E, E = .083 111 10. AT(°F) 9.0 r A: 30° B: 80° C: 130° 8.0 F 7.0 — 6.0 ~ 5.0 _ 4.0 F 3.0 — A 2.0 — B 1.0 - 0' 1 1 1 1 0. 0.2 0 4 0.6 0.8 1.0 u/Ue Figure 31. Velocity profiles for a heated flat plate. 06/000 = 1 - g, g = .118 o u nausea .Hamz on» um unmavmum >ufioon> can co wumam umam m mafiaooo pom mcwumoa mo uommwm one .mm ousmwm 112 w mH.O QH.O ¢H.O NH.O Hoo m0.0 00.0 G0.0 No.0 .0 q .o N.O v.0 0.0 3 :0 00ml m m.O 00 "Q 00m "0 com um oOMH "4 L . O H AboVBQ 113 Es 0.15” .L. e . % -100° -50° 0° 50° 100° 150° AT (=T -T )°F w on Figure 33. The effect of heat transfer on separation point. ue/um = 1 - I; 114 w-Hu mH.o 5H.o 8 0 D\ D .mumam “man 09:3: “96 mmocxoafi ucoamomHmmHo .3 9.26.3 vH.o NH.o 00.0 v0.0 No.0 oOMH com o0m oo cowl Armov B4 «saloon: N.o ¢.o 0.0 m.o 115 ma.0 wta 0H.0 8 o D\D vH.0 .mumam umam owumo: m um>o mmwcxowzp Educmaoz w NH.0 0H.0 00.0 00.0 v0.0 .mm musvflm No.0 coma com com o0 00ml Amove< a m 6 o m m0.0 0H.0 mH.0 0N.0 mN.0 0m.0 116 3.0 AT(°F) _ A: -80° 2'7 B: 30° C: 80° D: 130° 2.4 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0' l L L l 0. 0.2 0.4 0.6 0.8 1.0 G Figure 36. Temperature profiles for a heated flat plate. Ue/U0° = l - E, E = .047 117 3.0 AT(°F) 2.7 — A: -80° B: 30° C: 80° D: 130° 2.4 — 2.1 — A 1.8 ~ 1.5 — 1.2 *‘ C“ D 0.9 - O.6 - 0.3 L- O' 1 1 1 1 O. O 2 0 4 0.6 0.8 1.0 G Figure 37. Temperature profiles for a heated flat plate. Ue/U0° = l - E, E = .083 118 3.0 AT(°F) 2.7 b A: 30° B: 80° C: 130° 2.4 F- 2.1 *- 1.8 " 1.5 ~— A 1.2 -— B C 0.9 '— 0.6 _ 0.3 r 0' 1 1 1L 0. 0.2 0.4 0 6 0 8 G Figure 38. Temperature profiles for a heated flat plate. Ue/Uoo = l - E, E = .118 w , unanaga .Hmnfidz uammmsz HmooH may no mumam umam m mafiaooo 0cm mcflumwn mo uommmm one .mm muomwm 119 w wH.O 0H.O vH.O NH.O OH.O m0.0 $0.0 @0.0 No.0 .0 1 q q u — d d u q o O l N.O l ¢.O Q l 0.0 U m .« m6 cowl "Q 00m "U com um . oOMH "fl L O H Ah0v8< 120 4.0 AT(°F) 3.6 L’ A: -80° B: 0° C: 30° D: 80° E: 130° 3.2 ~— 2.8 ~— 2.4 7 2.0 b 1.6 _ 1.2 _ 0.8 F 0.4 — 0. Figure 40. Velocity profiles for a heated sphere. "Above critical" flow, a = 0°. 121 4.0 AT(°F) 3.6 _ A: -80° B: 0° C: 30° 3.2 __ D: 80° E: 130° 2.8 — 2.4 “ 1.6 0.4 u/Ue Figure 41. Velocity profiles for a heated sphere. "Above critical" flow, a = 47.5°. 122 4.0 3.6 _ AT(°F) A: -80° B: 0° 3'2 C: 30° D: 80° E: 130° 2.8 ‘ 2.4 _ 0 0.2 0.4 0.6 0.8 1.0 u/Ue Figure 42. Velocity profile for a heated sphere. "Above critical" flow, a = 67.8°. 123 .3oam :Hmowuwuo m>on¢= .Hamz on» no pcmfivmum wufiooaw> on» so mumsmm m mcwaooo.ocm mcflumws mo vommmm was .mv muomwm Amomummvv o coma omOH com ovm own 000 omq com ovN oNH oo 4’ 4’11’4/ ’/ 1 q d .— d d a — co , 1/ y ,, x I J m 1 O.H 3 =9 1 min .1 0.N 00ml m 00 Q 00m "U m com "m m. oOMH “d N $00.5 a o.m .3on :Hoowuanu o>on<: .uouoaouom coauoaum aflxm HouoH onu co ouonmm o mafiaooo 0cm mcfluoon mo uoommo one .vw ouamflm Amooumoovo coma omoa com ovm . one cow 00v own o¢m oNH o0 — _ 1 A _ — _ d d 124 coma com com o0 000! (“300W “moVeQ 0.0 N.H m.a v.m 33; HI) 5i( ) 125 3.0 AT(°F) 2.7 _ A: -80° B: 30° C: 80° D: 130° 2.4 _' 2.1 *- 1.8 _ 1.5 1- 1.2 ._ 0.9 - 0.6 — 0.3 ” 0' l l l l 0. 0 2 0 4 0.6 0 8 1 0 G Figure 45. Temperature profiles for a heated sphere. "Above critical" flow, a = 0°. 126 3.0 2.7 — AT(°F) A: -80° B: 30° 2.4 - C: 80° D: 130° 2.1 — 1.8 ~ 1.5 ~ 1.2 P' 0.9 - 0.6 - 0.3 — 0' 1 1 L 0. O 2 0 4 0.6 0.8 l 0 G Figure 46. Temperature profiles for a heated sphere. "Above critical" flow, a = 47.5°. 127 3.0 2.7 " AT(°F) __ A: -80° 2 4 B: 30° C: 80° D: 130° 2.1 h 1.8 " 1.3 _ 1.2 ” 0.9 — 0.6 _ 0.3 P 0' 1 1 1 0. 0.2 O 4 0.6 O 8 1.0 G Figure 47. Temperature profiles for a heated Sphere. "Above critical" flow, a = 67.8°. 128 .3OHM =Hmouuuuu o>onn: .muouosouom Howmcouu uoon HoooH on» no ouonmm o mcwaooo ono mcfipoon mo uoommo one .mv ousmam Amooumovv a coma omoa com ovm own 000 omv com ovm oma oo 4 . a _ q n . _ d .o 1 «.0 1 m.o Q 1 N.H 1 . U o H cowl «Q m 00m "0 4 00m um .1 Com oOMH "d “moved ¢.m X /nN 3y 129 4.0 3.6 _ AT(°F) A: -80° __ B: 0° 3'2 c: 30° D: 80° B: 130° 2.8 P 2.4 ~ 2.0 T 1.6 ”‘ 1.2 _' 0.8 " 0.4 *- 0. Figure 49. Velocity profiles for a heated sphere. "Below critical" flow, a = 40°- 130 4.0 3.6 _ AT(°F) A: -80° 3.2 '- B: 0° C: 30° D: 80° E: 130° 2.8 *‘ 2.4 ” 2.0 " 1.6 " 12L A B 0.8 F 0.4 P 0 1 1 1 0. 0.2 0.4 0.6 0.8 u/Ue Figure 50. Velocity profiles for a heated sphere. "Below critical flow, a = 62.6°. 131 4.0 3.6 P AT(°F) A: -80° 3.2 " B: 0° C: 30° D: 80° E: 130° 2.8 _ 2.4 " 2.0 P 1.6 — A 1.2 — B C 0.8 e- 0.4 “ 0‘ 1 1 1 L 0. 0.2 0.4 0 6 0 8 1 u/Ue Figure 51. Velocity profiles for a heated sphere. "Below critical" flow, a = 82°. 132 coma .3OHM omoa =HMUHuHuo 30Hmm= com .ouonmm ooumon o uo>o mmocxoflnu ucoEooonmwa Amooumovv d owe .Nm ousmflm oNH oo d coma com o0m 00 com! Aboved O. .0 O. .0 «moon: £1309!!! m.0 0.0 N.H m.H m.H 133 coma .3OHM omoa d coma cow 00m o0 00ml Amove< =Hooflueuo soaom: .ouonmm oouoon o uo>o mmocxoflnp Esunofioz .mm ousmwm Amooumovv 8 com ovm own com 000 com ovm oNH 00 a q q — — q — 14 dimUDFIJ H.0 N.0 m.0 v.0 m.0 0.0 134 .zoao =Hmuuuuuu onom: .Hao3 onu um ucofloouo mueooao> on» no ouonmm o mcflaooo 0cm mnfluoon mo uoommo one .vm ouswflm Amooumovv 8 coma omoa com com omh coo owv 00m ovm oma o0 d d 4/./ A u d a q q a go m.0 0.H m.a 0.N 00ml 00 00m 000 .N oOMH m $00.5 0.m :9 135 3.0 AT(°F) 2'7 _ A: -80° B: 30° C: 80° D: 130° 2.4 '— 2.1 b 1.8 " 1.5 F 1.2 — 0.9 F 0.6 h 0.3 — 0‘ 11 1 1 1 0. 0.2 0.4 0.6 0 8 1.0 G Figure 55. Temperature profiles for a heated sphere. "Below critical" flow, a = 40°. 136 3.0 2.7 _ AT(°F) A: -80° B: 30° C: 80° 2-4 D: 130° 2.1 L- 1.8 r 1.5 ” 1.2 F A 0.9 F B 0.6 P C \ D 0.3 ” 0. l J L l 0. 0 2 0 4 0.6 0.8 1 0 G Figure 56. Temperature profiles for a heated sphere. "Below critical" flow, a = 62.6°. 137 3.0 2.7 '- AT(°F) A: -80° 2 4 "' B: 300 C: 80° D: 130° 2.1 "' 1.8 “ 1.5 ” 1.2 "' 0.9 *- 0.6 - 0.3 *" 0' J l 1 0. 0.2 0.4 0.6 0 8 l 0 G Figure 57. Temperature profiles for a heated sphere. "Below critical" flow, a = 82°. 138 . .3on =Hooeuwuo .uouosouom Hommcouu noon HoooH on» no ouonmm o mcflaooo poo mnfiuoon mo poommo one .mm ousmfim BoHom: Amoouuocv a coma omoa com com one cow omv com ovm oma oo 2 41 q _ u q d _ _ .o 1 v.0 L m.o Q 1 NIH l 0.H 0 m 00ml ”0 com "0 d . com um I. O N coma an $00.2 v.~ nN e‘d/ Z///////////////I«/////// Y O Allu 1 Negative buoyanc cooling force with 7/////////////////////// 139 / /////////////////////// / m U T POSitive buoyancy force Wlth heating 1 w//////////////////////// (a‘) V. C n w m 0mg U uin eeO VCO lrc t0 Ifif O / ///////////////////fl//// x///////¢///////////////I m U Illlil. force with heating 1 Negative buoyancy AA7ULUL7UA7h/UhYULYULVHLVhLVH/UAYULVULV (d) (C) Figure 59. Definition of positive and negative buoyancy forces. 140 5.0 Buoyancy force A: negative zero C: positive (I! 3.5 3.0 2.5 Figure 60. The effect of buoyancy forces on the velocity profiles, when the sphere is cooled. "Below critical" flow, a = 82° AT = -80°F. 141 .mooml u e< .3oam :Hooflufluo 30Hom= .0oHooo we ouonmm onu conz .Haos on» um ucofiooum muwooHo> on» no moonom ecumeoon mo uoommo one .H0 ousmflm Amooumouv 8 coma omoa 000 000 one 000 owv 00m ovm oma 00 8 0 .J. j — _ _ - q _ :2 1 in, C, a 1 3.0 I; 1 om.0 m 1 mv.0 1 00.0 o>fiuomon “0 ouou ”m L mh.0 o>wuflmom "d oouom monoeosm 00.0 142 .moom u ac .3on zamofiuuuo 30Hmm= .nmummz me ouonmm on» nonB .Hao3 onu no ucoflnoum evaooHo> on» no moonom eonoeosn mo uoommo one .N0 ousmflm “moonmovv 8 coma omoa o0m ovm own 000 omv 00m ovm oma 00 n q .2 n 1 % _ n 4 q . .2 O .2 :1 : n . I L G O U m L 0.0 n 3 =8 1 N.H 1. 0.H o>Huomoc "0 ouon "m 1. 0.m o>wuwmom "m oouom eonomoom v.m 143 .ouonmm Uouoon maeuomwcs .mooe uuouo3e “boom u HHo3e .nemn nzoum mo muaomon Houcofifluomxo nuflz HonEdz uaommsz Hmooa mo acmflummfioo .m0 ouomwm Amooumonv 8 coma omoa 000 000 owe 000 00v o0m ovN oNH oo _ _ 1 1 _ _ _ . n 0 1 00m 00v 000 052 0 mean c3oum 0 000 x 000mm "U oomhm "m 1 000A 00000 “d Dom 00NH 144 H 00M3 o3 .ouonmm voumon washouflcs .moowa n e .mooma u Ha e .nemu nBoum mo muHSmoH Houcosfluomxo nuw3 nonEdz uaommoz HoooH mo comeuomsou .v0 onsoflm nooumocv 8 coma .moa com .qm one .08 .mv com .vm .NH .0 d n _ n — d d a — O L 03 1 00m we 1 00m 0 92 J. O nnmn n3oum o 1 00¢ x 00NHN "U ooeem ”m 000 ooeme an Dom 000 .145 coma .Hemn nzoum mo muaomou HoucoEeuomxo nue3 Houofionom Howmcouu poon mo nomHHomEoo 000a o0m ovw Aooumoov 8 owe o00 omv 00m ovm oNH .mo onswflm — _ n ne~1 czoum no n n q n _ a m.0 0.0 0.0 0.H 3 3 E/IJdZ/I 93/ “N Figure 66. Shadowgraph picture of the heated sphere. AT = 20°F Figure 67. Shadowgraph picture of the heated sphere. AT = 40°F Figure 68. Shadowgraph picture of the heated sphere. AT = 70°F AT=IOO F Figure 69. Shadowgraph picture of the heated sphere. AT = lOOOF 148 10.0 00 U(T) u/Ue Figure 70. The effect of treating viscosity constant and variable on the velocity profile. Ue/U0° = 1 — E, E = .083, AT = 80°F 149 3.0 1 u 'u 't 3 ,. E3 1.8 " 1.5 " 0.6 '- Figure 71. The effect of treating viscosity constant and variable on the temperature profile. Ue/Uoo = 1 - E, E = .083, AT = 80°F 150 m . .58 u S .0 .. H n 83 a Sm: onu um unoflnoum mufluoao> onu no oHanuo> 0cm unoumnoo euwmoomH> mcwuoouu mo uoomwo one .me ousmwm w m.0 0H.0 0H.0 va.0 ma.0 0H.0 00.0 00.0 00.0 No.0 .0 q q d u — j d _ d .0 E3 V 8 a. :1 :1 n u u :1 :1 :1 0 m o 1 8.0 L 04 =0 o boom u ed .0 1 a u 80\ 0 .uouosouom Hommcouu umon HoooH on» no oHanum> 0cm unnumcoo enamoomw> mnfiuoouu mo uoommo one .mn ousmfim w 0H.0 0H.0 ¢H.0 NH.0 0H.0 00.0 00.0 v0.0 No.0 .0 151 _ _ q _ n n _ n _ an a .m E: II :1 ‘12 BIBLIOGRAPHY [l] [2] [3] [4] [S] [6] [7] [8] [9] BIBLIOGRAPHY Chang, P. K., Separation of Flow, Pergamon Press, New York, 1970. Prandtl, L., fiber Flfissigkeitsbewegung bei sehr kleiner Reiburg. Proc. Third Intern. Math. Congress, Heidelberg, 1904, pp. 484—491. Prandtl, L., The Mechanics of Viscous Fluids, Aerodynamic Theory, Vol. III, Springer, Berlin, 1924, edited by W. F. Durand. Morduchow, M., "Review of theoretical investigation on effect of heat transfer on laminar separation," Morduchow, M., and Galowin, L., "The compressible laminar boundary layer in a pressure gradient over a surface cooled by fluid injection," Proceedings of the Iowa Thermodynamics Symposium, State Uni- versity of Iowa, Iowa City, Iowa, 1953, pp. 143-169. Morduchow, M., and Grape, R. 6., "Separation, sta- bility and other properties of compressible boundary layer with pressure gradient and heat transfer," NACA TN 3296 (May, 1955). Cohen, C. B., and Reshotko, E., "Similar solutions for the compressible laminar boundary layer with heat transfer and pressure gradient," NACA TN 3325 (February, 1955). Li, T. Y., and Nagamatsu, H. T., "Similar solutions for the compressible boundary layer equations," J. Aeronaut. Sci. 22, 607 (1955). Stewartson, K., "The Theory of Laminar Boundary Layers in Compressible Fluids, Oxford University Press, New York, 1964. 152 [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] 153 Illingworth, C. R., "The effect of heat transfer on the separation of a compressible laminar bound- ary layer," Quart. J. Mech. Appl. Math., VII, 8 (1954). Luxton, R. E., and Young, A. D., "Generalized methods for the calculation of of the laminar com- pressible boundary—layer characteristics with heat transfer and non—uniform pressure distribution," British Aeronautical Research Council, R & M 3233 (January, 1960). Law, G. M., "The compressible laminar boundary layer with heat transfer and small pressure gradient," NACA TN 3028 (October, 1953). Livingood, J. N. B., and Dononghe, P. L., "Summary of laminar-boundary layer solution for wedge-type flow over connection and transpiration cooled sur— faces," NACA TN 3588 (December, 1955). Gadd, G. E., "A review of theoretical work relevant to the problem of heat transfer effects on laminar separation," Aeronautical Research Council Paper 331 (June 13, 1956). Curle, N., "The steady compressible laminar boundary layer, with arbitrary pressure gradient and uniform wall temperature," Proc. Roy. Soc. (London), A249, 206-224 (1958). Poots, G., ”A solution of the compressible laminar boundary layer equations with heat transfer and adverse pressure gradient," Quart. J. Mech. Appl. Math. XIII, 57 (1960). Baxter, D. C., and Flfigge—Lozt, I., "The solution of compressible laminar boundary layer problems by a finite difference method," Stanford Univ., Div. of Engineering Mechanics, TRllO (October 15, 1957). Fannelop, T., and Flfigge—Lozt, I., "The laminary compressible boundary layer along a wave-shaped wall," Ing.—Arch., XXXIII, 24-35 (1963). Schuh, H.) "The solution of the laminar boundary layer equations for the flat plate for velocity and temperature fields for variable physical properties and for the diffusion field at high concentration," NACA TM 1275 (May, 1950). [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] 154 Hanna, 0. T., and Myers, J. E., "Laminary boundary layer flow and heat transfer past a flat plate for a liquid of variable viscosity," A.I.Ch.E. Journal, 7, 3, 437-441 (1961). Johnson, H. A., "Boelter Anniversary Volume Heat Transfer, Thermodynamics and Education, McGraw—Hill Book Co., New York, 1964, pp. 319-379 (chapter on "The Laminar Boundary Layer of a Liquid with Vari— able Viscosity" by R. A. Saban). Poots, G., and Raggett, G. F., "Theoretical results for variable prOperty laminar boundary layers in water," Inter. J. Heat Mass Transfer, 10, 597-610 (1967). Kaups, K., and Smith, A. M. O., "The Laminar Boundary Layer in Water with Variable Properties," Douglas Aircraft Company Paper, Long Beach, Calif. (August, 1967). Clutter, D. W., and Smith, A. M. 0., "Solution of the General Boundary Layer Equations for Compressible Laminar Flow, Including Transverse Curvature," Douglas Aircraft Company Report No. LB 31088, February, 1963. Kramers, H., "Heat transfer from spheres to flowing media," Physicia, 12, 61 (1946). Vliet, G. C., and Leppert, G., "Forced convection heat transfer from an isothermal sphere to water," Journal of Heat Transfer, Trans. ASME Series C, 83, 163-175 (1961). Brown, W. S., Forced Convection Heat Transfer from a Uniformly Heated Sphere to Water, Ph.D. Thesis, Stanford University, Stanford, Calif. (1960). Boltze, E., "Grenzschichten an Rotantionskarpern in Flfissigkeiten mit Kleiner Reiburg," Dissertation Gottingen (1908). Millikan, C. B., "The boundary layer and skin friction for a figure of revolution," Trans. Amer. Soc. Mech. Eng., Appl. Mech. Sec., 54, 29—43 (1932). Schlichting, H., Boundary Layer Theory, McGraw—Hill Book Co., New York (1962). [31] [32] [33] [34] [35] [36] [37] A [381 [39] [40] [41] 155 Tomotika, S., "Laminar boundary layer on the surface of a sphere in a uniform stream," British Aeronauti- cal Research Committee, R and M 1678 (1935). Hartree, D. R., and Womersley, J. R., "A method for the numerical or mechanical solutions of certain types of partial differential equations," Proc. Royal Soc. Series A, 161, 906, 353 (August, 1931). Howarth, L., Modern Developments in Fluid Dynamics-- High Speed Flow, Vol. 1, Chap. X, Oxford (1953). Howarth, L., "On the solution of the laminar boundary layer equations," Proc. Royal Society, 164, 547 (1938). Von Karman, T., and Millikan, C., "On the theory of laminar boundary layers involving separation," NACA Report No. 504 (1934). Hartree, D. R., "A solution of the laminar boundary layer equation for retarded flow," British R and M No. 2426 (1949). Smith, A. M. O., and Clutter, D. W., "Solution of Incompressible Laminar Boundary Layer Equations," Douglas Aircraft Company Report No. 40446 (July, 1961). Levy, 8., "Heat transfer to constant-property laminar boundary layer flows with power—function free-stream velocity and wall-temperature variation," Jour. Aero. Sci., 19, 15, 34-348 (1952). Leigh, D. C. F., "The laminar boundary layer equation: A method of solution by means of an automatic computer," Proc. Camb. Phil. Soc., 51, 320—332 (1955). Schraub, F. A., Kline, S. J., Hariry, J., Runstadler, P. W., and Little, A., "Use of hydrogen bubbles for quantitative determination of time-dependent velocity fields in low-speed water flows," J. of Basic Engi- neering, Trans. ASME, 429—444 (June, 1965). Fage, A., "Experiments on a sphere at critical Reynolds Numbers," British Aeronautical Research Committee R and M 1766 (1936). APPENDIX '1! 1] \II-lfl ll! IJIII) ‘I ‘ll lil1| z! I J ‘IIJ ll.‘ 4 . )1 1 ( 1.1.!11 till'li (111(1llt11 1’1. illll'llll 1.111 711.111 11 .11 4| 1 11.!- pt oznpposi mou;dmxozaa um oetuxa nczqo unnqum. omappnocnw> oaox; mmoxzaz 28__h—mezu ed mXDp_\moxqpom.>zom.>cmn.283nxm8\84518\~8\38~\ocxzozzC8 emu.qmao.mmao\n¢\znz\o¢\8 I . ibillllt.ll|llu 11 1". .¢»ooc.a~.mzm_._sm_.pmx_u_.gssun.8z.srz.sz.rz.sz.m ;wen.eranmamrmaxemxmmmo.aemVaN\Mm¢oqxm->44<~x< mm >003 th eqte mm SOIpw: wt» 20 zoneunxemmm wIe ;:i1i1-%lléiii-:- #NWWer%WDH;mwM¢4§HM¢QZ30maurn-mwdmw¢IQZmHmzcu(>0H000mn>1 wzmhdmah mOu CZd zzwb muaou >UZ<>020 wIe ozmoznuzn You mzcnnq 241001Q mIe omm—bzwlcxa CnaJm :24 mammzm8 :111.,,11@wnonwowo >334_p~z~ m4; zonpasom to corpoz orb .mzo_n<:cm zo><4 >z848848>u44418 lull"- oux p4 4443 #4 42848488 848n444188p midnm: 20 004 45 0x4 Jmmmmao 44nnnzn . II [‘0' ‘1 9...! "|.. 1.111. ‘1-1 ... . 1'5 ‘11 ll. .91 1| I? 1 1| ,I|l- 1 1 2 4443» p4 n24nmzoo 84 >8~88884> 28:4 nun 824p p4 ~24p8288 84 >p488884> 24:: u ,1- 8484444) 8848848288 >p4mo84> 28:3 ou84>4 mzxon >824>888.H8448481---;.:m-2--- 88884824 on on 8:48» >824>88x 28:3 4u>884 1 zonp84om 28np4800 2842028: >420 - 8n >4m88824e4824m 88>48m mm --1-! 1::. : -- - : op mzo-48am 28hzoxoz 824 >8mozo ream 28:3 nu828824 -..--11!- -.,.--.-- -i-S-4-m-a:8.ziii o- oopznaa mm on zo_p84om >8xmzo 8p4ncmzxopzn 28:3 .:.:;-4mwz -: pznal 82 u omeznma mm seize—F8488 28n2828z 8p4488248n24 28:3 nunz zonp848m 14442481282 ,1i:!-;; "sci-e:- 8ua~8881 mu zonp84om 2844 144~znm >428 28:3 nunz pqumzoo nu x24; ~40: 2013 a ' - '1 .F v Ilu.|I-1')11l|11l.11 1.11 11...! .- 1- 1111|i|- 0"»! (Ill 1!! It'll ' ' 8844188 »4 bz4nmz88 888n4zwa84p 28:3 op non 84 8» 414 8824 »2288 81F 000 000 5' 'l. ‘1'! 1} .\' ‘ll. III 11"] ‘I 158 ,-12::4pumamflwutdmequVoflmeflm\wn\4w~\wmxfidzwuw4448HamZHH\Hfl\mZdquVQMN~ ZZZ\ON\¢Z.MZ.NZ.~2\®N\11HIQ.3mrmozm4m.4rqhw.4dbm.Ia~.N3m~.~2mm.pmxnu_.3348H.UZ.4IZ.:S.IS.4z.m -UmHdJHMHmJflHflEMHVMM\mN£u¢Amu\dm\mwmud<43m1wZHHDOmem 02w zrbhmm .Kxxxxanoo¢mQG+4om 4440 m muazcuz~ 1;;odmxddfl A-dqde n m.dWHdmfiz gum—>H MDZHPZCU a -;1I:-fiadadowmoarmcq~ +ooovmo¢om.lo#mom8\.ooaotmoosmo.¢.m+cc¢mo¢on.#.MIm.~8¢OH.Avxm .o.z~m\.o.mooaau.~8aa Qmfijdqudddddl4fl13tagw333 war dw.Q3.uQflQ--m.11nfi¢-qnfld-4fl 3-343- x4zz.mu_ 4 oo .mu.~.ax a .Hu.~8¥m .11111i1113:-1;.defldAWaquflfiQOmmfidWAim8xdflflmwduwammvxmeN~¢~w4NN8xmmmmflouno—NVX uncoomu.cmvxmmhomoau.ou.xa~o¢m.~n.mu.xm0oom¢o«u.h~.xwmmm¢.~qu~8x mmm.~u.m~.x*~o>momua:~.xmcom~.~u.m~VXmomoo.mnamuvxa¢¢mnoou.-8x IINWNNQfiM4dafififldfidHNdw4fiqfim¢afldadu«flwfififlmflmM¢wanflagdfl¢¢on“ova, moo¢mou.mvxamam¢m.uacvxw3~¢maoou.mvK$¢m~o.on.m8x 8 .ou.~.x mmnxaa .mzmp2888 4N44448824 888 48428u4848 a 4u84 x4284u84 mm 8. 4448 on .84484841 8448148811 8.848 44~_~z_ 4881288 888 mm.om.mm.414hm_8ufi pm 488:8u4848 8 one 84 4448 mm l-IIII‘I ill'll-I' 1 1. --. 111 11-1-1... 1.9148 H.mzau.-.ZaH4~.z8_8482844>88cm 1111111111111111:11-1-1111111188uaqmu44482444HAJGM11-51- .88m4844m..ccmq.m~m..oom48mm..oom4848 zo~mzmz_8 444.884888.8.m.4.nm.m8.48.mm.mm.fim.za 288848848 4488co .: 1111111111:.884448448844884.m88.484azauwauumm144nqqq1s: -- p44uux4ox4hmm.>8om.>88_.2844\moxo44mox~8\88~\oo\zozz88 emu.4884.8884x44xz4xxo4xm -11.I11.-1|I11.1ilnllll11 0 1111'--- . .IID- -1lllg l-D 11...! . - .i. r.m8:844488888411444p8281.414:4~88:81m848448.~wqunmvfimzmm8mqrmi1: 4484.8.mx48.4.148248u4pmzp .1--1: .....z.::: -- - ;-;1 .-.. 88ZHHZC8 044 8444V4wmmmmndwwrmwwmmu8-z:1- o4zu\.4zxau.48za.u4248 m4" oN~ Op 00 1. .1 .111 YIIOIEC"II‘1' l I .. -1|- 1 1 I. I. - 1311i!" - .0 . .llun wzmna¢n4zu.4zara:44¥4..m8144m488u4448- ~21148448.4z:4.4444.448:44m44814488 844 -::;1-..---:. -.. .4844m8mmumzaza --.z.1-111-:44H4448M488mwanm-488 044 o» co .m4_¢m.mmu~zaxa - -. - . . w -- .--.44nm4484448w4284 884 mo4 ch 88 .44~.88.m84844 mo mo.mo.m44 44-48m4888 so {.---:11- 1..-.11-.--._----_- ...1 - -- - 1-- 1418-44-818m18fimjumm1. 09 oh so .mwmomo.¢wmvmu om co.ow.oo .38um m~.. ..E1111...--.--..---11-1---.1----.---1- - - .- -. -12-1. - --jMHIMNv-mflwdzmrm- .qlmmutmvthHZIQ ON ms 0» CO ---1- .----11 1-1.1111-1-111111. .1--1:111:11- .i -, .1- l 1.-- ------.. I-.- .1-.-1-1- 11:11dd-Qw4zmmm-tt 11 .4144444848u4218 m8 -4+-8p 88 .48~.88.484.u~ 88 111..- 11-1.1-- --.-1.--1.--1-1 111-11-111- 1111-11,- 2-1-1-11-11-..dNAI-3u1dd-1111. .1- o.oumdwc c.ou~amo mm 1.édmuuZMmMMMfifi1MHHZHMEHQIWMH4H42HGMQH#1MHjmflQU1I1UQU- ooommomma~ldhmmvun cm aQHIQuavaQ --1-1|.1-1.l11 111-2-111-1- - -11-- 11 111.11.11.- .- :1--111.11-.11.111111ll--1...111. C 1H-" «Nurm .111. -I11 CoauAvaa m¢ Hum -1-- . ._ - coou~HVJ¢ Anmwm oubmxuum 161 4m41au4m+444m 444rau4444m mod 04 co 1;. i - . - . .. . ... . a¢fin1u~m+~vmm 4m411u4m+44Nm 4N.Iau44144mm --f§ 111::1-1 1: .-.1. _;. -aavraudawwm- mo4 04 co 4:411u4n144mm 4m4rmuam+q4mm 4mvxau44444mm 4441Qn444mm -. . . .. .. .co~¢mmd.am4 4u24u~ .m234p343m zo4hqncm zapzmzoz mt» 24+4mquatooomld 4svraao.4~4+4mvrlnc.wmvao.oom\4oon¢zx4uacm mm4 a» cm i aom1414xau4141a 43:13:.x4uz4414m4.4.»:om14zx424m12x412cm nm4 oh ow 4.3m.444444u4 4 42x.::4mmuz<>ozm mmjm1qq- m~4 :4 oo 44. mz.>om4.u4 .ml.o.~+.N.IQ.#AQo~+.Nvtavvaz»_zquz_1aH HUIamHImuaaHIQ mmm 4443 H4. naHIa zo onHumzaou #242 000 11:15:11, .51- mam.mmm.mamAW>aHHMH. Gd“.-: . uanz -:-- QH.11 1-1 . HU+1QHIQHQQHIQ com o: O» 00 dualmHl1illi: oum>aH on 02 111lr1ll'l1 [ill- .11111IIIVIl-IE. HHHHIHIQI IHHIm%wmQHHmH .11 113M441 111-11 .1. «Hmu44hm HmHquH .21111111111211111111l111zilfiNfiflmm4flmltt1z!» aaHIQHHHIQ com 165 omm 0h 00 thHomvomeHHuH omm 1-11-11.11-11:111.111 - .1--.1111 .1 .11. . ; . . . .. , ; , ,. . agnjumm 2:- .131 3 . onm. on. omm .44.1H com HH1¢wH¢m4¢mquH11 co: -::11111i:-;..1- 1--1f-;;1oom 0H 00 AzHAIzfiH¢N4¢dezHrm.w.om¢-oh-oosa¢uaqhuorflfiqu.awar- HsmH4mHmquHm1 a»: 1on3 op 00 a c. on Hsza m as: ch co thH. mo.meH41H oc¢ 111111111 ..5 1913111111111ddfldwH5.fid4QdQflflfl479Jifiqdwi1 .mwuzumwuuHo 4HHZH1 >1 mw>HH<>H1uon up31zcu uuu oo¢.om¢.am: HH1oxmzm zHoum uuu uuu :11i-i-11111--3. .!1:;;+- ; . - ;. .; ;. . .;.;s-- -.MDZHHZOQ «1m mmoH 0H oo #41 444w : 1 1-11111 - , .... . . . - .. . _- ..11:,¢¢n-3H-:a HHnawomtduZHHMH. mazHHzou m¢m 1m-m o.H ow .H.om.zmH.1H 11-11111-11- .1- 111-. 1-1-- 1 1;- - - 11511:-.- .- 11 1 .---11--.--d144nm1.mmH11n.fi -1--.11 wazHHzou H¢m HHmH44cmwzw zwqucu mxuogm I I IIIII IIIIII.’ I II::. I III II. II III III-IIIIIIIIII II III adeo¢N.am~4szH.Ima maszzcu o¢¢ a» co m fiuzoqum A.m~vrao.-vzao.N.IQ.Nqu II:;IF .-: . .1. ,- . .. -, 31., I}? ado. HGOVI'JWWlPI .DOU0.0UOOOZO4<>NOQO N<4uN\¢0\mN>N\m0\3_ma.3aNOA\¢N\O<4mO\H¢\N oNflm\¢fl\NU\Ofl\4<flNw ..43N6mZ~N\~m\mZOUZN\OM\~ «Numo. «Nuzomequmo.NN.qu\o_\.oo¢vu.uzNaa..oo¢.N\m~\zozzcu o<4uo muomNzN 51...: I.-. -IE:..IA..I.|.-.I..I!I-.-V -.isi: II! -I. .1..--ii I!I--?-,_I . I: . I -. .4 I; EQHIIQZHIHDGMMNDM: czm zxaNma ooom.ooom.mm.xoauzw no mzoNNDJOm 93h od< uuuj om¢ oN co m mu-.~.Nmuz_ma omo 169 Imvteazwh.3#.hmomhho sodoaazubvtmahmqno oodlammm~.mmv\o. auaauo -3333.3IE§E33333-; 333-33 3: - --iSE. , -s:3uu1HVddoomfiJJflmH4MIZMHgo¢r o:.~¢.or Ama>uvu~ mufi a ocnHflmH..; ComuAfivU ON osmoaufi ON cc mm 33;; .... 3 .33332...:3:333 ,g-3.3;:-3-. -.- 33::13- 33..- :- :.;3 dda:0~-od..- quooz 4440 uma_\gooo¢+1zwp.nlzmh ,-.--.:w,, -33.;-,: :: - - w---,-,:mumax~4aw4aqaawm2uh .x_.u~ 170 Muxu\o.~oo~zu\o.~um~¥u 3; . . amdzmfim~¥m.\aazmum~xu A£~1u3c~xuv\mfizutu$~xu ANI~.m~ 9 Aladdm egg 3333333--. 3 3-13;3~4Hfl%UwaflfiU1Hmi AalfivhHAfivh 90¢.unfi Am 0c .333333szs3333-3333-333-3;-33s3333-ae3nn3aa3da3aoqem+mndqzaawa3dm:; an” ¢ umm»x..oo¢.uz_h.uazwp om mm o» o: a d.cuw a «page: 44cm anuuu.s.u xqz»m\. whdeMIKMhzm mhzmza wzuhaommbm math ZZZ\9N\.oNvIQ\-\hw~.~m~\®~\1wpz~ 4km nth mwp024c n 4bmc U m44>mw~2~ 4km no zmrsazn xm~ u mfidzmoxoou.4u~ooz Oh mzuhzozmnm x4hwo xu_.xx4pw\o~\zozzou - _.u— m4>~\mc\~ moxoM\x H -uz4m mt» 444 u>4x 4443 H4 aaHra no mm344> -m>Hmmmuu:m d3Haxva4zaou om H3Hra. aaHIQHmuzm44>H3cw smH.HmH\mH\m 4HmH\N4\MI aaHza.3nxm.3m4m.414HH.44Hu.I4Hm.era.4Hxao .--.:I .3mxH. HHnammwuwme.m>aH.m3mH H3mH. HmzHuH.3OJ4H. 4:. 412.33. 32.43.H. on: .41": ..EH .ImH\-Mlm\zm-NIV..H. HImuxom-Ndpwoa H ...-«Hquoa-xIHHHazz-QUI: 4Hux4z qu_:oxm:m -I-;- -I-I-I:-I I -IIIII-,II:;I:I,-I-I:II-I I..IIHI I H:zm I. III, I-.I.I-,--; I -. I- .. ;-II-zImJHMM-I§- uszzzcu om HNIHtevhmuHHI Htmvam IIII igdeHuwHEAai-E. HN¢HHmmt<+HNoHHNntHduHmoH:Im oH I IIIIIII II III III II IIII I I -IIIII I. wzmma Hma zuz uuouuuuu IdH+HHmm¢<+HHoHHNm¢fi¢M4AfianQIIII.- Hma sz UUUUUUUUU +4.4.HIH H: CO H4\Hx..mquH4 a H4\H34~mu 4 a IszmmIH¥HHmuH4 a NIwa-WIHun4m1:I Hoc4HmHm.HocoH.mm.HoooHHH4\m~\~uH. Humeszozzcu IIIIIIIII:-III IIII-.I- I- Hszmzw mzHHaoxmam iiIIIIii -Ei-III 3w?- zmnHmm ,I Hm. mHm.xm.m. mHm.I>HHzHu2H 024 4.H43 H4 m.uz:H4mmazmHI.on.H4zxou 4 ...... I I - - I--- ...-II I I: . HIMHIIHJHHIHIJIIHZHIHHWI HHHmHmIHHHH m .!IIIIIII:I.-.IIIIIIIIII-IIIIII-I- HuH.HnH n so III-II, III. III-I Iiflddfifim afiqt ooovam. HooonIm\Nm\~ImH . HmH\mH\Hoo¢H U3quza. H433. h\mH\zx\mH\zozzou hamuzw u7.:.301m3m ..II .1! lI'. .Ii‘IO-I II II. II |.I‘ -. in .‘i I’ll." I I «I ll IIIIIII ll}|"lnl-lr|0 ..Ill . I ..III: .. I .II I I .. In. 176 -: -I,I- IHIHHIHH zrnHma HHHMQH .:-- .. 444.44.4mmm HzHam HI4HmHqu ImHuuaH III-II .. I; -I- . fling-I...” IIHHJQH IxHuHIaH chH 4h 00 HIxHuHaH -w- . , - .I- II.Januumas JIIHHHUH JIIHHIQH JIa<< me.<_uQC.c<4m~VomHZ~ MZHPDOInzm ----I-II. II I ;I. . . . I . ZflHZHIMQwam. czm Ziapma ..... .. ., €5I25 ‘ , w _, 3:3.Séuizauzjuv. 1442mm 4440 mbzubzco mm -12 I .I . . . . . .AmIm~qud~.mom~u¢Kc~vwdzxcu ¢~ U..~.<.¢~ IUZDQ MA ._+.~Vmuu 5., ..II. .. ; ;I-EEIIIIIiéIT III, .-s.I;7 - - a ..--;I_ ELZuHmHIflHLGd NHII NN oh 00 NH O» co .o.3w.30~.u— ;.IIIIIIIIIIIIIIIII-II I II: :III-I- ..-..II . I. I. . - - .- .I-..II .IIIIdIIIInDdIHIIaIHanIMQIdHWI—qQHI ouDOH .~m.3u.am.mzm— ~2m~ hmxumu.304u~.uz 412 22.1: 42 m UIH AI:~.4¢~.IZ~\NN\ZX\M~\N IitltiITIII;!,25i-IIIT -lIz-.:I-. . I- II{-~dddum144dda~£Wo$43d+1b.~UDJm> meUZDQ uZMFDOI mHIh UUU IéI .. ~41 MZAHfidmnij 181 32w ZIDhmz ,>>4fl><< >>>< A>>< .><.xuox>< .><.xuox»< A>>< A> zoupuzam mfirh u o. odx4pwo m~_m aumm.mN~mmm man4HIIIINM u.m. 4 zo_m~ouxa ugmaoo . .u.m 4.4mo. 4.00m zo_huz:u zofimauuma u4n3co $22.. ----$22-2 9.2-2-... ozm Nam zxapwx mao.ofix44mo.4umcm -22299I92:99mw<.zonmwwummaquaoog .m.4puo.4.mou zcnpuzam zonm4umza u4m3oo ozm -9-IézastIiiiiiIIIIIIaass I29- Jsfinmm3-a- .m+o¢o.muu¢o.og.m¢o.o.«o.¢m\4hmo+4u4om a: pJHDm zwmm 0 m4: 4>4uo «:01 no wNHm amwmz m1; ammo m4432mquzzofih<40ammwzm- -q w.o.u.m < onmaumxa w4mbco ..Am.o.o.