THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Mechanical Engineering Heat Transfer and Thermodynamics Laboratory THE INFLUENCE OF TRANSVERSE HARMONIC OSCILLATIONS ON THE HEAT TRANSFER FROM FINITE AND INFINITE VERTICAL PLATES IN FREE CONVECTION Victor D. Plankenship John A. Clark ORA Project 02970 This report was also a dissertation submitted by the first author in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1962 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NO. G-9386 WASHINGTON, D. C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1962

ACKNOWLEDGMENTS The authors wish to express their appreciation to Professors Vedat S. Arpaci, Richard E. Balzhiser, Arthur G. Hansen and Herman Merte Jr., for their interest and cooperation in serving as members of the Thesis Advisory Committee. The research support of the National Science Foundation, Grant No. 9386, is gratefully acknowledged. The writers are grateful to Professor Wen-Jei Yang for his work on the early stages of the computer program and to Professor Franklin H. Westervelt who gave freely of his time throughout the computation stage. Special credit is due to William Boyd and Hussein Barakat for their fine work in connection with the experimental program. Gratitude is extended to Norlene Martin and the Industry Program of the College of Engineering for their excellent help and cooperation in the preparation of the manuscript. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS.....................ii LIST OF FIGURES........................ o........ooo............ vi NOMENCLATURE................................ xii ABSTRACT....................a............................ 0 xix CHAPTER I. INTRODUCTION AND LITERATURE SURVEY..................1 II. FIRST AND SECOND ORDER PERTURBATIONS OF A LAMINAR FREE CONVECTION BOUNDARY LAYER DUE TO TRANSVERSE HARMONIC OSCILLATIONS OF A THIN PLATE OF FINITE LENGTH....................12 Statement of the Problem.12 Fundamental Equations and Transformation of Coordinates...................... 14 The Potential Flow Response......... 19 Non-dimensionalization of the Equations................. 29 Perturbation of the Equations......................... 31 Zero Order Equations..............55.......... 33 Approximated Free Convection Functions................ 43 FirSt-Order Velocities................1....... 49 First-Order Temperatures.......... 56 First-Order Shear-Stress and Nusselt Number............ 67 Second-Order Velocities................................. 75 Second-Order (time-steady) Shear Stress and Nusselt Number.................................. 83 Solutions up to Second-Order..................... 89 Discussion of Results.............................. 95 III. FIRST AND SECOND ORDER PERTURBATIONS OF A LAMINAR FREE CONVECTION BOUNDARY LAYER DUE TO TRANSVERSE HARMONIC OSCILLITIONS OF AN INFINITE PLATE 97 Introduction...................................... 97 Formulation of Problem............... 97 Perturbations of the Equations...........102 Zero-Order Solutions.................................. 105 First-Order Solutions...... 106 Second-Order Solutions..123 Effect of Oscillations on Heat Transfer and Shear Stress.............. 130 Discussion of Results............................. 140 iii

TABLE OF CONTENTS CONT'D CHAPTER Page IV. EXPERIMENTAL WORK............................. 143 Introduction..................................... 143 Experimental Apparatus............................. 144 Discussion of Results............................. 165 Discussion of Other Experimental Results............ 190 V. SUMMARY OF RESULTS,,..................... 203 LIST OF REFERENCES......e.....ee v................. 206 APPENDIX I.o COMPUTER EVALUATION METHOD............................ 214 II. EXPERIMENTAL HEAT TRANSFER DATA........................ 221 IIA. Smoke Studies Data.............................. 222 III. TABULATED RESULTS OF OSTRACH (C-8).................... 223 ~IIIA Tabulated Results of t(n1) from Schoenhals (A-34). 223 iv

LIST OF FIGURES Figure Page 2-1 A Sketch of Heated Thin Plate of Finite Length Oscillating Transversely in an Infinite Incompressible Fluid at Rest................................... 13 2-2 Sketch Showing a Thin Plate of Finite Length Vibrating Transversely in an Infinite Incompressible Fluid at Rest................................. 21 2-3 The Oscillating Potential Flow Streamline Pattern for a Plate Vibrating Transversely from Schoenhals (A-34)....... 23 2-4 The Oscillating Potential Flow Velocity Distribution for a Plate Vibrating Transversely from Schoenhals (A-34)... 24 2-5 The Oscillating Potential Flow Velocity Distribution for a Plate Vibrating Transversely from Schoenhals (A-34).................................. 25 2-6 The Oscillating Potential Flow Velocity Distribution for a Plate Vibrating Transversely from Schoenhals (A-34).................................. 26 2-7 The Oscillating Potential Flow Velocity Distribution for a Plate Vibrating Transversely from Schoenhals (A-34)................................................. 27 2-8 The Oscillating Potential Flow Along a Plate Vibrating Transversely from Schoenhals (A-34)..................... 28 2-9 Velocity Profiles for Laminar Free Convection Boundary Layers as Reproduced from Ostrach (C-8)............. 36 2-10 Temperature Profiles for Laminar Free Convection Boundary Layer as Reproduced from Ostrach (C-8).............. 37 2-11 Velocity Measurements of Schmidt and Beckman (C-ll) and Comparison with Theory as Reproduced from Ostrach (C-8).................................. 38 2-12 Velocity Measurements of Schmidt and Beckman (C-ll) and Comparison with Theory as Reproduced from Ostrach (C-8).. -.......0...... -—...... 39

Figure Page 2-13 Temperature Measurements of Schmidt and Beckman (C-1l) and Comparison with Theory as Reproduced from Ostrach (C-8). ee... a.. e....a.........e. * a e e..ee ee 40 2-14 Temperature Measurements of Schmidt and Beckman (C-ll) and Comparison with Theory as Reproduced from Ostrach (c-8).............................................. 41 2-15 Comparison of ~(Q)/5(O) from Schoenhals (A-34) to that of Equation (2-43).................................... 44 2-16 Comparison of -H'(T) from Ostrach (C-8) to that of Equation (2-45).................. 00..... 46 2-17 Comparison of F'(r) from Ostrach (C-8) to that of Equation (2-47)..................................48 2-18 Magnitude and Phase of the First Order Velocity, ulA..... 53 2-19 Magnitude and Phase of First Order Velocities ulA and Ulco... 6o.........................*.... 57 2-20 Magnitude and Phase of First Order Velocities ulA and lC...................................................... 58 2-21 Magnitude and Phase of the First Order Velocity, ulA..... 59 2-22 Magnitude and Phase of First Order Temperatures of T1A and TiC...............................................62 2-23 Magnitude and Phase of First Order Temperatures of T]A and T1......63 2-24 Magnitude and Phase of First Order Temperatures of T1A and TiC.............0.................... 64 2-25 Magnitude and Phase of First Order Temperatures of T1A and Tc.. 65 2-26 Magnitude and Phase of First Order Temperatures of T1A and T1C............................................66 2-27 Magnitude and Phase of T1B.............................. 68 2-28 Magnitude and Phase of the First Order Temperature Derivative Divided by the Zeroth Derivative of Temperature.... 71 vi

Figure Page 2-29 Magnitude and Phase of the First Order Temperature Derivative Divided by the Zeroth Derivative of Temperature............................................. 72 2-30 Magnitude and Phase of the First Order Temperature Derivative Divided by the Zeroth Derivative of Temperature............................................. 73 2-31 Magnitude and Phase of the First Order Temperature Derivative Divided by the Zeroth Derivative of Temperature............................................. 74 2-32 Values of:'(t) of Equation (2-97)...................... 79 2-33 Values of 5(t) of Equation (2-100)...................... 81 2-34 Root-Locus Plots for Determination of the Streamlines of Equation (2-102).................................... 84 2-35 Time-Independent. (Secondary) Streamlines in the Top Inner Region of the Plate............................... 85 2-36 A Representative Plot of Time-Independent Flow Streamlines................................................... 86 2-37 Magnitude of the Time-Independent Temperature Derivative Divided by the Zeroth Derivative of Temperature.... 90 2-38 Magnitude of the Time-Independent Temperature Derivative Divided by the Zeroth Derivative of Temperature.... 91 2-39 Magnitude of the Time-Independent Temperature Derivative Divided by the Zeroth Derivative of Temperature.... 92 2-40 Magnitude of the Time-Independent Temperature Derivative Divided by the Zeroth Derivative of Temperature... 93 3-1 Laminar Free Convection Boundary Layer Adjacent to a Heated Wall with Transverse Vibrations................. 98 3-2 Magnitude and Phase of First Order Velocities with Comparison to that of Schoenhals and Clark (A-33)....... 110 3-3 Magnitude and Phase of First Order Velocities........... 111 3-4 Magnitude and Phase of First Order Velocities.......... 112 3-5 Magnitude and Phase of First Order Velocities.......... 113 vii

Figure Page 3-6 Magnitude and Phase of First Order Velocities........... 114 3-7 Magnitude and Phase of First Order Temperatures......... 118 3-8 Magnitude and Phase of First Order Temperatures......... 119 3-9 Magnitude and Phase of First Order Temperatures.......... 120 3-10 Magnitude and Phase of First Order Temperatures......... 121 3-11 Magnitude and Phase of First Order Temperatures......... 122 3-12 Magnitude of Second Order, Time-Independent, Velocities. 127 3-13 Magnitude of Second Order, Time-Independent, Velocities. 128 3-14 Magnitude of Second Order, Time-Independent, Velocities. 129 3-15 Magnitude and Phase of First Order Velocity Derivatives Divided by the Zeroth Derivative of Velocity............ 132 3-16 Magnitude and Phase of First Order Velocity Derivatives Divided by the Zeroth Derivative of Velocity........... 133 3-17 Magnitude of the Second Order, Time-Independent, Velocity Derivatives Divided by the Zeroth Derivative of Velocity.......................................... 135 3-18 Magnitude of the Second Order, Time-Independent, Velocity Derivatives Divided by the Zeroth Derivative of Velocity................................................ 136 3-19 Magnitude and Phase of the First Order Temperature Derivatives Divided by the Zeroth Derivative of Temperature............................ 138 3-20 Magnitude and Phase of the First Order Temperature Derivatives Divided by the Zeroth Derivative of Temperature............................................ 139 3-21 Magnitude of the Second Order, Time-Independent, Temperature Divided by the Zeroth Derivative of Temperature.............................. 141 3-22 Magnitude of the Second Order, Time-Independent, Temperature Divided by the Zeroth Derivative of Temperature.............................. 142 viii

Figure Page 4-1 General View of the Experimental Apparatus............... 145 4-2 Rear View of the Experimental Apparatus................. 145 4-3 Construction of the Test Plate.......................... 146 4-4 Electrical Circuit for Measurement of Energy Input and Control of Heater Elements............................. 148 4-5 View of Test Plate.................................. 150 4-6 Sketch Showing the Arrangement of the Vibratory Components of the Apparatus............................ 152 4-7 Galvanometer Circuit for Monitoring Guard Heater Elements 154 4-8 View of Some of the Instruments......................... 156 4-9 View of Some of the Instruments....................... 156 4-10 Linear Differential Transformer Circuit for Measurement of Amplitude of Vibration.......................... 157 4-11 Output Signal of the Linear Differential Transformer with the Core Displaced from the Null Position as Displayed on an Oscilloscope.......................... 159 4-12 Output Signal of the Linear Differential Transformer Under Vibratory Conditions as Displayed on an Oscilloscope...........o.............................. 159 4-13 Sketch of the Smoke Generator........................... 162 4-14 View of the Smoke Injectors............................ 164 4-15 Sketch Showing One of the Four Rods Attached to the Corners of the Heated Section........................... 167 4-16 Steady-State (Non-Vibratory) Heat Transfer Correlation.. 169 4-17 Experimental Potential Flow Velocity Measurements and Comparison with Theory from Schoenhals (A-34)....... 171 4-18 Experimental Potential Flow Velocity Measurements and Comparison with Theory from Schoenhals (A-34)....... 172 ix

Figure Page 4-19 Experimental Velocity Measurements Made in the Boundary Layer on an Adiabatic Plate Vibrating Transversely and the Comparison with Theory from Schoenhals (A-34)................................ 174 4-20 Experimental Velocity Measurements Made in the Boundary Layer on an Adiabatic Plate Vibrating Transversely and the Comparison with Theory from Schoenhals (A-34)..................... 175 4-21 Experimental Time-Independent Heat Transfer Data for a Heated Plate Oscillating Transversely............ 178 4-22 Experimental Smoke Study Data for Determining the Condition of Transition................................ 179 4-23 Experimental Smoke Study Data for Determining the Condition of Transition.............................. 180 4-24 Experimental Smoke Study Data for Determining the Condition of Transition................................. 181 4-25 Smoke Photograph for a 0, = 0 O. GrPr = 1.08 x 107, a/A=92 BTU aw 2 0 ReV2 0. 184 hr ft2 g Gr(a/L) 4-26 Smoke Photograph for a = 0.002 in., = 50 cycle/sec., GrPr = 1o08 x 107 aw= 051, g/A = 92 BTUI g hr ft2 e V 37................................o........ 184 Gr(a/L) 4-27 Smoke Photograph for - 0.01 ino, w = 50 cycle/sec., Grgr = 1.08 x 107, a /g = 2.54, q/A = 92 BTU/hr ft, Re v/Gr(a/L) = 18.4............... o................. 185 4-28 Smoke Photograph for a = 0.016 in., w = 50 cycle/sew., GrPr = 1.08 x 107, aw2/g = 4.09, q/A = 92 BTU/hr ft, Re2v/Gr(a/L) = 29.5.................................... 185 4-29 Smoke Photograph for a = 0.022 in., w = 50 cycle/sec., GrPr = 1.08 x 107, aw2/g = 5.62, q/A 92 BTU/hr ft2, Re2v/Gr(a/L) = 40.7.................................... 186 4-30 Smoke Photograph for a =.03 in. w = 50 cycle/sec GrPr = 1.08 x 107, aw /g = 7.69, q/A = 92 BTU/hr ft., Re2v/Gr (a/L) = 55.7 -........0 186

Figure Page 4-31 Smoke Photograph for a = 0.068 in., w = 50 cycles/sec., GrPr = 1.08 x 107, aw2/g = 17.4, q/A = 92 BTU/hr ft2, Re2v/Gr(a/L) = 126............................... 187 4-32 Smoke Photograph for a = 0.03, w = 25 cycles/sec., GrPr 1.08 x 107, aw2/g = 1.92, q/A = 92 BTU/hr ft2, Re2v/Gr(a/L) = 1.9..... 187 4-33 Experimental Heat Transfer Data Taken from Shine (A-35) 192 4-34 Photographs Showing Waves in the Fringes Taken from Shine(A-35)^ *- * *^*.......................... 193 4-35 Comparison of Heat Transfer Data at Constant Difference in Temperature. For a Horizontal Heated Cylinder Subjected to Mechanical and Acoustical Vibrations in Air Taken from Fand and Peebles (A-9).................. 195 4-36 Photographs Representing Acoustic Streaming Taken from Fand and Kaye (B-4).................... 197 4-37 Experimental Heat Transfer Data of Martinelli and Boelter (A-22) For Vertical Vibrations of a Horizontal Cylinder in Water...4........................... 198

NOMENCLATURE a amplitude of wall vibration C length of 1/2 the plate, C = L/2 Cp the specific heat per unit mass of the fluid -4 __> 3u 3v div the divergence of the vector V, div V = + -- ax ay D/DT* the substantive derivative operator for the stationary coordinate system D/DT the substantive derivative operator for the moving coordinate system e base of the natural logarithm F(ri) defines the free convection velocity distribution GrL the Grashof number, gL3MAi//v2 GrX the Grashof number, gX3A0o/ v2 g body force per unit mass acting parallel to the wall H(r) defines the free dimensionless convection temperature distribution, To i the square root of -1 K thermal conductivity of the fluid Ko dimensionless term WG L Kloo K4 functions of Prandtl number L the length of the plate n the exponent in ~n NuL the Nusselt number, hL/K NuX the local Nusselt number, hX/K Pr the Prandtl number of the fluid P pressure xii

2x-1 P pressure from potential flow, 2 ~x(x-l) T dimensionless temperature, G - G /0. - 9 To, T25 the steady components of the temperature distribution T.To + e T2s T1 TI = Real (T1 ei t) t dimensionless time, V _GRL T L2 T U* velocity parallel to the wall in the stationary coordinate system u dimensionless velocity parallel to the wall in the moving coordinate system UI velocity parallel to the wall in the moving coordinate system ul uI = Real (ul e ) u2s steady velocity normal to the wall in the moving coordinate system U potential flow parallel to the wall in the moving coordinate system u component of the complex velocity Vo velocity of the plate normal to the wall in the stationary coordinate system V* velocity normal to the wall in the stationary coordinate system v dimensionless velocity normal to the wall in the moving coordinate system VI velocity normal to the wall in the moving coordinate system v1 vI = Real (v1 eiat) xiii

V2s steady velocity normal to the wall in the moving coordinate system V component of the complex velocity W complex potential, for potential flow distance parallel to the wall in the moving coordinate system distance parallel to the wall in the stationary coordinate system X dimensionless parallel distance, X = =t/L distance normal to the wall in the stationary coordinate system distance normal to the wall in the moving coordinate system y 4 Y dimensionless normal distance, Y = - TGR L RL Z coordinate normal to the X-Y plane X0 subscript indicating a condition far from the wall GREEK LETTERS Ca the thermal diffusivity of the fluid K/p Cp coefficient of thermal expansion, p = po[l-'-o] AGO temperature difference, Lo T G - G0O V2 the Laplacian Operator in the moving coordinate system betL2 the Laplacian Operator in the stationary coordinate system 6 the boundary layer thickness dimensionless distance, y 0w/2 the stream function for ~2s xiv

E E = acw/L (Chapter II), E = (a/L) -2o (Chapter III) (() f5(n) = /f NH'(d)d 00 the similarity variable, q = y/(4x)l/4 id the dimensional similarity variable temperature the absolute viscosity of the fluid v the kinematic viscosity of the fluid p the density of the fluid T time Ts shear stress I0 phase angle of velocisty and temperature oscillations defined by, for example Real ulei~t = Julj cos (wt + 0) r.- the free convection stream function V2s the secondary flow stream function a frequency of wall vibration L2 W dimensionless frequency, w0 = v GRL X acoustic wave length = - (4X) 1/2 xv

NOMENCLATURE FOR CHAPTER IV All discussion directly involving the analysis of Chapter II and III is done in terms of symbols used in those chapters. Additional symbols used elsewhere in Chapter IV are given below. A the total area of the heated section of the test plate a the amplitude of plate vibration D the diameter of stainless steel rods where they attach to the corners of the heated section. See Figure 4-15 h the free convection heat transfer coefficient under nonvibratory conditions at a particular value of 2GO. hco the free convection heat transfer coefficient under nonvibratory conditions at a particular energy input to the heated section corresponding to a temperature difference of ~o' hk the conduction heat transfer coefficient hL the heat transfer coefficient for losses hLo the heat transfer coefficient for losses at a particular energy input corresponding to a temperature difference of ZO hr the radiation heat transfer coefficient defined by Equation (4-4) h the convection heat transfer coefficient for the rods attached to the corners of the heated section. See Figure 4-15. h the increase in the convection heat transfer coefficient due to vibration at a particular ~s kR the thermal conductivity of the rods attached to the corners of the heated section, See Figure 4-15. L the distance between the thermocouples mounted on the rods attached to the corners of the heated section, See Figure 4-15 mr see Equation (4-7). mr = 4 hs/Dkr xvi

q the total rate of energy input to the heated section heat transfer loss due to conduction through the rods attached to the corners of the heated section. See Figure 4-15 the combined heat transfer lossed due to conduction and radiation, qL = (qk + qr) qR heat transfer loss due to radiation aw L Rev vibrational Reynolds number, - Tp absolute temperature of the heated section of the test plate Ta absolute temperature of the ambient Xr distance along the rods attached to the four corners of the heated section, GREEK LETTERS the emissivity of the polished aluminum plates forming the heated section G temperature 90 temperature of the heated section of the test plate e0 temperature of the ambient 0r temperature at a point on the rod shown in Figure 4-15 temperature difference between the heated section of the test plate and ambient,,Z = (90 - G,) 90o temperature difference CZ corresponding to steady state under nonvibratory conditions at a particular energy input to the heated section. Zlr difference between the temperature at a point along the rod and ambient. LSr = (Gr - @@) a the Stefan-Boltzman constant. a = o1713 x 10-8 hft2(R)4 xvii

ABSTRACT Two problems are investigated in this thesis on the effects of transverse mechanical vibrations on the heat transfer from vertical plates. These include plates of both infinite and finite extent. This study examines the case in which high frequency vibrations are imposed as perturbations on a free convection flow. The case of the infinite plate is investigated theoretically and the finite plate is studied both theoretically and experimentally. Time-independent modifications of heat transfer and the transition region from laminar to turbulent flow are investigated experimentally. Analytical studies employ a perturbation type solution. The zeroth order solution is that of free convection, the first order solutions are periodic and the second order solution includes terms which are both doubly periodic and time-independent. These two problems differ in their mechanics in that the finite plate is subjected to a potential flow while the infinite plate is not. For the first order (periodic) solutions the phase angle and magnitude of velocity, temperature, shear stress and Nusselt number are determined. In the case of the second order solution time-independent velocity, shear stress and Nusselt number are found. The solutions were numerically evaluated on an IBM 709 digital computer for two Prandtl numbers, 0.72 and 10. xix

Air was the fluid in the experimental measurements. Heat transfer data were correlated in the form of Nusselt number as a function of a vibrational Reynolds number and the Grashof-Prandtl number product. The Nusselt number was found to be approximately constant, essentially uninfluenced by vibration for laminar conditions. A maximum decrease of approximately 2% was found for smaller values of vibration Reynolds number. For larger values an abrupt increase in the Nusselt number. was obtained experimentally. The theoretical results for laminar flow predicted a very slight decrease in the time-independent heat transfer coefficient with vibration, which is confirmed by the experimental data. The sharp increase in the Nusselt number was concluded to be a result of a flow transitional phenomenon. Smoke studies were made to investigate this more thoroughly. These studies clearly confirm that this transitional phenomenon occurs. The transition condition is correlated in terms of a vibrational Reynolds number, amplitude ratio and Grashof number in the form Rev2/GRL(a/L). In summary, the significance of the results of this research may be given as follows. Vibrations and flow oscillations are capable of causing permanent (time-independent) alterations in both the velocity and temperature profiles in the laminar boundary layer of systems having pressure gradients in the flow direction. This provides a secondary, time-independent flow, and permanently alters the wall shear stress and heat transfer rate. These changes are small and are detected from the analysis only when solutions are obtained to at least the second order perturbation beyond the solution for the steady free convection problem. xx

Furthermore, while the alternation in the wall shear stress is such as to increase its value, the opposite effect is both predicted and observed for the heat transfer rate under laminar flow conditions. This emphasizes an important character of this new class of phenomena, namely, that the traditional heat transfer - momentum transfer analogy is no longer a useful guide, at least for oscillating systems in laminar flow. However, increases in heat transfer rate (and probably wall shear stress also) are to be expected to result from imposed vibrations and flow oscillations but only if such disturbances bring about a change in the character of the boundary layer flow. From the experimental results of this research this change appears to be a transition from a laminar to a turbulent flow at a critical condition of a vibratory disturbance, The exact source of this disturbance and the mechanics of its propagation into the boundary layer remains to be resolved by future research, xxi

CHAPTER I INTRODUCTION AND LITERATURE SURVEY This thesis treats the influence of transverse harmonic oscillations on the heat transfer from finite and infinite, vertical plates in free convection. It attempts to resolve some of the questions pertaining to steady effects produced on free convection flows from purely periodic distrubances, mainly that of the time-independent modification of the heat transfer coefficient. Both analytical and experimental methods have been used to achieve this result. The literature relating to this particular field is reviewed, owing to its diversification it was expedient to divide it into three groups designated as A, B, and C. Group A includes papers on unsteady motions of all kinds with emphasis on oscillatory flows and the related heat transfer problems. Group B includes references on unsteady motions resulting from sound waves, and the related heat transfer problems. Group C includes references and books concerned primarily with heat transfer which general nature was helpful for a theoretical solution to this problem. Papers from the field of acoustics were investigated because they are related to oscillating boundary layers. When plane acoustic waves interact with a stationary object a steady, i.e., time-independent rotational flow occurs. Also, a steady rotational flow occurs in an oscillating fluid or is generated in a quiescent fluid where solid boundary oscillate. These phenomena are sometimes called acoustic streaming or secondary flow. Examples of thesephenomena are flow in a Kundt's -1

-2tube, that near vibrating cylinders and spheres, the flow near vibrating plates and membranes, and that near orifices transversed by sound. Flow in the vicinity of vibrating gas bubbles, and that associated with standing waves in tubes and between walls also are examples in which secondary flows exist. For more details concerning these phenomena the work of early observers can be found in the bibliographies of.:I.:oltsmark, Johnsen, Sekkeland, and Skavlen (B-8), Nyborg (B-15,16), and Westervelt (B-24). The fluid dynamic problem in the absence of heat transfer will be considered first. Schlichting (A-32) used successive approximations to solve the non-linear momentum equation, and therefore, illustrated streaming mathematically. He showed that streaming is a consequence of the non-linear characteristics, i.eo, the convective inertia terms, of the momentum equations for a viscous fluid. Absence of the inertia terms will result mathematically in no streaming. Therefore, streaming is believed to be the result of a physical interaction between inertia and viscous effects in a fluid. Schlichting (A-32) considered the case of a plane wall executing harmonic oscillations in an otherwise quiescent fluid and applied his solution to a circular cylinder, using Prandtl's boundary layer simplifications of the Navier-Stokes equations. He also made experimental verifications of streaming. Holtsmark, et al., (B-8) considered the flow around a circular cylinder in two dimensions for an oscillating incompres.sible fluid with the axis of the cylinder normal to the direction of oscillation. They conducted both experimental and analytical studies. Using successive approximations they solved the complete set of Navier-Stokes

-3equations. Higher order approximations and discussions of these two papers are in Andres and Ingard (B-l, 2), Ramez, Corelli and Westervelt (B-17). In the case of a cylinder, two regions of streaming were predicted in each quadrant around a cylinder. Next to the cylinder an inner vortex system which has been called the DC boundary layer was found. The thickness of the DC boundary is defined as the distance from the surface to the streamline = O0. Next to this inner system is an outer vortex system called the AC boundary layer. In limited space the core of the AC vortices is located at a finite distance. In unlimited space the outer vortex cores are an infinity. Schlichting took photographs of a cylinder oscillating in water with fine metallic particles in it to render the motion visible and found good agreement with the theoretical patterns of stream lines. Holtsmark, et al, (B-8) used a Kundt's tube arrangement and MgO smoke in air oscillating around a circular cylinder and took photographs. Their agreement between theory and experiment is excellent for periodic flow and very good for streaming. For limited space both Schlichting and Holtsmark found DC and AC systems which were similar in general nature, but Schlichting's case gives a much thinner DC system and the velocities in the outer system are higher. For unlimited space Schlichting's analysis predicts no DC system. Raney, et al, (B-17) pointed out that Schlichting's analysis is characterized by small values of vibration amplitude therefore is not necessarily valid for large values of vibration Reynolds number. They also present experimental curves showing the existence of a universal curve for the DC boundary-layer thickness.

-4Rott (A-30) and Glauert (A-ll) both used the method of successive approximations to study cases of viscous incompressible stagnation flow onto an infinite plate which is oscillating in its own plane where the first approximation is Hiemenz flow. Glauert (A-ll) used his results in applying them for a cylinder of arbitrary cross-section in the following cases. 1) cylinder fixed, stream oscillates in magnitude; 2) cylinder fixed, stream oscillates in direction; 3) stream constant, cylinder oscillates in the stream direction; 4) stream constant, cylinder oscillates in the transverse direction; and 5) stream constant, cylinder oscillates about its axis. Rosenblatt (A-24) using successive approximations considered torsional oscillations of a plane in a viscous fluid. He found that a true representation of this flow can only be obtained through inclusion of the convective inertia terms, which is important in determining the character of the flow at large distances from the plate. This is another example of this streaming or secondary flow phenomena. Lighthill (A-20) has considered the cases of incompressible steady flow on an arbitrary cylinder which is oscillating with small amplitude. This can be said to be the case of low-frequency small perturbation analysis of Blasius flow. Cheng and Elliot (A-4) considered arbitrary and specific time-dependent motions of a semi-infinite plate. They compared one of their cases, which corresponded physically, to that of Lighthill. Lighthill used the momentum-integral technique and they used a more rigorous mathematical procedure. It is significant that the momentum-integral technique was checked because both solutions for the case of wall shear stress were found to be in agreement in form and numerical constants. Lighthill (A-15) also discussed the temperature fluctuations using successive approximations.

-5Lin (A-22) has devised a linearized method to study boundarylayers over a flat plate subjected to rapidly oscillating external flows. He has shown that high-frequency oscillations behave like shear waves of the Stokes type regardless of the mean flow. He reasoned that if the frequency of oscillation is very high, then local acceleration is much larger than the time-dependent part of the convection of momentum. Then the time-dependent part can be treated by linear theory which is then related to the problem first studied by Rayleigh and Stokes. Karlsson (A-16) has considered an unsteady turbulent boundary layer by studying the non-linear effects of low speed parallel flow, harmonically oscillating in magnitude about a constant mean over a flat plate. It was found surprisingly enough, that the effect due to nonlinear interaction was quite small for fluctuation amplitudes as large as 34% of the free stream velocity. Tennekes (A-38) used the results of Karlsson and suggested the possibility of constructing the response of the boundary layer to any input transient by a linear Fourier synthesis of the response. This theory compared favorably with Karlsson s experimental results. Hill and Stenning (A-15) have made a study of the effects of free stream oscillations on laminar boundary layers of the Howarth type and made experimental measurements. They found that the existing solutions of Lighthill (A-20), Lin (A-22) and Nickerson (A-26) for low and high frequencies are adequate for describing the flow. For the intermediate frequency range they solved for a theoretical solution which accounted for the flow in that region.

The following papers include heat transfer effects along with the fluid dynamic problems. Stuart (A-36) solved the Navier-Stokes equations exactly for incompressible flow for the case of a flat plate with constant suction and a fluctuating free stream velocity. By including viscous dissipation he obtained the exact solution of the energy equation for the case of zero heat transfer between the wall and fluid. Ostrach (A-27) extended the Stokes solution by analyzing the temperature distribution in a fluid over an oscillating infinite surface with heat transfer by including compressibility and viscous dissipation. It was found that the heat transfer for the oscillating surface can be considerably different from that of conduction alone, and that oscillation can alter the fluid temperature appreciably. Moore (A-25) analyzed the laminar compressible boundary layer over an insulated semi-infinite flat plate moving with time-dependent velocity. Ostrach (A-27) extended the results of Moore representing a more exact treatment and included the effects of heat transfer. For the particular case of a plate oscillating about a steady velocity the unsteady effects can alter the boundary-layer characteristics appreciably. This paper did not settle the question what effect oscillations have on mean skin-friction and heat transfer. Kestin, Maeder, and Wang (A-17) have considered the case of a flat plate on which the free stream is a flow wherein a sinusoidal velocity disturbance is carried downstream with the mean velocity. They obtained a solution by expanding the velocities and temperatures in terms of series expansions in increasing powers of the small relative amplitudes (X) of

-7the sinusoidal disturbance. This procedure will yield an equation to solve for expansions up to Xn, The zero-order solution is the case of Blasius. By considering another series expansion for the first and second-order equations they were able to obtain the time-averaged skin friction and heat transfer coefficient. This second expansion limited their study to low frequencies and the first expansion to small amplitudes. Their analysis predicts a slight increase in the skin friction coefficient and a slight decrease in the heat transfer coefficient. It is significant to point out that the terms that contribute to the timeaveraged coefficient in this analysis are part of an infinite series and that only the first term was used. The part that contributes is from the second order solution and is the steady part of that solution. Nickerson (A-26) has considered a flat plate executing simple harmonic oscillations in a steadily moving fluid with zero pressure gradient. Using series expansions for stream and temperature functions he was able to obtain successive approximations and make experimental observations. He carried his analysis far enough to obtain steady secondorder velocity and temperature solutions which are the result of oscillations. He theoretically obtained the result that there would be a slight steady increase in heat transfer and skin friction coefficients. The increase in heat transfer coefficient was within the range of experimental error and therefore could not be checked. Velocity profiles obtained using a hot-wire anemometer showed good agreement with theory. Chung and Anderson (A-6) analyzed the laminar free-convection boundary layer on a vertical plate or a horizontal cylinder under unsteady conditions. For a Prandtl number of 0.72 they found that if the

-8wall temperature increases with time there is an increase in the heat transfer whereas the heat transfer is decreased when the acceleration of gravity field is increased with time. Schoenhals and Clark (A-33) have considered the response of velocity and temperature of a laminar incompressible fluid to a vertical plate oscillating harmonically in a horizontal direction. They considered the harmonic oscillations of a semi-infinite plate with free convection and the fluid response of a finite adiabatic plate to harmonic oscillations. Using successive approximations streaming was found to exist for the case of the finite plate. They did not theoretically consider the time-averaged heat transfer coefficient, but studied the temperature and velocity response of a free convection boundary layer within the first two approximations. Experimentally they studied the effects of oscillation on free convection on a finite plate and presented results showing the increase in heat transfer coefficient. In the case of an infinite vertical plate oscillating vertically, considered by Rott (A-30) and Glauert (A-ll), should the oncoming stream be of sufficient strength to permit ignoring the buoyancy effect of the temperature field, then oscillations have no effect when the wall is maintained at a constant temperature. Analytical results for the increases of heat transfer rates have not been as fruitful as the considerations of the fluid dynamics of adiabatic systems. This is confirmed by experimental observations. Experimentally there has been confirmation that oscillation of a solid or oscillation of the flow over a solid increases the heat transfer rate. Also, it has been experimentally confirmed that various types

-9of sound fields increase the heat transfer rate. These facts have been demonstrated by the papers (A-1,7,8,9,22,33,34,38,40,41) and (B-4,5,6, 18,19). For flow through tubes the increases have been demonstrated by (A-12,18) and (B-10,20). Also the phenomena of transition from laminar to turbulent flow and flow separation have been experimentally observed. In the case of sound field effects on the free convection boundary layer of a horizontal cylinder, flow visualizations and shadowgraph pictures were studied by Fand and Kaye (B-4), Holman (B-5), Holman and Mott-Smith (B-6), Soehngen and Holman (B-18), and Sprott, Holman and Durand (B-19). A critical or threshold sound pressure level of 136 db was found below in which no increase in the heat transfer coefficient was found and above which it increased. The point was made that streaming is the physical mechanism causing the increase and that its interaction is evident at the critical sound pressure level at onset of the increase of heat transfer coefficient. Soehngen, et al, (B-18) found that the heat transfer coefficient could be increased three times that of no sound for the same temperature difference. Sprott, et al, (B-19) and Holman, et al. (B-6) found that the heat transfer coefficient could be increased by 100%. Fand, et al, (B-4) found that the heat transfer coefficient could be increased by 300% by high intensity stationary and progressive sound fields on the free convection boundary layer of a horizontal cylinder. Fand and Kaye (A-7) investigated the influence of vertical mechanically induced simple harmonic vibrations upon the rate of heat transfer by free convection from a heated horizontal cylinder to air. For intensities of vibration less than 0.3 ft/sec they reported negligible

-10change in the heat transfer coefficient. Above this value the coefficient changes significantly. A flow-visualization method employing smoke as the medium indicated that vibrationally induced turbulence is the mechanism which caused the increase. They also reported that above 0.9 ft/sec and AT > 100~F there is a "fully developed turbulent flow." Fand and Peebles (A-8) made an investigation of the comparison of the influence of mechanical and acoustical vibration on free convection from a horizontal cylinder. They found that although there is a ten-fold difference in frequency and a difference in amplitude, the heat transfer correlation for horizontal acoustical vibrations is also valid for horizontal mechanical vibrations and the character of the boundary layer is the same. They found a critical intensity of 0.36 ft/sec. Shine (A-35), Tsui (A-39) and Schoenhals, et al, (A-34) studied experimentally the effects of a vertical plate oscillating horizontally with free convection. Shine and Tsui both used Zehnder-Mach interferometers in their studies. Tsui claimed that the flow seemed to be transitional on the basis of his observations. In his experiment the maximum percentage increase was found to be 24%. Shine defined a "critical intensity'" (amplitude times frequency) peculiar to his system, and showed experimentally that above it heat transfer is increased and below it no change was observed. He noted that coincident with the occurrence of change in heat transfer coefficient there was an inception of waviness in the fringes of the boundary layer, which was intensified by increases in vibration intensity. He pointed out this waviness indicates a flow transition from laminar flow and may represent the physical mechanism causing the increases. Schoenhals, et al, (A-335) data do not show

-11evidence of a critical intensity value. They experimentally found an increase in the heat transfer coefficient of 46% due to horizontal oscillations of a finite vertical plate. Martinelli and Boelter (A-17) have investigated the effect of vertical oscillation upon the rate of heat transfer from a horizontal tube immersed in otherwise quiescent water. They found increases up to 400%. It was found that the coefficient of heat transfer was unaffected for low values of vibration Reynolds number. Teleki (A-38) has investigated a case physically similar to that of Martinelli, et al, (A-22) except the media was air. He found increases up to 200% and did experience a critical intensity in his heat transfer coefficients. Lemlich (A-19) has investigated electrically heated wires subjected to both vertical and horizontal vibrations in air and observed an increase in heat transfer coefficient irrespective of the direction of vibration. He did not experience a critical intensity in his heat transfer coefficient. He found increases as high as 400%.

CHAPTER II FIRST AND SECOND ORDER PERTURBATIONS OF A LAMINAR FREE CONVECTION BOUNDARY LAYER DUE TO TRANSVERSE HARMONIC OSCILLATIONS OF A THIN PLATE OF FINITE LENGTH Statement of the Problem Figure 2-1 is a sketch of the physical system to be investigated. Consider a heated thin plate of finite length L with the stationary coordinate system fixed at the base of the plate. The plate is undergoing transverse harmonic oscillations normal to the X* - Z* plane, which is in the stationary coordinate system. The analysis is restricted to two dimensional incompressible flow in the X* - Y* plane. From a compressibility standpoint it is necessary that the velocity of the plate be of a lower order than the velocity of sound in the fluid. The harmonic oscillation of the plate is VO = a Q cos QT where a is the amplitude of vibration, Q is the frequency of vibration, T is time. The product a Q represents the maximum velocity of the plate and VO represents the velocity of the plate with respect to time. The plate is heated to a uniform temperature, 90, and is maintained in contact with an otherwise quiescent fluid at temperature A.. The fluid has constant viscosity A~, constant thermal diffusivity a, and constant coefficient of thermal expansion A. There exists in the fluid a uniform force field g which in the absence of the harmonic oscillation would generate a free convection flow only. The solutions of the governing equations of momentum, energy and mass treat the case where buoyancy forces of free convective flow -12

-13INFINITE INCOMPRESSIBLE FLUID AT REST U V 8,p Vo-= a ncosfT I -(XY) STATIONARY COORDINATE SYSTEM - (X,Y ) MOVING COORDINATE SYSTEM Figure 2-1. A Sketch of Heated Thin Plate of Finite Length Oscillating Transversely in an Infinite Incompressible Fluid at Rest.

-14predominate over the effects of the transverse harmonic oscillations. The effects of the transverse harmonic oscillations are treated as first and second order perturbations of a free convection boundary layer flow. These first and second order perturbations result mainly from a potential flow pattern at the outer edge of the boundary layer. Fundamental Equations and Transformation of Coordinates At first it is necessary to look at the governing equations of conservation of mass, momentum, and energy. The flow analysis is restricted to incompressible flow, although in free convection the generating force term is due to thermal expansion of the fluid. Therefore it is necessary to investigate what restrictions are necessary to be able to use an incompressible analysis. This has been discussed by Ostrach (C-8) and Hellums (A-14). Presented below are the governing equations for viscous compressible flow with K, t and Cp being assumed constant and viscous dissipation and the work of compression assumed negligible. A stationary coordinate system will be considered first with the origin located at a datum plane at the bottom edge of the plate, therefore ado -9 He 8+90XU+32( _r — dap' cV] (2-1) P.[

and DO -_ K For air one can consider (2-2) where for an ideal gas XIq = }(2-3) 6oo Consider pressure to be composed of the following: ~~~p ~~~~~~p ~~~~~p ) (2-4) where P = pressure in the surrounding fluid, and 00 Pi = pressure due to motions, therefore, we obtain (2-5) aP_ -- P' Since the quantity P(@ - GO0), from Ostrach, (C-8), is,,ofs o,>oz, \6

then Equation (2-2) can be expressed as P /,t(OyOo) 4iiA( =f-rjX (EA(-60 —)I (2-7) and by defining T as a dimensionless variable _T-0-__ 0 —C- (2-8) Gw- 0DC L o, therefore Equation (2-7) is also expressed as 4J3 /# j TX Oo (2-9) and since /2 7-TL o < O ( / ) then / \ ~)d - / (2-10) and we have before reducing: 3 oz, -F (2-11) 4' i#AB z 7-'24o ]V jT P -;f o([lK A -( ZD t I A -I 2 Te

-17The set of equations have reduced to: PCO7C~ C)O* f37tioAiage v(2-12) IP 2I _ _ It is convenient to analyze the problem in a frame of reference fixed in the plate, i.eo, a coordinate system moving with the plate. As pointed out by Lighthill (A-20) in the hydrodynamics of an incompressible fluid, frames of reference are equivalent if they are in a uniform, though not necessarily constant, translational motion to one another. This is because any uniform inertial force per unit volume is automatically canceled by a uniform pressure gradient. Therefore, it would appear that equivalent frames of reference could be employed on this system without changing the equations. This would be true except for the fact that there is relative acceleration between the moving system and the stationary system in the Y direction. This alters the Y-momentum equations and has been treated by Schoenhals (A-34). The alteration resulting from the shift of reference systems produces a d'Alembert or body force due to the vibratory acceleration of the fluid coupled with its own inert ia. By the adoption of a coordinate system attached to the

-18accelerating plate, Equations (2-12) become: PD - I C)P _ + v,7,qz r.L-.,,AVVo7..A'D" (2-13) d0 dO --- D 7T According to Ostrach, (C-8), the free convection flow along a vertical plate is of the boundary-layer type for large, sub-transitional Grashof numbers. This is the case in the present investigation. For free convection boundary layer flow without oscillation the Y-momentum equation is zero, but with oscillatory perturbations it becomes: a p =V = _ O __ (2-14) By integrating Equation (2-14) from a distance Y1, with respect to Y, outside the boundary layer, the pressure distribution is given as P P(XZ, j- )- -x- (I-TA C) Ycr (2-15) y7 Differentiation with respect to X gives the pressure gradient along

-19the wall to be d~P = @P cA 3TX cSY- (2-16) The first term gives the pressure gradient due to the potential flow just outside the boundary layer, and the second term gives the pressure distribution due to the inertial force term. It should be noted here that in this case the thermal variation of density provides the effect necessary to effect the pressure gradient. If there were no thermal variation of density then there would be no difference between a moving coordinate system and a stationary one. Therefore the final equations to solve with boundary-layer simplifications are d 7 (2-17) D x — 0 2 The Potential Flow Response Schoenhals and Clark (A-33) have obtained the potential flow response of a thin plate of finite length vibrating transversely in an

-20incompressible fluid at rest using Lamb's treatment to obtain the potential flow about an elliptical cylinder. They approximated the thin plate of finite length by letting the semi-minor axis of the elliptical cylinder approach zero. Only the highlights of this development will be given here, more detailed information may be found in References (A-33) and (A-34). The coordinate system was fixed in the body as shown in Figure 2-2. Therefore the velocity of the fluid at large distances from the plate is -aQ cos fT and at the plate it is zero. A complex potential W(Z) was utilized the Z being a complex variable. In this representation the complex velocity is written as dw V (2-18) The complex potential was determined with the appropriate boundary conditions so that the components of the complex velocity are _2 X En1 Ji4(T/4((Z)PC (2-19) 7 Ev,, L-, j%"Z47")2]

-21+ =2+C INFINITE INCOMPRESSIBLE FLUID AT REST 0- 0 1-aQ cos at Figure 2-2. Sketch Showing a Thin Plate of Finite Length Vibrating Transversely in an Infinite Incompressible Fluid at Rest.

-22and (2-20) +2V CC) (C7)~ 7 ()2 (Z) The oscillating potential flow streamline pattern, taken from Schoenhals (A-34) is shown in Figure 2-3. Also values of the velocity profiles, U and v, are shown as function of x/c and y/c in Figures 2-4 through 2-7. For small values of Y, that is, the boundary layer regions, the above equations reduce to (2-21A) V0,(/2 - - - - X C 5J 2 )(2-21B) The:values nf the bracketed term in Equation (2-21B) are given in Figure 2-8. There are singular points at the edges of the plates where the theory would be invalid for a viscous fluid. Schoenhals, (A-34) have found good agreement with theory by using a.hot-wire anemometer, except close to the singular points~ This will be discussed in more detail in Chapter IVo

-231.7 1.6 1.5 1.3 1.2.01.125 1 5.75 I. I 1.0.9 9.8.7.6.5.3.2.1 0.1.2.3.4.5.6.7.8.9 1.0 1.1 1.2 1.3 X/C Figure 2-3. The Oscillating Potential Flow Streamline Patter for a Plate Vibrating Transversely from Schoenhals (A-34).

1.0.8.6.4.2 0 2 3 4 X/C Figure 2-4. The Oscillating Potential Flow Velocity Distribution for a Plate Vibrating Transversely from Schoenhals (A-34).

1.2 /C 1.0.6.4 0 I 2 3 4 y/C Figure 2-5. The Oscillating Potential Flow Velocity Distribution for a Plate Vibrating Transversely from Schoenhals (A-34).

1.4 r~~=0 1.2 Y=y/ =0 1.0 0 23.0 4.6 x/C Figure 2-6. The Oscillating Potential Flow Velocity Distribution for a Plate Vibrating Transversely from Schoenhals (A-34).

-27c Q m Q C D A oN OA/^ - Figure 2-7. The Oscillating Potential Fiw Velocity Distribution for a Plate Vibrating Transversely from Schoenhals (A-34).

1.8 X/C U/ xI I I I -L/C 1.6 _ j 1.4 1.2 1.0 o:.8.6.4.2 0..2.3.4.5.6.7.8.9 X/C 2-8 The Oscillating Potential Flow Along a Plate Vibrating Transversely from Schoenhals (A-34)......................

-29Non-Dimensionalization of the Equations Non-dimensionalization of the equations is fashioned after Ostrach (C-8). The potential flow was determined with the reference system fixed in the middle of the plate. In this problem it is more convenient to fix the coordinate system at the bottom edge of the plate, the same as the reference system of the governing equation. By defining O/- _-.mZ — (2-22) we have by shifting the coordinate X to (X-l) /p~~ (o~~~ T /C ) / ~~~(2-23) The pressure distribution is determined from the potential flow by utilizing the Euler's equations. Euler's equations are I dP =_ T@7 r 6 aL (2-24) -,-Y- = - and by utilizing the Equations (2-21) at the outer edge of the boundary

-30layer we have,P 0P 3Ui (2-25) / - O The dimensionless terms used in the non-dimensionalization are X = - - L6- L y= 4s,/3ino 42 /~t __ ZvX'X The dimensionless equations from Equations (2-17) are therefore represented as D = -yt tt Co -wt E r__, ~Vr, 4 (2-27) t 1 /'~ Ct) c y /S/N X t \v7/ a

-310y' 7" / -' d7"Perturbation of the Equations As stated in the preceding discussion, this problem treats the case where buoyancy forces predominate. In other words, where the flow is mainly that of free convection and the oscillations are imposed as perturbations on the free convection velocity and temperature field. This is expressed by the following expansion in terms of E (.an s wrttn t (2-28) and is written in the form UI =- So+ 5W IZ+' d (2 -29A) The uo is the free convection velocity. This expansion is advantageous because the first term is the velocity of the free convection boundary layer which has been solved analytically by Ostrach (C-8). The expansions are therefore taken as L/ C1- +6+ &b +'E 0Z U: " 2 "a (2-29A) wl- 17 j E 6zv 4 ( 2 +,'' (2-29B) 7=,7'0 + + 7- # 62 Tz + (2-29c)

-32By substituting these expansions into Equations (2-27) and by separating terms according to the powers of n, n = 0, 1, 2, 1, the following equations are found Zero-order equations, co dx UO US O U2O t=o (2-30) o~a T Uo 1 c) dX t Pe odY 0ik8% _._L ~2 7 First-order equations, e1 d X cY (2-31A) +~ + --- tU t ko t dTy w 3 U1Z U a-O dx1JO _ o)1 (2-32B) c)Y ~lrawl+/%u __~T y s_ / t d — -= t i' ~ s w -r Q 2~ 7~ - / cJTz (2-50)) X'"t.X ax y c.)X - PR dyZ Second-order equations, E2 ax 0 (2-32A)

v33dt 6 XX t la t dY gY P)d Y - -t Y" - a y l_, + ( m de p o nt e (2-52C) The value of e is chosen small so that the first three terms of the expansion will approximate the physical problem. By choice of small e, terms such as cuI + e2 U+.... will always be small. The quantity (a/L) must always be considered small to stay within compressibility requirements. Since e includes w, then as frequency increases (a/L) must decrease proportionately. Zero-Order Equations Ostrach's (C-8) solution and tabulated values were used for the zero-order equations. The non-dimensionalization of the equations was made so that the zero-order equations would yield the form used by Ostrach.

He utilized a stream function such that u = d Po (2 -*33'-(A) y and T_ _ 3t (2-33B) so that continuity is satisfied. This produces o 2 ~Zd o_ TY) _) d y d6XY X Yz ay/3 (2-34) * YT6O 8 cY7; I 62T7 6Y 3X X 6 Y PA ) Y/ The boundary conditions are (&ko) (d)o = w o (2-35) o(7r)na un i(T Oto Final simplification of the equations was made by transforming the coordinates using a "similarity transformation"'(- (2-36)

-35and a stream function =(4+</i4F(T7) (2-37) and a temperature function -7 =/o(7) * (2-38) and -/ /. Pp FH' ) (2-40) The boundary conditions are,~() =- F'o) = O (2-41) A/(- C = / and F'~(c) =/7/(o>)- O Some of the results obtained by Ostrach are shown in Figures 9 through 14. His results agreed with measurements of Schmidt and Beckman (C-ll), and Eckert (C-1). Schmidt and Beckman performed experiments on free-convection flows in which velocity measurements at various points along the plate were made by means of a quartz-filament anemometer and

-36_.2 J- - L. 6 8 I0 12 14 16 18 20 22 24 77.65 II/ I I I 1 1...... 1... 1.4 - X IL.72l i - - C" 0 1 2 3 4 5 6 7 7) Figure 2-9. Velocity Profiles for Laminar Free Convection Boundary Layers as Reproduced from Ostrach (C-8).

-37-.6.2 4 r.O I 6 8 10 12 14 16 18 20 22 24 1.0I I.6 I.2 3 4 6 Figure 2-10. Temperature Profiles for Lamina;r Free Convection Boundary Layer as Reproduced from Ostrach (C-8).

-38-.32 0 x 2d> (IN.) 2V C I I 0 0.394 0 0.787 o 1.575 EXPERIMENTAL v 2.756.24 - A 4.331 CD THEORETICAL FOR Pr=0.72.20 0.16 0.12.08.04 0 1.0 2.0 3.0 4.0 5.0 6.0 Figure 2-11. Velocity Measurements of Schmidt and Beckman (C-ll) and Comparison with Theory as Reproduced from Ostrach (C-8)

-39-.32 X.28 0 (IN.) 0 1.969 3.939 a~ Abl 1 v 7.878 EXPERIMENTAL F0 A o ^ 8.817.24 *A - * 15.756 I L I I _ - THEORETICAL FOR Pr-=0.72.20.16 l~ a.12.08 0 1.0 2D 3.0 4.0 5.0 6.0 Figure 2-12. Velocity Measurements of Schmidt and Beckman (C-ll) and Comparison with Theory as Reproduced from Ostrach (C-8) Comparison with Theory as Reproduced from Ostrach (C-8)

-401.2 X 1.0 (IN.) 0 0.394 0.787 o I.575 EXPERIMENTAL v 2.756.8 A 4.331 -l - - THEORETICAL FOR Pr=0.72 _.6.4 0 - 0 1.0 20 30 4.0 5.0 6D Figure 2-13. Temperature Measurements of Schmidt and Beckman (C-ll) and Comparison with Theory as Reproduced from Ostrach (C-8).

-411.2 ZL x (IN.) ~~~~1.0 ~~o 1.969 o 3.939 v 7.878 EXPERIMENTAL 8.817.8 ~ 15.756 \ v --— THEORETICAL FOR Pr-0.72.6 El.4 0i O_ 0 1.0 2.0 3.0 4.0 5.0 6.0 Figure 2-14, Temperature Measurements of Schmidt and Beckman (C-11) and Comparison with Theory as Reproduced from Ostrach (C-8).

-42the temperature measurements were obtained by means of manganeseconstantan thermocoupleso Eckert made similar measurements by means of a Zehnder-Mach interferometer. The lines shown in the figures are the theory of Ostrach and the data is that of Schmidt and Beckman. The results for the small plate are nearly identical to that of theory especially for small values of A. This was attributed to the fact that for the small plates the flow was entirely laminar except near the outer edge of the boundary layer where some slight turbulence from the room air disturbed the measurements. As can be seen, in the figures for the large plate the scatter of the data is more than that of the small plate. Large periodic oscillations of the flow near the downstream edge were reported for the large plate in addition to room turbulence at the outer edge of the boundary layer. It is well known that by increasing the Grashof number the transition region from laminar to turbulent flow will move down a plate, Heat transfer data correlate well with the theoretically computed values except for those of oil. The Prandtl number of oil in the experiment varies approximately from 90 to 500. The maximum error was reported to be approximately 20 percent. The deviations were attributed to either end effects in the measurements or to the fact that viscosity changes in oil are large for even small temperature differences. As a result of the good correlation of the data of Schmidt and Beckman (C-l1) to the theory of Ostrach (C-8) for velocity, temperature and Nusselt number, Ostrach's tabulated results were used for the zeroorder solution.

Approximated Zero-Order Functions In the first-order and second-order equations most terms depend upon functions of solutions of the zero-order equations. The zeroorder solutions are tabulated values which do not lend themselves to convenient integration. The reason for their presence is due to physical interaction of the small effects of the oscillations on the free convection field arising from inertia and viscous forces in the boundary layer. Mathematically it is represented by the non-linearity of the governing partial differential equations. The last term in the first-order momentum equation can be calculated as () i2P y= (4X) ) = (1i7) Y(7) (2-42) Schoenhals (A-34) has done this, and the solid line in Figure 2-15 is values of his result. His tabulated results can be found in Appendix IIIA. In order to solve the first order momentum equation a simple but satisfactory analytical expression has been found in the present study for:(n) which adequately fits the data. These are shown as the dotted line in Figure 2-15. The equation which fits approximately the data for two Prandtl numbers is <9t) =,t,(JPR)7e /P)V, e3 ~(2-43) This normalized representation is exact at r = O0 This curve approximates very well 5(Q) for low values of r1 and for extremely high values of i,

1.0 Pr C(O) 72 L25.9 \\,o10.5I 2.22 0.8 0.6 Pr=10 Pr =0.72 OAf 0.41 1' 0.2 0 ~ 0 0 I 2 3 4 5 6 7.0 Figure 2-15. Comparison of t(r)/t(O) from Schoenhals (A-34) to That of Equation (2-43).

where it approaches zero. More terms or a higher order polynomial with exponential terms would give an improved fit but the effort required compared to the additional accuracy gained would be excessive. The value of 5(r) at any point in the boundary layer indicates the order of magnitude of the oscillating pressure gradient produced by wall vibration arising from thermal variation of density. In the next section the order of magnitude of this quantity will be discussed with respect to the pressure gradient arising from the potential flow. Since the derivatives of the free convection temperature can be expressed as \ ('4A) (2-44) a simple but satisfactory expression was found in this present study which adequately approximated the curve of H'(rQ). The solid line in Figure 2-16 represent the tabulated results of Ostrach (C-8) and the dashed lines are the values of r ~~~~ - (l ~ k,(B )l _ (/7 0jv} V (2-45) All the coefficients K1 and K2 are constants for a particular Prandtl number. K1 and K2 are given on Figure 2-16 for two Prandtl numbers, 0~72 and 10. The values are exact for j = 0 since here it is equal to H' (O)o The curves for both Prandtl numbers fit well for small values of and for large values of B The intermediate values of r are slightly

1.2 1.0 Pr H I(O) KI j K2.72 -.5046.9.5046 10 -1.169 2.22 3.1 0.8 \ Pr IO % 0.6 ~Pr0.72 0.2 01 0 1.0 2.0 3.0 4.0 5.0 6.0 Figure 2-16. Comparison of -H'(r) from Ostrach (C-8) to that of Equation (2-45).

-47higher than the theoretical results of Ostrach (C-8). The approximated expression was made specifically to fit closely for small regions of r since this region is of more interest because it determines the Nusselt number and shear stress. Since free convection velocity and stream function in terms of the "similarity transformation" are (2-46) - (~:A)K>) it was decided to fit F'('q) also to a simple but satisfactory expression. The result can be seen in Figure 2-17. This expression was found to be ~';/k''~ = X 3 ( )i~ aekr ACRE t (2-47) This expression fits extremely well for small values of r and also represents the peaking of F'(r) in a very exact manner. For large values of i, the Pr =.72 curve is higher and for Pr = 10 the curve is lower than that of Ostrach's (C-8) results. This expression is a better fit of the small plate data for large values of r but as reported previously the flow was entirely laminar except near the outer edge of the boundary-layer where slight turbulence from the room air disturbed the measurements. Therefore it is well to say that F'(q) for Pr = 0.72 is high for large values of A. This expression is not as important as the others, as will be brought out in the next section.

-48-.32 X X x x X ____THEORETICAL RESULTS OF.28.28 X ~OSTRACH X X X EXPERIMENTAL MEASUREx MENTS FOR A SMALL x I \O ~~PLATE.24 0 —-~ F(17) K$~-7 \\ K =.75 X 3 ~~~Pr -.72 K4 = 1.0.20 K3=.45 (4 ~X Pr 10.0 i~~i~~- I I \a ~~~ X 4, 14.12 ~K Pr = 10 0. 1 X X x1 X~~~.0 i2 3 4 5 7) Figure 2-17. Comparlson of F?(rl) from Ostrach (c-8) to that of Equation (2-47). ~~\Pr-.72X I X~~~~~~~~ ~~~~~ IO! 0

-49First -Order Velocities The governing equations of momentum and energy are 9~r 31 dJ7 dc d - t-Uo ~ u —-r f:r J2r2 d -t dx c( d)y dX A y2(2-48B) It is convenient to adopt complex notations and to write. (x~', yt) - Ei L 1 (z, y ) wt (2-49) f: Y)SI Vt = R:~ L L —' f(:) e t.I:I (X., Yt) - -- (:, yR)e)' A e ( The momentum Equation (2-48A) becomes t' +7, t U,,-,, yf,~? d=

where i = =l and only the real part of ul(x,y)ei1t has significance. This can also be written as E'AL, e' //Cos (t- 4) or (2-51) REL U, e = R FgL a, Cals t - Imps. 1, sJV/w where 0 is the phase angle and lull is the absolute magnitude of the vector quantity ul. For large frequencies all convective terms and the temperature term are negligible with respect to the terms containing frequency as a factor in Equation (2-50). Therefore the terms retained are the inertia terms, the pressure gradient due to the potential flow and the forcing function 5(r) which all have frequency as a factor. The highest order derivative also is retained. This has been done by Lighthill (A-20) and is in agreement with the theory of differential equation with large parameters. By dropping the vector superscript, Equation (2-50) becomes _3i _ tY t X W U-% 4)3 < ) (2-52) It is convenient to find solutions in terms of the similarity variable of the zero-order equations. By the change of variable the following equation is obtained (dkX 4su, =-X a) P. ~(, (2-53)

with the boundary conditions 1, ( 0) =O (2-54) U, (~)= P We now have an ordinary differential equation which requires a particular and a homogeneous solution, The solution to this is U ( JV('e iw)-e (2-55) where kAo' W f1() k, (2-56) V~ -Ko VSSi).2 k, VL ), =.(4X)I4 ( 6');o i'Xi") (__,__, ( ~4x Now the relative effect of the pressure gradient due to potential flow and the pressure gradient due to thermal variations can be investigated~ The ratio of the pressure gradient due to potential flow to that due to thermal variations of density is (2-57) Aof(o) (vsrw o) This quantity has very large values for all values of X except where X = 0.5 where P 0= O The reason for the large values is that (2-58) Gr > o(J)

The expression (Qw - @y) is a small quantity. The coefficient of volumetric expansion for gases is $ = 1/T and in general it is of the order of magnitude between 10-2 and 10-4 The analysis of Ostrach (C-8) points out the fact that free convection flow is of the boundary layer type only for large Grashof numbers. The value of P, which is given as a dimensionless term in Equations (2-26), can become large. This suggests that Equation (2-57) would certainly assume large values. This was checked by numerical evaluation of the solution to ul. The resulting values of ul for both finite and zero Ko were essentially identical. It is concluded that the effect on ul due to the thermal variation of density arising from the inertial force term is insignificant compared to the effects from an oscillating potential flow. Therefore all results given in this chapter are for Ko = O. Figure 2-18 shows some representative values of wul, where Ko = 0O For high frequencies mu, is not a function of Pr number but only a function of X and wo This is due to the fact that ul is responding to the potential flow which is only a function of the dimensions of the thin plate. The ul was multiplied by w in order to show the frequency dependency. The multiplicative term comes from e, E = amo/L. With Ko = 0 the first order velocity is identical to the periodic velocity of an adiabatic plate, treated by Schoenhals (A-34). The first part of the solution and all subsequent solutions have been programmed on the 709 IBM computer at the University of Michigan Computing Center. Two representative methods of evaluation are presented in Appendix I.o

-538 _ _. 4 i -. 25 w 3A- 3 -IO 3 2IILl <_ 5 = C, Z X=.2 4 4 4 3 28- - x.75 0 0 I 2 3 4 5 6 r1 Figure 2-18. Magnitude and Phase of the First Order Velocity, ulA.

-54For intermediate frequencies, the whole momentum equation should be integrated with the exception of T1. For small frequencies, the total momentum equation and the energy equation would have to be integrated together since they would be coupled. To ascertain where the high frequency case describes the flow the intermediate case was solved. This was done by assuming that the high frequency case is the first approximation to the intermediate frequency case, using the following U I= -,C U+L4 (2-59) The quantity ulA is ul of the preceding case of high frequency. The conditions here on the approximation is, of course (2-60) The problem is to decide where (2-61) When the above condition holds this is a cut-off point and the high frequency solution describes the problem. Otherwise the solution holds for intermediate frequencies.

To find the governing equation for ulB, the superscripts having been dropped for convenience, consider Equation (2-52) where ul = UlA and also substitute ul = ulA + UlB into Equation (2-48A). By subtraction it is found that: Iw322~ du,, godx 6'i d+ Cdy -(2-62) Therefore, the following is obtained. c ---- Li43 C #id L7} t6 C_ (2-63) Since UlA > UlB, ulC and vlC are replaced by ulA and vlA in the right hand side of the above equation. This represents a type of forcing function. By change of variable from y to q we have a-?~c('(IX - -- -I(4 ti ), y 2 d Y (2-64) with the boundary conditions (2-65) L,e (o) -I so that (2-66) )ll 1 / He-/D

-56This is carried out by using the continuity equation to find vlA and then evaluating the forcing function on the right hand side of Equation (2-64). Some characteristic results are shown in the form of graphs in Figure 2-19 through Figure 2-21. At X = 0.5, uI is zero. For X > 0.5 the phase angle leads by 45 degrees at r = 0 and for large values of r it is 0 degrees. For X < 0.5 the phase angle leads by 235 degrees at r = 0 and for large r it is 180 degrees. If the phase angle were referenced to the oscillating potential flow as has been done in Schoenhals (A-34) and Schlichting (A-34) the phase angle would be 45 degrees at the wall and 0 for large values of i. The significance of the method of plotting phase angles as used here is to preserve the absolute phase angle relationships for the first order complex vectors in the second order perturbations. The graphs reveal that for w = 10 and greater, the high frequency solution is valid. High frequency analysis is discussed in the next section in more detail. Schoenhals (A-34) using a hot wire anemometer found good agreement for periodic velocities of an adiabatic plate. For high frequencies, as discussed previously in this section, the first order velocities with Ko = 0 is identical with the velocities in the region of an oscillating adiabatic plate. Schoenhals experimental results are discussed in more detail in Chapter IV and graphs of his data are shown. First Order Temperatures The energy equation with the adopted complex notation is c)X axdX +'ds = By2 (2-67)

-570,I =10 8 1 I I I Pr =.72 X = 625 ~ UIC 6 5 4 0 2 3 5 6 f71 0 0 I 2 3 4 5 6 Figure 2-19. Magnitude and Phase of First Order Velocities ulA and ulC.

-5810 ~~~~~~~~9 ~~Pr =10 ( I I0 27 6 _ _ _ _ rO 3 0 2 3 4 5 6 7 Figure 2-20. Magnitude and Phase of First OrderVelocities u and u Figure 2-20. Magnitude and Phase of First Order' Velocities UlA and U1C.

-5920.. 18 -_ W 30o 16 Pr:=.72 14 XX =.25 I W 0I 8 1 2 3 4. 6 7 8 9 Figure 2-21. Magnitude and Phase of the First Order Velocity, ulA.

From the preceding analysis it "is known that Uo 0- o(1 ), ~O(i) (/ vo( C(/) Therefore it is found that c)x x )ty (2-69) o(/:~) o(/,'>) O(/ o(//) it is apparent then that LI,- t dTo > UC) T; 7 r 0< T7 (2-70) c)X o)y — X ~oc for the case of high frequencies. The governing equation is therefore _T__27 p; do 3 4X (2-71) which becomes, after the change of variable from Y to, Tj7, T)7o a, T7 d-X S2AIi =(4X) &= Pe(<~~x 20t(2-72) with the boundary conditions (2-73) 77(X)o k

The integration has been performed and the results are shown in Figures2-22 through 2-26. To find the cut-off point between intermediate and high frequencies it is necessary to perform the same type of process of determining where the criteria of high frequency is valid as was done for velocities therefore 1/C;L~~~ = /}87~~~ t v~ ~(2-74) The governing equation is i@% V2C) t B x t d t y (2-75) Since TjA > T1B, TlC is replaced by,T1A in the above equation. By change of variable from Y to rl, Equation (2-75) becomes U27; BU~ 7;+ W Ate d x 7 (2-76) with zero boundary cond itions at = and = with zero boundary conditions at vj = 0 and rl =o Graphs of TiA show that the phase angle at j = 0 approaches 90 degrees. The phase lag for large values of r is asymptotic to 90 degrees. For intermediate T the phase lag increases and then decreases. For the same values of w and X the temperature profiles for Pr = 0.72 peak at a higher point than those of Pr = 10. The temperature profiles for Pr = 10 peak for smaller values of ir than those of Pr = 0.72.

-62lT/Xc Xl.75.6 4 _ -TXc X=.7,5 1 2 3 4 5 6 7 Figure 2-22. Magnitude and Phase of First Order Temperatures of TA and T

I.6 1.4 W 15 i.2 Pr= 10 1.0 _ I I eoT XI.25 0.8 ~3 ____1~ ____,T,c X:.75 0.6 0. 2 3 4 5 6 7 U1 2 o 1 2 3 4 5 6 7 Figure 2-23. Magnitude and Phase of First Order Temperaturescxof T1A and TiC.

646 1.4 W= F 10 1.2 Q2L I Pr=.72 6 oe X -.25 0.2X 0.6 0 T X.25 Z'i IC 5[ I o ll a I 0 I 2 3 4 5 6 7 Figure 2-24. Magnitude and Phase of First Order Temperatures of TiA and TiC.

-651,8 1.6 W 15 Pr =~72 1.4 U~21 d r \ I - Tic X=.75._,.o TcX=.25 o.e -,.- X =.75 o.6 0.2! z 6 0-..4 2 0 1 2 3 4 5 6 7 Figure 2-25. Magnitude and Phase of First Order Temperatures of TiA and Tlc.

-661.4 ____ =t 30 1.2 PrJ2 Pr =.72 1.0 - X=.25 u 3e of T ics _ 0.6 0.4 0.2 06 0 I 2 $ 4 5 6 7 8 Figure 2-26. Magnitude and Phase of First -Order Temperatures of TiA and TiC.

This is because H'(r), see Figure 2-15, has a smaller thermal boundary layer thickness for larger Prandtl numbers. Figures 2-22 through 2-25 show the convergence of T1A to that of TlC for high frequency. The circles and squares are values of Tlc and the solid lines are values of TiA. As can be seen in these figures, convergence is better for large values of X, Pr and ac. The graphs indicate the rate of convergence for increasing X, Pr and W. The criteria for TiA to be used in place of TlC was found from the results to be peO 2 1jX(X)/ j (2-77) This represents the fact that for the high frequency solution, TlA, to be valid the above group cf parameters must be approximately greater than 25. The values of frequency which permit TiA to describe the periodic temperature solutions are greater for low Prandtl numbers than those of high Prandtl number. Also if X becomes very small then X would have to become very large to satisfy Equation (2-77). Figure 2-27 shows characteristic values of wTlB for two Prandtl numbers. First-Order Shear-Stress and Nusselt Number The shear stress will be evaluated first. The derivative of ul without the term f(r), described by Ko = O, which was found to be negligible, is (, ) _: I =Ar Pe (W t z-L (2-78)

-680.7 0.6 X=.75 0.5I / \ Pr =.72 0.4 0.3 0.I 3 -e- 2 0 I 2 3 4 5 6 7 Figure 2-27. Magnitude and Phase of TlB.

The expression for the zeroth and first order (periodic) velocities is C' = I0 + 6k8Ea1 LI, e t (2-79) which can be written as (/l- S ZL f,/ s(w&&+c/) 7 (2-80) The expression for the zeroth and first order (periodic) shear-stress is (2-81).By,,~ (4,x) aZ which can also be written as -= (PV"W sd'?)t5) E?~t" "' (2-82) The first-order contribution in the shear stress is not a function of Prandtl number, as may be seen in Equation (2-82). It is, however, a function of viscosity. It leads the disturbance by an angle of x/4o At X = 0.5 there is a line of symmetry for P, but the X-dependency of the first order (periodic) shear stress is not symmetrical around X = 0.5 because of the influence of the factor (4X)1/2. The frequency dependency of the first order (periodic) shear stress is w5/2 because e is proportional to w. The derivative of the first-order temperature distribution is presented in the form of graphs in Figures 2-28 through 2-30. Figures 2-28 and 2-30 for Pr = 0.72 have the same ordinate values. The difference in the plots is in the abscissa so that the effects of frequency

-700.3 Pr=.72 0.2 ~0 O iI _\ _rrX=.8 0.11 I I I X. 0 W -90 =....4 X CD -110 5 10 15 20 25 30 35 4.0 45 50 Figure 2-28. Magnitude and Phase of the First Order Temperature Derivative Divided by the Zeroth Derivative of Temperature.

* a ^admaj Jo GA$T4ATJaG t[o4OZ aT Xq PaPTAT(I AT1WAT.IOqG 9n4-emaj aapao SIqta An J0 aseO d pus apnI.TuswI 6Z-& qamT1! m~~~~~~06. OS St Of S~ o 0~ S Z' OZ 01 SR.ZO 9I0 301 m m m m m I ~ O= — _' ___ __ 90' -TL

-720.3 Pr=.72 0.2 o.1 I 10 Q: -90' 1 -4O 0 -— C~30 0.2 0.3 0.4 0.5 0.6 0.7 L. n- -9o0 4___ 30_____, a~ -,100 20 -1T -~5 Figure 2-30. Magnitude and Phase of the First Order Temperature Derivative Divided by the Zeroth Derivative of Temperature.

073Pr= 10 0.16 0.14 i 0.12.r~t 4 -' I' 0,n. 12 E 0lo 0 Z4 5 30 00 10 w 2_______0_ Figure 2-31. Magnitude and Phase of the First Order Temperature Derivative Divided by the Zeroth Derivative of Temperature.

-74and X can be shown separately. The same is true for Figures 2-29 and 2-31 for Pr = 10. The expression for the zeroth and first order temperature is J=T~R * t E 9L L e j+- (2-83) and the derivative is dt7 Im T / CO t t (2-84) c)Y - IX) 4 dY The derivative was represented only in the limits from X = 0.2 to X = 0.8. Figures 2-30 and 2-31 of the first order temperature derivative versus X show the asymmetrical character of the derivativeo The limitations on X are necessary because of the existence of singular points at X = 0 and X = 1.0, These singularity points originate in the potential flow and the solutions do not apply at or very close to the singularity points. For increasing frequency the phase lag approaches 90 degrees for both Prandtl numbers and the absolute magnitude of the first order temperature derivative decreases. For large frequency the absolute magnitude decreases as -lo The absolute magnitudes and phase angles of first order temperature derivatives as a function of frequency versus X for two Prandtl numbers are shown in Figures 2-30 and 2-31. The local Nusselt number, in terms of dimensional quantities of Equation (2-26), is defined as -- = - l - o (2-85)

which, in terms of the zeroth and first order temperature derivatives, results in 1, (,x,>, e (C (2-86) By simplification and division the following relationship is obtained l__d C Di () 7/ (2-87) Second-Order Velocities The governing equation of momentum is Equation (2-32.B) with Ko= O0 It is seen in Equation (2-32B) that certain terms will contribute terms with cos2ot. These terms are o<) ~U XcX wd (2-88) These relationships can be reduced to terms which contain only cos2wt and steady terms, as )l h _ r Pr ZC RrlJ p - C LA CYI dL ff L' J C o S'd L. 2 r d Y (2-89) pi51cLsP 6/

-76By Equations (2-89) the result can be seen why oscillations of a body in a fluid at rest induce characteristic secondary flows or streaming flows, whose nature are such that a steady motion occurs in the fluid from a purely periodic disturbance. The only conclusion that can be made is that secondary flow is a result of viscous terms and convective terms, i.e., the non-linear terms. Therefore, it may be assumed that the solution has the following form of time-dependent and time-independent solutions If _ REL (z e. j) (2-90) /ok = %2S + RE/, (%2 e'"t) The two following equations, which are obtained by substituting Equations (2-89) and (2-90) into Equation (2-32B) and separating according to timedependency, are 2 x d 12 dX + dy (2-91AB) __ dX d aey2+ 7 Ur t7/;~~~-I \L

-77The bar over the symbols denotes the respective conjugate complex quantities. The order of magnitude of the quantities above have been investigated previously, Equations (2-67), with the exception of u2s and v2s. The unsteady solution u2t was not solved since it is doubly harmonic. The steady part u2s was solved because of its importance to a time-steady shear stress and Nusselt number. The order of magnitude of u2s cannot be greater than O(W1). The forcing terms in Equation (2-91B), the underlined terms, have the order of magnitude 0(1). Each integration will decrease the order of magnitude by JW and since two integrations are required the resulting order of magnitude of u2s will be O(o-1). Then u2s and v2s are treated as having 0(w-1) and it is found that dudX d_ XL{ d- 5l X Xr d/J (2-92) O(1o/) O(L/'/) O(C f/ S ) (2-92) reduces to d \Z _d (2-93)?I C) d yJ d cX Equation (2-93) was found by inspection to be that first solved by Schlichting (A-32) for an adiabatic plane executing harmonic wall motion. He applied the result to a circular cylinder. Equation (2-93) was

-78also solved by Schoenhals (A-34) for an adiabatic finite plate executing harmonic transverse motion. The potential flow is, of course, different for both cases. Their similarity results from the fact that the solutions to each are restricted to high frequency. Schlichting obtained u2s as the product of a x-dependent function (potential flow) and a ydependent function, as shown in Equations (2-94) and (2-95). Although Equation (2-93) was not solved here in this way, the y-dependent results are the same as given by Schlichting. The form of Equation (2-94), particularly convenient for plotting the stream function, is Cl=Ipi dX?') (2-94) where (2-95) r(r) satisfies Z"=a44'4 (tozo "i x7') (2-96) where p,o is the first-order velocity. The solution in terms of I' and ~ is (2-97) oeur conto-S ani) This is plotted in Figure 2-32. The steady velocity can only satisfy boundary conditions at the wall. At distances outside the boundary layer

-7902 0.1 2 3 4 5 6 7 -0.1 -0.6 -0.7 - 0.8 Figure 2-32. Values of [(~) of Equation (2-97).

-80it is possible to make the velocity finite but not zero. The value of j' is (~ = 4 (2-98) Schlichting (A-34) was able to verify secondary flow movement by photographing the movement of fine metallic particles about an oscillating cylinder in a tank filled with water. The stream function is Y2 = Zz- P (2-99) By integrating i the result was found to be _ 1_ 3, -, 3gr _, 8 - 3 8 -f (2-100) which is plotted in Figure 2-330 The previous discussion leaves the stream function in a very convenient form for finding constant values of r, the streamlines. A new variable is defined as V-= if;>u,2~ (59/ 7(2-101) By shifting the coordinate of P from the bottom edge of the plate to the middle and by letting X/L = X/C for this development Equation (2-99) reduces to X Xl) (2-102)

0.2 0.1 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 -0.I - -0.2 -0.6 -0.4 -o. I 1 I5 I -0.6 Fr -0.6. of 5 5 -0.7' -0.7-0.8 -0.8 -0.9 -0.9'' -I.0 -I.0 -I.I -I. VleooEqa0. Figure 2-33. Values of 5(5) of Equation (2-100).

-82which is written as /-?Xi-X4 = X;(W/t/Y (2-103) For a chosen there are particular values of X and ~ which satisfy this equation. The solution of Equation (2-Q34) can be done in several ways. One way is to select values of t and * and then solve for the fourth order roots of X. This method is difficult. A more convenient way has been ascertained by realizing that the root-locus method from automatic control theory can be applied here. It was realized that Equation (2-103) has the form of a closed-loop transfer function for a feed-back system. A brief account of root-locus theory will be given here as explanation. The root-locus method represents, through simple and analytic techniques, the changes in position of the poles and zeros of the closed-loop transfer function as the gain of the open-loop transfer function is varied. From this point on X is replaced by Z, a complex variable, where only X, the real part, has significance. The closed-loop transfer function can be written as, _ -IGc D X LiZ-X2 (2-104) where D represents the gain. Root-locus will yield where W- /G: 0 (2-105) G has a zero at Z 0 O, double order poles at Z = + 1 and a third

order zero at co. With this known and by utilizing the rules for the asymptotes of the root-locus, the root-locus plots were made and are shown in Figure 2-34. As can be ascertained from the root-locus plots there will be, on one side of the plate, four regions of 4r. In Figure 2-34 is shown one of these areas and in Figure 2-35 is shown the right half side of the plate. The streamline, o2s' of Equation (2-99) decreases by -3/2 and u2s of Equation (2-44) decreases by w-1. Since the second-order time-independent contribution to velocity is E2u2s (e = aw/L) the frequency dependency of velocity is w. The inner vortex of Figure 2-36 on the upper quadrant goes up and the one on the lower quadrant goes down, near the plate. The result is that free convection (time-independent) velocities are increased in the upper quadrant and decreased in the lower quadrant near the wall due to time-independent second-order perturbations. Also for increasing frequencies the vortices move inward toward the plate. The streamlines touch at X = 1 and ~ = 0 and 2.63 due to the approximations imposed on the potential flow. In reality these would not touch but would be close together in these regions. Second-Order (Time-Independent) Shear Stress and Nusselt Number Taking the expression from the steady velocity and differentiating it, the time-independent derivative is found to be d Y (L2 w) (2-106)

ROOT-LOCUS PLOTS +r/c z4- z2- 5-I ]Z+ /+ Z = x/c +1i r/c PLOT "A"': NEGATIVE /4 /// =/-o 0 m 0 + x/c -4.0 -3.0 -2.0 -0.O 1 +2.0 +3.0 ODD\ a\\ I \\ /itc -2.0;1.0 Xl.O +2.0 +3.0 +4.0 +5.0 ~/ /~ |PLOT UB": POSITIVE // Figure 2-34. Root-Locus Plots for Determination of the Streamlines of Equation (2-102).

1.0 0.8 O..8 0.7 0.6.10 0.5 0.4 __.0_ __ 0.2 0.2 0.002.01 0.1.002 0 0.2 0.4 0.6 0.8 1.0 1.2 i.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Figure 2-35. Time-Independent (Secondary) Streamline in the Top Inner Region of the Plate.

-86I 4 2 3 4 Figure 2-36. A Representative Plot of Time-Independent Flow Streamlines.

-87Therefore the second-order or time-independent contribution, T2s, can be written as UP 1 p I g Z / 8 X (2-107) This result is not a function of the Prandtl number since for high frequency the non-negligible terms in Equation (2-92) are independent of Prandtl number. The result in Equation (2-106) decreases for increasing frequency as -1/2. T2s, however, increases as 3/2 since E = ae/L. Except for the (4X)l/2 in the denominator, the contribution to Ts/pw (4x)1/2 for X < 0.5 is equal and of opposite sign to that of X < 0.5. It is appropriate at this point to investigate the secondorder energy equation. The governing Equation (2-31C) will yield terms which contribute cos2wt terms. These were reduced to terms which contained only cos 2nt and steady terms exactly like the procedure used for the steady velocities. The two resulting time-dependent and timeindependent equations are a 2 SC Te) a TC2 TO7 dx LdY=Pe d, Z (2-108) U 3- 7 - _ Us (I Y ) -% 7,

-88where the bar over the symbols denotes the respective complex conjugate quantities. Only the steady equation is treated because of its importance as a steady contribution to the Nusselt number. The order of magnitude is ci7(,-'-L'dx dY,'d c.)eY'. 0(1 2 o(t ^l) 2 (i ) O(W-' a) (2-109) Therefore we conclude that T2s will be approximately of the order of magnitude of w-3/2. By change of variable from Y to r and retaining only higher order terms Equation (2-309) reduces to d l _-_,pa (,4,)~2L7 I ~ / yh.... d~o P v e 1tv(2-110) This was integrated to (6T2s/8)0. By one simple integration and evaluation at ~ = 0 some of the burdensome algebra was eliminated. Furthermore the steady-state contribution to the Nusselt number is the prime objective and (6T2s/6,)0 yields this information. The zeroth and second-order Nusselt number is A-(/2 C) (2-111) (~~~~~~~~~~(-A/'Za)

-89The steady-state contribution is shown in Figures 2-37 through 2-40. As can be seen in these figures, there is an asymmetrical nature to the gradient along the X-direction. This asymmetry is due to the fact that the potential flow is symmetrical about the middle of the plate and the free convection (zero-order) part is similar from the leading edge up and the net effect is a non-symmetrical result. The gradient is positive and therefore the second-order contribution is to decrease the Nusselt number. The gradient (6T2s/6)0 increases approximately as -1 as frequency is increased. Since e = an/L the second order perturbation dependency on frequency is c which decreases as c increases. Also the second order dependency on amplitude is (a/L)2 and the Nusselt number decreases as (a/L)2. The Nusselt number in Equation (2-111) decreases at higher Prandtl number. Solutions Up to Second Order The velocity in dimensional quantities is 2t z _f'() +(~4 COa(wzjt/)t U#6s (2-112) To express (6U/6Y)O in terms of the known functions, there is obtained by substitution of these functions into Equation (2-112) the following equation'I // ~ t5 213 (2-113) d~hd

-9o3 0 I Pr=.72 4 2 = 5 10 15 20 25 30 35 40 45 50 Figure 2-37. Magnitude of the Time-Independent Temperature Derivative Divided by the Zeroth Derivative of Temperature.

-9116 15 / 14 13 12 I0 0 i 8 cm 0 _ _ o. 7. P.. r.72 ~5~~~ v 2 W = 20 0 ~ =30 Fu2.r. 4.5 of6.7 Tm 8 Figure 2-38. Magnitude of the Time-Independent Temperature Derivative Divided by the Zeroth Derivative of Temperature.

-92x..8 Pr =10 3 "pof? 0 _! X:.2 X=.4 5 10 15 20 25 30 35 40 45 50 Figure 2-39. Magnitude of the Time-Independent Temperature Derivative Divided by the Zeroth Derivative of Temperature.

16 14 13 12 I I 10 9 1 1 I ~ Pr =10 8 _ 3 =5 _ W =10 4 w = 20 W - 30 2 1 =40.2.3.4.5.6.7.8 x Figure 2-40. Magnitude of the Time-Independent Temperature Derivative Divided by the Zeroth Derivative of Temperature.

By definition the shear stress is (3~`7i4 6 (2-114) so that substitution of Equation (2-113) into Equation (2-114) will yield (2-115) C (@5) (Alix ). The steady-state shear stress is Equation (2-115) without the second term on the right hand side of the equation. The periodic middle term will contribute nothing when integrated over a whole cycle. The local Nusselt number is defined as 1 - /X..:.-o (2-116) and to express (9O/6Y)o in terms of the known functions, there is obtained by substitution of these functions into the derivative of the zeroth, first and second order temperatures the following equation o __ a6 6: —) F/,/ (2-117) +eR (d/~)

so that substitution of Equation (2-117) evaluated at j = 0 into Equation (2-116) yields (2-118) Again the middle term contributes nothing to the steady-state value. The values for all of these expressions are presented in the previous sections either in the form of graphs or the expression itself. Discussion of Results In the preceding sections an analysis of the effects of transverse vibrations of a finite, vertical flat plate in an otherwise free convection field was presented. The case treated was that of free convection with a perturbing effect from a potential flow of small a2 and high frequency. The external periodic potential flow induces oscillations of temperature and velocity in the boundary layer. The phase angle and magnitude of these oscillations as well as the secondary or steadystate alterations of the free convection boundary layer were obtained. These were present in the second-order perturbations and arise from the inertia terms of the governing equations of momentum and energy. Also studied were shear stress and Nusselt number changes due to the external potential flow oscillations. These effects were both periodic and steady-valued. The effect of oscillations produced a slight

-96decrease in the heat transfer coefficient. Experimental observations, discussed in Chapter IV, support this result and explain a phenomenon which is prevalent in experimental data in the literature. This phenomenon is reported to be a transition from a laminar to a turbulent flow at a critical condition of vibration.

CHAPTER III FIRST AND SECOND ORDER PERTURBATION OF A LAMINAR FREE CONVECTION BOUNDARY LAYER DUE TO TRANSVERSE HARMONIC OSCILLATIONS OF AN INFINITE PLATE Introduction This chapter is an extension of work on the subject of vibrational effects in heat transfer completed recently by the Heat Transfer and Thermodynamics Laboratory (A-34), Schoenhals and Clark reference (A-33) treated the first order perturbation. The present results disclose the existence of a steady streaming or secondary flow phenomena. Shear stress and Nusselt number are presented in the form of series solutions. A revision is made in the analysis from that of reference (A-33) in that mathematical relations are fitted to the important variable functions and the derivates of the free convection boundary layer. This revision is found to allow more freedom in the solution. The present results and those found earlier (A-34) are identical in these regions where both are applicable. This revision was found also to be more convenient for utilization in the second order perturbation. Formulation of the Problem The physical problem is that of a vertical heated flat plate in an otherwise quiescent fluid. The surface of the plate is maintained at a constant temperature 0 for X > 0, Figure 3-1, In the absence of oscillations the problem would be that of free convection previously -97

-98INCOMPRESSIBLE FLUID i1, K, - BOUNDARY MOTION VoO aW SIN wt e > co ao BOUNDARY LAYER THICKNESS 8" 7Yb~ /1 MOVING COORD SYSTEM STATIONARY COORD. e SYSTEM Figure 3-1. Laminar Free Convection Boundary Layer Adjacent to a Heated Wall with Transverse Vibration.

-99treated by Schmidt and Beckman (C-ll) and Ostrach (C-8). Their results will be called herein the zeroth-order solution. For transverse sinusoidal oscillation the first-order approximation has been determined (A-34), and has a periodic character. The plate is considered to be of infinite extent. In a practical case this would require that edge effects be negligible in the central portions of the plate. If the plate were of finite extent the problem would be that of Chapter II. This chapter is concerned with the effects of transverse wall vibration on a free convection flow at high frequencies for an infinite plate. The perturbation method of solution used linearizes the governing of equations. Therefore by the nature of the solution the effects of vibration are expected to be small. It is in this way the problem can be rendered mathematically amenable. The analysis is concerned with incompressible flow. The question of compressibility is examined in reference (A-34) and is repeated here. The motion of the plate is VO = a Q sinQT. If the plate is considered to be the source of a sound wave propagating normal to its surface with velocity Vs and frequency Q, then the acoustic wave length normal to the wall is X = Vs/2T E. It was found that compressibility is not important as long as ao0 is small compared to the velocity of sound of the surrounding fluid. For example with air let ao = 1 inch and E = 30 cps then the relative compressiblity is about 0.1 per cent.

-100The problem is formulated in a frame of reference fixed to the oscillating plate. The governing equations of momentum, energy and mass are (A-34) Pu _ 0P bets e y (3-1A) (3-1D) If there were no oscillations the last term of Equation (3-1B) would be zero. Utilizing the usual boundary layer assumptions these equations reduce, from reference (A-34), to -P - -,n -z-Can~-s ( — l (3-2A) 9fiVVc)2"z(3-2C) with t C co(3-2D) 22 7 with the boundary conditions

-101y=O ~F- 7 = a 0= (33) (3-3) y-oo ==Oj )= -oEquation (3-2) introduces the influence of the oscillation. Should the frequency be zero then Equations (3-2) would describe the laminar free convection heat transfer. In Chapter II the method is described for treating the pressure distribution due to the thermal variations of density. This is not repeated here and it is seen from Chapter II that is the governing equation for momentum. The governing equations are non-dimensionalized by the same relationships used in Chapter II. The result is CtO DL = Tga) dx 4X 0 d/ ~(3-5) Id ZPRdYZ with the boundary conditions y=' u'3-r=o - 0 =/ (3-6) yAc c /-T=O

-102The quantity in parenthesis in Equation (3-5) is defined as e -,- -)cj __''=(3-7) This chapter treats small c and high frequencies. It can be seen that c will be small even if o is high because of the extremely small magnitude of the coefficient of c2 in Equation (3-7). It should be noted that ao/L has to be small because of the compressibility restraint. Considering the requirements of compressibility and free convection flow it can be stated that (o/L) <o (I) ORL. << ol) (1) a<0-IL~~ <K C:2 O~(/ ) ~(3-8) A.TS- << o (/) G, >>O(/) The first three are related with compressiblity restraints. The last two are conditions that are necessary for the treatment of free convection as used by Ostrach (C-8) which are inherent to this problem as previously explained in Chapter II. Perturbations of the Equations Equations (3-5) are coupled, non-linear partial differential equations. Owing to these complexities a solution is sought using the perturbation technique. The solution is a convergent series expansion in the following form

-103/ - L/c 6 + 6 6 L 6 C2 LZZ 7L. (3-9A) T,- o h th H t;'' (3-9B) T= <% +t T Z tx 2 r 2,,, (3-9c) These expansions are discussed in Chapter II in the section concerning the perturbations of the equations. The n-number of differential equations are found by substituting Equations (3-9) into Equations (3-5) and separating according to powers of E. The continuity equation is, of course, linear. The equation found with &' as the common coefficient is called the first-order equation, that with e2 the second-order approximation, etc. The results up to second-order are as follows: Zeroth order, E0 UO b+o _ y~ = To + d u ~o X 1bY P2 b/A (3-lo) d Lo o =0 with the boundary conditions (3-11).=,0,jO ~

first order, c' d+ UZ --- U Iit7 T l- U0= -Now (3-12A),U,37L UI 1 (3-12B) d x )t Y y- O (3-12C) with the boundary conditions >' =O0 Uz - 7 - 7- L O (3-14) second-order, E d Ut__ UoO tr UIi. )2 + ~0 >0 tl 6 cZ y (3-15A); Y - X y 72 —r-~,5X C'

-105at 6UX Ur r To T 4 d-= 7 _ / c327-Lz (3-l5B) 7LE ayizr_ O (3-15c) with the boundary conditions (3-16) Except for the zeroth-order equations, the solution of the governing nonlinear equations, valid only for high frequencies are formed from a convergent series of linear solutions. To have convergence it is necessary that Uo> e Ur > 6 2, > ~ > ~ (3-17) Zero-Order Solution The zero-order solutions are exactly those of Ostrach (C-8) and were discussed in Chapter II. It is appropriate here to recall the similarity variable

7 - / I ( 4 X) 2+ (3-18) and 9 ( d t3/4 F(2 7)(3-19) where 0o is the stream function. F(QI) and H(r) are tabulated values of Ostrach (C-8) and are listed in Appendix III. First-Order Solutions Two solutions desired are of a steady-periodic nature. This is facilitated by the use of a complex variable time function as was done in Chapter II. By realizing the trignometric relationship COS &At - /&) eL 2 (3-20) we can therefore say t /1 -- / ( 6-1/, t z t! (3-21) where uI is a complex quantity. Then the solution can be written in the form L1=z / S / c(&Cas # @ t -i ) (3-22) where 0 is the phase lag and IUII is the absolute magnitude of the vector quantity uI. Therefore we have 3A~~~~~t t (3-23) 6t

-107By substitution of these two relationships into Equations (3-12) it is seen that the exponential term drops out. For large frequencies all convective terms are negligible in the momentum equation for a first approximation. In view of this the terms retained are the inertia terms involving u, the highest order derivative and the forcing function (r). This method has been used by Lighthill (A-20) and is in agreement with the theory of differential equations with large parameters. For the energy equation the two underlined terms of Equation (3-12B) are retained along with the inertia term involving X and the highest order derivative. The two underlined terms represent a type of forcing function for the energy equation. This was done because the subscripted zero terms have approximately the same order of magnitude and ul and vl are larger than T1. This results in 8 2~ U,, 11, = —(4 X) r (V) (3-24) and Xp a > —~ ~~ 8 (3-25) By changing from the variable y to rj they become' _ -, 5 x5 U, = -j".Y) (3 -26A) 4 x2 _ 37;' —/ _ ~ _ a )/2; A, I (3-27B) / a*'_,T, (~xafu c)

whe re 4-6{ (51z() t (3-27) The solution to the first-order momentum equation is. =/ % _ eX t e k (3-28) By introducing the variable _1 _'__._.L_'/ (3-29) V1 and V2 are complex functions which can be described as, ( ) [i t f (3-30) The solution of reference (A-33) for u is (U4' __,0,.ing 5(11),7l,0) by t e 5f&i''l curve in Chapter II this (3-31) By replacing (Tj)/5(O) by the fitted curve in Chapter II this function is given as -.e +/F, e" (3-32) ~~i~ ~9-/x7

-109It was found that solution (3-28) and that obtained by Schoenhals (A-33) are identical for large values of y and hence small values of;, as is easily seen by letting X-+0 in Equation (3-29). In solution (3-28) V1 and V2 are evaluated completely by a numerical method and are not approximated by the series of Equations (3-30). The series were used only to compare solutions (3-28) and (3-31). The solution of reference (A-33) treats large y and w therefore moderately large values of X. The present solution treats large w and therefore almost any value of y and X. Although X cannot be too small, because the non-oscillating free convection boundary layer solution is not valid in that region and (nj) is dependent on its results. The reason the present solution, Equation (3-28), was derived was to have a form which is not dependent upon the tabulated results of the free convection solution. Solution (3-28) is in closed form and has been evaluated for numerical results on an IBM 709 digital computer. Two typical programs for the computation are given in Appendix I. Figure 3-2 shows howEquations (3-28) and (3-31) compare for a moderate value of y = 10. The agreement is quite good for small values of 1 and it is fair for large values of A. For smaller values of X the comparison would not be as favorable, but for moderately large values of r the two solutions coincide within the limits of the fitted curve and the assumption for the solution of reference (A-33). For this value of y the phase angle appears to oscillate. This is due to the fact that (rq) is actually zero but the fitted curve is not. Figures 3-2 through 3-6 show some representative values of the first-order velocities. They are represented in the form of e2 times

2,0 -1 EL 0 C.'ES W 5.0 r 1.r X= 1.0 0 AND a FROM' REF(A-33) i. 12 Pr =.72 - 0.8 04 -20 -40 -60 -80 --- Pr =.72 Pr =10 -120 -140 0 I 2 4 77 Figure 3-2. Magnitude and Phase of First Order Velocities With Comparison to that of Schoenhalsand Clark (A-33).

'salITOOTA IJ0ap.0 %s.I, JOo aSuqd pu~ GpniuuW'%-g a.rnmT & 9 G ~ g 0 T' I lOil001369:l'lO 089=0'7 OLo9'~' 09O~_' O -TT0 TZL'O:= Jd 8-0 -~1~TTT -

-1123.4 _ 3.2 3.0 2.6 l l | \ | Pr 0.72 2.4 X: 1.0 2.2 2.0 I A 1.2= ____ 1.6 0.8 0.6 0.4 0.2 -40 -60 W=4 -80 120 I 2 3 4 5 6 7 Figure 3-4. Magnitude and Phase of First Order Velocities.

-1132.0_ W-o Pr =lO X =.25 W=6 1.0 ":3 i -10 -20 -40 -60 -80 W=6 W PC8P -100 -120 -140 (0 =4 -160 -180 - 200 — 220 0 I 2 3 4 5 6 7 Figure 3-5. Hagnitude and Phase of First Order Velocities.

1.0 Pr IO X 1.0 0.8 W=8 0.6 0.4 0.2 0 -30 -60 -90 -120 -150 /-"C=4 -180 0 2 3 4 5 6 7 Figure 3-6. Magnitude and Phase of First Order Velocities.

-115absolute magnitude and phase angle. The absolute magnitude was multiplied by the m2 taken from e in order to show the whole effect of frequency. The motion of the plate is sin JLT and the oscillating pressure distribution has a timewise variation of coslLr. The solution is represented as ReL4 e Rexu,( (3-33) so that in this case phase lag refers to the oscillating pressure gradient. The phase lag of Pr = 0.72 some distance out in the boundary layer has a value of 900 for sufficiently large values of u and X. For smaller values of X for the same frequency the phase lag is greater and for the same value of X with increasing frequency the phase lag is approximately 90o. A similar phenomena is observed for Pr = 10 except that larger values of frequency and X are required for the phase angle to approach 90~ as an asymptotic value. It can be seen in Figures 3-2 through 3-6 that the absolute magnitude of velocity decreases as X increaseso This is explained by considering the fact that the oscillating pressure distribution is dependent upon thermal density variations which are a result of the nonoscillating free convection boundary layer temperature distribution. The derivative of the free convection temperature distribution decreases with increasing X and therefore the oscillating pressure distribution decreases. Since this serves as a type of forcing function for the first-order momentum equation the velocity will decrease with increasing

X. With everything else being constant it was found from Figures 3-2 through 3-6 that the magnitude of velocity decreases for increasing Prandtl numbero To solve the energy equation of Equations (3-20) use was. made of the simple curve of the derivative of the free convection temperature distribution which is given in Figure 2-16 of Chapter II. Rather than fitting a curve to the free convection temperature and then having to differentiate it which has a large possibility of error, a curve was fit to the derivative of free convection temperature, H'(1). This is very convenient because (3-34) ~To I ~~ FX)'4,' which requires only one fitted curve for both derivatives. By the use of this fitted curve it was possible to get a solution for all values of j whereas in reference (ca-33) the solution for temperature was valid for small values and large values of r only. Then with these two regions an intermediate region of 1 was approximated. The present solution applies to all values of I. By performing the necessary operations on the energy Equation (3-26B) the following solution was found to be w7= i~ eV +e( 4 V+ (~/) e6 +v, P7 yL 2)e('<*)7 9 / t t

-117The V's in this equation are complex quantities due to the exponential time factor and consist of real and imaginary parts. The boundary conditions are zero at Tl = 0 and T = oo so that V3 = V4 - V6 - V9. This solution is evaluated numerically in the computer program which is discussed in Appendix I. Figures 3-7 through 3-11 show representative values of the first-order temperature distribution. Figure 3-7 shows the relationship of the temperature distribution for two different Prandtl numbers, Figures 3-7 through 3-11 show the relationship of the first-order temperature distribution for different X, X and Prandtl number. The first-order temperature perturbations show that for higher Prandtl numbers, with everything else constant, there is less temperature response to the same oscillation. This is due to the fact that both viscosity and thermal diffusivity are having an effect. A more viscous fluid cannot respond with the same velocities as that of a less viscous fluid and therefore the temperature field is less. The thermal diffusivity being lower in higher Prandtl numbers may also keep the temperature response low. The smaller response of first-order temperature for higher Prandtl numbers actually originates in the non-oscillating temperature field which causes the thermal density variations thereby producing with the oscillations an oscillating pressure distribution, which decreases for increasing Prandtl numbers. The quantity W21T1I increases with frequency for the values of frequency given in Figures 3-7 through 3-11. The quantity a2IT1, for larger values of frequency approaches an asymptotic constant value~ The ratio of peak values of temperature to velocity

-11812 (0=10 X=.75 10.09.08.07 3.06.05.04.03 /l_.02.01 Prz 10 -120 - 1 30 -140 -150 -160 Pr 10 -170 Pr= 0.72 0 1 2 3 4 5 6 7 Figure 3-7. Magnitude and Phase of First Order Temperatures.

-1190. 8 W-8 0.7 (d =6 0.6 o~~s I I / / /t~ (d = 4 Pr =.72 0.5 X =.25 0.4 cm3 0.3 0.2 0.1 - 120 -130 W=4 - 140 -8- ~~~~~~~L() 66 150 8 - 160 - 170 -180 0 I 2 3 -4 5 6 7 Figure -8 agn7tude and Phase of First Order Temperatures Figure 3-8. Magnitude and Phase of First Order Temperatures.

0,08 _________Pr =O.72 o. 07 -b)=6 4 XX= 1.0 (,d=4 0.06 -0.05 cN 0.04 0.03 0.02 0.o01 0 -120 -130 /-(.=4 -140 W 6= -150 -160 -170 -180 0 1 2 3 4 5 6 7 77Figure 3-9. Magnitude and Phase of First Order Temperatures. Figure 3-9. Magnitude and Phase of First Order Temperatures.

.16.14 Wc 8 Pr= 10.12 W=tt 6 X=.25 W=4.10 - 0.8 i- o.s 0.6 0.4 0.2 0 - 110 CII= -0 w4 "' -—' W=6 -130 =8 -140 -i5O -150 - 160 -170 - 180 7 0 I 2 3 4 5 6 7 rFigure 3-10. Magnitude and Phase of First Order Temeratures Figure 3-10. Magnitude and Phase of First Order Temperatures.

-1220.02... 0.018 W=8 o.on - -W z 6 Pr = 10 0.016 W= X =1.0 0.014 0. 012 0.008 0.006 0.004 0.002 0 -110 -120 W =4 W=6 -130 =8 -140.-150 - isO -170 0 I 2 3 4 5 6 7 77 Figure 3-11. Magnitude and Phase of First Order Temperatures.

-123magnitudes for Pr = 0.72; co = 8 and X =.25 is less than 10 per cent, for X = 1.0 the ratio is less than 3 per cent. The ratio of peak values of temperature to velocity magnitudes for Pr = 10, c = 8 and X = 1.0 is less than 3 per cent. For X = 1.0 the ratio is less than 2 per cent. This proves the original assumption that at high frequencies the momentum equation and the energy equation would not be coupled. First-order temperature decreases with increasing X due originally to the fact that the oscillating pressure distribution decreases, Phase angles for the first-order temperature solutions at i- 0O are around -150~ to -180~. As'r increases the phase angles increase for a while, peak and then decrease again. As X and X are both increased the values of 0 at r = 0 and large j both asymptotically approach -180~o The peaking of the phase angles is higher for larger Prandtl numbers. Second-Order Solutions Utilizing the exponential time factor with double harmonics and the theory of differential equations with large parameters, the second-order momentum Equation (3-15A) reduces to (3.36) wt

-124It is useful to repeat that only the real part of the solutions considered here have significance. Therefore it is necessary to note that RPanLk iZe cwost = "-'L- 1L Xz' e < R l u R@L y e[w/e. w i / E (3-37) where the bar denotes the complex conjugate. This is true also for all first-order products. Realizing this, the second-order momentum equation is separable into two equations, a time-dependent equation and a timeindependent equation. The time-independent solution of the momentum equation is what has been called the secondary-flow velocity or "streaming" velocity. This has its origin in the convective terms and is due to the interaction between the effects of inertia and viscosity. Therefore it is plausible to write (3038) The following time-dependent and time-independent equations re sut from

-125Equation (3-15A) with the change of variable from y to j d?/2 - i'>f- %(3u39) The convective terms which were omitted from Equation (3-15A) are those containing u2 and u2so A solution was found only for u2s because of its time-independent nature and therefore the alteration of the steady velocity of free convection. For this to be true u2s should decrease in accordance with w-2 for increasing frequency. This was found to be the case. This solution will consider large X and large frequency. The boundary conditions, zero at the wall and at infinity, for the second harmonic perturbation can be satisfied at the wall and at infinity, The steady-state contribution can be satisfied at the wall, and at large distances the value can be made finite but not zero, This is the manner of Schlichting (A-32) who was the first to give this type of flow an analytical treatment and also observed this flow experimentally. The steady-state Equation (3-39) was solved completely. The solution is in the form 3, 7C)e'21

-126where the Vn quantities are complex quantities with real and imaginary parts. Figures 3-12 through 3-14 show how the streaming velocity varies with frequency, position and Prandtl number, Figure 3-12 shows the relationship of w u2s for two different Prandtl numbers. Figures 3-13 and 3-14 show how w4u2 varies for different values of X and frequency for two different Prandtl numbers. 4u2s was plotted since this represents completely the frequency dependency, The larger the Prandtl number the smaller the streaming velocity as Figure 3-12 represents. Large values of X result in small values of velocity, Performing the same operations on the energy Equation (3-15B) as were done on the momentum equation the following equations result -c''~- xL & c) _ P'O ty ( (-4A ) _Pa d)\/ 2- (3-41B) OTo The terms left out were the terms containing T2s in theconvective terms. Since T2s < v2s < u2s the two terms left out will be negligible compared to the first two terms on the right hand side, As in the case of the momentum equation only the steady equation was considered, The steady equation becomes, after the change of variable from Y to 1 01' - Ox

-127Pr =.72 200 100 w =I O X =.75 3 10 Pr= 10 0 I 2 3 4 5 r t Figure 3-12. Magnitude of Second Order, Time-Independent, Velocities.

(D 0 ~ ~ ~ ~ ~ 0 Cl(D 0 CfD CD o N PII (D CD~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~O CD PI~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I CD H (D C) CD (D

-12940 W=8 I0 W=4 Pr = 10 0X 0.25 C) cJ Pr =10 X I.0 0.1 0 I 2 3 4 5 6 F1 Figure 3-14. Magnitude of Second Order, Time-Independent, Velocities.

This equation was solved to the first derivative at r = O. It was not carried further due to the excessive amount of computation that would be necessary. Effect of Oscillation on Heat Transfer and Shear Stress For free convection alone the shear stress by definition is T/Q _ (3-43) By utilizing the dimensionless parameters and the tabulated results in Ostrach (C-8) this relationship of shear stress is / >r i - -?"~(:)o> (3-44) (# iKI T. (Y ct 7 With oscillations and the perturbation type of solution the local shear stress (time-dependent) derived by taking the derivative of Equation (3-9A) and making the terms dimensional is 6/ u,J (3/45) raids) J& 4x 1 (5X)i2 Fka) (Fx) 2,t)(1)

-131The time-averaged local shear stress from Equation (3-45) is /& / 7( _ ___ —J, I (51+6) /A The values of (dl)'l)'2 and 0 are shown in Figures 3-15 and 3-16 for different values of frequency9 position, and Prandtl number. The plots of (i_.1i2 LA2 in Figures 3-15 and 3-16 show that the local, oscillating s.hear stress increases with frequency more than linearly but not as fast as c2 The oscillating shear stress will increase linearly with amplitude if frequency is held constant, If the oscillating pressure distribution is held constant corresponding to a particular value of e then the amplitude of the oscillating shear stress will decrease with increasing frequency. Since e is a function of both vibration amplitude and frequency, increasing the latter requires a corresponding decrease in the former in order to maintain constant.0 The local oscillating shear stress is largest near the leading edge, The analysis is still valid in regions close to the leading edge The smaller the Prandtl number the greater the magnitude of the oscillating (time-dependent) shear stress, The phase lag for increasing frequency for both Prandtl numbers approaches 45', The phase lag is greater f:lc. smaller Prandtl numbers. The time-independent contributions to the shear stress to be used in Equations (3-45) and (3>46) are shown in Figures 317 and 1i8, The values given in these figures are found by the computer evallation method descrlbed in Appendix:I

-132500 450 Pr =.72 400 350 300 i.~ 250 2- O0 /. o50 50 40 X=.5 X=.25 45.... 40 20 30 40 50 Figure 3-15. Magnitude and Phase of First Order Velocity Derivatives Divided by the Zeroth Derivative of Velocity.

-1 33300 280 260 240 — Pr= I0 220 200 -0 180 X 1 60 1 140 20 30 40 Zeroth Derivative of Velocity. 80 60 20 —.25 e -X 1.0 -50 I0 20 30 40 50 Zeroth Derivative of Velocity.

-13 4X=.25 103 -0. Figure 3-17. agnitude of the Second Order, Time-ndependent, Derivative Veloci t I 0 I0 20 30 40 50 Figure 3-17. Magnitude of the Second Order, Time-Independent, Velocity Derivatives Divided by the Zeroth Derivative of Velocity.

-135X x.25 3 0 3V""% Pr = I0 I0.. 10 20 30 40 50 Figure 3-18. Magnitude of the Second Order, Time-Independent, Velocity Derivatives Divided by the Zeroth Derivative of Velocity.

-136The values of (-'l2J' in Figures 3-17 and 3-18 show that the local time-independent contribution to the shear stress increases with increasing frequency, The multiplicative factor 4 comes from e2 The derivative /. decreases approximately the same as cw As the Prandtl number decreases increases. The local steady shear stress also is greater the closer one is to the leading edge. Consideration of the fact that e contains amplitude asserts that the local steady contribution increases with the square of the amplitude. It should be kept in mind that the c4 was included in the plot to show the effect of frequency and comes from e2 which is restricted to small values, For free convection alone the local Nusselt number is defined as Mf -( ( 3-47) By utilizing the dimensionless parameters and the tabulated results in Ostrach (C-8) the relationship is NL( [4 )() (3-48) The solution for the temperature distribution is T -= To 7 C T t ~%;t T (3-49) where H is the zeroth order solution due to Ostrach (C-8), T1 is the solution to Equation (3j-26B) and T2 is the solutions to Equations (3-41)o

-137Therefore with oscillations and the perturbation-type of solution the local Nusselt number, time-dependent and time-independent, is,,v. = I +TE ~O V l~ dos ("5t, t-) + The time-independent local Nusselt number is A())(6z/)- (-H//D)) (-51) The plots of the derivative of T1 and its phase angle are shown in Figures 3-19 and 3-20 for two different Prandtl numbers. It is seen that the magnitude of the oscillating Nusselt number multiplied by ct2, taken from c, increases with increasing frequency for small values of X and remains essentially constant for larger values of X. This holds true for both Prandtl numbers. Without the multiplicative factor of 2 then the derivative decreases with increasing frequency. In Figures 3 -19 and 3-20 it can be seen that the smaller the Prandtl number or X the greater is the magnitude of the Nusselt number. The phase lag function of X, Prandtl number and frequency are shown in Figures 3-21 and 3-22 for both Prandtl numbers. The steady-state contribution to the Nusselt number is seen to decrease as compared to the non-oscillating

-1381.0 0.9 -25 0.8 0't 0.7 Pr =.72 I 0.5 0.4 0.3 0.2 - X=.7 5 0.1 X =1.0 150 9-x -.25 160 -5 lX =75 170 180 10 20 30 40 50 Figure 3-19. Magnitude and Phase of the First Order Temperature Derivatives Divided by the Zeroth Derivative of Temperature.

-139-.i6 X=L25.14.1 2.10.08 Pr = 10.06 X=.5 a.0 X=.75 - 140 -150 X=.5 -16 0 1.1 1X=.75 — 180 I0 20 30 40 50 Figure 3-2(0. Magnitude and Phase of the First Order Temperature Derivatives Divided by the Zeroth Derivative of Temperature.

-140free convection case, From these results it is at once evident that Reynold's analogy does not hold for:- oscillating flows with heat transfer. The physical reason for the decrease, is believed to be a result of the phase lag of the boundary layer. If conditions were such that the temperature were oscillating as a primary effect then temperature would be similar to the streaming velocity which creates an increase in the shear stress, Then it seems probable an increase in the Nusselt number would be found. Also the: order of magnitude of the decrease in the 2 Nusselt number is small because of the multiplicative factor e. The fact that ( 7) (/) decreases less with increasing frequency can be seen in Figures 3-20 and 3-21. For smaller Prandtl numbers the magnitude of ( ) ) /-i//)) is greater. This would correspond to a greater decrease in the Nusselt number, Discussion of Results Schoenhals and Clark, (A-33) with a physical system approximating the mathematical model of this present investion could find no effect on the time-averaged Nusselt numbers, The analysis of this chapter supports these experimental findings in that the decrease of the time-independent Nusselt number is so small that it would seem to be practically impossible to measure. As an example, for Gr = 108, Pr = 0.72, AT = 100~F and (a /L) = 0.01, the percentage decrease in the non-oscillating Nusselt number for o = 20, 50 and 100 are less than 0,01, 0.03 and 0.1 percent, respectively.

0P~~~ ~ ~~~~~ - -I4I0 %N -2 = 10 10 20 30 40 50 (C Figure 3-21. Magnitude of the Second Order, Time-Independent, Temperature Divided by the Zeroth Derivative of Temperature.

-1422 I0 Pr =10 I =.25 I0 0, I0 10 20 30 40 50 Figure 3-22. Magnitude of the Second Order, Time-Independent, Temperature Divided by the Zeroth Derivative of Temperature.

CHAPTER IV EXPERIMENTAL WORK Int roduction This chapter describes additional experimental work on the subject of vibrational effects in heat transfer and extends research completed recently by Schoenhals (A-34). The experimental apparatus constructed by Schoenhals (A-34) was adapted to these particular experiments, His results included data obtained from velocity measurements made with a hot wire anemometer under adiabatic conditions and some heat transfer data taken to determine the effect of wall vibration on the time averaged heat transfer rate. Air was the fluid used for all experimental work. In the latter part of this chapter the combined results obtained by Schoenhals (A-34) and those obtained here will be compared to other experimental work which have appeared in the literature, These measurements also will be compared to the theoretical work of the first two chapters. The results of this chapter involve heat transfer data in the form of nusselt number as a function of a vibrational Reynolds number and Grashof-Prandtl number product. The Nusselt number was found to be essentially uninfluenced by vibration for laminar conditions, A maximum decrease of 20 was found for smaller value of vibration Reynolds number. For large values of Reynolds number a sharp increase in the Nusselt number was obtained experimentally. The experimental results confirm the theoretical results of an almost negligible decrease in -143

-144time-independent heat transfer. The sharp increase in Nusselt number was concluded to be a result of a flow transition from a laminar to a turbulent flow. Smoke studies were made to investigate this possibility more thoroughly. These clearly showed that this phenomena exists. Some photographs are presented of different flow conditions. A correlation in terms of a vibrational Reynolds number, amplitude ratio and Grashof number was also made. Experimental Apparatus The experiment is designed to produce a two dimensional boundary layer on a vertical, finite, flat plate undergoing transverse vibrations. General views of the test apparatus are shown in Figures 4-1 and 4-2. The test section is a 10 inch square plate 3/4 of an inch thick with a 6 inch square heated section in the center. During test conditions this heated section is maintained at a constant temperature in a manner which will be described later, The heated section consists of two highly polished aluminum plates on the outside with a heater contained between them. The heater is made up of a heating element which is chromel ribbon 1/8 of an inch wide by 0,002 inches thick wound on a thin mica sheet, A mica sheet was placed on each side of the heating element to protect it from the aluminum plates, These plates are clamped together by four hex nuts at the corners which thread onto four support rods which hold the plate in a vertical position. Construction of the test plate is shown in Figure 4-3, As can be seen in this figure, the

Allowed ~Ji!~~ ~ m~.... v,V. in ~ ~'!a:!rivi. ~i tll. ha aI..!t —?. As ~. i~~'

ALUMINUM PLATE - AIR GAPS U - SHAPED SECTION TOP GUARD HEATER (CHROMAL RIBBON WOUND ON TRANSITE STRIP) TRANSITE SPACER MAGNESIUM STRIP ~ ~...1MICA SHEETS SIDE GUARD HEATER HEATING ELEMENT. (CHROMAL RIBBON WOUND ON MICA SHEET) BOTTOM GUARD HEATER LOCATING STRIP GLASS CLOTH BOTTOM SECTION Figure 4-3. Construction of the Test Plate.

-147center section is supported independently of the rest of the apparatus except for very small transite spacers which maintain an air gap of approximately 1 inch. Surrounding this air strip are thin magnesium strips which have been glued to transite strips, A thin mica sheet insulates the magnesium strips from the guard heater elements directly behind them. There are four guard heaters and they are made of thin chromal ribbon wound on transite strips. There also are air gaps behind these guard heaters. The purpose of the guard heaters is to prevent heat loss from the edge of the plate by maintaining the temperature of the magnesium strips equal to that of the edges of the plate. Each magnesium strip, since it has low thermal resistance, achieves essentially an isothermal condition and is controlled by the power input to the guard heaters, The electrical circuit for measurement of energy input and control of the heater element is shown in Figure 4-4, This has been achieved by utilizing differential thermocouples between the plate and the strips and the details of the circuit for doing this will be discussed later in this chapter. These air gaps, on both sides of the heater, provide a high thermal resistance, The housing or remaining portion of the test plate consists of two sections. The top section is U-shaped while the bottom section is straight. To reduce the weight of these parts they were made out of magnesium. The U-shaped section is supported by 4 rods and the bottom section is supported by 2, thus making the whole test section have 10 support rods. The top section is connected to all 4 of the transite strips, thereby causing its temperature to increase slightly during operation.

GUARD HEATER ELEMENTS MAIN HEATER E~LEMENTI FUSE AA7[MODVEL NOLME WESTON A.C. 00 IAC MODEL NO.94 AMMETER ---- -J ~~~~WESTON A.C. VOLTMETER LAMP 110 V- At Figure 4-4. Electrical Circuit for Measurement of Energy Input and Control of Heater Elements.

-149The bottom section, which is connected to the top section by only two locating strips on each side, experiences a very slight rise in temperature during operation. This is desirable since the thermal boundarylayer starts at this point and the experiment should approximate as closely as possible a step change in temperature. In order to have a flat, continuous surface which is impervious to the flow of air, thin glass cloth with a thermosetting adhesive was applied over the air gaps. This prevents the flow of air through the test section and maintains the high thermal resistance of the air gaps. Since we are interested in two dimensional considerations 2 magnesium side plates are attached to the sides of each test plate, as shown in Figure 4-5. The temperature of the side plates increases only slightly during operation. The 4 rods supporting the aluminum test section were made of 3/8 inch diameter stainless steel tapered down to a 0.14 inch diameter for a distance of one inch from the plate. This was done to decrease the amount of energy that could be lost by conduction through the rods, The 6 rods supporting the outside section, composed of the U-shaped section and the straight section, were made of 3/8 inch diameter magnesium similarly tapered down before the plate. These 10 rods maintain the test section at a distance of about 16 inches from the wall. This distance was provided so that a symmetry of flow can exist.' At the wall these 10 rods enter a stationary magnesiumi plate which supports them with teflon bearings, On the other side of this plate the 10 rods are connected to another plate, which is connected to the vibrator, The whole test

i;CV CD

-151section is contained in a celotex enclosure as shown in Figure 4-2, This enclosure was necessary to prevent stray currents of air from disturbing'the experiment. To vibrate the test section a MB Vibration Exciter was used. It is an electromagnetic force generator. The vibration exciter is a model C5B and its control unit is a model T51-Do This system consists of an oscillatory power amplifier and oscillator to supply the variablefrequency power to the driver coil of the vibration exciter. The DC field supply furnishes the power for the vibration exciter's field coilo The control cabinet is operated from a 480 volt, three-phase, sixty-cycle lineo Control circuits are operated from a 115 volt, single-phase, sixty-cycle lineo In this system the frequency is set on the control unit and the amplitude is raised to the desired amount, The arrangement of the components of the apparatus are shown in Figure 4-6, The rod support system which is carried in the bearing wall by the teflon bushings are bolted together to one piece. This piece is then connected to the vibration exciter by means of small threaded rod, The test plate was adjusted until it was perpendicular to the floor and its top was level, The bearing wall fits into the wall of the celotex enclosure, The large overhang of the rod support system into the enclosure is necessary so that symmetry can be obtained. On the side of the rod support system there are 8 support rods inside the 2 side plateso Ideally the plate should oscillate in an infinite space without any supports, Experimentally this is almost achieved. Seven thermocouples were used in each aluminum plate. They were imbedded to within 3/32 inch of the polished surface by drilling

VIBRATION EXCITER ROD SUPPORT SYSTEM TEFLON BUSHING'i'ii~'~I~',~~ I II VIBRATORY MOTION TEST PLATE BEARING WALL AMPLITUDE PICKUP Figure 4-6. Sketch Showing the Arrangement of the Vibratory Components of the Apparatus.

-153holes in the back of the plate and gluing the thermocouples with epoxy thermosetting resin. Slots were drilled on the back of the plates to provide space for the wires to be taken out of the bottom of the test section. These wires were connected to a system of rotating switches so that a single thermocouple could be read at one time. The thermocouples were made from number 24 gauge copperconstantan wire. A Leeds and Northrup Model 8662 Portable Precision Potentiometer was used to read the indicated output voltageo Distilled ice water was used for the reference junction. The thermocouple wire has been calibrated in the laboratory at the steam and tin points. A Honeywell Visico.rder Model 1012 was used to monitor instantaneous temperature excursions. In the range of experimental values of amplitude and frequency it was found that the surface temperature oscillated only very slightly. This corroborates what R. J. Schoenhals (A-34) assumed in constructing the apparatus, This is due to the large thermal inertia and low thermal resistance of the aluminum plates and the relatively low heat transfer coefficient at the surface, The power input to the heater between the two aluminum plates is controlled by a variaco The current and voltage drop are measured to obtain the power supplied, as shown in Figure 4-4. The four guard heaters are also controlled by variacs, Calibration of the ammeter and voltmeter was done in the calibration laboratory of the Electrical Engineering Department to an accuracy of Ool0.1 The galvanometer circuit for monitoring the guard heater elements is shown in Figure 4-7, A thermocouple was imbedded in each

-154-THERMOCOUPLE ON MAGNESIUM STRIP THERMOCOUPLE IMBEDDED IN ALUMINUM PLATE CONSTANTAN -I, COPPER rd ~ri 0 r. 0.Ip) ~) 0~

-155magnesium strip. These four thermocouples and four thermocouples from the aluminum plates were used to form four differential thermocouples. By closing one knife switch at a time the deflection of a sensitive galvanometer indicates whether the strip or the plate is at the higher temperature. If a null is indicated both have equal temperature. With this monitoring system and the ten guard heaters heat loss at the edges was largely eliminated, Two thermocouples were imbedded in each of two stainless steel support rods, as shown in Figure 4-7. These thermocouples were used to determine the magnitude of the energy lost by conduction down the rods. In Figures 4-8 and 4-9 are shown some of the instruments used during operation. Frequency of oscillation is controlled by setting the desired value on the dial, and was calibrated by using a General Ratio Type 631-B strobotac. The oscillation amplitude was measured in two ways. One means made use of a Schaevitz Linear Differential Transformer Model 100-AS-L. The transformer was placed on a platform which has a stationary mounting and a fine adjustment which allows a sliding member to be controlled by a threaded shaft, The movable core of the transformer system was attached to the plate by means of a brass rod. This can be seen in Figure 4-10. The differential transformer is composed of a primary coil and two secondary coils spaced on a hollow ceramic cylinder. The movable core moves axially within the cylinder in response to a mechanical input. When the movable core is in the center or null position, the voltage induced in the secondary coils will be equal.

=^|~~~~~~~~~~~~~~~~~~~~~~~~~...: |E i':~:':: ~d 00S:iA:::! 1~~~~~~~~~~ J I I' _ 7-; (.;. Ii-~~~~~~~~~ i:::!:l I _ ~~~~~~iFigure 4-8 View of Some of the Instruments. Figurge 4~.9 View of Some of the Instruments.

SCHAEVITZ LINEAR DIFFERETIAL MOVABLE CORE TRANSFORMER MODEL 100-AS -L I I I0 KILOCYCLES I I — e HEWLETT- PACKARD DUMONT MODEL 304 -H MODEL 200 CD OSCILLOSCOPE AUDIO OSCILLATOR BRASS ROD ATTACHED TO TEST PLATE Figure 4-10. Linear Differential Transformer Circuit for Measurement of Amplitude of Vibration.

-158If the core is moved away from null position a net AC output voltage will be produced with the amplitude proportional to the displacement from null. For the differential transformer it is known that the output voltage is a highly linear function of core displacement. This was experimentally verified. A Hewlett-Packard Model 200 CD Audio Oscillator provided 10 Kilocycles carrier frequency for input to the transformer, The output was displayed on a Dumont Type 304-H Oscilloscope. The linearity of this oscilloscope was determined by using a General Radio type 1800-A electron-tube voltmeter and variac. To measure amplitude it is necessary to find the null position by adjusting the position of the transformer. This would be seen as a thin line on the oscilloscope. For a specific amplitude a feeler gage of this thickness is used with the fine adjustment to displace the transformer and thereby cuase a blurred signal in the form of a rectangle on the oscilloscope screen, as shown in Figure 4-11. In this way a specific amplitude corresponds to apredetermined displacement on the oscilloscope. During vibration the output voltage from the transformer is varied so that the modulation envelope accurately portrays the waveform frequency and amplitude of the applied vibration. The carrier frequency in all cases was maintained at least 50 times the highest frequency measured. When motion of the core occurs equally on both sides of null position of the transformer the modulated output voltage appears as the envelope of a sine. This can be seen in Figure 4-12. The amplitude of vibration is the calibrated distance on the oscilloscope screen. The other method for determining amplitude was to accurately measure two converging lines

-159Figure 4-12,, Output Signal of the Linear Differential Transf ormer Under Vibratory Conditions as Displayed on an Oscilloscope. Figure 4-11. Output Signal of the Linear Differential Transformer with the Core Displaced from -the Null Posi~tion as Displayed on an n I I gj! -...l.f.}!~i.!!!!!:i.,,.;i..,o,,,; i!ii~,i i:::: ~~~i C Figure 4-12o Output Signal of the Linear Differential Transformer Under Vibratory Conditions as Displayed on an Oscilloscope....... E. ~:, W..i~iii!:iiii~:it i!11i ff.??000?!!!?:- ~'ii'S S:iii i {r00: N''.? i~t i A:. t Ci'Sii:?. 7!;r,~i ~,~ i~!.!,', ii!i:fi!!!!i!! 0:: W:i Figure 4-11. Output Signal of the Linear Differential Transformer with the Core Displaced from the Null Position as Displayed On an Oscilloscope.

shown.in Figure 4-5 on the side plate and during vibration the image of where the lines crossed indicated amplitude. One of the heat losses in this experimental apparatus is that of radiation. To evaluate the emissivity of the polished aluminum plate in order to find the heat loss by radiation a type 810C69 Cenco Thermopile was used. A black body constructed from 1/8 inch thick copper plate formed into a cone of square cross-sections, was used to calibrate the thermopile. Soldered onto this were additional copper plates 1/4 of an inch thick. The open end of the black body was 6 inches square and the included angle was approximately 12 degrees. The inside of this cone was blackened with carbon deposit from an acetylene torch, This type of geometry at the opening simulates a black body at the inside temperature. The cone was insulated with a layer of asbestos paper. High resistance wire was wound around the cone on this paper and was connected to a 110 VAC variac for heater control. Heavy fiberglas surrounded the cone and its heating element, A Leeds and Northrup Model 8662 portable Precision Potentiometer was used to read the indicated output voltage of the thermopile. A Honeywell Visicorder Model 1012 was used to find the minimum response time of the thermopile to steady-state values. The black body can be seen in Figure 4-1, By experimentation the maximum distance was found where the thermopile could be placed away from the black body and. the aluminum plate. This distance occurred when the solid angle viewed by the thermopile was entirely subtended by the radiating surface of the plate.

-161A maximum distance was determined so that the possibility of convection currents entering the thermopile was elminated. To determine emissivity the indicated output voltages of the thermopile for that of the plate divided by that of the black body at the same temperatures were evaluated. The emissivity was found to be essentially constant, 0.065 for a range of temperatures from 800 to 210~F. The following procedure was used during the course of the experiments. A specified voltage and amperes were supplied to the test section by adjusting the variac. A period of usually three or four hours was required to establish steady-state, non-oscillating conditionso After steady-state conditions were attained the desired frequency was set on the control unit. Next, the differential transformer was calibrated and the test section was vibrated at the desired amplitude. During a period of from one to two hours the variacs were adjusted periodically until a steady-state oscillatory condition was obtained. This condition was determined to be steady-state when the temperature of the thermocouples in the plates were constant and equal and the differential thermocouples of the guard heaters and the plates were null. At that time voltage, ampere, four temperatures of the support rods, temperatures of the plate and ambient temperature were recorded. Periodically during operation a Model 11-A Televiso Vibration Meter with a manual vibration pickup was used to check the uniformity of vibration of the test plate. The smoke studies made use of a smoke generator. A schematic diagram of the smoke generator is shown in Figure 4-13. The smoke from

-162LOW PRESSURE COMPRESSED AIR REGULATING VALVE OIL SOAKED CIGARS TO INJECTORS STEEL WOOL F TO EXHAUST ICE BATH WAT ER Fl LT ER SMOKE GENERATOR -163

-163 - cigars impregnated with oil was used as the indicating medium. The smoke was generated by blowing compressed air down over two cigars in a steel tube. Each cigar was loaded with approximately 6 cc of light lubrication oil. The smoke from the cigars was then bubbled through about 2 inches of water in the bottom of a 1000 cc Erleyemeyer Flask. This flask was in an ice bath in order to settle out some of the unburned oil and tar from the cigarso The smoke then filtered out through a steel wool filter to go to the injecting nozzles located on the test section, as shown in Figure 4-14. The injectors consisted of ten small nozzles of approximately O.C32inches diameter, attached to the plate by clamps. The edges of the nozzles were adjusted to coincide with the beginning of the thermal boundary layer. The smoke was adjusted so that its velocity was as low as possible and yet great enough so that the smoke filaments could be visualized. The experimental procedure consisted of permitting the plate to achieve different temperatures for different power inputs. Once the plate temperature had stabilized the smoke generator was started. An optimum smoke velocity was found after ignition of the cigars by adjusting the compressed air and the various valves, to make the smoke visual to an observer looking transversely at the injectors through a sight glass which was inserted into one of the side plates. An observer, by sitting next to the test section and looking through the sight glass, observed changes in the flow as frequency and amplitude were varied. This was done to observe the transitional phenomena. Visualization was made possible by emnploying a General Radio Type 631- B Strobotac.

- i~~~~~~~~~~~~~~~~iD -~~I l~ l! Figure 4-!4. View of t- t':"'~+l~t nceto s. ~,:i~oeinjions.:

-165The strobotac, when its frequency matched that of the plate, froze the transverse motion. When the strobotac was not employed only a fuzzy image was observed due to double images. Discussion of Results As mentioned previously the undesired losses in the system are by conduction and radiation. The method used for correlating the data is by accounting for four heat transfer quantities, natural convection, radiation, conduction, and the heat transferred due to the vibration. For steady-state conditions and no vibrations this would be #To 6?Co t Lo (4-1) and for steady-state conditions under the effect of vibration this would be 9r = +c~LL)~V tl(4-2) Non-vibrational conditions have a temperature difference which is called ago and the temperature difference under vibratory condition is SQ. The emissivity of the aluminum plates was 0~065 for the range of experimental temperatures used. The heat lost by the plates due to radiation is =6 TA Csp- 7efi) (4-3) For purposes of convenience in reducing the data a coefficient of heat transfer is defined as

xand=hR Go (4-4) and therefore } ~ -_ ( E-f )7...-_ —- (4-5) For conduction losses of the support rods attention is focused on Figure 4-15o Points 1 and 2 are where thermocouples are located on the support rods. This can be considered to be a- fin problem with known boundary conditions at two ends. The first law of thermodynamics yields dCa aft ~ 0 m (4-6) where -R =,'"'-' P(4-7) The heat lost at the base of one of the support rods is found by the first derivative of the temperature distribution as (ci}k) -/ T z ~fD /)(4-8) =Y~04 ( /XR For convenience in reducing experimental data a heat transfer coefficient is defined as whe fAor A (4-9) where for all rods, 2rD ~/h^sk,0-/g( r),C 58 L - (^W)o |,, Ax

h 2 BTU ESTIMATED FROM McADAM'S HR.F.F EQUATION (7-7a) REF HEAT ED SECTION 1 2 _ _ __ _ __ _ __ _ __,140 Dia. ______________ * - STAINLESS STEEL 10 ROD Xr Kr= 10 BTU HR. FT.6F *(DIMENSIONS IN INCHES) Figure 4-15. Sketch Showing One of the Four Rods Attached to the Corners of the Heated Section.

An assumption made here is that hs is the same under vibratory and nonvibratory conditions. There is no potential flow along the length of the rod and therefore there would be no appreciable change in the timeaveraged heat transfer coefficient. A cylinder vibrating along its axis will not have a pressure distribution due to potential flow. This type of problem would be similar to that in Chapter III where no significant change in heat transfer was found. Therefore, hL, = 82? /k (4-11) and, for steady-state ii, -1C hi.'(4-12) A steady-state correlation was experimentally determined, as h>,, -I y y()a. "-14 (4-13) Schoenhals (A-34) found for this case and experimental apparatus a value of 0~4 instead of 0.385. The value of 0.385 is slightly higher than Equation (7-4B) of McAdams, (C-5). A plot of this correlation and the experimental points are shown in Figure 4-16. The power input is the same for vibratory and non-vibratory conditions, therefore (ho L) = h L + v (4-14) realizing (4-15)

5.0 4.0 3,.0 2.0 LL 0.:1.0 4- ~ ~ ~ ~ ~ ~ ~ ~ ~0.9:~ 0.8 ~:": 0.7o c. hc0.385 (8) I/4, 00. o.~ 0. ---- EQ. 7-4b, REF.(C-5) 0.4 0.3 0,2 3 4 5 6 7 8 I0 20 30 40 50 60 80 I00 200 300 400 A e(OF) Figure 4-16. Steady-State (Non-Vibratory) Heat Transfer Correlation.

-170so hC L_ 1i X (__ no (4-16) The ratio h,/hc was found to have a value approximately constant at 0.11. The ratio h:o/hc was approximately equal to 1. The advantage of reducing the data in this way is that it is dependent primarily on temperature differences only. Temperature can be measured accurately and temperature difference is a very convenient way of determining the heat transfer coefficient quickly and accurately. Schoenhk:ls, (A-34) made hot wire measurements of transverse vibration of the test section used in this experiment for adiabatic conditions. His potential flow measurements are reproduced in Figures 4-17 and 4-18. For the potential flow measurements fair agreement was found between theory and experiment, as can be seen. The discrepancies may be, due to four possible causes. First, theory assumes an infinitely thin plate which is the degenerate case of an ellipse while the plate is actually 3/4 of an inch thick. Second, viscous effects are ignored at the edges. Third, the assumption of two dimensional flow may not hold at the edges, and fourth, the probe and mounting structure have some effect on the flow. It is believed that in a vibrating plate such as this there probably exists small vortices which may influence hot wire readings. The experimental values tend to be lower than that of theory for higher values of X/C. But if vortices were large it would seem impossible to get the results that Schoenhals (A-34) obtained. Standing vortices may exist at the edge of the plate but apparently

2.4 2.2 2.0 C0 EXPERIMENTAL Uo: X/C 1.8 1.6 ( 0o =.050 INCHES f =15 CYCLES/SECOND 0.9 1.4.2 X ~ H/C Figure 4-17. Experimental Potential Flow Velocity Measurements and Comrparison with Theory from Schoenhals (A-34). 1.0...-.8.....6 -.4...2~~~~~~. 0.I.2.$.4.5 ~~~.6.7.8.9 x/C Figure ~4-1l7. Experimental Potential Flow Velocity Measurements anI Comparison with Theory from Schoenhals (A-34i).

2.4 2.2 2.0 0 EXPERIMENTAL Uo _ X/C 1.8 ~ ~~Vo'_(/) 1.8 6 1.6 - 00a =.100 INCHES f 30 CYCLES/SECOND 1.4 / 0 o 1.2 1.0.6.4.2 0 =I I I I I I I~~~~~ ~ ~ ~ i I f i 0.I.2.3.4.5.6.7.8.9 X/C Figure 4-18. Experimental Potential Flow Velocity Measurements and Comparison with Theory from Schoenhals (A-34).

-173are insignificant~ The agreement of theory and experiment can be described as good for most values of X/C. The potential flow and its agreement with theory of Schoenhal (A-34) were carefully considered since this potential flow is identical to that used in Chapter IIo Oscillating velocities in the boundary layer were also obtained by Schoenhals using a hot wire anemometer for adiabatic conditions. The experimental values agree very well with that of his theory, as shown in Figures 4-19 and 4-20. The l1 in these figures is dimensionless, l =Xy —. The lu]/vo, the ordinate of Figures 4-19 and 4-20 is the absolute magnitude of the periodic velocity for an adiabatic finite plate. The significance is that the oscillating potential flow and the oscillating boundary-layer flow of Schoenhals experiments agree with his theory for an oscillating adiabatic finite plate. Also of significance is that this potential flow is identical to the potential flow of the case treated here, Chapter II, of an oscillating finite plate with heat transfero It is well known that one of the reasons for the neglect of experimental free convection boundary layer transitional data is that it is virtually impossible to probe and measure the flow for various quantities of interest. Schoenhals (A-34) was able to probe the oscillating velocity and oscillating potential flow as related previously, but this was without heat flow. The analysis of Chapter II is directed to a problem in which the main consideration is free convection heat transfer with first and second order perturbations of velocity and temperature

1.2 1.0.8, ) 0 EXPERIMENTAL f' ( 71.6 0 =.100 INCHES f =20 CYCLES/SECOND X/C =.55.4 0 2 3 4 5 6 7 Figure 4-19. Experimental Velocity Measurements Made in the Boundary Layer on an Adiabatic Plate Vibrating Transversely and the Comparisor with Theory from Schoenhals (A-34).

12 1.0.8 0 EXPERIMENTAL F.6 3ao =.075 INCHES — _,__f = 20 CYCLES/SECOND X/C =.55'J.4.2 0 I 2 3 4 5 6 7 Figure 4-20. Experimental Velocity Measurements Made in the Boundary Layer on an Adiabatic Plate Vibrating Transversely and the Comparison with Theory from Schoenhals (A-34).

-176having been determined. Neglecting the free convection velocities and temperatures for the moment, the problem of trying to measure an oscillating temperature and an oscillating velocity with a hot wire anemometer will be examined. A hot-wire depends upon the heat loss between itself and the environment. The rate of heat loss depends upon geometry, flow conditions and physical properties. Schoenhals (A-34) was- able to make measurements because he was dealing with an isothermal system which allowed calibrations to be made. In this case there are both varying and steady velocities and temperatures. The presence of a hot-wire vibrating back and forth in its own path in the boundary layer may very well affect the results. For these reasons a hot-wire measurement was not attempted. Two possible methods left to investigate the boundary layer are smoke and an interferometer. The heat transfer results are plotted as Nusselt number versus a vibrational Reynolds number with the Grashof-Prandtl number product as a parameter. These results are given in Figure 4-21. A critical vibration Reynolds number was observed. Up to a certain critical value of this Reynolds number the data follow the theoretical results of Chapter II, i.e., an almost negligible influence but a definite decrease in the heat transfer rate with vibration. With the limits of the perturbation theory used, the theory is confirmed by these results. The conclusion, therefore, from analytical and experimental results is that the effect of oscillations on laminar free convection boundary layer heat transfer is negligible if the boundary layer remains laminar.

-177However, from an inspection of the heat transfer data in Figure 4-21 it is apparent that vibrations and flow oscillations do cause a significant increase in the steady rate of heat transfer. Since the theory assuming laminar flow conditions does not disclose this behavior and, in fact, even predicts a decrease in the heat transfer rate, it is evident that a completely new influence is being introduced by the oscillating flow, It seems probable that this new influence is in the form of a disturbance in the flow which causes a transition from a laminar to a turbulent condition in the boundary layer. That such is likely the case is reasonable from the standpoint of the stability of boundary layer flow where it is known that disturbances can be amplified, in an existing laminar boundary causing it to become turbulent, Furthermore, it can be expected that such a condition of transition would depend on the mechanics of the laminar force convection boundary layer- namely, its state of stability as indicated by the Grashof number or Grashof-Prandtl number product in a manner similar to that describing the stability of a forced convection laminar boundary layer in terms of ir's Reynolds number, This threshold condition was found for the transition of the free convection laminar boundary layer t-o a turbulent boundary layer and is described in terms of a critical vibratory Reynolds number dependent on the Grashof-Prandtl number of the laminar boundary layer, The experimental results of this transition phenomenon are given in Figures 4 )22A 4 -23 and 4) oi_ The transition region is very difficult to ascerta-in from heat transfer data alone. At this point in

~621 iGr Pr x16 58 0 14.9 013~~~~ 1 2 ~TURBULENT E] 12 54 x 5.5 REGION a4~ —+ —1X 5.5 REFERS TO FIGURE No.4-26 50 ~-' FOR 10.9 50 -- AYSMPTOTIC VALUE (W-O) 46 = 42 3 8 X 4:'.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-0, / -... -e -—.. _x H _~~~~~ ~ ~ _ ___/__1_ cx 38~ -1 __ __ El 0I ~ 7 (26) ~ I __ X 34 x 30 26 —.- - -- 4 5 7 022 3 4 5 7 103 3 4 5 7 04 Re VIBRATIONAL Figure 4-21. Experimental Time-Independent Heat Transfer Data for a Heated Plate Oscillating Transversely.

-17916 REFERS TO FIGURE NO.4-25 15 Is 14..... TURBULENT LAMINAR REGION REGION 13 12 (4~~~0~ ~\0 C) go G 0 III0 Z W =81III~ I 0' /~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~',. / - o wo ~~= ~oRev 9 9LS / 3~-~ ZI-. 8~~~~~~~~~ 0 ~ ~ 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ IIOC456 Figure 4-22. Experimnental Smoke Study Data for Determining the Condition of Transition.

'uoT;Tsu,I, o uoTq.TpuoD aq q wJ a loUTUp M.1aa(a ~o,(I pnrS am[omS'[equgmaTTacxa'EE-, ris9Ta V aM D D 01 6 8 L 9 S t ~ I 0 L 0,_,. _ _ I T T > ts~~~~~~~~bLI 01'"CDt~~tI *0

15 o o\ 14 |O \.~ ~ 51) REFERS TO FIGURE NO. 4-25 TURBULENT |kLArINAR R~GIREGION 12 6'_ 126 II 0 x x x x x. 10 LAMINAR REGION 7 0 10 20 30 40 50 60 70 80 Re VIBRATIONAL Gr (9/L) Figure 4-24. Experimental Smoke Study Data for Determining the Condi~ton of Transition.

-182the investigation another experimental method of attack was taken. Smoke studies were used to determine the transition region of laminar flow to turbulence. The smoke filaments were observed and when under different values of amplitude and frequency the pattern in the boundary layer changed, the pertinent data were taken. The velocity of the smoke was attempted to be excluded as a factor in this experiment by taking each set of readings at a low smoke velocity. The condition of change in the smoke pattern seems to be the onset of turbulence. This condition occurred for different values of amplitude, frequency and AT (Grashof number). The data correlate with GrPr product versus aw2 not as ax as appears to be indicated from the heat transfer data. This can be interpreted to mean that the instability of a laminar free convection boundary layer is dependent upon, for mechanically induced vibrations, an acceleration of the boundary layer or a disturbing force per poundmass of boundary layer fluid, Considering again the governing differential Equation (2-27) for x-momentum of Chapter II and again disregarding thermal variations of density as being negligible we see that L are the terms due to the effects of oscillation, The first term of the above has the proper terms to cause the transition and most likely causes the transition to occur. The data also show that for higher GrPr products the flow is less stable to the disturbances. This ties in with the well-known experimental fact that the higher the GrPr

-183product the free convection flow is less stable and less disturbance would be required to cause transition. This experimental fact also suggests that if the non-oscillatory free convection flow were initially turbulent due to large GrPr product then oscillation would show an immediate effect. Unfortunately the test apparatus could not produce sufficiently large Grashof-Prandtl numbers for initially turbulent flow without the danger of burning out the electric heaters. Air was the only fluid used so no significant change in Pr was made. The equality of the Grashof number establishes dynamic similarity for different flows. Equality of Grashof and Prandtl numbers establish similarity for velocity and temperature distributions of free convection flow for geometrically similar bodies and therefore the Nusselt number, On this basis it was found desirable to use GrPr product in reporting data. Figure 4-25 through 4-32 are photographs of the smoke studies, These photographs are representative of what was seen without a camera but with the aid of a strobatac to freeze the motion, These pictures were taken through the sight window in the side plate, shown in Figure (4-5). Two spotlights were focused on the smoke, Onelight was focused at a small angle from the line of sight of the camera, The other light was placed above the smoke at a slight angle to a line parallel to the smoke injectors pointing toward the camera, This lighting arrangement was found to yield the best pictures, A 4 x 5 Crown Graphic Graflex Optor camera was used with a setting of 1/400 second at f/4,7, The film was Polaroid Polapan 200 type 52,

Figure 4-25. Smoke Photograph for a = 0, o = 0, Figure 4.-26. Smoke Photograph for a = 0.002 in., GrPr = i.o8 x i07, q/A = 92 BTU/hr ft2, )= 50cycle/sec. GrPr = 1.08 x &D2/g = 0, R 2V/Gr(a/L) = 0. w2/g = 0.51, q/A = 92 BTU/hr ft2, Re2/Gr (a/L) = 15.7

Figure 4-27. Smoke?hotograph for a 0.01 in., Figure 4-28. Smoke Photo raph for a 0,06 in., M = 50cycle/sec, Grr 1.08 107, a) = E50cyce/ see. GrPr =1.C8 X aw2/g = 2.54, q/A = 92 BTJ/hr ft2, aw2g = 4.09, q/A = 92 BTU/hr ft2, Re2/Gr\a/L) = 18.4 Re2r/Gr(a/L) = 29.5

Figure 4-29. Smoke Photograph for a = 0.022 in., Figure 4-30. Smoke Photograph for a = 0.03 in,, " = 50cycle/sec., GrPr =1.08 x 107, = 50cycle/sec.,G = 1.08 x 107,aa2/g = 5.62, /7.6A = 92 BTU/hr ft2= R 2'Grv a/L= 4o7 R 2 /Gr(a/L) = 5540.7 7 eV~~TM5.

Figure 4-31. Smoke Photo raph for a 0.068 in., Figure 4-32. Smoke Photograph for a = 0.05 in. = 50cycle/sec., GrPr = 1.08 x 107, - 25 cycle/sec., GrPr =1.08 x 107 a82/g = 17.4, q/A = 92 BTU/hr ft2, aw2/g = 1.92, q/A = 92 BTU/hr ft2, Re2v/Gr(a/L) = 126. Re2v/Gr(a/L) = 13.9

These photographs for different values of the pertinent parameters, listed on the figures, have also been related to Figures 4-21, 4-22, and 4-24. The conditions corresponding to the photographs are shown on these figures. The photographs) correlation data and this discussion attempt to disclose the source and mechanics of propagation of the vibratory disturbance. The significance of these results is that they definitely show the gross effect of a transitional phenomenon. The method for obtaining data for Figures 4-22, 4-23 and 4-24 was to vary both amplitude and frequency for a given heater input to the plate. The inner smoke filaments were observed closely. When the inner filaments, approximately in the boundary layer, began to mix with each other. This was considered to be the start of transition. It was observed from the photographs that a disturbance originated outside the boundary layer in the potential flow region and propagated in toward the plate causing transition in the boundary layer. Also it was observed that this transition process depended on the Grashof-Prandtl number which is a rr;e:e i; iKJe stability of the free convection flow. The critical condition was always found to.rigiAnXtt.at Jhvs ttop tJ. the plate where the flow stability may be expected to be less. Figures 4-26 through 4-31 are not to be interpreted as streamlines, pathlines or streak lines. What is seen in these photographs is a rather complicated phenomenon. In these photographs the smoke has an upward velocity which is superimposed on the velocity of the fluid. Outside the boundary-layer the effects of the potential flow can be seen.

-189The potential flow is oscillating with time at different angles, depending on location, with respect to the velocity of the flow. A smoke particle outside the boundary layer moving upward, due to its initial velocity is subjected to accelerations and decelerations from the potential flow thereby causing the smoke patterns. The flow in the boundary layer is predominately oscillating up and down and causes patterns in the boundary-layer to be different from those outside of the boundary-layer in the potential flow. This difference can be seen in the photographs. The flow was considered to be laminar as long as the smoke pattern in the boundary-layer was regular. When the pattern in the boundary-layer became irregular and smoke filaments mixed with each other, this condition was considered to be the critical condition. As the photographs indicate the disturbance originated outside the bounary-layer and propagated inwards causing a transional phenomenon to occur,. The laminar boundary layer can be distinguished in Figures 4-25, 4-26, 4-27, 4-28 and 4-32 because the patterns are still visible. Figure 4-29 represents what can be described as the beginning of this critical condition. The finer filaments of smoke are seen to start mixing with each other and the smoke outside the boundary-layer is irregular. Figures 4-30 and 4-31 are definitely in the turbulent region. Figure 4-30 indicates a large degree of turbulence outside the boundary-layer. The inner smoke filaments appear very turbulent. Figure 4-31 represents a high degree of turbulence and indicates apparently a churning action.

-190In the experimental work performed in this study a critical condition of transition was found. A correlation was experimentally found to determine the transition region from a laminar flow to what appears to be a turbulent flow. This condition was determinedin two ways, first, by being indicated in the heat transfer experimental results and secondly, by the smoke studies, The transition region was found to be a function of the stability of the free convection boundary layer and a disturbance which propagates inward from the potential flow. It is of value at this point in this study to investigate some other experimental results from the literature. Discussion of Other Experimental Results Kestin, Maeder and Sogin, reference (A-44) discussed the influence of turbulence on the heat transfer to cylinders near the stagnation point. They observed from experiment that when the boundary layer is laminar near the forward stagnation point there are large increases in heat transfer rate when the free stream velocity is large and when there exists a large downstream gradient in the amplitude of oscillation. A net decrease in the heat transfer rate for this type of oscillating system was predict by Kestin, Maeder and Wang, reference (A-17). In view of these conflicting results it is apparent that their theoretical model imperfectly represents the physical system, Two possibilities are suggested. First, the system of equations solved (A-17) does not properly represent the physical phenomena in the stagnation region in that normal-direction momentum effects are ignored, Second, turbulence may be encountered in the boundary layer which is not accounted for in the theoretical description.

Shine (A-35) used an Zehnder-Mach interferometer to study the boundary layer on a vertical, heated plate as it vibrates. His results showed that as the product of amplitude times frequency called, intensity of vibration, is increased, the heat transfer coefficient remains unchanged until a critical intensity of 0.52 ft/sec is reached, He found that coincident with the occurrence of a change in heat transfer coefficient there was an inception of waviness in the boundary layer. Also he found that this waviness was intensified by increases in vibration intensity, He pointed out that this waviness signifies a flow transition away from the laminar flow and presented this as a mechanism to explain the change in heat transfer rate. He presented some of his data as the ratio of the temperature differences between the plate and the ambient air for the vibrating conditions and a stationary plate condition. This is shown in Figure 4-33. In Figure 4-34 taken from Shine (A-35), are shown some typical photographs of waves in the fringes, The waviness was found to be increased regardless whether frequency or amplitude was increased as long as the intensity of vibration exceeded the critical point, The top two photographs have an intensity of 0 and.88 respectively. The remaining four interferograms have an intensity greater than critical and a waviness can easily be seen, Also the degree of waviness can be seen as the intensity is increased. In this way Shine predicted the critical intensity value, Shine did not take into account the conduction and radiation losses of the system, Also the test sections used were flat plates without restrictions to prevent three dimensional effects, This is different, of course, from the two dimensional case considered in the

-1921.0~ 0 1.2 0.6 YI AMPLITUDE.9 WHEN f FREQUENCY.5 STATIONARY er-Tw-T WHEN 0.4' l _ _ _ _ T - o 0.4 Y, f Figure 4-33. Experimental Heat Transfer Data Taken From Shine (A-35)

-194experiment. Another difference is that his plate was completely heated while the test apparatus of this study was heated only in the middle, Despite the differences these results support the findings of the previous section in that a critical condition occurs. Some other pertinent experimental cases are those of round cylinders undergoing both mechanical and acoustical vibration. Both Westervelt (B-25) and Lighthill (A-20) pointed out that the physical mechanism of interaction between free-convection from a heated horizontal cylinder and horizontal vibration is essentially the same, whether the vibrations are acoustically or mechanically induced. This was demonstrated by Fand and Peebles (A-9). They compared their findings, which treated horizontal vibration, to that of Fand and Kaye (B-4) which treated the influence of sound. Their results can be seen in Figure 4-35. They found a critical intensity to be approximately 0.36 ft/sec, ascertained from the heat transfer data. They tried smoke and photographic procedures, but found the photographs not to be of publishable quality. As can be seen in their results, Figure 4-35, at 0.-7 ft/sec the data makes a bend. Above this value Fand and Kaye have made the following empirical correlation equation. h - O 72 2 L AT ( ) 2 F (4-17) -- 69, 7;?, 2(4-17) The factor F is a geometrical weighing factor defined in reference (B-4). They found that a flow visualization study indicated the observed increases in the heat transfer coefficient to be that of vibrationally induced

I0 8 - HORIZONTAL CYLINDER SUBJECTED TO HORIZONTAL ACOUSTICAL 7t.8r r Don VIBRATIONS: Do = 3/4",f=1496cps 6 (Ref. B-4) DIRECTION OF ()HORIZONTAL CYLINDER SUBJECTED VIBRATION TO HORIZONTAL MECHANICAL VIBRATIONS, (Ref. A-9) 0 F At = 150~F 504F of, f t /sec Figure 4-35. Comparison of Heat Transfer Data at Constant Difference in Temperature. For a Horizontal Heated Cylinder Subjected to Mechanical and Acoustical Vibrations in Air Taken from Fand and Peebles (A-9).

-196turbulence. They pointed out that this turbulent type of boundarylayer flow differed radically from the vortex type of flow which develops near a horizontal cylinder in the presence of acoustically induced transverse vibrations. This vortex type of flow has been called thermoacoustic streaming. Two photographs of the acoustic streaming, from Fand and Kaye (B-4), are shown in Figures 4-36. These photographs were taken by employing smoke as the indicating medium. The smoke studies in this present work appear not to be that of a vortex type of flow. The results of Martinelli and Boelter (A-22) are shown in Figure 4-37 for vertical vibrations of a horizontal cylinder in water. As can be seen in their results, the heat transfer coefficient experienced a critical condition. It should be noted however, that Boelter reported that later measurements did not agree with the original measurements. It is appropriate at this point in the discussion to mention briefly some work concerning boundary layer oscillations because of their relevance to the problem of transition from laminar to turbulent flowo Eckert and Soehngen (A-7) conducted an experimental study of the stability of laminar flow in free convection by using interference photographs. They found that the instability region, the region where small oscillations do not damp out, began at a Grashof number of 4 x 108. Also they found that the velocity of the disturbance was approximately 73 per cent of the maximum flow velocity in the boundary layer at the

o 00~~Z = V O: i,:::.'?', - -L 6tg i~~ UG31-e 9U.lea| T'~nOT0q LE| oo 0~~~~L1

-198100 1 Re o- ~ o0 o o10500 o ON, 7630 _0-0 0o ~ 7080 -0omo 0Ido-)o-' ~ 6690 =Z 0 0 ooso- 4500 3380 Mo-m 302730 ( Nu)o = 0.50 (Gr Pr)'1/4 1i06 IO?7 GrPr Figure 4-37, Experimental Heat Transfer Data of Martinelli and Boelter (A-22) for Vertical Vibrations of a Horizontal Cylinder in Water.

-199point of observation which was in the range of the value of the velocity at the point of inflection which is o.683 Umax. They concluded that the boundary layer filtered out fluctuations of certain critical frequencies and amplified these until turbulent waves were produced. Birch (A-2) studied critical boundary layer frequency in more detail by introducing controlled disturbances into the flow with an electrically pulsed wire. Byutilizing an interferometer he determined wave lengths and amplitudes at the outer edges of the thermal boundary layer. He found that for certain disturbance frequencies higher amplitude oscillations were produced than for other disturbance frequencies. Because of this the conclusion was that the boundary layer absorbed energy more readily at these frequencies. Birch (A-2) made the following correlations e;= /. as (Gr98)o ~8 0(41 8) and e — 8 ( o s~ (4-19) where f is the disturbance frequency expressed in cycles per second and X is the wave length expressed in inches. Since these frequencies and wave lengths represented some sort of resonance phenomena they were called "natural"values. It should be noted that f is not the frequency of the boundary layer oscillation but rather the pulse frequency. The boundary layer may vary well and does oscillate at a frequency different from that of the pulse frequency.

-200Holman, J. P., Gartrell, H. E., and Soehngen (B-7) investigated the physical processes involved in the propagation of oscillations in free convection boundary layers. A Zehnder Mach interferometer and motion picture camera were used to witness various boundary layer phenomena on a vertical flat isothermal plate. Disturbances were created in the boundary layer by introducing a fine wire into the boundary layer at various distances from the heated surface. The wire was pulsed with a square-wave electrical signal which heated it periodically and created pulses in the boundary layer. The pulse energy was, in general, of the order of one percent of the boundary layer energy. The pulse energy which is dissipated from the small heated wire creates a decrease in the air density of a small cylindrical volume around the wire so that through buoyancy forces an additional local velocity component is superimposed on the free convection profile. As this energy pulse moves up its energy is diffused throughout the boundary layer until it has finally imposed a velocity pulse on the whole field due to the transient heating process. This pulsing process is repeated periodically and waves advance up the plate in a complicated fashion. These waves become unstable and break up into more waves at higher frequencies. This instability occurs in a region of the plate which is characterized by a certain local Grashof number. These authors also measured wave length, wave speed, and wave amplitude in plane free convection as functions of the distances from the leading edge and plate surface. Use was made of a wave energy parameter. The wave energy flux was written as Boundary layer wave energy -x /- r. (4-20) 7

-201where a is the amplitude of the wave and wI is the boundary layer frequency. They chose a wave energy parameter which represents the wave energy flux to be Av 3o4a (4-21) since the absolute temperature varied only by eight percent. It was found that wave energy was zero at the wall and increased towards the outer layers of the boundary as the boundary layer became progressively thicker. It was postulated that when the wave energy in the outer portion of the boundary layer is large compared to that of the region near the wall then transition turbulence begins. Experiments furnished some confirmation of this postulate. It was also found that there is disturbance frequency which will produce a maximum wave amplitude. The above papers were investigated because of the existence of a critical condition for laminar free convection flow. Since the Grashof number is a type of Reynolds number and since the Prandtl number affects the temperature field and thereby the Grashof number it was logical to suspect that the critical condition was dependent in some way on both of these numbers. Physically, the stability of a laminar free convection flow decreases for increasing Grashof number. If the Grashof number is large enough, of course, the flow is turbulent and small oscillations or disturbances would immediately cause an increase in the heat transfer coefficient. Therefore it was postulated that for a constant Gr Pr product the critical intensity is constant and for increasing values of Gr Pr the critical intensity decreases and approaches zero when Gr Pr is

-202large enough for the flow to be turbulent to begin with without disturbance or oscillations. The data of Peebles and Fand (A-9) and Fand and Kaye (A-8), in Figure 4-35, seem to show this effect.

CHAPTER V SUMMARY OF RESULTS In Chapter II the system of partial differential equations governing fluid motion and heat transfer were solved by a perturbation technique for the case of a finite vertical flat plate in free convection oscillating normal to its own plane. The behavior of the boundary layer due to oscillations of the finite plate were investigated. Two influences were found which alter the boundary layer flow. These influences are first, the potential flow outside the boundary layer and second, the existence of an inertial force within the boundary layer. The influence of the potential flow was found to predominate over the influence of the inertial force. The problem treated is for large Grashof numbers (boundary layer type of flow) subjected to the perturbing effect of potential flow for low vibrational amplitude and high frequency. The perturbation technique was used up to the second order solution in order to show time-independent effects. First order, periodic, and second order, time-independent, solutions were found for velocity, temperature, shear stress and Nusselt number, Calculated results are given for Prandtl numbers of 0.72 and 10. The Nusselt number was found to be decreased slightly for both Prandtl numbers, This theoretical result was confirmed in the experimental findings of Chapter IVo The work of Reference (A-33) was extended in Chapter III to show time-independent effects resulting from transverse oscillations of an infinite, vertical plate in free convection, The alteration of the boundary layer is due to an oscillating pressure gradient which -205

-204arises from a coupling of the thermal variations of density with the acceleration (inertial force) of the oscillatory motion. A perturbation technique due to Reference (A-33) was used and extended by finding the second order solutions. A different method from that of Reference (A-33) was used in solving the first order solution in order to facilitate the second order solutions. Both first order solutions were found to agree within their range of applicability. First order and second order solution were found for velocity, temperature, shear stress and Nusselt number. Calculated results are given for Prandtl numbers of 0.72 and 10. The time-independent Nusselt number was found to decrease while the shear stress was found to increase under the influence of oscillation. In both Chapters II and III it was shown that vibrations and flow oscillations are capable of causing permanent (time-independent) alterations in both the velocity and temperature profiles in the laminar boundary layer of systems having pressure gradients in the flow direction. This provides a secondary, time-independent flow, and permanently alters the wall shear stress and heat transfer rate. These changes are small and are detected from the analysis only when solutions are obtained to at least the second order approximation beyond the solution for the steady free convection problem. Furthermore, while the alteration in the wall shear stress is such as to increase its value, the opposite effect is both predicted and observed for the heat transfer rate under laminar flow conditions. This emphasizes an important character of this new class of phenomena, namely, that the traditional heat transfer - momentum transfer analogy is no longer a useful guide, at least for oscillating systems in laminar flow over flat surfaces, Similar phenomena associated

-205with curved surfaces, especially at stagnation points, may be quite different owing the necessity to consider a simultaneous influence of momentum effects in a direction normal to the surface. This is suggested by the experiments of Kestin, et al. (A-44). Experimental findings are reported in Chapter IV. The heat transfer rate from a finite plate was found to decrease a small but definite amount due to being oscillated normal to its own plane for small values of a vibrational Reynolds number, This confirms the theoretical results of Chapter II. However, as the Reynolds number increased, sharp increases in the heat transfer rate were found. It would appear that a critical condition such as this represents transition from a laminar to a turbulent flow. Smoke studies indicate this transition is occurring. This critical condition was found for the transition of the free convection laminar boundary layer to a turbulent boundary layer and is described in terms of a critical vibratory Reynolds number dependent on the Grashof-Prandtl number of the laminar boundary layer.

A. UNSTEADY PAPERS 1. Anantanarayanan, R., Ramachandran, A., "Effect of Vibration on Heat Transfer from a Wire to Air in Parallel Flow," ASME Paper No. 57-A-100, July 1957. 2. Birch, W. D., "On the Stability of Free Convection Boundary Layers on a Vertical Flat Plate," M.S. Thesis, Air Force Institute of Technology, Wright-Patterson AFB, Ohio, 1957. 3. Carrier, G. F., DiPrima, R. C., "On the Unsteady Motion of a Viscous Fluid Past a Semi-Infinite Flat Plate," Journal of Mathematics and Physics, January 1957, pp. 359-383. 4. Cheng, S. I., Elliot, D., "The Unsteady Laminar Boundary Layer on a Flat Plate," Proc. Heat Transfer and Fluid Mechanics Institute (1956), Trans. ASME, Vol. 79, (1957), pp. 725-753. 5. Cheng, S. I., "Some Aspects of Unsteady Boundary Layer Flows," Quarterly of Applied Mathematics, (January 1957), pp. 359-383. 6 Chung, P. M., Anderson, A. O., "Unsteady Laminar Free Convection," Journal of Heat Transfer, Series C., No. 4, November 1961. 7. Eckert, E. R. G., and Soehengen, F., "Interferometric Studies on the Stability and Transition to Turbulence of a Free Convection Boundary Layer," Proc. General Discussion on Heat Transfer, Inst. of Mech. Engr., London, 1951. 8. Fand, R. M., Kaye, J., "The Influence of Vertical Vibrations on Heat Transfer by Free Convection from a Horizontal Cylinder," Second International Heat Transfer Conference, Boulder, Colorado, (1961) Paper A-17. 9. Fand, R. M., Peebles, F. M., "A Comparison of the Influence of Mechanical and Acoustical Vibrations on Free Convection from a Horizontal Cylinder," To be published. 10. Goldstein, R. J., Eckert, E. R. G., "The Steady and Transient Free Convection Boundary Layer on a Uniformly Heated Vertical Plate," Int. J. Heat Mass Transfer, Vol. 1, (1960), pp. 208-218. 11. Glauert, M. B., "The Laminar Boundary Layer on Oscillating Plates and Cylinders," Journal Fluid Mech., Vol. 1, (1956), pp. 97-110. 12. Harrje, D. T., Croke, E. J., "Heat Transfer in Oscillating Flow," Princeton Aero. Eng. Rept. #483, October 3, 1959. -206

-20713. Hassan, M. A., "On Unsteady Boundary Layers," Journal of Fluid Mechanics, Vol. 9, Part 2, October 1960. 14. Hellums, J. D., "Finite Difference Computation of Natural Convection Heat Transfer," Ph.D. Thesis) University of Michigan, September 1960. 15. Hill, P. G., Stenning, A. M., "Laminar Boundary Layers in Oscillatory Flow," (59-A-265) Journal of Basic Engineering, Vol. 82, Series D, No. 3, (1960), pp. 593. 16. Karlsson, S. K. F., "An Unsteady Turbulent Boundary Layer," Journal of Fluid Mechanics, Vol. 5, Part 4, (1951), pp. 622-636. 17. Kestin, J., Maeder, P. F., Wang, H. E., "On Boundary Layers Associated with Oscillating Streams," Appl. Sci. Res., Section A., Vol. 10, 1960. 18. Layton, J. P., "Heat Transfer in Oscillating Flow," Princeton Aero Engr. Rept. #266A, May 15, 1956. 19. Lemlich, R., "Effect of Vibration on Natural Convection Heat Transfer," Industrial and Eng. Chem 47, (1955), pp. 1175-1180. 20. Lighthill, M. J., "The Response of Laminar Skin Friction and Heat Transfer to Fluctuations in the Stream Velocity," Proc. Roy. Soc. Ser. A., Vol. 224, (1954), pp. 1-24. 21. Lin, C. C., "Motion in the Boundary Layer with Rapidly Oscillating External Flow," Proc. IX Int. Cong. Appl. Mech. (1956) Vol. 4, pp. 155. 22. Martinelli, R. C., Boelter, L. M. K., "The Effect of Vibration on Heat Transfer by Free Convection from a Horizontal Cylinder," Proc. 5th Int. Cong. of Appl. Mech., (1938), p. 578. 23. Martini, W. R., Carr, J. W., Churchill, S. W., "Machine Computation of Natural Convection Inside a Horizontal Cylinder," Mimeograph Sheets. 24. Mickelson, W. R., and Laurence, J. E., "Measurements and Analysis of Turbulent Flow Containing Periodic Flow Fluctuations," NACA RM E53 F19, 1953. 25. MoDre, F. K., "Unsteady Laminar Boundary-Layer Flow," NACA, TN 2471, 1951. 26. Nickerson, R. J., "The Effect of Free-Stream Oscillations on the Laminar Boundary Layers on a Flat Plate," WADC Tr 57-481, ASTIA Doco No. AD 210477, Wright Air Development Center, December 1958.

-20827. Ostrach, S., "Note on the Aerodynamic Heating of an Oscillating Surface," NACA Tn 3146, 1954. 28. Ostrach, S., "Compressible Laminar Boundary Layer and Heat Transfer for Unsteady Motions of a Flat Plate," NACA TN 3569, 1955. 29. Rosenblat, S., "Torsional Oscillations of a Plane in a Viscous Fluid," Journal of Fluid Mechanics, Vol. t, (1959) pp. 206-222. 30. Rott, N., "Unsteady Viscous Flow in the Vicinity of a Stagnation Point," Quarterly of Applied Math., Vol. 13, No. 4, 1956. 31. Rozin, L. A., "The Growth of a Laminar Boundary Layer on a Flat Plate Set Impulsively into Motion," Jour. of Applied Math. and Mechanics, Trans. of the Soviet Journal, Prilkaoniaia Matematika 1 Mekhanika, Pergamon Press, Vol. 22, (1958), pp. 568-575. 32. Schlichttng, H., "Berechung ebener periodischer Grenzschicht Stroemungen," Phy. Zs. 33., (1932), p. 327. 33. Schoenhals, R. J., Clark, J. A., "The Response of Laminar Incompressible Fluid Flow and Heat Transfer to Transverse Wall Vibrations," ASME Paper No. 61-WA-174, 1961. 34. Schoenhals, R. J., "The Response of Laminar Incompressible Fluid Flow and Heat Transfer to Transverse Wall Vibrations," Ph.D. Thesis, University of Michigan, January 1961. 35. Shine, A. J., "The Effect of Transverse Vibrations on the Heat Transfer Rate from a Heated Vertical Plate," M.S. Thesis, Air Inst. of Tech., Wright-Patterson Air Force Base, Ohio, (1957), ASME Preprint 59-HT-27. 36. Siegel, R., "Transient Free Convection from a Vertical Flat Plate," ASME Paper No. 57-SA-8, 1957. 37. Stuart, J. T., "Solutions of the N. S. and Energy Equations Illustrating the Response of Skin Friction and Temperature of an Infinite Plate Thermometer to Fluctuations in the Stress Velocity," Proc. Royal Soc., Vol. 231, Series A, (1955), p. 116. 38. Teleki, C., "The Influence of Vertical Vibrations on the Rate of Heat Transfer from a Horizontal Cylinder in Air," WADC Technical Note 59-357, Aeronautical Research Lab., Oct. 1960. 39. Tennekes, H., "Another Unsteady Turbulent Boundary Layer," John Hopkins Univ. Contract AF 49(638) - 496, May 1959.

-20940. Tsui, Y. T., "The Effect of Vibrations on Heat Transfer Coefficients," Ph.D. Thesis, Ohio State University, 1952. 41. West, F. B., and Taylor, A. T., "The Effect of Pulsations on Heat Transfer," Chemo Eng. Progress, Vol. 48, 1953. 42. Wu, T. Y., "Small Perturbations in the Unsteady Flow of a Compressible Viscous and Heat-Conducting Fluid.," Journ. of Math. and Physics, (January 1957), pp. 13-27. 43. Wang, K. T., "Unsteady Laminar Boundary Layers in an Incompressible Stagnation Flow," Jour. of Basic Engineering, Vol. 25, No. 4, (1958), p. 421. 44. Kestin, J., Maeder, P. F., Sogin, H. H., "The Influence of Turbulence on the Transfer of Heat to Cylinders Near the Stagnation Point," Jour. of Applied Math. and Physics (ZAMP) Vol. XII, Fasc. 2, (1961), pp. 115-132.

-210B. RELATED PAPERS FROM THE FIELD OF ACOUSTICS 1. Andres, J. M. and Ingard, U., "Acoustic Steaming at High Reynolds Numbers," Journal Acoust. Soc. Am. 25, 928, 1953. 2. Andres, J. M., and Ingard, U., "Acoustic Steaming at Low Reynolds Numbers," Journal Acoust. Soc. Am. 25, 032, 1953. 3. Eckert, C., "Vortices and Streams Caused by Sound Waves," Phys. Rev. 73, 68, 1958. 4. Fand, R. M., Kaye, J., "The Effect of High Intensity Stationary and Progressive Sound Fields on Free Convection from a Horizontal Cylinder," WADC Tech. Note 59-18, ASTIA Doc. No. AD 209532, March 1959. 5. Holman, J. P., "The Mechanism of Sound Field Effects on Heat Transfer," Jour. of Heat Transfer, Tech. Brief, Vol. 82, Series C., No. 4, (November 1960), p. 393. 6. Holman, J. P., Mott-Smith, T. P., "The Effect of High Constant Pressure Sound Fields on Free Convection Heat Transfer from a Horizontal Cylinder," WADC Technical Note 58-352, ASTIA Doc. No. AD 206906, Wright Air. Dev. Center, December 1958. 7. Holman, J. P., Gartrell, H. E., Soehngen, E. E., "A Study of Free Convection Boundary Layer Oscillations and Their Effects on Heat Transfer," WADD Tech. Rep. 59-3, Wright Air. Dev. Div., December 1959. 8. Holtsmark, J., Johnsen, I., Sekkelanc, T., and Skavlen, S., "Boundary Layer Flow Near a Cylindrical Obstacle in an Oscillating Incompressible Fluid," J. Acoustical Soc. Am. 26, 1954. 9. Hunt, F. V., "Notes on the Exact Equations Governing the. Propagation of Sound in Fluids," J. Acoustic Soc. Am. 27, 1019, 1955. 10. Jackson, T. W., Harrison, W. B., and Boteler, W. C., "Free Convection, Forced Convection, and Acoustic Vibrations in a Constant Temperature Vertical Tube," ASME Paper No. 58-HT-6, September, 1958. 11. Lane, C. A., "Acoustical Streaming in the Vicinity of a Sphere," J. Acoust. Soc. Am. 27, 1082 (1955). 12. Mawardi, O. K., "On the Thermo-Acoustic Transduction in Potential Flow," J. Acoustic Soc. Am., 28, 239, 1956. 13. Mawardi, O. K., "On Acoustic Boundary Layer Heating," J. Acoustic Soc. Am. 26, 766, 1954.

-21114. Medwin, H., and Rudnick, I., "Surface and Volume Vorticity in Acoustic Fields," J. Acoustic Soc. Am. 25, 1953. 15. Nyborg, W. L., "Acoustic Streaming Due to Attenuated Plane Waves," Jour. Acoustic Soc. Am. 25, (1953), p. 68-75. 16. Nyborg, W. L., "Acoustic Streaming Near a Boundary," Jour. Acoustic Soc. Am. 30, (1958), p. 329-339. 17. Raney, W. P., Corelli, J. C., and Westervelt, F. J., "Acoustical Streaming in the Vicinity of a Cylinder," J. Acoustic Soc. Am. 26, 1007, 1954. 18. Soehngen, E. E., Holman, J. P., "Experimental Studies on the Interaction of Strong Sound Fields with Free Convection Boundary Layers," ARL Tech Rep. 60-232, October 1960. 19. Sprott, A. L., Holman, J. P., Durand, F. L., "An Experimental Study of the Interaction of Strong Sound Fields with Free Convection Boundary Layers," WADC Tech. Rep. 59-717, Wright Air Dev. Center, November, 1959. 20. Spurlock, J. M., Jackson, T. W., Purdy, K. R., Oliver, C. C., Johnson, H. L., "The Effects of Resonant Acoustic Vibrations on Heat Transfer to Air in Horizontal Tubes," WADC TN 59-330 Contract No. AF 33(616)-5742, Wright Air Dev. Center, June 1959. 21. Walker, C. T., Adams, a. E., "Thermal Effects of Acoustic Streaming Near a Cylindrical Boundary," Jour. Acoustic Soc. Am. 31, (1959) pp. 813-814. 22. Westervelt, F. J., "The Theory of Steady Rotational Flow Generated by a Sound Field," Jour. Acoustic Soc. Am. 25, 60, 1953. 23. Westervelt, F. J., "Hydrodynamic Flow and Oseen's Approximations," J. Acoustic Soc. Am. 25, 951, 1953. 24. Westervelt, F. J., "Acoustic Streaming near a Small Obstacle," Jour. Acoustic Soc. Am. 25, (1953), p. 1123. 25. Westervelt, F. J., "The Effects of Sound Waves on Heat Transfer," Jour. Acoustic Soc. Am., Vol. 32, No. 3, (March 1960), pp. 337-338. 26. Zartman, W. N., "Heat Transfer from Acoustically Resonating Gas Flames in a Cylindrical Burner," University of Michigan, Ph.D. Thesis, 1960.

C. RELATED PAPERS AND BOOKS 1. Eckert, E. R. G., Soehngen, E. E., "Studies on Heat Transfer in Laminar Free Convection with the Zehnder-Mach Interferometer," Tech. Rep. No. 5747, ATI No. 44580, Air Materiel Command (Dayton, Ohio) December 27, 1948. 2. Ku, Y. K., "Boundary Layer Problems Solved by the Method of Nonlinear Mechanics," Proc. IX Int. Cong. Appl. Mech., Vol. 4, (1956), 132. 3. Levy, S., "Effect of Large Temperature Changes (Including Viscous Heating) Upon Laminar Boundary Layers with Variable Free-Stream Velocity," J. of the Aero. Sciences, Vol. 21, No. 7, (1954), pp. 459-474. 4. Martini, W. R., "Natural Convection Inside a Horizontal Cylinder," Ph.D. Thesis, University of Michigan, August 1956. 5. McAdams, W. H., Heat Transmission, Third Edition, McGraw-Hill Book Company, Inc., 1942. 6. Ostrach, S., "An Analysis of Laminar Free-Convection Flow and Heat Transfer About a Flat Plate Parallel to the Direction of the Generating Body Force," U.S. NACA TN2635, 1952. 7. Ostrach, S., "Laminar Natural Convection Flow and Heat Transfer of Fluids with and without Heat Sources in Channels with Constant Wall Temperatures," NACA TN 2863, December 1952. 8. Ostrach, S., "An Analysis of Laminar Free-Convection Flow and Heat Transfer About a Flat Plate Parallel to the Direction of the Direction of the Generating Body Force," NACA TR 1111, 1953. 9. Ostrach, S., Thornton, P. R., "On the Stagnation of Natural-Convection Flows in Closed-End Tubes," ASME Paper No. 57-SA-2, 1957. 10. Schecter, R. S., Isbin, H. S., "Natural Convection Heat Transfer in Regions of Maximum Fluid Density," AICHE Paper No. 570HT-25, 1957. 11. Schmidt, E., and Beckman, W., "Das Temperatures-und Geschwundegkeitsfeld von einer Warme abgehenden senkrechter Platte bei naturlicher Konvektion," Tech. Mech. U. Thermodynamick, Bd. 1 Nr. 10, Oxt. 1930, pp. 341-349; cont. Bd. 1, Nr. 11, Nov. 1930, pp. 391-406. 12. Sparrow, W. M., Gregg, J. L., "Laminar Free Convection From a Vertical Plate with Uniform Surface Heat Flux," Trans. ASME, Vol. 78, (1956), pp. 435-440. -212

-21313. Sparrow, E. M., Gregg, J. L., "Similar Solutions for Free Convection from a Nonisothermal Vertical Plate," ASME, Paper No. 57-SA-3, 1957. 14. Sparrow, E. M., Gregg, J. L. "The Variable Fluid-Property Problem in Free Convection," ASME Paper No. 57-a-46, 1958. 15. Sparrow, E. M., Gregg, J. L., "Prandtl Number Effects on Unsteady Forced Convection Heat Transfer," NACA TN 4311, 1958. 16. Sparrow, E. M., Eichhorn, R., Gregg, J. L., "Combined Forced and Free Convection in a Boundary Layer Flow," The Physics of Fluids, Vol. 2, No. 3, May-June 1959. 17. Tomotika, S., Yosinobu, M., "The Flow of a Viscous Liquid Past a Flat Plate at Small Reynolds Numbers," Jour. of Math. and Physics, (January 1957), pp. 1-12.

APPENDIX I COMPUTER EVALUATION METHOD The method of two of the programs are presented herein by which solutions to the two analytical cases of Chapters II and III were numerically evaluated. This method was found to be lengthy due to the algebra involved but very advantageous from an accuracy of results standpoint. It should be noted that the similarity of this method for the analytical cases lies in the general flow of the programming and not in the detailed algebra of each specific case. The programs were written in MAD Language, Michigan Algorithm Development, for the IBM 709 digital computer at the University of Michigan Computing Center. Each analytical case was divided into approximately five programs which were dependent, except for the first program, on the preceeding program. The punched output and printed results from each program were checked for errors and discrepancies. The punched output were then entered as data in the subsequent programs. To make these points clear two of the smaller programs are included in this section. They come from Chapter III. The following symbols were used. A(O) = 5(o) A(1) = K1 A(2) K2 A(3) = H'(O) A(4) = PR A(5) = w -214

-215A(6) = x B(I) = r X(1) = Real V1 X(2) = Real V2 Z(1) = Imag.V1 z(2) = Imag. V2 2V1 U () = Real (-) 2X u(2) = Real (2v2) 0 ~ 2X V(1) = Imag. (2X) V(2) = Imag. (2V2) 2X For example the first order velocity is u, Ve -v} e, - 7e whe re - _ _o. )@ (x 2 /, 22 i The program called part 1 on page 217 is evaluated by the computer first. The punched output of part 1 is then entered as data into program ANA 1A. The printed results of ANA 1A are first order velocity and temperature versus TI for each specific value of Prandtl number, X and frequency. For higher values of Vn the derivative with respect to X became very difficult because of the large amount of algebra involved. These

-216Vn were evaluated by taking finite difference derivatives in the program by - mU()x)A ( To make certain that LbX was small enough for each case, different values of LX were evaluated and compared to the exact derivative. Four place accuracy was the criteria.

-217VICTOR D. BLANKENSHIP S141F 003 015 200 PART IA VICTOR D. BLANKENSHIP S141F 003 015 200 PART IA S PRINT OBJECT9 PUNCH OBJE CT PART1000 $ COMPILE MAD. EXECUTE9 DUMP 001 DIMENSION A(100) X(100) Y(100) Z(100) START READ DATA YIO)-4.*A(6) Y(1)=SQRT*(YIO)) *07 Y(2)=SQRT.(Y(1)) 008 Y(3)=A(1)*A( 1) 009 Y(4)=Y(3)*Y(3) 010 Y(5)=(3)*Y(4) 011l Y(6)=Y(4)+A(5)*A(5)*Y(O) 012 Y(7)=Y(6)*Y(6) 013 X( )=(-3.*A(O)*Y(5)+A(0)*Y(3)*A(5 )*A(5)*Y(O))/(Y(2)*Y(6)*Y(6) 014 1) 015 Z(1)=(-5.*A(O)*A(5)*Y(2)*Y(4)-A(O)*Y(2)*Y(1)*Y{()*A(5) 016 1*A(5)*A(5))/(Y(6)*Y(6)) 017 X(2)=-A(O)*A(l1)*Y('3)/(Y(2)*Y(6)) 018 Z(2)=-A(O)*All{)*Y{2)*A(5)/Y(6) 019 Y(8)=Y(5)-3-+Y(3)*A(5)*A(5)*Y(O) 020 Y(9)=-3*Y(4)*A(5)*Y(1)+A(5).*A(5)*A(5)*Y(1)*Y(1)*Y(1) 021 Y(10)=A(O)*(3e*Y(4)/(Y(2)eP.5)}-3*A(5)*A(5)/Y(2)) 022 Y(11)=-14.*AtO)*Y(3)*A(S5)/(Y(2).P.3.) Y(12)=Y(6)+*Y'(6)*Y(6) 024 X(O)=(Y(10)*Y.(8)+Y{l)*Y(9))/Y(12) 025 ZO)=aY(11)*Y(8)-Y(10)*Y(9))/Y(12) 026 Y(13)=YI4)-A(5)*A(5)*Y(0) 027 Y(14)=-2~*Y(3)*A(5)*Y(l) 028 Y(15)=A(O)*A(1)*Y(3)/(Y(2).P,5) Y( 16)=-A(0)*36*A(5)*A(1)/(Y(2) *P.3) X( 24)=Yf 15)*Y( 13)+Y( 16)*Y( 14) )/Y(7) 03.1 Z(24)=(Y(16)*Y(13)-Y(15)*Y(14))/Y(7) 032 Y(17)=-A(3)*Y(2)*A(4)/(SQRT. (2**A(5)1) 033 X(5)=Y(l7)*(X(0)+Z(0)) 034 Z(5)=Y(17)*(Z(O)-X(O)) 035 Y(18)=-A(2)*A(4)/Y( I) 036 X(7)=Y(18)*X(1) 037 Z(7)=Y(18)*Z(1) 038 Y(19)=-A(3)/A(2) 039 X(6)=Y(19)*X(7)+X(5)/Y(19) 040 Z(6)=Y(19)*Z(7)+Z(5)/Y( 19) 041 Y(20)=Y(19)/A(1) 042 Y(21)=A(3)*Y(1)*A(4)/A(l) 043 Y-(22)=A(3)*A('4)/(Y(3)*Y( 1 ) ) 044 X(3)=-X (5)-Y(20)*X(7)-Y(21)*(X(O)+X(24)/A( 1))-Y(-22)*X:.2) 045 Z(3)=-Z(5)-Y(20)*Z(7)-Y(21)*(Z(0)+Z(24)/A( 1))-Y.22)*Z(2) 046 X(4)=X(3)/Y(19) 047 Z(4)=Z(3)/Y(19) 048 Y(23)=2**A(2)*A(4)/Y(1 ) 049 X(13) =0. Z{13)=0. Y(24)=0.5+Y(19) 052 Y(25)=A(2)*A(4)*Y(1)/A(l1) 053 X(12)=-Y(25)*X(24)-Y(23)*X(2)/(2,*A(1)) Z(12)=-Y(25)*Z(24)-Y(23)*Z(2)/(2.*A(1)) X(8)=-X(3)-X(5) 056 Z(8)=-Z(3)-Z(5) 057 Y (26)=-2.*YY' ( 19) 058

-218Y(27)0.*5*Y(20) 059 X(11 )X(8)/Y(19)+Y(19)*X( 12) Z(11)Zt8 )/Y( 19)+Y(19)*Z( 12)0 X(9)=SQRT~(A(5)*Y( 1)/2*) 062 Z(9)=X(9) 063 X(10 )A( 1 )+X(9) 064 Z(10) -Z(9) 065 Y(28)=X(10)*X(10)-Z(10)*Z( 10) 066 Y(t 29.)=A ( 5 )*A (4)*Y( 1 ) 067 Y( 30)' 2 *X( 10)*Z( 10)-Y( 29) 068 Y(31 )=Y( 4 )+Y(29)*Y(29) 069 X(16)-=(Y(3)*X(4)-Y(29)*Z(4))/Y(31) 070 iZ16)=(Y(3)*Z(4)+Y(29)*X(4) )/Y(31) 071 X(15)=(Y(3)*X(3)+2.*A( 1)*Y(3)*X(16)-Yt29)*(Z(3)+2.*A(1)*Z (16) 072 1))/Y(31) 073 Z(15)=(Y(3)*Z(3)+2**A(l)*Y(3)*Z(16)+Y(29)*(X(3)+2.*A(1)*X( 16) 074 1 ))/ IY(31) 075 Y(32 )Y( 28')*Y(28)+Y(30)*Y( 30) 076 X(19)=(X(7)*Y(28)+Z(7)*Y(30))/Y(32) 077 Z 19).Z( 7)*Y(28)-X(7)*Y(30))/Y(32) 078 X18 ) =tX6)*Y(28)+Z( 6 )*Y(30)+4*Y(28)*(X( 10)*X(19)-Z(10 ) *Z(19 079 1)) +4.*Y(30)*(X ( 19)*Z(10)+X1('10)*Z(19) /Y( 32) 080 Z(18)=.(Z(6)*Y(28)-X(6)*Y(30)+4**Y(28)*(X(19)*Z(10)+Z(19)*X(10 081 1))-4*Y(30)*(X(10)*X 19)-Z( 10)*Z ( 19 ) ) )/Y( 32) 082 X( 17)=X ( 5)*Y(28)+Z(5)*Y( 30)-2.*X( 19)*Y(28)-2.*Z(.19)*Y( 30) 083 1+2**Y( 28)*(X'(10)*X(1:8 )-Z(10)*Z ( 18 ))+2-*Y(30)*(Z(10) 084 2*X( 18 )+X:10) *Z(18 )) )/Y(32) 085 Z(17)w(Z 5 )*Y(28 )-X( 5)*Y(30)-2 *Z( 19)*Y(28)+2.*X(19) 086 1*Y0(30)+2,- * - Z.,,,(olO 10 18J+X( 1 *Z(18 ) -2**Y(30)*(X(10) 087 2*X;( 18)-Z( 10) *Z( 18. ) IJY ( 32 ) 088 Y( 33 )l16.*Y4 )+Y(29)*Y(29) 089 Y(t 34 )=4.*Y( 3 ) 090 X(23)=(Y(34)*X(13)-Y(29)*Z(13) )/Y(33) 091 Z( 23 ) (Y 34)*Z( 13 )+Y(29)*X ( 13) )/Y(33 ) 092 X(22)=Y( 34)*X( 12)+48 *A( 1)*Y(3)*X(23)-Y ( 29)*tZ( 12)+12 ~ 093 1*A(1)*Z(.23t)))/Y(33) 094 Z(22)wY( 34)*Z(12)+48 *A (-1 )*Y(3)*Z 23)+Y( 29) 095 1*(X(12)+12.*A(1)*X( 23) ))/Y(33) 096 X(21 ) =(Y 34 )* ( X(11)-6**X(23)+8**A ( )*X(22) )-Y29)*(Z(1 1) 097 1-6.*Z(23).+8*A( 1 )*Z(22 ) ))/Y( 33) 098 Z(21)=tY(34)*(Z(11)-6** Z(23)+8*A( 1 )*Z(22))+Y(29) 099 1*X( 11 )-6 *X.( 23)+8*A ( 1)*X(22 ) )/Y ( 33) 100 Y(35)X( 8 )-2.*X( 22) Y('36)=Z(8 )-2.e**Z(22)***t X(.20) (Y34)*(Y( 35)+4*A ( 1 )*X(21 ) -Y(29)*(Y(36)+4+*A (1) 1*Z'(21)))/Y(33) 102 Z (20)(Y(34)*(Y(36)+4*A ( 1 )*Z(21 ) )+Y29.)* ( Y35)+4.*A(1) 1*X(21)5:);)/Y(33) 104 X(14)=-X( 15)-X(17)-X(20) 105 Z(14)=-Z ( 15)-Z(17)-Z(.20) 106 PRINT FORMAT DATApA(O),*..A(6) 004 VECTOR. VALUES DATA=$6E2.+4*$ 005 WHENEVER CHECK.NE.O, PRINT RESULTS Y(O).**Y(34) PRINT COMMENT $ X-O)*...X(24)$ PRINT FORMAT RESUL T#X( O )**X(24 ) PUNCH FORMAT INPUTX(O)*.*.X(24) PRINT COMMENT $ Z(O)*.**Z(24)$ PRINT FORMAT RESULTtZ(0).,*Z(24)

-219PUNCH FORMAT INPUTZ(O)***Z(24) VECTOR VALUES INPUT=$6K12*$ VECTOR VALUES RESULT=$lHO5EE15.5*$ TRANSFER TO START 109 VECTOR VALUES CHECK=0O END OF PROGRAM 110 $ DATA VICTOR De BLANKENSHIP S141F 005 010 090 ANA 1A VICTOR D. BLANKENSHIP S141F 005 010 090 ANA 1A $ COMPILE MAD, EXECUTE. PUNCH OBJECT. PRINT OBJECT. DUMP DIMENSION A(100)B(100)X X(100), Z(100)# 1YC(70),YD(70) YE(70),YF(70) YG(70), 2YH(70)*YH(70),UR(70),UI (70),UT(70) 3UA(70)tYI (70) tYJ(70),YK(70) YL(70) YM(70), 4YN(70),YO(70),YP(70) YQ(70).YR ('70).VR(70).VI (70) t 5VT470) VA(70) INTEGER I PRINT COMMENT SlANALYSISS START READ DATA READ FORMAT I-NPUT.X { O0) X (24 ) READ FORMAT IMPUT#Z(O). eZ(24) VECTOR VALUES INPUT=$6K12*$PRINT FORMAT DATAtA(O).**A(6) B( 1).:*B(45 ) X(O..**X(24), 1Z(0)***Z( 24) THROUGH BETA. FOR I = 1 1 G.39 YC('I)EXP~(-A(1)*B I YD.( II=X.-(9)*B I YEt( i =Z(9)*Bi YF(l -EXP ('-YD(l:)) YG t)SIN,( YE( I ):) YH( I)COS ( YE(I)) UR(I)=(X(1)+X:(2)*B I )*iYC(I)-X(1)*YF(I)*YH(1) 1 -Z ) *YF( l )*YG( I ) UI({i)Z(Z(1)+Z(2)*B I )*YC(I)-Z(1)*YF(I)*YH( ) 1 +X(l) *YF(I )*YG() UT(I )=SQRT. (IUR ( ]., P.2+UI ( I ) *P2 )*A (5) *A ( 5 ) UA4 I )ATANtUI ( I )/UR( I)) YI() IEXP*(-2.*A(1)*B I YJ(I)uX(9)'8 I *SQRT.(A(4)) YK(II)Z(9)*8 I *SQRT.(A(4)) YL ( I ) mEXP ~ (-tJ ( I ) YM(I)* SIN.*(YK( t)) YN ( ).sCOS * (YK( I.) ) YO('f IEXP( -XflO)*B I ) YP(l)=Z(l.0)*B I YQ(I)MSIN.(YP(I)) YR( I) COS. (YP(I)) VR(.fI)t=(X15).+X. 16)*B I )*YC(I)+(X(20)+X(21)*B I 1 +X(22)*B I *8 I +X(23)*B I *B I *8 I )*YI(I) 2 +X(14)*YL(I)*YN( )+Z(14)*YL ( I ) *YM ( I ) 3 +(X(17)+X(18)*B I +X(19)*B I *B I )*YO(I) 4 *YR(l)+(Z(17)+Z(18)*B I +Z(19)*B I B* I ) 5 YO( I )*YQ(I) VI(I.)u( Z(15)+Z(16)*B I )YC(It )+(Z(20)+Z(21)e*B I 1 +Z.(22)*B I *B. +Z(23)*B I *8 I *B I )*YI(I) 2 +Z(1.4)* YL(I)*YN( I)-X(14)*YL( I )*YM(I) 3 +(2(.171)+Z(18)B I +Z(19)*B I *8 I )*YO(I)

-2204 *YR(I)-(X(17)+X(18)*B I +X(19)*B I *B I ) 5 *YO( I)*YQ( ) VT(I)=SQRT.(VR( I).P2+VI ( I )P2)*A5)*A(5) VA( I )=ATAN, (V I ()/VR(I ) ) BETA PRINT FORMAT RESULT,UR(I),UI(I ) UT(I),UA (I),VR(I),VI(I),VT( 1 1)-VA(I VECTOR VALUES DATA=$6E12,4*$ VECTOR VALUES RESULT=$lHO,8E14.4*$ TRANSFER TO START END OF PROGRAM $ DATA

APPENDIX II EXPERIMENTAL DATA Heat Transfer Data a w aw Tf Tamb h GrLPr ReV Nu in. cycle F F BTU 10-6 10-3 sec. hrftF 0.011 20.8 0.228 216.1 81.5 1.306 14.85 0.28 40.31 0.0105 41.5 0.435 216.1 80.9 1.298 14.9 0.536 40.06 0.025 20.8 0.52 216.3 80.6 1.292 14.9 0.662 39.87 o.oo65 20.8 0.135 216.3 81.1 1.298 14.9 0.172 40.06 0.025 31.2 0.775 216.5 80.5 1.287 15 0.gg99 39.72 0.05 26 1.30 204.6 77.2 1.336 14.7 1.685 41.2 0.025 52 1.30 212 83.6 1.336 14.2 1.655 40.8 0.025 78 1.95 198.8 74.4 1.432 13.4 2.55 44.3 0.025 26 0.65 209.6 77.6 1.29 14.9 0.8375 39.6 0.05 52 2.6 187.6 76.6 1.554 13.6 3.47 48.6 0.025 104 2.6 187.4 78.7 1.57 13.2 3.45 48.9 0.043 78 3.74 210.4 89.2 2.02 13.43 4.775 61.8 0.05 26 1.3 204.6 77.2 1.33 14.62 1.69 41.05 0.o01 20.8 0.208 187.6 79.4 1.228 11.9 0.275 38.38 0.0095 41.5 0.394 188.0 79.4 1.224 11.9 0.501 38.25 0.oo005 20.8 0.104 187.6 79.4 1.23 11.9 0.132 38.43 0.0275 20.8 0.57 188.4 79.2 1.215 12.0 0.726 37.97 0.025 31.2 0.779 187.8 79.0 1.223 12.0 1.029 38.22 0.058 41.6 2.42 176.7 81.6 1.415 11.45 3.27 44.1. 055 52 2.75 108.7 81.5 1.05 4.53 4.o6 34.3 0.055 72.8 3.85 105.2 83.2 1.32 3.67 5.7 43.3 0.075 52 3.9 189.2 91.4 1.808 11.8 5.08 55.8. o48 78 3.75 187.7 91.6 1.812 11.2 4.89 54.9 -221

-222APPENDIX IIA Smoke Studies Data a w aw aw2 aw2 T R PR T Rev in. cycles in-cycle ft ~F 10-6 GRL ) sec. sec. 2 sec 0.012 50 o.6 98.75 3.06 133.8 14.45 217.6 56 0.01 50 0.5 82.25 2.55 133.8 14.45 2.7.6 11.6 0.02 50 1.0 164.5 5.11 84.4 11.15 163.8 140.8 0.023 50 1.15 189.5 5.875 84.4 11.15 163.8 162 0.018 50 0.9 148.5 4.59 110.4 13.2 190.6 99.2 0.018 50 0.9 148.5 4.59 108.4 12.95 188.6 101.2 0.008 70 0.56 129 4 110.4 13.2 190.6 86.4 0.009 70 0.63 145 4.5 110.4 13.2 190.6 97.2 0.017 50 0.85 140 4.34 114.6 13.7 193.2 90.4 0.016 50 0.8 132 4.083 114.6 13.7 193.2 84.8 0.008 70 o 56 129 4 118.2 13.95 196.8 81.6 0.007 70 0.49 113 3.5 118.2 13.95 196.8 71.4 0.054 25 1.35 111.5 3.45 118.2 13.95 196.8 70 0.0194 50 0.97 160 4.96 51.6 7.8 131.6 236 o.009 70 0.63 145 4.5 51 7.7 131.4 215.6 0.01 70 0.7 161.5 5.0 50.8 7.69 130.8 240 0.027 50 1.35 222 6.9 0 0 80 0.03 50 1.5 247 7.65 0 0 80 0.094 25 2.2 182 6.o6 0 0 80

APPENDIX III TABULATED RESULTS APPENDIX III The tabulated results of Ostrach Reference C are found on -the following pages, APPENDIX IIIA This section was taken from Schoenhals Reference A-34. The forcing function is defined as Schoenhals obtained tabulated results by integrating (V7) =/uaz)dy on tH(y) fIrown pa The results are found on the following pages. -223

-224Prandtl number, 0.01 F [ F~ F" H H~ ~ F 1~t E" H H'...... 0 O. 0000 O. 0000 O. 9862 1. 0000 — 0. 0812 4.16 1. 9319 O. 4037 -~-0. 0615 O. 6741 — 0. 07Z3. 1. 0048. 0936. 8868., 9919 u. 0812 4.18 1. 9400. 4024 —. 0614. 6726 u. 0722 ~ 2. 0184. 1774 ~ 7891. 9838 —. 0812 4. 20 1. 9480. 4012 ~. 0612. 6712 ~. 0721. 3. 0399. 2516. 6942. 9756 ~. 0812 4. 24 1. 9640. 3988 ~. 0609. 6683 —. 0720 ~ 4. 0684. 3164. 6030. 9675 —. 0811 4. 28 1. 9799. 3964 —.'0605. 6654 M. 0718 ~ 5. 1029. 3723. 5163. 9594 ~. 6811 4. 32 1. 9957. 3939 —. 0602. 6626 —. 0716 ~ 6 ~ 1426. 4198. 4349. 9513 u. 0811 4. 36 2. 0114. 3915 —. 0599. 6597 —. 0714 ~ 7. 1866. 4595. 3595. 9432 ~. 0811 4. 40 2. 0270. 3891 ~. 0595. 6568 ~. 0713 ~ 8 ~ 2342. 4919. 2906. 9351 —. 0810 4. 50 2. 0657. 3832 ~. 0587. 6497 ~. 0708 ~ 9. 2848 ~ 5178. 2285. 9270 ~. 0809 4. 60 2.1037 ~ 3774 u. 0579. 6427 ~. 0704 1.0. 3376. 5379. 1734. 9189 ~. 0809 4. 70 2.1411 ~ 3716 —. 0572. 6357 —. 0899 1.1. 3922. 5527. 1253. 9108 —. 0808 4. 80 2.1780. 3660 ~. 0564. 6287 ~. 0695 1.2. 4480. 5631. 0839. 9028 ~. 0807 4. 90 2. 2143. 3604 ~. 0556.. 6218 —. 0690 1.3. 5047. 5697. 0489 ~ 8947 m. 0806 5. O0 2. 2501. 3548 —. 0549. 6149 —. 0686 1.4. 5618. 5731. 0198. 8866 ~. 9804 5.10 2. 2853 ~ 3494 ~. 0541. 6081 —. 0681 1.5. 6192. 5739 —. 0039. 8786 —. 0803 5. 20 2. 3200. 3440 ~. 0534. 6013 —. 0676 1.6. 6765. 5725 ~. 0229. 8706 —. 0801 5. 40 2. 3877. 3335 —. 0520. 5878 ~. 0667 1.7. 7336. 5694 ~. 0379. 8626 ~. 0800 5. 60 2. 4534. 3232 ~. 0506 ~ 5746 ~. 0657 1.8. 7904. 5650 ~. 6493. 8546 u. 0798 5. 80 2. 5170. 3132 —. 0493. 5615 —. 0648 1.9. 8466. 5596 ~. 0579. 8466 —. 0796 6. O0 2. 5787 ~ 3035 ~. 0479. 5487 ~. 0638 2. 0. 9023. 5535 ~. 0641. 8387 ~. 0794 6. 20 2. 6384. 2940 ~. 0467. 5360 ~. 0628 2.1 ~ 9573. 5469 —. 0685. 8307 —. 0792 6. 40 2. 6963. 2849 ~. 0454. 5236 ~. 0618 2. 2 1. 0117. 5399 ~. 0714. 8228 u. 0789 6. 60 2. 7524. 2759 —. 0442. 5113 ~. 0608 2. 3 1. 0653. 5326 ~. 0732. 8150 ~. 0787 6. 80 2. 8067. 2671 ~. 0430. 4993 —. 0598 2. 4 1.1182. 5253 ~. 0742. 8071 ~. 0784 7. O0 2. 8593. 2586 ~. 0419. 4874 ~. 0588 2. 5 1.1703. 5178 ~. 0746. 7993 u. 0782 7. 40 2. 9594. 2423 ~. 0397. 4643 —. 0568 2. 6 1. 2217. 5104 —. 0745. 7915 —. 0779 7. 80 3. 0532. 2269 ~. 0376. 4420 ~. 0547 2. 7 1. 2724. 5029 ~. 0741. 7837 ~. 0776 8. 20 3.1411. 2123 ~. 0356 ~ 4205 ~. 0527 2. 8 1. 3223. 4956 ~. 0735. 7760 ~. 0773 8. 60 3. 2232 ~ 1984 ~. 0337. 3998 ~. 0508 2. 9 1. 3715. 4882 —. 0728 ~ 7682 ~. 0770 9. O0 3. 2999. 1853 —. 0319. 3799 ~. 0489 3. 0 1. 4200. 4810 ~. 0719. 7606 u. 0766 9. 40 3. 3715 ~ 1729 ~. 0302. 3607 ~. 0470 3.1 1. 4677. 4739 ~. 0711. 7529 —. 0763 9. 80 3. 4383. 1612 ~. 0285. 3423 ~. 0451 3. 2 1. 5148. 4668 ~. 0701. 7453 ~. 0760 10. 20 3. 5005. 1501 —. 0270. 3246 —. 0432 3. 3 1. 5611. 4598 —. 0692. 7377 —. 0756 10. 60 3. 5584. 1396 ~. 0254. 3069 ~. 0413 3. 4 1. 6067. 4530 ~. 0683 ~ 7302 ~. 0753 11. O0 3. 6123. 1297 ~. 9241. 2915 t. 0397 3. 45 1. 6293. 4496 u. 0678. 7264 —. 0751 11.40 3. 6622. 1203 ~. 0228 ~ 2759 ~. 0380 3. 5 1. 6517. 4462 —. 0674 ~ 7227 —. 0749 11.80 3. 7086. 1114 ~. 0215. 2610 t. ~364 3. 55 1. 6739. 4428 ~. 0669. 7189 ~. 0747 12. 20 3. 7514. 1031 —. 0203. 2468 ~. 0348 3. 6 1. 6960. 4395 —. 0664. 7152 —. 0745 12. 60 3. 7911. 0952 —. 0192. 2332 ~. (~332 3. 65 1. 7179. 4362 ~. 0660 ~ 7115 —. 0743 13. O0 3. 8276 ~ 0877 —. 0181. 2202 ~. 0317 3. 7 1. 7396. 4329 —. 0655. 7078 —. 0741 13. 60 3. 8771. 0773 ~. 0165. 2018 —. 0'296 3. 75 1. 7611. 4296 —. 0651. 7041 —. 0739 14. 20 3. 9206. 0679 —. 0151. 1847 —. 0276 3. 8 1. 7825. 4264 —. 0646. 7004 —. 0738 14. 80 3. 9587 ~ 0592 ~. 0138. 1687 M. 0257 4. 85 1. 8308.4232 —. 0642.6967 ~. 0736 15. 40 3. 9918 0513 ~. 0126.1538 —. 0239 3. 9 1. 8249. 4200 ~. 0638. 6930 ~. 0734 16. O0 4. 0204. 0441 ~. 0114. 1399 ~. 0223 3. 95 1. 8458 ~ 4168 ~. 0633. 6893 ~. 0732 17. O0 4. 0591. 0335 ~. 0097 ~ 1189 ~. 0197 4. O0 1. 8665. 4136 ~. 0629. 6857 —. 0729 18. O0 4. 0880. 0246 ~. 0082. 1003 ~. 0175 4. 05 1. 8871. 4105 ~. 0625 ~ 6821 —. 0727 19. O0 4.1088. 0171 ~. 0069. 0839 —. 0154 4.10 1. 9076 ~ 4074 ~. 0620. 6784 —. 0725 20. O0 4.1226. 0108 ~. 0057. 0694 —. 0137 4.12 1. 9157. 4061 —. 0619. 6770 —. 0725 21. O0 4.1308 ~ 0057 0046. 0565 —. 0121 4.14 1. 9238. 4049 ~. 0617 ~ 6755 ~. 0724 22. O0 4.1343 ~ 0015 —. 0037 ~ 0452 ~. 0107 Prandtl number, 0.72 Prandtl number, 0.733 0 O. 0500 O. 0800 O. 6760 1.0050 — 0. 5046 0 O. 0050 O. 0000 O. 6741 1.0300 — 0. 5080, 1.0032. 0627. 54~. 9495 —. 5045. 1. 0032. 0625. 5767. 9492 —. 5079 ~ 2. 0122. 1159.. 8991 —. 5037. 2. 0122. 1155. 4849. 8984 —. 5071 ~ 3. 0261. 1602. 4007. 8488 —. 5016. 3. 0260. 1597. 3990. 8478 —. 5050 ~ 4.0440. 1962. 3210. 7989 —. 4979. 4. 0438. 1955. 3194. 7975 —. 5012 ~ 5. 0651. 2246. 2479. 7493 —. 4921. 5. 0649. 2238. 2465. 7477 —. 4953 ~ 6. 0887. 2460. 1817. 7005 —. 4840. 6. 0884. 2451. 1804. 6985 —. 4870 ~ 7. 1141. 2612. 1224. 6526 —. 4735. 7. 1137. 2601. 1212. 6503 —. 4763 ~ 8. 1407. 2708. 0700. 6059 —. 4607. 8. 1402. 2695. 0691. 6033 —. 4632 ~ 9. 1681. 2754.0246. 5606 —. 4456. 9. 1674.2741. 0237. 5578 —. 4478 1.0. 1957. 2759 —. 0143. 5168 —. 4284 1.0. 1949. 2745 —. 0149. 5139 —. 4303 1.1. 2231. 2728 —. 0468. 4749 —. 4095 1.1. 2222 2713 —. 0473. 4718 —. 4110 1.2. 2501. 2667 —. 0734. 4350 —. 3891 1.2. 2490. 2652 —. 0737. 4317 —. 3902 1.3. 2764. 2583 —. 0~5. 3972 —. 3676 1.3. 2752 2568 —. 0946. 3938 —. 3684 1.4. 3017. 2480 —. 1106. 3615 —. 3453 1.4. 3003.2465 —. 1106. 3581 —. 3458 1.5. 3260. 2363 —. 1224. 3281 —. 3227 1.5. 3244.2348 —. 1222. 3246 —. 3228 1.6. 3490. 2236 —. 1302. 2970 —. 3000 1.6. 3473. 2222 —. 1299. 2935 —. 2998 1.7. 3707. 2104 —. 1347. 2681 —. 2775 1.7. 3688.2090 —. 1342. 2647 —. 2771 1.8. 3910. 1968 —. 1363. 2415 —. 2556 1.8. 3891. 1954 —. 1358. 2381 —. 2549 1.9. 4100. 1832 —. 1356. 2170 —. 2344 1.9. 4079. 1819 —. 1350. 2136 —. 2335 2. 0. 4277. 1697 —. 1331. 1945 —. 2141 2. 0. 4254. 1685 —. 1324. 1913 —. 2130 2.1. 4440. 1566 —. 1290. 1741 —. 1949 2.1. 4416. 1554 —. 1283. 1710 —. 1937 2. 2.4590.1440 —. 1238.1555 —. 1768 2. 2.4565.1429 —. 1230.1526 —. 1754 2. 3. 4728. 1319 —. 1178. 1387 —. 1598 2. 3. 4702. 1309 —. 1170. 1359 —. 1584 2. 4. 4854. 1204 —. 1113. 1235 —. 1441 2. 4. 4827. 1195 —. 1104. 1208 —. 1427 2. 5. 4969. 1097 —. 1044. 1098 —. 1296 2. 5. 4941. 1088 —. 1035. 1073 —. 1281 2. 6. 5074.0996 —. 0973. 0976 —. 1163 2. 6. 5045. 0988 —. 0965.0952 —. 1148 2. 7. 5168. 0902 —. 0903. 0865 —. 1041 2. 7. 5139. 0895 —. 0895 ~ 0843 —. 1026 2. 8. 5254.0815 —. 0834.0767 —. 0930 2. 8. 5224.0809 —. 0826.0746 —. 0916 2. 9. 5332. 0735 —. 0767.0679 —. 0830 2. 9. 5301. 0729 —. 0759. 0660 —. 0816 3. 0.5401.. 0661 —. 0703.0601 —. 0739 3.0.5370.0657 —. 0695.0583 —. 0725 3.1. 5464. 0594 —. 0642.0531 —. 0657 3.1. 5433. 0590 —. 0635. 0514 —. 0644 3. 2. 5520.0533 —. 0585. 0469 —. 0584 3. 2. 5489. 0530 —. 0578. 0454 —. 0571 3. 3. 5571. 0477 —. 0531. 0414 —. 0518 3. 3. 5539. 0475 —. 0524. 0400 —. 0506 3. 4.5616.0427 —. 0481.0365 —. 0459 3. 4.5584.0425 —. 9474.0352 —. 0448 3. 6.5692.0339 —. 0392.0284 —. 0359 3. 6.5660.0339 —. 030~2.0273 —. 0350 3.8.5735.0269 —. 0317.0220 —. 0281 3. 8.5720.0269 —..0211 —. 0272 4. 0.5801.0212 —. 0255.0170 —. 0219 4. 0.5769.0213 —. 0250.0163 —. 0211 4. 2.5838.0166 —. 0204.0132 —. 0170 4. 2.5807.0169 —. 0200.0126 —. 0164 4. 4.5868.0130 —. 0162.0102 —. 0132 4. 4.5837.0133 —. 0159.0097 —. 0127 4. 6.5891.0101 —. 0128.0078 —. 0102 4. 6.5860.0105 —. 0125.0074 —. 0038 4, 8.5908.0078 —. 0102.0060 —. 0080 4.8.5879.0082 —. 0099.0057 —. 0076 5. 0.5922.0060 —. 0079.0046 —. 0061 5. 0.5893.0065 —. 0077.0044 —. 0058 5. 2.5933.0046 —. 0063.0035 —. 0048 5. 2.5905.0051 —. 0061.0033 —. 0045 5. 4.5941.0034 —. 0048.0027 —. 0037 5. 4.5914.0040 —. 0047.0025 —. 0035 5. 8.5951 0019 0029.0015 —. 0022 5. 6.5921.0032 —. 0036.0019 —. 0027 6. 2.5957.0010 —. 0018.0009 —. 0013 5. 8.5927.0025 —. 0028.0015 —. 0021 6. 8.5960.0003 —. 0008.0003 —. 0006 6. 0.5932.0021 —. 0022.0011 —. 0016 7. 3.5961.0030 —. 0004.0051 —. 0003 6. 4.5938.0014 —. 0013.0006 —. 0009 6.8.5943.0010 —. 0008.0003 —.0006 7. 2.5946.0007 —. 0004.0001 —. 0003 7.6.5949.0006 —. 0002.0000 —.0002 8. 0.5951.0005 —. 0051.0080 —. 0001

-22..5Prandtl number, 1 Prandtl number, 2 n F F' F" H H' ] F F' F" H H' 0 0. 0800 0. 0000 0. 6421 1.0003 — 0. 5671 0 0. 0000 0. 0800 0. 5713 1.0000 — 0. 7165. 1 ~ 0030.0593. 5450. 9433 —. 5669. 1.0027.0523. 4749.9284 —. 7161 ~ 2. 0115. 1092 ~ 4540. 8867 —. 5657. 2. 0101. 0952. 3861. 8569 — ~ 7135 ~ 3. 0246. 1503. 3694. 8302 —. 5627. 3. 0215. 1297 ~ 3049. 7858 — ~ 7069 ~ 4. 0413. 1833. 2916. 7742 —. 5572. 4 ~ 0358. 1565 ~ 2318. 7157 —. 6949 ~ 5. 0610. 2089. 2208. 7189 —. 5488. 5. 0525. 1763. 1666. 6470 — ~ 6768 ~ 6. 0829. 2'277. 1572.'~45 —. 5371. 6. 0709. 1901. 1095. 5805 —. 6523 ~ 7. 1064. 2406. 1008. 6116 —. 5221. 7. 0904. 1985. 0602. 5168 —. 6215 ~ 8. 1308. 2481. 0516. 5602 —. 5038. 8. 1104. 2024. 0185. 4564 —. 5852. 9. 1558. 2511. 0093. 5109 —. 4826. 9. 1307. 2024 —. 0161. 3999 —. 5443 1.0. 1809. 2502 —. 0263. 4638 —. 4589 1.0. 1508. 1994 —. 0440. 3476 —. 5002 1.1. 2058. 2461 —. 0557. 4192 —. 4330 1.1. 1705. 1938 — *0659. 2999 —. 4543 1.2. 2300. 2393 —. 0793. 3772 —. 4056 1.2. 1895. 1864 —. 0824. 2568 —. 4077 1.3. 2535. 2304 —. 0975. 3381 —. 3772 1.3. 2077. 1775 —. 0942. 2183 —. 3619 1.4. 2761. 2199 —. 1110. 3018 —. 3484 1.4. 2250. 1677 —. 1020. 1844 —. 3178 1.5. 2975. 2083 —. 1203. 2684 —. 3197 1.5. 2412. 1572 —. 1063. 1547 —. 2763 1.6.3177.1960 —. 1260.2379 —. 2915 1.6.2564.1465 —. 1078.1290 —. 2380 1.7. 3367. 1832 —. 1287. 2101 —. 2642 1.7. 2706. 1357 —. 1071. 1069 —. 2032 1.8. 3543. 1708 —. 1288. 1850 —. 2382 1.8. 2836. 1252 —. 1046. 0882 —. 1721 1.9. 3707. 1575 —. 1268. 1624 —. 2136 1.9. 2956. 1149 —. 1008. 0724 —. 1446 2.0.3859.1450 —.1233.1422 —.1907 2.0.71(}66.1050 —.0960.0592 —.1207 2.1. 3997. 1239 —. 1185. 1242 —. 1695 2.1. 3166. 0957 —. 0906. 0482 —. 1001 2. 2. 4125. 1213 —. 1127. 1082 —. 1501 2. 2. 3257'. 0869 —. 0849. 0391 —. 0826 2.3.4240.1104 —.1064.0941 —.1324 2.3.3340.0787 —.0789.0316 —.0677 2.4.4346. 1031 —.0997.0817 —. 1164 2.4.3415.0711 —.0729.0254 —.0553 2. 5. 4441. 0904 —. 0928. 0708 —. 1020 2. 5. 3483. 0641 —. 0671. 0204 —. 0450 2. 6. 4527. 0815 —. 0859. 0613 —. 0892 2. 6. 3543. 0577 —. 0614. 0164 —. 0364 2. 7. 4604. 0733 —. 0791. 0529 —. 0777 2. 7. 3598. 0518 —. 0560. 0131 —. 0294 2. 8. 4673. 0657 —. 0725. 0457 —. 0676 2. 8. 3643. 0465 —. 0509. 0105 —. 0237 2. 9. 4736. 0588 —. 0662. 0392 —. 0587 2. 9. 3691. 0417 —. 0461. 0083 —. 0190 3.0. 4791, 0524 —. 0602. 0339 —. 0509 3.0. 3731 ~ 0373 —. 0416. 0066 —. 0152 3.1. 4841. 0467 —. 0546. 0291 —. 0441 3.1. 3766. 0333 —. 0375. 0053 —. 0121 3.2. 4885. 0415 —. 0493. 0250 —. 0381 3.2. 3798. 0298 —. 0338. 0042 —. 0097 3. 3. 4924. 0368 —. 0444. 0215 —. 0329 3. 3. 3826. 0266 —. 0303. 0033 —. 0077 3.4. 4959. 0326 —. 0399. 0185 —. 0283 3.4. 3851. 0237 —. 0272. 0026 —. 0061 3.. 5016. 0254 —. 0321. 0136 —. 0210 3.6. 3893. 0188 —. 0218. 0017 —. 0038 3.8.5061.0197 —.0255.0009 —.0155 3.8.3927.0149 —.0174.0010 —.0024 4.0. 5096. 0151 —. 0202. 0072 —. 0115 4.0. 3953. 0018 —. 0138. 0007 —. 0015 4. 2. 5122. 0116 —. 0158. 0053 —. 0084 4.2. 3974. 0094 —. 0109. 0004 —. 0009 4.4. 5143. 0087 —. 0124. 0038 —. 0062 4.4. 3991. 0074 —. 0086. 0003 —. 0006 4.6. 5158. 0064i —. 0096. 0027 —. 0045 4. 6. 4004. 0059 —. 0068. 0002 —. 0004 4. 8. 5169. 0049 —. 0075. 0020 —. 0034 4. 8. 4015. 0047 —. 0354. 0001 —. 0002 5.0. 5177. 0035 —. 0057. 0014 —. 0024 5. 0. 4023. 0037 —. 0042. 0031 —. 0031 5. 2. 5183. 0025 —. 0044. 0010 —. 0018 5. 4. 4035. 0024 —. 0026. 0001 —. 0031 5.5.5189.0014 —.0029.0006 —.0011 5.8.4043.0015 —.0016.0000.0003 6.0.5194.0004 —.0014.0002 —. 0005 6.4.4049.0008 —. 0008.0000.0000 6. 25. 5194. 0800 —. 0010. 0003 —. 0004 7. 0.4053. 0005 —. 0004. 0000.0003 8.0.4056.0002 —. 0001.0000.0000 9.0.4058.0001.0000.0000.0000 10. 0. 4059. 0001. 00(g). 0000. 0000 11.0.4059..0001.0000.0000.0000......... Prandtl number, 10 Prandtl number, 100 n F F' F" H H',/ F F' F" H H'.. 0 00000 00000 04~92 10000 -L1694 o ooo 00000 00000 02517,0000 -,19,, ooo 037, 32~ ss32 -1,071 075 oool 0060 7274 9457 -:,o,.2.0071.0654.2428.7670 — 1.1521.050.0003.0114.2044.8905 — 2.18g 0147 os61,7~ ~4 -,,155 075 oooo 0,02,s~s s350 -21so 024,,oo3,,3~.~4s -,o576,oo oo,, o~o5,6~ 7s,5 -:,~.5.0346. 1091.0655.4437 —. 9640. 125.0017.0244. 1438.7276 — 2.144.6.0458. 1137.0279.3527 —.854,5.150.0023.0277.1263.6744 — 2.113 ~ 7.0573. 1150 —. 0011.2733 —. 7322. 175.0031.0307. 1101.6221 — 2.071 s 06s7,137 -oaa, ~064 -606, ~ 0030 0332 095~ 5709 -,os.9.0800. 1107 —. 0367. 1519 —.4849.225.0047.0355.0816.5213 — 1.954 1.0.0908. 1066 —. 0462. 1090 —. 3753.250.0056.0373.0692.4733 -- 1. 880 1.1. 1012. 1016 —. 0518.0763 —. 2813.275.0066.0389.0580.4273 — 1. 786 1.2.lnl.0963 -.0545.0522 -.2045 ~300.0076 0402.0479.3836 -1.704 1.3.1205.0908 —. 0552.0349 —. 1445,325.0086.0413.0389.3422 — L604 1.4. 1293.0853 —. 0544.0228 —. 0993.350.0096.0422.0309.3034 — 1 498 1.5.1376.0799 —.0527.0146 —. 0665.375.0107.0429.0238.2674 — 1.388 1.6. 1453.0748 —. 0505.0092 —. 0435.400.0118.0434.0176.'2341 — 1. 276,7,52,, 0,~9~ -04s0 005,~ -0~,7s 4~ 0,~ 043s 0,~ ~36 -,,~ 1.8. 1593.0652 —. 04g{.0034 —. 0174.450.0140.0440.0076. 1759 — 1.052 I. 9. 1656.0608 —. 0427. i)020 —. 0107.475.0151.0442.0036. 1510 —. 9430 2.0. 1714.0567 —.0400.0012 —.006,5.500.0162.0442.0002. 1287 —.8393 2.1. 1769.05o-,8 —.0375.0007 —.0038.52.5.0173 ~0442 —.0026. 1089 —. 7404 2. 2. 1820.0491 —. 0351.0004 —. 002"2.550.0184.0441 —. 0050.0916 —. 6478. 1868.0458 —. 0328.0002 —. 0013.575.0195.0439 —. 0070 ~ 0765 —. 5621 22.3.~.o4~ -.o~o~.ooo~ -.ooo~.~oo.o~o~.o~ -.oo~.o~ -.~ 2]65.1953.0396 —. 0286.0001 —. 0004.65.0227.0432 —.0111.0427 —. 3496 ~.,~.0.~ -.0~6~.0000 -.000~.~0.0~.0~ -.0,~.0~0 -.~0 ~.~.~.0~.~ -.0~.0000 -.000,.~.0~70.0~ -.013~.0,~ -.1~ ~.~.~.0~,~ -.0~.0000 -.000,.~.0~.0~1~ -.0,0.0,, -.,0~ ~.~.~.0~ -.0~,~.0000.0000.8~.0~1,.0~0~ -.0,~.00~ -.0~ ~.0.0~.0000.~.0~.0~ -.01~.00~ -.0~.~,~ -.0~.0000 ~.~.~.0~ -.0~.0000.0000.~.0~,.03~ -.0,,.00.~ -.0~ 3.2.2170.0238 —.0174.0000.0000 1.00.0371.0385 —. 0140.0012 —.0149 3.3.2193.0221 —. 0162.0000.0000 1.10.0408.0371 —. 0136.0004 —. 0046 3.4.2215.0206 —. 0151.0000.0000 1.20.0445.0357 —. 0132.0001 —. 0013 3.6.2253.0178 —. 0131.0000.0000 1.30.0480.0344 —. 0128.0000 —. 0003 3.8.2286.0153 —. 0113.0000.0000 1.40.0514.0332 —. 0124.0000 —. 0001 4.0.2314.0132 —. 0098.0000.0000 1.50.0546.0320 —. 0119.0000.0000 4.2.2339.0114 —.0084.0000.0000 1.60.0578.0308 —.0116.0000.0000 4.4.2360.0098 —. 0073.0000.0000 1.70.0608.0297 —. 0112.0000.0000 4.6.2379.0085 —. 0063.0000.0000 1.80,0637.0286 —. 0108.0000.0000 4.8.2394.0073 —. 0054.0000.0000 1.90.0665.0275 —. 0104.0000.0000 5. 0.2408.0063 —. 0047.0000.0000 2. 0.0692.0265 —. 0101.0000.0000 5. 4.2430.0047 —. 0035.0000.0000 2. I.0718.0255 —. 0097.0000.0000 5. 8.2446.0035 —. 0026.0000.0000 2. 2.0743.0245 —. 0094.0000.0000 6. 2.2458.0026 —. 0019.0000.0000 2. 3.0767.0236 —. 0091.0000.0000 7. 0.2474.0014 —. 0011.0000.0000 2. 4.0790.0227 —. 0088.0000.0000 8. 0.2484.0007 —. 0005.0000.0000 2. 6.0834.0210 —. 0082.0000.0000 9. 0.2489.0003 —. 0002.0000.0000 2. 8.0874.0195 —. 0076.0000.0000 10.0.2491.0002 —. 0001.0000.0000 3.0.0912.0180 —. 0071.0000.0000

-226Prandtl number, 100(-Concluded Prandtl number, 1000 7 F F' F" H H' | | F F' F" H H' 3. 2 0917 0. 0166 -0. 0066 0. 0000 0.0000 0. 0.0000 0.0000 00.1450 10000 -3.966 3.4.0979.0154 -.0061.0000.0000.025.0000.0033.1212.9009 -3.962 3.6.1008.0142 -.0056.0000.0000.050.0002.0061.0999.8021 -3.933 3.8.1035.0131 -.0052.0 (0.0000.075.0003.0083.0811.7046 -3.861 4.0.1061.0121 -.0049.0000.0000.100.0006.0102.0647.6096 -3.731 4. 4.1105.0103 -.0012.0000.0000.125.0008.0116.0506.5186 -3. 538 4. 8.1143.0088 -.0036.0000.0000.150.0012.0127.0387.4332 -3.283 5.2.1176.0074 -. 0031.0000. 000.175.D015.0135.0289.3549 -2.975 5. 6.1203.0063 -. 0026.0000.0000.200.0018.0142.0209.2847 -2.628 6. 0.1226.0053 -.0022.0000.0000.225.0022.0146.0146.2236 -2.261 6. 6.1254.0041 -.0018.0000. 0000.250.0026.0149.0096.1717 -1. 893 7.2.1276.0032 -. 0014.0000.0000.275.0029.0151.0059.1288 -1.541 8. 0.1297.0022 -. 0010.0000.0000.300.0033.0152.0032.0944 -1.220 9. 0.1315.0014 -.0007.0000.0000.325.0037.0153.0012.0675 - 9381 10.0.1326.0008 -.0005.0000. 0000.35.0041.0153 -.0003.0471 -. 7012 11.0.1332.004 -.0000.0000.375.0045.0152 -.0012.0321 -. 5093 12.0.1335.0002 -. 0002.0000.0000.400.0018.0152 -.0019.0213 -.3596 13.0.1336.0000 -.0001.0000.0000.425.0052.0152 -.0023.0138 -. 2467.450.0056.0151 -.0026.0087 -.1645.475.0060.0150 -. 0027.0053 -.066.500.0063.0150 -.0028.0032 -.0672.525.0067.0149 -.0029.0019 -.0412. 550.0071.0148 -.0029.0011oo -.0245.575.0075.0147 -.0029.0006 -.0142.600.0078.0147 -.0029.0003 -.0080.62b.0082.0146 -.0029.0002 -.0044.800.0107.0141 -.0028.0000.0000 1.000.0135.0136 -.0027.0000.0000 1.40.0187.0125 -.0025.0000.0000 1.80.0235.0115 -.0023.0000.0000 2.20.0279.0106 -.0022.0000.0000 2.60.0320.0098 -.0020.0000.0000oooo 3.0.0358.o0 -.oo0019.0000.0000 3.6.0409.(180 -.0017.0000.000() 4. 2.0454.0070 -.0015.0000.0000 5. 0.0505.0060 -.0012. 0000.0000 5. 8.0549.0050 -. 0011.000.0000. o.0603.0039 -.0008.0000.0000 8.0.0638.0032 -.0007. 000.0000 10.0.0691.0(022 -. M000.0000.0000 12.0.0727.0015 -.0003.0000.0000 14.0.0752.0011 -. 0002.0000.0000 16.0.0771.0008 -.0001.0000).0000 18. 0.0786.0007.0000.0000 o.00o 20.0.0798.0006.0000.0000.0000 22.0.0809.0005. 000.o000.0000 23.6.0816.0005.0000.0000.0000

-227Prandtl Number =.01 TI f('1) TI (-f)dT i T fj T(n)drn 0 O' 0 8.25 0 4.2 7.57 33.68.2 8.25 1.65 4.4 7.51 35.18.4 8.24 3.30 4.6 7.45 36.68.6 8.23 4.95 4.8 7.38 38.16.8 8.22 6.59 5.0 7.31 39.63 1.0 8.21 8.24 5.8 7.03 4.537 1.2 8.19 9.88 6.6 6.71 50.87 1.4 8.17 11.51 7.4 6.39 56.11 1.6 8.13 13.14 8.2 6.o4 61.08 1.8 8.12 14.77 9.0 5.70 65.77 2.0 8.og 16.39 9.8 5.34 70.19 2.2 8.o6 18.00 10.6 4.98 74.32 2.4 8.02 19.61 11.4 4.64 78.17 2.6 7.98 21.21 12.2 4.30 81.74 2.8 7.94 22.80 13.0 3.96 85.o4 3.0 7.89 24.39 14.2 3.48 89.51 3.2 7.85 25.96 15.4 3.02 93.40 3.4 7.80 27.53 16.0 2.80 95.15 3.6 7.74 29.08 18.0 2.13 100.1 3.8 7.69 30.62 20.0 1.55 103.7 4.0 7.63 32.15 22.0 1.03 100.3 24.0 107.5 Prandtl Number -.72 Ti'(I).fT( T)di T 1 (T) I (fT (T)d o o0 0 1.252 0 2.4.396 2.144.1 1.250.125 2.5.363 2.181.2 1.242.250 2.6.332 2.216.3 1.230.373 2.7.302 2.248.4 1.212.496 2.8.275 2.277.5 1.190.616 2.9.250 2.303.6 1.163.733 3.0.227 2.327.7 1.132.848 3.1.206 2.349.8 1.097.960 3.2.187 2.368.9 9 159 1.067 3.3.167 2.386 1.0 1.017 1.171 3.4.152 2.402 1.1.973 1.271 3.6.124 2.430 1.2.927 1.366 3.8.100 2.452 1.3.880 1.456 4.0.081 2.470 1.4.832 1.542 4.2.065 2.484 1.5. 783 1.622 4.4.052 2.496 1.6.735 1.698 4.6.042 2.505 1.7.687 1.769 4.8.033 2.513 1.8.641 1.836 5.0.025 2.519 1.9.596 1.898 5.2.021 2.523 2.0.52 1.955 5.4.o16 2.527 2.1.510 2.008 5.8.010 2.532 2.2.470 2.057 6.2.00oo6 2.533 2.3.432 2.102 6.8.002 2.538 7.3.001 2.538

-228Prandtl Number = 10 5 (T) f l(TI))drl I 5(,) T(fr)dj o 0 0.510 0 1.2.075.378.1.504.051 1.3.053.384.2.487.100 1.4.037.388.3.458.148 1.5.025.391.4.420.192 1.6.017.394.5.375.232 1.7.011.395.6.325.267 1.8.007.396 ~ 7.273.296 1.9.004.396.8.223.321 2.0.003.397.9.177.341 2.1.002.397 1.0.137.357 2.2.001 ~397 1.1.102.369 Prandtl Number = 1000 TI t(T) ~t (n) dd 0.148 0.025.147.0037.050.143.0073.075.137.0108.100.129.o014.125.118.0172.150.107.0201 o 175.094.0226.200.081.0247.225.o68.0266.250.0o56.0281.275.044.0294.300.034.0304.325.026.0311.350.019.0317.375.014.0321.400.010.0324 425.007.0326.450.oo004.0327.475.003.0328.500.002.0329.525.001.0329

UNIVERSITY OF MICHIGAN i 9110115 l 111 111111111 111111111 3 9015 02229 2430