THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE EFFECT OF LONGITUDINAL OSCILLATIONS ON FLUID FLOW AND HEAT TRANSFER FROM VERTICAL SURFACES IN FREE CONVECTION i ~..v... Siavash Eshghy May, 1963 IP-621

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ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to Professor Vedat S. Arpaci, Chairman of the Doctoral Committee, under whose direction this work was performed. Gratitude is extended to Professor John A. Clark, project supervisor and Director of the Heat Transfer and Thermodynamics Laboratory, The writer is grateful to Professors Arthur G. Hansen. Robert C. Juvinall and Ruel V. Churchill for their interest and cooperation. The research reported herein is part of the research program on heat transfer of unsteady flows of the Aeronautical Research Laborator-, Office of Aerospace Research of the United States Air Force whose support is acknowledged. The assistance of Dr, Mo Go Scherberg of this Laboratory is appreciated. The research support of the National Science Foundation, Grant Noo 9386, and the fellowship support of the Iranian Ministry of Education, are gratefully acknowledgedo Many thanks go to Mrs. Norlene Martin, Mrs. oAlice DeRienzo, and to the Industry Program of the College of Engineering for their help in the preparation of the manuscript. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS o............................ i.i LIST OF FIGURESo................................o...... v NOMENCLATURE................................................ ix LIST OF COMPLEX QUANTITIES.....................................0 xiii CHAPTER I INTRODUCTION......................................... 1 II THEORETICAL ANALYSIS...........e.................. 5 Statement of the Problem........................ 5 Formulation of the Problem.......................... 5 Perturbation of the Differential Equations........ 8 First Approximation, c............... 9 Second Approximation, ~..................... Third Approximation, E2........................... 10 Solutions of the Differential Equations............. 18 Large c Solutions for the Second Approximation..... 18 Small w Solutions for the Second Approximation;... 21 Comparison of the Two. Limiting Solutions............ 36 Solutions to the Third Approximation................ 56 III DISCUSSION OF THEORETICAL RESLULTS.................... 62 IV STEADY LAMINAR FREE-FORCED CONVECTION PHENOMENON..O.... 66 -v EXPERIMENTAL WORK.................................... 69 Introduction...................................... 69 The Test Apparatus............................... 70 Instrumentation of the Test Section................ 77 Measuring Instrument s."........................ 77 Procedure and Results of Heat Transfer MeasurementsOO.........OO......OO..QO............................ 84 Correlation of the Experimental Data............... 98 Summary of Experimental Results..................... 108 REFERENCES o o o.... o...... o.... o ----—.....1.............o 112 iii

TABLE OF CONTENTS ( CONT'D ) Page APPENDIX I EQUIVALENCE OF THE CASE OF AN OSCILLATING PLATE IN A STAGNANT AMBIENT TO THE CASE OF A FIXED PLATE IN AN OSCILLATING POTENTIAL FLOW................ 114 II TABULATION OF THE RESULTS OF THE STEADY STATE INTEGRAL SOLUTION..................................... 116 III DEFINITION OF a's AND's........................... 118 IV TABULATION OF Yn AND Xn FOR DIFFERENT VALUES OF PRANDTL NUMBER..................................... 120 V COMPUTATION PROCEDURE................................. 121 VI EXPERIMENTAL DATA..................... 126 iv

LIST OF FIGURES Figure Page 1 Analytical Model................................... 6 2 The Steady State Nusselt Number as a Function of Prandtl Number............................. 25 3 The Steady State Maximum Velocity as a Function of Prandtl Number.....o o.oe.. e..... P 26 4 Amplitude Ratio of Friction Factor as a Function of W*, for Pr = 0.757...... e... ooooo... e,...... 41 5 Amplitude Ratio of Nusselt Number as a Function of W*, for Pr = 0.757..........................* 42 6 Phase Angle of the Friction Factor as a Function of W0'* for Pr = 0 757........................ 43 7 Phase Angle of the Nusselt Number as a Function of u*9 for Pr = 0o757...... O.................. 44 8 Amplitude Ratio of the Friction Factor as a Function of w*~ for Pr = 10........................ 45 9 Amplitude Ratio of the Nusselt Number as a Function of w*, for Pr = 10........................ 46 10 Phase Angle of the Friction Factor as a Function of C*, for Pr = 10........e..o...o........O..O.... O 47 11 Phase Angle of the Nusselt Number as a Function of w*, for Pr = 10....,.........-........ 48 12 Amplitude Ratio of the Friction Factor as a Function of w*, for Pr = 100...................... 49 13 Amplitude Ratio of the Nusselt Number as a Function of w*, for Pr = 100.,.......o............ 50 14 Phase Angle of the Friction Factor as a Function of W* for Pr = 100,......,,........., o..... 51 15 Phase Angle of the Nusselt Number as a Function of w*, for Pr = l00.......................... 52 16 Quasi-Steady Friction Factor as a Function of Prandtl Number.............................. 53 v

LIST OF FIGURES (CONT'D) Figure Page 17 Quasi-Steady Nusselt Number as a Function of Prandtl Number...................................... 5 4 18 Critical Frequency Parameter as a Function of Prandtl Number.................. o.........-.0.. 55 19 The Steady Decrease in Nusselt Number as a Function of Prandtl Number..................... o..... 59 20 The Steady Increase in Friction Factor as a Function of Prandtl Number...................... 60 21 Physical Model for the Zero-Frequency Case.......... 67 22 A Sketch of the Test Section Assembly................ 71 23 A View of the Test Section and End Cylinders......... 74 24 General View of the Test Apparatus.................. 75 25 Sketch of the Smoke Generator...................... 76 26 Electrical Diagram for Measurement and Control of Energy Input to the Heater Elements...... o......... 79 27 A View of the Amplitude Pick-Up................. 80 28 Linear Differential Transformer Circuit for Measurement of Vibration Amplitude..o..o............ 81 29 Output Signal of the Linear Differential Transformer with the Core Displaced from the Null Position as Displayed on an Oscilloscope........... 83 30 Output Signal of the Linear Differential Transformer Under Vibratory Conditions as Displayed on an Oscilloscope................. o.....o......... 83 31 A View of Some of the Instruments..................... 85 32 Steady Free Convection Data,.........,,..a.......o 87 33 Typical Data Shwoing Nusselt Number Versus Vibration Amplitude Ratio.......................*.. 88 vi

LIST OF FIGURES (CONT'D) Figure Page 34 Experimental Data Showing Per Cent Change in Nusselt Number Resulting from Oscillations..,...., 90 35 Smoke Study Data Showing the Nature of Flow Regime Along the Length of the Cylinder for Various Heat Transfer Measurements................... 92 36 Smoke Study Data Showing the Nature of Flow Regime Along the Length of the Cylinder for Various Heat Transfer Measurements................... 93 37 Smoke Study Data Showing the Nature of Flow Regime Along the Length of the Cylinder for Various Heat Transfer Measurements.................,. 94 38 Smoke Photograph for Tw - To = 1390F, and No Oscillation..... e,ooo, a................ 95 39 Smoke Photograph for Tw - T = 139~F,A = 0.05 Inches and fr = 16 cps.................... 95 40 Smoke Photograph for Tw - = 1760F, A = 0.11 Inches and fr = 16 cps........oo........ o...... 96 41 Smoke Photograph for Tw - T. = 1760F, A = 0.07 Inches and fr = 45 cps....4................ 96 42 Smoke Photograph for Tw T = 1760~F A = 0.005 Inches and fr = 200 cps....... o....O..o.......... 97 43 Per Cent Change in Nusselt Number Resulting from Oscillations as a Function of Amplitude.............. 99 44 Smoke Photograph for Gr. Pr = 3.56 x 107, and No Oscillation,..................................... 100 45 Smoke Photograph for Gr. Pr = 3.56 x 107, A/2 = 2.9 x 10-3 and fr = 12 cps..................... 101 46 Smoke Photograph for Gr. Pr = 3.56 x 107, A/2 = 5.7 x 10-3 and fr = 12 cps................ 101 47 Smoke Photograph for Gr. Pr = 3.56 x 107, A/2 = 8.1 x 10'3 and fr = 12 cps..................... 102 48 Smoke Photograph for Gr. Pr = 3.56 x 107, A/ = 9.5 x 0l3 and fr = 12 cps..................... 102 vii

LIST OF FIGURES (CONTsD) Figure Page 49 Smoke Photograph for Gr. Pr = 3056 x 107, A/2 = 11.3 x 10-3 and fr = 12 cps........,....... 103 50 Smoke Photograph for Gr. Pr = 3.56 x 107, A/2 = 14.7 x 10-3 and fr =12 cps..o............ 103 51 Smoke Photograph for Gr. Pr = 3.56 x 107, A/2 -18.1 x 10-3 and fr = 12 cps,,..,......ooo...o 104 52 Correlation of Turbulent, Oscillatory, Free Convection Data...o..o........o........... 0...oooo..O 106 53 Correlation of Laminar Datao.........O......O...ooo 109 54 Correlation of Laminar Data...........0oo.,...0....o 110 @ * @

NOMENCLATURE a Thermal diffusivity of the fluid A Amplitude of oscillation A1.. A5 Functions of Prandtl number AT Total heat transfer area B Coefficient of r in the second approximation velocity profile B1 Coefficient of Aq in the second approximation temperature profile Cl...C5 Functions of Prandtl number Dl... D3 Functions of Prandtl number En Notation defined as being'it when n is odd and'1' when n is even f Friction factor fr Frequency, cycles per unit time F(q) j dependency of steady free-convection stream function g Gravitational acceleration G(A) A polynomial function of A Grx Grashof number, pgGwx3/v2 Gre Grashof number, wgBwe3/v2 h Coefficient of heat transfer H(i) Dimensionless temperature function in steady free convection i Positive square root of -1, f-1 k Thermal conductivity K A constant 2 Characteristic length n Dummy integer in the series ix

Nux Local Nusselt number, hx/k Nu2 Average Nusselt number, 1 f h dx p Fluid pressure Pr Prandtl number of the fluid q Rate of heat transfer Rej Reynolds number, U x/v Rei Reynolds number, UIe/v S(A) A polynomial function of A t Time variable T Temperature of the fluid. u x-directional velocity of the fluid U(t) Potential flow function Uo Coefficient of E cos wt in the expression for U(t) UjO Uniform velocity outside the boundary layer v y-directional velocity of the fluid V(A) A polynomial function of A VW(x) Velocity function in the steady-state integral solution, PgSW82/6v V -o A constant defined as PgGw/6v W(A) A polynomial function of A x Distance along the plate X(A) A polynomial function of A y Distance perpendicular to the wall Y(A) A polynomial function of A Z(A) A polynomial function of A

Greek Letters %'a 9all ThFunctions-of Prandtl number Coefficients of volume expansivity at constant pressure. 1.... Functions of Prandtl number 7n Coefficients of the series solution for B 6 Velocity boundary layer thickness 59 Thermal boundary layer thickness bij Kronecker delta A Ratio of boundary layer thicknesses, 6g/5 e I U(t) I/uomax Em Emissivity of the radiating surface Steady-state similarity parameter, y/5 Ad Steady-state similarity parameter, y/56 Temperature difference, T - T~ Xn Coefficients of the series solution for B1 Viscosity of the fluid v Kinematic viscosity of the fluid Dimensionless boundary layer thickness, 6/2 tl1 A dimensionless number, Pg@wI2/vUO "2 A dimensionless number, PgGw52/vUo p Fluid density a The Stefan-Boltzmann constant Tw Shear stress at the wall c@ ~ Phase angle ~V ~ Steady-state stream function xi

W Circular frequency, radiants per unit time Frequency parameter, 862/v Q Dimensionless frequency, w2/v Subscripts W Condition at the wall (y = 0) X0 Condition outside the boundary layer o First approximation 1 Second approximation 2 Third approximation 20 Steady component of the third approximation 21 Oscillating component of the third approximation xii

LIST OF COMPLEX QUANTITIES B B1 fl f21 hi Nu1 U1 v1 V21 @1 @21 T Wl xiii

CHAPTER I INTRODUCTION Much work has been published on the analytical treatment of problems in steady as well as unsteady laminar free convection. Schmidt and Beckmann, 1930,(1) S. Ostrach, 1952,(2) E. M. Sparrow and J. L. Gregg, 1956(3) and 1958,(4) Finston, 1956,(5) Millsaps and Pohlhausen, 1958(6) and some others have obtained analytical solutions for steady heat transfer from vertical plates and cylinders, each using a different condition at the wall. Yang, 1960,(7) lists possible similarity (exact) solutions for the steady free convection flow and heat transfer as follows: 1. From a vertical flat plate with a. Prescribed constant surface temperature bo Surface temperature varying with any power of a linear function of xo c.'Surface temperature varying with an exponential function of x. 2~ From a vertical circular cylinder with surface temperature varying linearly with xo In the above four cases, the restrictions are imposed only on surface temperature, however, for each case, one can find the similar restrictions on the surface heat flux. Sugawara and Michiyoshi, 1951(8) obtained approximate solutions, valid for small time, for momentum and energy equations in the case of a step change in the surface temperature of a vertical flat plate. Siegel, 1958(9) treated both the above case and that

of a step change in surface heat flux; he used integral techniques on the boundary-layer form of momentum and energy equations. Their work was followed by a few unsteady exact solutions in which similarity techniques were used exclusively. Yang,(7) presents possible unsteady exact solutions obtainable by the similarity technique. These are: 1. For the case of a vertical flat plate. ao Unsteady free convection with surface temperature varying inversely as a linear combination of x and t. b. Surface temperature varying directly with a linear function of x and inversely with the square of a linear function of t. c, Uniform but unsteady surface temperature variation at large distance x. 2. For the case of a vertical circular cylinder, uniform but unsteady surface temperature at large values of x. In all of the above cases, the cause of the transient is some unsteady thermal condition at the wall. Chung and Anderson, 1961(10) treat the case of free convection from a vertical plate and a horizontal cylinder in which Grashof number is an arbitrary function of time. The solution obtained applies when either the uniform wall temperature or the acceleration field is arbitrarily time dependent.. Their analytical approach is a combination of' similarity and successive approximation techniques. In recent years much attention has been focused on unsteady convection heat transfer caused by flow oscillations. This can be achieved either by mechanical oscillation of the heat transfer surface or the

-3super-position of a sound field on the external flowo Lighthill, 1954(11) calculated the response of laminar skin friction and heat transfer from an oscillating arbitrary cylinder. Illingworth, 1958(12) treated the case of a heated flat plate with zero incidence in a low speed stream containing a progressive sound wave. His calculations are for a compressible fluid with a potential flow of the following form (T(x,' -. [E U. E,c - t- Cos where M is the Mach number of the mean flow. He carries only two approximations and obtains l- imit-ing- solutions for both large and small frequencies. Kestin, et al., 1961(13) analysed the same problem for an incompressible fluid with a potential flow of the form Uf(xL). =;,~Ft 6 Cos( - ) They calculate the first three approximations for the case of small frequency cX Considering the third approximation in their' calculation, they obtain the net change in heat transfer and skin friction to be negative and positive, respectively. In the above two cases, the effect of gravity has been neglected such that the heat transfer mechanism is only forced convection. Schoenhals and Clark, 1962(14) present an analytical treatment of the effect of transverse wall vibrations on free-convective flow and heat transfer from a vertical flat plate. They carry the first two approximations for the case of an incompressible fluid. Blankenship(l5) extended Schoenhal's work and carried the third approximation for both finite as well as infinite plate.

-4-_ The problem of free-convective flow and heat transfer from a vertical plate oscillating in its own plane and in the direction of a force field, has apparently remained untreated. A limited number of experimental measurements have been made of the average heat transfer coefficient. Chamberlin(l6) oscillated an open-end vertical cylinder in air and found no noticeable change in the free convective heat transfer. Spratt, et al.(17) studied the effect of strong sound fields on. free convective heat transfer from a horizontal cylinder and found substantial increases in heat transfer. Holman, et alo,(18) introducing artificial oscillation in free convective flow along a flat plate, found no change in the steady heat transfer. Soehngen and Holman,(l9) studying the effect of sound fields on free convective heat transfer from a horizontal cylinder, conclude that for low intensities there is no change in heat transfer but for high intensities substantial increases are noticed. In the present research, the problem of free-convective heat transfer and fluid flow from a vertical flat plate oscillating in its own plane has been given analytical treatment. An experimental program has been conducted. Measurements of the mean coefficient of heat transfer from an oscillating cylinder with large diameter have been made. This system adequately models a flat plate.

CHAPTER II THEORETICAL ANALYSIS Statement of the Problem The analytical model is shown in Figure 1. It consists of as, thin flat plate extending from 0 to oo in x direction and from -0 to + o0 in z direction. The plate has a uniform temperature Tw which is greater (for g being in the -x direction) than that of the ambient at rest, signified by T. 0 The plate oscillates harmonically in its own plane and in the direction of an external (gravitational) force field. It is desired to determine analytical expressions, for the rate of heat transfer and the wall shear stresso Formulation of the Problem The following assumptions have been made in the formulation of the problem described above: a. The flow is laminar - this will restrict the upper limit of Reynolds number~ b. The boundary-layer approximations are valid - this will restrict the lower limit.of Reynolds number~ co The properties of the fluid are taken to be constant except for the thermally induced density variations which give rise to a buoyancy. term in the momentum equation (c.fo Reference 1) d, Viscous dissipation and other compressibility effects are neglected~

HEATED PLATE 1 Too DIRECTION OF PLATEjU (t) | INFINITE FLUID MOTION | AT REST Xi I Figure 1. Analytical Model.

-7Based on the foregoing assumptions and with respect to a reference frame fixed to the plate, the momentum, continuity and energy equations, and the appropriate boundary conditions are: t _ u_ axL t e +A U(X,ot') -' (X,,o t) = o t, o t) U(t) "r1+ a= o(2) -t + uL S- c v 9Z eS(xc,,t) = a O(x, o, t) - O The same formulation would be obtained for a fixed plate in an oscillating potential flow. This fact has been demonstrated in Appendix I. Equations (1), (2), and (3) must now be solved simultaneously for u, v, and @. Considering that a similarity solution does not exist* and Equation (1) is nonlinear, a linearization process must be carried out if the problem is to be approached analytically. Assuming E to be a positive number less than unity, the velocity of the potential flow can be specified as U(t) = U,Cos t (4) *-The possibility of a similarity solution, in its classical sense, was investigated but no suitable transformation was found that would reduce the partial differential equations to ordinary differential equations.

-8where Uo is assumed a constant independent of frequency of the plate oscillation. The order of magnitude of Uo, as a reference velocity, is assumed the same as the maximum velocity of steady free convection at a distance x = Xe Introduction of E into the expression for U(t), is the first step towards perturbation of the governing differential equations, which is a well-known technique of linearization. The quantities u, v, and G are perturbed around their steady state values ioe., the case where C = 0.'Perturbation of the Differential Equations Considering the nature of U(t) expressed in Equation (4), the following forms may be assumed for the functions involved. u1( A ) =,( x,, + - U 1 ( ~xX. t) ) (~X, J. t ) (E 6) Sousitt:) ~,,(:~,,) ( -O, ((6,,t() -, — and () (7) Substituting Equations (5), (6), (7), and (4) into Equations (1), (2), and (3), and arranging according to the powers of E and neglecting terms of the order of.3 and higher, results in

First Approximation, c~ -O —- + — a = o0 +9 (8) Lo +;\ t a) elo-sx + U; aX a~ (lo) The above equations which are the formulation of steady laminar free convection from a vertical flat plate, have been solved and tabulated for eight different Prandtl numbers by Ostrach. (2) Second Approximation, 1 t [: -tUoC COS + L9$ + ~-o k (11) U, (,o,t) =?(x,o t) -o; U1 (X, ot) U. Cos t (12) jt1, )t

(13) do (xo, ot), (x, o t) = o Third Approximation, E2 -- LU 2 - + C + + - u 13 2 --- +~ - 1 -- LA o1 (14) U2(X,O,t) = ~',X o, t) — o dtA2 aLv7 -1- o (15) In all the above equations, u1, u2, 91 and 92 are functions of time t, as well as x and y. If solutions are sought for large values of time

where the effects of initial transients have died out, the initial conditions become immaterial and the time dependency of the functions may be predicted. The periodic term Uo cos ct appearing in Equation (11) suggests that all the first approximation functions are also time wise periodic functions with the same frequency w. In effect, the solutions thus sought will be steady periodic in nature. For mathematical convenience, use is made of the imaginary exponential functions in the following manner. [J(tz —RjCi'Tte $ - U, cEs I t7 l,(xjl -.; a{ L(/'J)5 C * id (17) where t denotes the real value of the complex quantity in the brackets. The new functions ul, v1 and 91 are now complex quantities and independent of t. Substituting (17) into (11), (12), and (13), and observing that the symbol appears in front of every term, it is concluded that since L is a linear operator (the sum of real values is the real value of the sum), it can be dropped out of the equationso All terms are now linear in eict which can also be dropped out. In this manner, time dependency is omitted from the differential equations and time-independent equations and the boundary conditions are obtained for the second

-12approximat ion Lw(L A -U6) tL A~ --- - 4 --- -~ A- -- 3Oj 2 O (18) (X, O) - u; (,O)o; 1( X, m )t U(, --, ). = ~ (19) ax o. o; 1 i + tOL -.. +eo.............. V +X -+.... %-0 (20),1(XO)-= e,( A) w)- o Substituting Equations (17) into (14), results in, - Lt- z. u-~ U. +L' A +lz g i'="2ft L Uta (V1){LL p t)

-13 - The terms on the right of Equation (21), can each be reduced into a time-independent term and a harmonic term in the following manner. 1Rrt, e. - L,1R CoC 5 t - UlrL:t. v XIR 1I where ulR and ulI are the real part and the imaginary part of ul respectively. vlR and vlIp similarly, correspond to the real part and imaginary part of the complex quantity vl. Then, g ~{ e t} tW { U t~)SI C z Q i. I b u, i. (Lij t 3 Cos:~' i -L'. S L t - (21a) 2 ( 1 R a X 1 1 r )X ) S Ln tT Considering that, cSnos t -- u1 (1 -Caos2Lt) Cos Ut -' + CosaS ot) 2

-14Equation (21a) reduces to ( ~L kir e-*<-)St;' tA1 a Lk1L ) CosZt) - This can be further reduced by the use of the following identities ~~~~~'ttto its f inal form, 7 { t 9L~L LI ( L 1a -,) t * ) - e ( UA - L1.;4Dx 2x + Ix,}' x C S2 t?,... e=,, -R'. _,, ) cos 2 toz to its final form ax 1 ax~~~~D

-15 - where the superscript bar (-) denotes the complex conjugate. The term can also be reduced, in a similar manner, to its final form (21c) Substituting Equations (21b) and (21c) into (21) results in ~ —''t;'' (22) 13J0 i i ){e where uio -— 5-('tX (u, J + V. + J -, and, Inspection of Equation (22) reveals that i f it is to be satisied Inspection of Equation (22) reveals that, if it is to be satisfied uniquely, for large values of time, then the quantities u2, v2 and @2

must have the following forms L1( t = z(X1 ),2 (t CX K'" C' and, V:Xj XqL) ( vl I (X (23) and, -} JK ( x -t-) e ( e2= tli) -i<- j 92 (X are in order to comply with u10 + l,{ulle2iwt} in Equation (22). It is obvious from Equation (22) that unless u10 is identically zero, the functions u2, 92 and v2 cannot be purely harmonic. Conditions of Equations (23) are such that Equations (15) and (16) are also identically satisfied in time domain. Substituting Equations (23) into (14), (15) and (16) and separating the terms with the symbol (R and without, the following sets of equations are obtained for the third approximation. a. Non-Oscillatory Components -(1 d t_ _ a- )V=Io (24) Ub0(X ) I 0:( X) ) = 0

-17Cxot D = (25) U _J620 t- w —s0 —a- - J2- + Itzo-)x u t 7J -- 1 ( us -itT7, + ) (26) 02(X' = 0 b. Oscillatory Components Zitw(,I )Lta at, Ul, ~, tt ~- u z L2 (A L( 2 2Lw =0 t V _o (27) Ul LA, (A X~-, oo) -~~~~~ o ~~~~~~(28) +x aX I 0 (29) A A I tin1(,00=

-18Solutions of the Differential Equations Analytical treatment of Equations (18) - (20), (24) - (26) and (27) - (29), in their present form, is very unlikely to produce the solutions desired. It is, however, possible to obtain solutions valid for limited ranges of some of the variables and parameters involved. One such parameter, is the frequency w, It is mathematically possible, for large values of this parameter, to evaluate the order of magnitude of the terms in the differential equations. This would suggest that solutions be attempted for two regions, namely, the large w region and the small w region. In each of the above mentioned regions, it is sometimes necessary to subdivide the already limited range of w, into subregions of small y and large y. This subdivision, however, does not affect substantially the accuracy of the results obtained for heat transfer and shear stress at the wall, considering that these quantities themselves are evaluated at the wall. Large w Solutions for the Second Approximation It can be shown that in Equation (18) which is linear in u1 but nonhomogeneous, au, -O U Au- O[Uil when X becomes substantially large. Rewriting Equation (18) with the

-19order of magnitude of each term written beneath it, one obtains, Lll Lt LA, t q; <la Comparing the linear terms and retaining the ones with the highest order of magnitude namely [wul], results in Lb~:-J —~-tO = ~6, t ~1Uo + The nonhomogeneous terms, however, should be compared not to the linear terms but to themselves, It is now physically obvious that since the cause of transient (which produces 01) is mechanical, in nature, rather than thermal, gOg1 may be neglected, for large c, compared to iwUo. Therefore, = LLU U1()(x - o) (50) u: (X<,) - Uo I is the governing momentum equation within the limits of the foregoing assumptions. The solution of Equation (30) may readily be obtained in the form u- Uo( -e ) (31)

-20 - and furthermore, vl = 0 from continuity considerations. Evaluating the order of magnitude of terms in Equation (20), in the same manner as above, one obtains the following [tu'0 t M, -- [t "ra (%'10 Neglecting the terms whose order of magnitude is smaller than [G1l] and considering that vl = 0, one obtains the governing energy equation and its appropriate boundary conditions for the second approximation as 9- - a)cX (32) o1( XO) 61( K I = o The general solution to Equation (31) is readily obtainable, however, the particular solution will involve the function GO(x,y) which is available in tabulated form and therefore cannot be operated on analytically. This may be overcome by approximating the exact value of GO by its tangent at the wall, for small values of- y, in the form then, adO _;~, n Ap\

-21Considering that ( )w is independent of y, expressed in (y ixsien Equation (33) is now explicit in y and can readily be operated on with respect to y. Then the solution of Equation (32), near the wall, is et(X,7) = Si) eS<ax) -S( lCA 1~e (34) 7a77 \12- egg Away from the wall (near the edge of the boundary layer), as Lighthill(ll) points out, W2 may be neglected compared to iwg1 and thus a solution is obtained for @l(x,y) valid for large values of y, as U'O~~~~~~U Small X Solutions for the Second Approximation In the preceding section asymptotic solutions were obtained for large values of frequency w. In order to estimate the range of validity of these asymptotic expressions, it seems essential to attempt solutions that would be valid for the other extreme, namely small values of frequency. When such solutions are compared with each other, it is usually possible

-22 - to estimate, by observation, the vicinity of the point of transition of one limiting solution to the other. Solutions will be obtained for Equations (18), (19) and (20), for the case of small values of frequency w, using integral techniques. The integral procedure adopted in this work requires an integral solution of the steady state free convection. Although some integral solutions do exist in the literature for the case of steady free convection (c.f. References 20 and 21), an integral solution was derived using fourth order profiles both for temperature Go and velocity uo. U0 V(X) ~~~3 ~(36) where 7 Y and a Equations (36) already satisfy the appropriate boundary conditions, second order smoothness on the edge of the boundary, and the following conditions obtained from Equations (8) and (10) for y = 0 (at the wall)

-23 - and namely, aa)( ag'')w = ~e3J (37) Defining A -= 0/6, there are two unknowns to be found, 6 and A. These will be found by satisfying the integrated form of Equations (8) and (10) over the thickness of the boundary layer, 9 o = ta( t taeo (38) 0 0 and d 1~a 2( aJ D*(39) 0 The following value is obtained for 6. G = 5.9 [(1.8t -X) 49ge j (40) Here A satisfies 378 A(71.8A-I)G() -- --- (41) Pr

-24where Grh ) = as h,5 t+ 2980 51 (42) 20~ 30 /O Io -o The values of A have been tabulated as a function of Prandtl number Pr in Appendix IIO Figure 2 shows a plot of Nux/(Grx)l/4 versus Prandtl number Pr and a similar plot has been presented in Figure 3 for Umax / gOwx The agreement of these calculations with the exact similarity solution (2) is very good for Prandtl numbers ranging from 0.7 to 200. A somewhat similar procedure is used for obtaining solutions for Equations (18), (19) and (20). Here it was found convenient to use fifth order profiles for the functions involved. The result is Uo-1- + i (7- I57 ~37 ) + ~. —-(-7 +37-? ) + -3-(7 -27 +7 ) + (43) 65 Ta A (-7 +27 -7 )

5 4 3.8.08 o.06.04.03 L.01.r I 10 100 1000 Pr Figure 2. The Steady State Nusselt Number as a Function of Prandtl Number.

EXACT SOLUTION.8 --— INTEGRAL SOLUTION.6 Q'.3.080.06.04.03.02.01.1 I 10 100 1000 PrFigure 3. The Steady State Maximum Velocity as a Function of Prandtl Number.

w -8[ Is-Y1 t", "Y 3 7 - (44) The velocity profile u1 already satisfies the following conditions: at y = 6: 0 a1 - = 8 =l = C) at y=O: 6,= o; ) (45) a~ - 9 7~ 7 - where the last two conditions have been derived from Equation (18). On the other hand, the temperature profile @1, satisfies, at y =6:'o at y=O: -1 (26) where the last two conditions have been derived from Equation (20). The unknowns B and B1 will be determined by satisfying the integrated forms of Equations (18) and (20). The velocities vo and vl have been

-28eliminated by the use of Equations (9) and (19). O O (47) 00 00 O2 oiLo', tl - UO ~(7 o o;w 1 d t t( d- t Ax ( 6~8 + eo u1 7 -~ o (48) Substitution of Equations (43) and (44) into Equation (47) results in the following differential equation: I - -. 33 ~U -2 Uo (g3 dX ( T ) h'" (5 -s + 3'' ( G ) t L7)Z (3/ lx _ ( I)8 ) 36o t ( 3o ( B) ~~~~~~~~~~~~~where ~~3 1) = ( T' ~'/-O-,o31 c 3 I O 1(26C1 dX 126o 6X (49) -a where V ~ (x)

-29Equation (49) is the new form of the governing momentum equation. Similarly, the new form of the governing energy equation can be obtained by substitution of Equations (43) and (44) into (48), and namely -1 Lu9h t 51 ( 6t0- 5 c h 2 ) + t O(ab) Bx + Uox [ Z o W~ - tB +VS(), t (50) 6 Pr- Y() t) t, 8 - 31 Uo V(L),=-~_'-...... =o where s' ( t- ( 2 _6 h e 1 2 e 2 3 (51) I)( ( 3 6

-30Now, Equations (49) and (50) must be solved simultaneously for B and B1. However, before attempting this, new dimensionless parameters are introduced, for convenience, and equations are rearranged accordingly. The independent variable x is replaced by the boundary layer thickness 8(x) and non-dimensionalized by using a characteristic length, B. The new dimensionless quantities are: 2 n=3~ 9t-~~~~ (52) By substitution and rearrangement, Equations (49) and (50) become a d -l 2 3 (It<i'Sj ) ~(" + c<FiLA -<6. ) 13 F3 2 B 3 ( and (I+AySX ) dZ t (d, + 6 Q -tn 1 )B13 I r [(~1 43 + 11 A ) if (3 33 t 486Jl!) + (54) -3 -1 ~42 +3sLhIa j

-31where a's and W's are functions of Prandtl number alone and are defined in Appendix IIIo Since solutions are desired for the case of small values of u, it seems appropriate to attempt a series expansion of B and B1 around their quasi-steady values (a = 0), The quasi-steady case describes the case of a vertical heated flat plate moving parallel and in the direction of gravitational field, with a uniform velocity cUOo The two velocities will be additive and increases are anticipated both in the rate of heat transfer and the shear stress at the wallo This fact was the criterion for improving the profiles involved. The present profile for velocity ul gives a friction factor that when plotted against Pr, will cross the Pr axis around Pr = 500. This will limit the upper limit of Prandtl number for which the following solution will holdo The quasi-steady friction factor and Nusselt number are plotted versus Pr, in Figures 16 and 17 respectively. In expressing B and B1 in their corresponding Taylor series expansion, one can write 7 sn=O B1(!ft) = X no a BI3:) ni n=o This assumes that B and Bl are continuous functions of Q with continuous derivatives and that the order of differentiation with respect to Q and 6 can be interchanged, By differentiating Equations (53)

-32and (54), n times with respect to 2 and setting 2 = O. The following generalized equations are obtained:* 3 d 0() fd () A fl aflfIoo 6 a b'(,o) c6- o n - nt v- 8 ~ =C n (n3 6 X 0Ro) (55) ( 1( ~ dJ [/ ) + -1 3 0on a) 3 C2 n 0 10 where 5ij is the Kronecker delta defined as, 1, s - *~ It is understood that a() of any quantity is zero, when y < O and, the quantity itself, when y = 0.

-33 - and 01) ~ d (r n-1)) (n -J) t nR dQ- kH J n(n-n)!3 ta a n-., g a 5f,,, Go ) = ~ 1 (56) l/3 8 13 d.2 + 3 a (., l. (r) rln~l d B(io) + 13 In order to satisfy Equations (55) and (56) simultaneously for all n, the n-th derivatives of B and B1 with respect to,. must have the following forms, (n) * 6(Avo) E 2n (57) () n)TT n where 1, when n is even En = i, when n is odd

-34When expressions in Equation (57) are substituted into (55) and (56), powers of r are eliminated from the equations for n > 2 and general expressions are derived for the coefficients of the series solution. When n = 0, 1 and 2, one should refer to Equations (55) and (56) and solve the special form of these equations for the values of n < 2. These general expressions are: ~1Xn -'2 (A tz X/n-9 L/9 Xn3 ) A 1< A- C~~/93c3 l / cy~ C1/\ FCza,\) t (58) _,~.~ a c9h, C+ h. c XH _ 3( 0 Xn 2 1 ) I where A =n- rj ) c<r ~ a ( n-) i- A, = Y (n-I)46 q3 = 9.(~-l)o t ia ~ n a > (59) /9 = (- t) c<: ~cK

-55and C- _n,-. t ) 2 _ (- ),,'(-, _ t t~ |1, ( ) ( (60) Cs)7 -' 8]'8 In terms of yn and Xn, then, the quantities B and Bi take the following forms. O/=0 2~(~'~75 ) E,7.j t - o' za, 3 6 (61) i\ - o n o ~. Q A + 2 Z /I The coefficients 7n and Xn have been evaluated for Prandtl numbers 0o757, 1, 10 and 100, using the University of Michigan IBM 7090 Digital Computer. The first few of these coefficients are listed in Appendix IV.

-36Comparison of the Two Limiting Solutions The combination of O, 2, and 1l in the preceding calculations, gives rise to two new dimensionless parameters which are independent of j and namely, and 2-2( The quantities chosen for comparison are the frequency response of the Nusselt number and the friction factor. These quantities are defined as: For large values of frequency o, Nul and fl are expressed as: Blr - -w Nu I +__._' D., (62) ~/_3:Z

-37and for small values of frequency, n=o OEn From the above equations, the quasi-steady values can be deduced as (64) )=0 U The quantities in Equations (63) are complex whereas the ones in (64) are real. Therefore, it is appropriate to normalize the quantities in Equation (63) with respect to their corresponding values in (64) and plot the amplitude and phase angle of each quantity as functions of frequency W*, for different values of Prandtl number. One of the terms in Equations (62) is the steady function 2o ( 0) that can be obtained from the numerical solution of Ostrach(2) ~yx w in the following manner. In Reference 2, go is expressed as o= is H(U7) where -a/c and C = (436Li I

-38Then it follows that, ( 9 2~ ) _ C Ow H (~) \ (z~ x w- -c0,H( ) SI w'/"/ t9 _H (o) (GyX) X After necessary substitution and rearrangement, the following expressions are obtained for the normalized Nusselt number and friction factor frequency responses. io(',, v.) e., for large values of o* (N1) - W=O (65) En for small values of w* n=o ( ) = X ov e, for large values of o* (66) I -n(~ U, for small values of w*,

-39where a11 is a function of Prandtl number and is defined in Appendix III. Defining cpfl as the phase angle of the friction factor and'PNul as the phase angle of Nusselt number and the amplitude of each quantity by placing it between "absolute value" signs (I I), they can be expressed as follows: (a) for large values of co* 0 l~ = q 5 (t1)w=o (67) Ik,~/u, -1 -$ Nul - and (b) for small values of w* \ n+v 2n+I t n=o n=o (68) _ 1) = gYzln *lzg t q.+an I (n — - F arj z

-40 - 00 nzn+l =3 Nut~~1 00~o |2f~ * (68 cont'd)! a 2 ) JNut) _ ) ___ ___ _ l=o O _____ _21,+ L NU,) AO (2n)! n~=O j Y1=o ro' Figures' 4 and 5 show plots of amplitude ratios versus u* for Pr 0.757. This is the Prandtl number that makes A = 1 and is the lower limit of Pr in the small frequency analysis. Figures 6 and 7 show the variation of phase angles with the frequency parameter wc*o In Figure 6 the two limiting curves intersect at a frequency o* and in Figure 7 at These two frequencies, however, are identical for Pr = 0.757 where the two boundary layers (velocity and thermal) are equal and different for other values of Pro Careful inspections of Figures 4 through 15 indicates that the frequency w* is a good criterion for estimating the range of 0 validity of the two limiting solutions. It appears that 2 w* may be used as the lower limit of the large frequency solution and 1/2 wt as the upper limit of the small frequency solution. Figure 18 shows that c* is a decreasing function of Prandtl number. This is the most important conclusion drawn from the small frequency calculations. This criterion will be extended to the large

30 520....... _ ___Pr 0.75 20 1 5 I0 - - -. -- f 8i 9,',,/o........... 0 10 20 30 40 60 70 80 90 100 110 120 30 140 ISO 160 W, 7 i F F c *............... F.5 4/ /~~~~~~~~~~~~~~~ 2.......,..,~ , ~ ~ ~ ~ w /.................. Amltd.ai fFito atra ucino w...

10.5 4 3 5 4 3 2 INu1I __ 10 ( NU1)W=O 5 4 3 2 -2 10 5 4 3 2 I 0 0 10 20 30 40 60 70 80 90 100 110 120 130 140 150 160 * 1/ Figure 5. Amplitude Ratio of Nusselt Number as a Function of w*~.

80 Pr =0Q 757 70 60 UI) W 50 40 20 10HA r n 0 10 20 30 40 *. 60 70 80 90 100 110 120 130 140 150 160 Wa ~* w s2 W3 W Figure 6 Phase Angle of the Friction Factor as a Function of w*.

+20 - - - -- -... - - - +10 Pr: 0.757 -I0 -20 C,) -30 W w (C -40 w LI.I o~~~~~~~~' \ IIi~ -50 Z ~ I -6C -70 -80 -90 - -- -100 -7c \ - -.......-.0 10 20 30 40 60 70 80 90 100 110 120 130 140 150 160 170 0 I0 20 30 40 (u~~~~~~~~~~~18 *( w2 Figure 7. Phase Angle of the Nusselt Number as a Function of c*.

-4550 40 Pr io 30 20....... ic _________* ________ o0 I 1 I C 8 ( fl)CO 7 6 2 = - 0 10 2t 0 30 40 50 60 *... Figure 8. Amplitude Ratio of the Friction Factor as a Function of w*.

8 7 6 5 4 3 2 Pr io 8 6 5 INull 4 (Nu1)wuo 3 2 10 8 6 5 4 3 2 1 0 0 2 4 6 8 10 12 14 18 20 22 24 26 28 30 32 34 Wo +Ws m/ Fi~gure 9.. Amplitude Ratio ofl the Nusselt Number as a Funiction of w*c.

90 80 r =io I r r~l 70 60 W 50 w 40 30 20 10 0 2 4 6 8 10 12 14 18 20 22 24 26 28 30 32 34 Figure 10. Phase Angle of the Friction Factor as a Function of wu*.

60 40 Pr io 20 0 -20 IAJ Id (Z -40 of -60 -80O -100 -120 -1 4 0 0 2 4 6 8 10 14 16 18 20 22 24 26 28 30 32 34 (A) Figure 11. Phase Angle of the Nusselt Number as a Function of cw*.

-4970 60 50 40, 30 20 Pr = OO f, (fi)w.d.. ___ 8 7 6 5 4 2 I - - - W..- - 0 2 3 Fgure4 5 6 8 9 0 IIactor 12 Figure 12. Amplitude Ratio of the Friction Factor as a Function of w*.

Pr = I'oo 5 (I),,o I= 4 I2 34 4 6 7 8 9 -ge3Apteao 0 Figure 15. Amplitude Ratio of the Nusselt Number as a Function-of w*.

100 901 T I Pr= Ioo 80 70 60 (I) w LUJ 50 40.. 30 20.I' 0 2 3 4 5 6 8 9 (A) Figure 14. Phase Angle of the Friction Factor as a Function of' w*.

60 40 r =ioo 20 0 Lj -20 LLJ Uj -4 0 -60 z -80 -100 -120 -140 0.2.4.6.8 1.0 1.2 1.4- 1.6 I. 8 2.0 2-.4 2.6 2.8 3.0 3.2 3.4 wP A Wa Figure 15. Phase Angle of the Nusselt Number as a Func~tion of w*.

.8 e0 0.7 I 2 5 0 50 100 500 1000 Pr Figure 16. Quasi-Steady Friction Factor as a Function of Prandtl Number.

6 6 C0 \J1 Cb D 431.7 I 2 5 10 50 100 500 Pr Figure 17. Quasi-Steacly Nlasselt Number as a Aunction of Prandtl Number.

80 60 40 30 20 I0 8 Wo 6 4 - -._ __ 3_ _ 2.5.1.2.4.6.8I 5 O 50 IOO 500 Pr Figure 18. Critical Frequency Parameter as a Function of Prandtl Number.

-56frequency solutions of the third approximation; this would eliminate the necessity of carrying out a separate integral procedure. Therefore, only large frequency solutions will be obtained for the third approximationo Solutions to the Third Approximation As it was demonstrated in Equations (23), third approximation velocities and temperature consist of a term which is only a function of x and y, and another term which oscillates harmonically with a frequency of 2co. The oscillating component that integrates to zero with respect to time, is of less importance compared to the non-oscillatory component which contributes to the steady phenomenon of flow and.-.heat transfer. It is this component that results in a net change in friction and the rate of heat transfer. Therefore, attention is focused on these components that are governed by Equations (24), (25) and (26)~ Solutions are sought for large values of frequency and the region near the wall which allow simplification of the differential equationso Substituting Equation (31) into (24) and (26), gives, (A 2-o 2- de) Lto )J8~+ a-y + 020 X+ 70-d ix ax ro d~~~~~~~~~ (' c~S0 _ __

-57It should be noted that near the wall (small y), Uo X ax 2 - aU2o a D where ZXO signifies proportionality. Near the wall and for large values of frequency, the differential equations can be simplified by retaining only the highest order terms. Neglecting terms of the order of y2 and smaller, the following differential equations are obtained. z (69),()," ) =0 O and, X__ = yI ( - aX (70) sol (X, O ) - O Equations (69) and (70) plus the condition of the boundedness of general solutions of the differential equations, are sufficient to obtain expressions for the rate of heat transfer and shear stress at the wall.

-58The value of 91 (x,y) used in Equations (70) was expressed in Equation (34). Defining Nklocx —._8- x a w 0) and __ = ( __ auzo ) the final results of the third approximation are expressed as: Nb'zo.~ - 3 H( (l+) Pr)( ) (6r) N Ut H (0\)m2 (Gry) (14c) and, (71) 7e5 _ 5'0' ( *) ((' ) /(_ Figures 19 and 20 show the Prandtl number dependency of N20x and f20. Summing up the results for large values of frequency, NuX N LAoX - C (P Nt + C ~. Nllzl ~ (72) and fa z t 220

-I 10 -2 3,10 I0 Z6 4 - -- _ 1 0 2 3 4 5 __ _3 ~~~Ix~~~~~~~~~P r Figure 19. The Steady Decrease in Nusselt Number as a Function of Prandtl Number. t o I0-I 23$4{5 I 23$4,5 10 23$4.5 102 2 34 5 I0 $ 1 0 Pr Figure 19. The Steady Decrease in Nusselt'Number as a Function of Frandtl Number.

5 Nt 2'C 3 -2 10.02 3 45 I 2 34 5 10 2 34 5 102 2 3 45 10Q4 P r Figure 20. The Steady Increase in Friction Factor as a Function of Prandtl Number.

-61Substituting each term with its corresponding expression and retaining the first three terms, the following equations will be obtained. Nu. = H(o) (Gr,) 1 t UO Cos (C - ) VN t - - (IfW) I.T - 3a (1 Pr)(lt-Pv ) o, J ___UOX~ ) ( X [ /+ KS( j; C E UO X CS(Xt ) 28L ( i-G - r (OX' /L*JX\ L H12H(o) P r) (r ) Where F(r) is the r dependency of the stream function in Ostrach's similarity solution(2) defined as -U(oX Lf r)= C X F(v7) ~ ~UoX co..s(~+ )~~~~~~

CHAPTER III DISCUSSION OF THEORETICAL RESULTS In the preceding chapter, the problem was stated, formulated, and treated analytically. The differential equations were perturbed for small oscillations and the first three terms calculated. The first term is the steady free convection for which exist both numerical and integral solutions. However, an improved integral solution was obtained which agrees very well with the exact solution in the range of Pr from.7 to 200. According to these calculations, the Prandtl number at which the thermal boundary layer thickness equals the velocity boundary layer thickness is 0.757. This solution was necessary for the integral procedure used on the second approximation. The differential equations for the second term were solved using the classical simplifications for large values of frequency. Expressions thus obtained for Nul and fl, show that heat transfer lags behind the plate oscillations by 90 degrees with an amplitude which decreases with frequency, and the shear stress at the wall leads the plate oscillations by 45 degrees with an amplitude which increases with increasing frequency. The heat transfer is a decreasing function and shear stress an increasing function of Prandtl number. In the small frequency region, the shear stress increases its. amplitude with increasing frequency and always has a phase leado This is the same trend that it possessed for large values of frequency. The heat transfer, however, experiences an increase in its amplitude and possesses a small phase lead at the beginning of the small frequency region. The phase angle after reaching a

peak lead, decreases to a lag for slightly higher frequencies. The amplitude of heat transfer should also begin to decrease but the present series solution fails to demonstrate this fact for Pr = 0.757 and 10. For Pr = 100, Figure 13, this decreasing effect does fall within the range in which the series solution is valid. As mentioned in Chapter II9 twice the value of wo is a good estimate of the lower limit of large frequency range. For an eight inch plate at 200F, placed in air at 80~F, the boundary layer thickness at x = 8" is 0o17 inches, o* - 51 5 and the critical frequency Wcrit beyond which the large frequency solutions are valid, can be calculated as -2 Ct)I 2.)Ct,) _ - 1,1 rcLJ/ epr; which corresponds to 16 cycles per second. Therefore the assumption of large frequencies does not present too much of a restriction and can cover a wide range of frequencies. The perturbation procedure shows that the functions of the order of E2, or the third approximation functions, must be composed, each, of two terms, a time-independent term, and an oscillating term. Calculations were carried out only for the time-independent, or the steady terms. The perturbation procedure alone predicts a steady change in heat transfer and shear stress due to a purely harmonic disturbance (plate oscillations). The question is, whether this steady change is zero, positive or negative. The results of the calculations presented in Equation (71) show that this steady change is positive for the shear stress and negative for the rate of heat transfer.

Any steady change in heat transfer coefficient, whether it is a decrease or an increase, will have valuable engineering applications provided the change is substantial. Although perturbation procedure involves small changes, it will be helpful to estimate the order of magnitude of these changes and compare them with relative intensity of the disturbance and namely, E. For an eight inch plate at 200~F placed in air at 80~F, 7....G _ M 5'?_, (N:?,'6 and L.o - i i.i ~t/Set Taking c 0o2 (this corresponds to an amplitude of 0.022" for a frequency of 20 cps) the steady change in the local Nusselt number will be 2 C Nul~rz"-. O -), a20 which is a decrease of 0o014%o This change compared to 20% is negligible. For all practical purposes, one may conclude that in the laminar region the rate of heat transfer is not disturbed by the oscillations of the plate. It is interesting to note that the per cent decrease in Nusselt number is only a function of Prandtl number and By definition, (n xA 0x(74) where A is the amplitude of plate oscillationso But by (74)9 - - = A C(X X ioeo, the per cent decrease in Nusselt number is only a function of Pr and A

-65Per cent increase in friction factor, however, is an increasing function of local Grashof:- number as well as Prandtl number. Although the expressions derived for the rate of heat transfer and shear stress at the wall are restricted to narrow regions and approximate, they have the important advantage of being in closed form. Furthermore, these expressions are simple and explicit with respect to all the parameters involved. This will best serve the study of qualitative behavior of the phenomenon in questiono

CHAPTER IV STEADY LAMINAR FREE-FORCED CONVECTION PHENOMENON This chapter is based on a portion of the analysis in Chapter II herein discussed in detail. During the course of obtaining a solution for small values of frequency, expressions were derived for Nusselt number and friction factor for the case of zero frequencyo These expressions added to their corresponding steady terms are solutions to the problem of steady free-forced convection from a vertical flat plate. Figure 21 shows the physical model where Ua is the free stream velocity and could be in either + x or x direction~ Since the quasi-steady solutions are perturbations around steady free convection, the results discussed in this chapter will only represent the effect of a small free stream velocity on steady free convection heat transfer and will not cover the regions where forced convection predominates, The expression for local Nusselt number will be,t.(75) where 9 and Values of D1 are tabulated in Appendix IIo The second term on the right of Equation (75) can be integrated over a finite length of the plate giving the average change in Nusselt numbero Nuke - __ - 1/9 -1/Li (76) -66

HEATED STATIONARY I PLATE AT TEMPERA- TURE TW > Tro I FLUID AT / TEMPERATURE Too I U0 / X i / Figure 21. Physical Model for the Zero-Frequency Case.

-68The above expression will hold as long as (e ) Re DD3((Pr) g where D3(Pr) = (Ugowx) uOmax has been plotted versus Pr in Figure 3, and also tabulated in Appendix II. For a gas, (Pr = 0.76), in free convection, with Gr2 = 108, a free stream flow with Re = 103, will give rise to a change of 3.8% in the average Nusselt number - an increase if UO is in + x direction and a decrease otherwise. The use of Equation (76) is recommended for predicting changes in average Nusselt number up to about 10%.

CHAPTER V EXPERIMENTAL WORK Introduction An experimental program has been conducted as suggested by the analysis of Chapter II.o Before the final design of the test section, a number of preliminary isothermal tests were made in water of different geometrical configurations, The purpose of these tests was to seek an experimental model that would closely approximate the analytical model of Chapter IIo These visual studies were carried out on a plate 1/16 of an inch thick and on a cylinder one inch in diameter. The cylinder was oscillated once with its spherical end open and once with its end projected through a large plate. It was noted that vortices produced in the wake of the plate and the open-end cylinder, would disturb the flow around the test section. By introducing the large plate mentioned above, these disturbances were eliminated. A slight disturbance, however, was introduced at the intersection of the plate and the cylinder but was of a local nature. Based on these observations it was decided to use a cylinder, with a diameter many times greater than the steady boundary layer thickness, and to project it through two large plates one on each end, heating it only in a section away from the two end plates. All the experiments have been made in an enclosure containing atmospheric air. Measurements were made of the mean coefficient of heat transfer. For small values of velocity amplitude, slight decreases were noticed which confirm the results of the analysis in Chapter IIo At -69

-70higher velocity amplitudes increases were observed which were assumed to have been caused by transition to turbulent flow. This was confirmed by a series of smoke studies. The maximum decrease observed in the coefficient of heat transfer is 1.2% at a frequency of 10 cycles per second, which is of the same order of magnitude as the prediction of the theoretical analysis. The Test Apparatus The assembly drawing of the test section and other mechanical components of the test apparatus are shown in Figure 22. At the bottom of the drawing, the vibrator has been shown schematically. The vibration equipment is built by the MB Manufacturing Company. The vibrator is a model C-5B and its control unit is a model T-51D. The vibrator together with its control unit can be seen in Figure 24. As it is shown in Figure 22, the moving horizontal plate is connected to a steel rod (item 4) through a flexible connector (item 11). This is the driving rod which in turn is connected to the bottom end cylinder (item 2) and the test section (item 1). This driving mechanism will cause the oscillation of test package (items 1, 2 and 3) through two bearings provided by teflon O-rings. These O-rings serve both as bearings and seals. The test section is made of Aluminum tube, eight inches long which is highly polished on the outside. The outside diameter is 4.870 inches and the wall is 0.310 inches thick. Ring heaters are provided at either end of the test section and serve as guard heaters. The ends are closed by aluminum disks fastened to flanges welded to the cylinder wall. All the threaded joints are secured by double-nuts or slotted nuts held with wire. The

CD Ci. 0 I-3 (D CD 0 CD ct 1 ~ ~ Y vl J -t;I~~t~L~~37 I ~I1Y _I b~~trlt ~ ~ \\ \\ A

-72List of Items in Figure 22 1 Test section 2 Lower cylinder 3 Upper cylinder 4 Driving rod 5 Upper rod 6 O-Ring supports 7 Lower bearing support 8 Upper bearing support 9 Lower cover 10 Upper cover 11 Flexible connector 12 O-Rings 13 Insulating tape 14 Supporting structure 15 Vibration exciter 16 Supporting rods 17 Enclosure walls 18 Ring heaters 19 Electric tape 20 Heater ribbon 21 Thermocouples

-73 gaps on either end of test section are covered with a strip of electrical tape to secure a continuous surface on the outside. A view of the test section and the end cylinders is shown in Figure 23. Also shown in this figure and in Figure 24, is the masonite enclosure 4 feet by 4 feet by 105 feet surrounding the test area. Its function is to insure that the measurements are not influenced by drafts and air currents produced by the ventilating system of the building or the cooling fan on the vibrator. The energy input to the main heater and the guard heaters is supplied by a 120 volt AC line. The line power is passed through a voltage regulator and then to three separate variacs each controlling the voltage across their corresponding heater element.. The voltage regulator is a Sornesen 0O5 KVA Model 500. This device will minimize any voltage changes that may occur in the lineo The smoke studies made use of a smoke generator built by Blankenship.(15) A schematic diagram of the smoke generator is shown in Figure 25. The smoke from cigars soaked in oil was used as the indicate ing medium. The smoke was generated by blowing compressed air down over two cigars in a copper tube. The smoke from the cigars was then bubbled through a layer of water in the bottom of a glass container. This layer of water would filter the unburned oil and tar from the cigars. The smoke then filtered out through a steel wool filter to go to the injecting nozzles located on a fixed plate at the bottom end of the test section. The injectors consisted of ten small nozzles of approximately 0.032 inches diameter, Hoowever. only three of them were used for visual studies.

-74Figure 25. A View of' the Test Section an. End Cylinders.

-75Figure 24. General View of the Test Apparatus.

-76LOW PRESSURE COMPRESSED AIR REGULATING VALVE OIL SOAKED CIGARS TO INJECTORS STEEL WOOL FILTER \ l- I TO EXHAUST ICE BATH WAT ER FI LT ER Figure 25. Sketch of the Smoke Generator.

-77 - Instrumentation of the Test Section, The test section is heated by Chromel-A heater ribbon 1/4 of an inch wide by 0~002 inches thick wrapped uniformly on the inner surface of the test sectiono The ribbon. is sandwiched between two layers of Scotch Noo 69 electrical tape to provide both electrical insulation from the wall and maintaining the relative position of sections of the ribbono'This tape is thermosetting and can withstand a maximum temperature of 356~Fo The resistance of the main heater is about 18 ohms. Two guard heaters are provided on either end of the cylinder~ The purpose of guard heaters is to prevent the leakage of any part of the main heater energy from the ends. These are Chromalox ring heaters each with a capacity of 100 watts at 120 volts. Chromel-Constantan thermocouples. have been used for temperature measurements consisting of 30 guage twin wires insulated. individually and placed in fiber glass insulationo The thermocouples used for measuring surface temperatures are installed in 1/32 of an inch holes in the wall and are secured in place by filling the holes with aluminum cement such that the tip of each thermocouple is almost in the same level as the outer surface of the cylinder. A total of eighteen thermocouples have been used, thirteen of which are installed in the test sections four in the walls of the end cylinders and one within the masonite enclosure for measuring ambient temperature. Measuring Instruments Thermocouple wires are connected to a system of rotary switches making it possible to switch to any single thermocouple and read its

-78indicated output voltage using a Leeds and Northrup Model 8662 portable precision potentiometer. An ice bath is used for the reference junction and the thermocouple wire has been calibrated in the laboratory. Sometimes a Sanborn four-channel recorder was used during the initial transient period in order to observe the approach to equilibrium. However, all the actual temperature measurements were made by the potentiometer mentioned above. These measuring devices can be seen in Figure:24. Figure 26 shows a diagram of the heating circuit. The energy input to the main heater is measured by reading the current and voltage drop across the heating element at all times. The current is measured by a Westan Model 904 AC ammeter and the voltage by a Westan Model 155 AC voltmeter. In evaluating heat transfer measurements it is important to know the magnitude of the radiation losses from the outer surface of the test section. For this reason the emissivity of the polished aluminum cylinder was measured using a Model 810C69 Cenco Thermopile. The output of the thermopile was measured by a precision potentiometer. A black body constructed by Schoenhals(23) was used as an essential device for measuring emissivities. This and the thermopile can be seen in Figure 24. The procedure for obtaining the emissivity of the aluminum surface was to focus the thermopile on the surface of the aluminum cylinder and to record the output of the thermopile for each surface temperature. This was done after every run. This process was repeated for the black body in a wide range of black body temperatures. The ratio of the thermopile outputs at a given temperature, then, would give the emissivity of the aluminum surface. The emissivity of the polished aluminum surface of the test section

GUARD HEATER ELEMENTS MAIN HEATER ELEMENT FUSE 0 I 1 ~~~~~VOLT METERR VA R I A CI I LAMP a 0 VOLTA GE 120 VLA-C. 120v c ~REGULATOR, Figure 26. Electrical Diagram for Measurement and Control of Energy Input to the Heater Elements.

-80 - Figure 27. A View of the Amplitude Pick-up.

SCHAEVITZ LINEAR DIFFERETIAL MOVABLE CORE. TRANSFORMER MODEL 100-AS -L 10 KILOCYCLES HEWLETT- PACKARD DUMONT MODEL 304-H MODEL 200 CD OSCILLOSCOPE AUDIO OSCILLATOR BRASS ROD ATTACHED TO TEST PACKAGE Figure 28. Linear Differential Transformer Circuit for Measurement of Vibration Amplitude.

-82varied between 0.062 and 0OO74 for surface temperatures ranging from 120-270~Fo. The frequency of oscillation is set on the frequency dial of the vibrator control unit ranging from 5 to 500 cycles per second. This frequency was measured by a Model 631-BL Strobotac manufactured by General Radio Companyo The results of strobatac measurements agreed closely with the numbers indicated on the frequency dial of the control unit' The amplitude of oscillation is monitored by means of a Schaevitz Linear Differential Transformer Model 100-AS-L and a movable core attached to the top of the top end cylinder with a small brass rod. The transformer has a stationary mounting, built by Schoenhals,(23) provided with fine position adjustment by means of a sliding memeber controlled by a threaded shaft. The transformer and movable core system can be seen in Figure 27~ A Hewlett-Packard Model 200 CD Audio Oscillator is used for an imput to the transformer. The output is displayed on a Model 304H DumontOscilloscope~ A diagram of this system is shown in Figure 28~ In order to calibrate the system, the transformer is first placed in its null position with respect to core, by moving it until a zero signal is shown on the scope. Then the sliding member of the transformer mounting system is moved against a spacer having the same thickness as the amount of the desired amplitude to be measured. The spacer is placed in a slot provided for this purpose. This displaces the transformer by the amount desired, with respect to the core, away from the null position so that the high carrier frequency causes a blurred signal on the scope as shown in Figure 29. The oscillator output or scope gain

Figure 29. Output Signal of the Linear Figure 50. Output Signal of the Linear Differential Transformer with the Core Differential Transformer under VibraDisplaced from the Null Position as tory Conditions as Displayed on an Displayed on an Oscilloscope. Oscilloscope.

is then adjusted until the signal occupies the full scale of the scope screen. This calibrates the system so that the full scale corresponds to the same displacement as the thickness of the spacer. During vibratory conditions the sweep frequency on the scope is set equal to the frequency of vibration. The signal appears as the envelope of a sine wave as shown in Figure 30. Then the amplitude is increased by turning the amplitude knob on the control unit such that the image on the scope covers full scope; now the vibrator is oscillating with the desired frequency. A view of some of the instruments is shown in Figure 31. The black body is shown on the lower left corner. Procedure and Results of Heat Transfer Measurements The procedure for temperature measurements and the determination of the total heat input have been discussed earlier. Based on these two measurements and knowing the heat transfer area (0.85 square feet), a total heat transfer coefficient can be defined as (s/AT))'' - w- Twwhere q is the main heater wattage in its proper units. In order to correct for radiation losses, a radiation heat transfer coefficient hr may be defined as hr- Cmn Tw - T' Tw - To

-85-.. Rg B s B~~ ~~a eR RBB ~~r B;- nB Be Bs Ug g g g g~~~~~~gR.f.>s.>.@ X.;-f~~~~f-,............. R; f l f; B - g........................... -.>.................. Figure 31. A View of Some of the Instruments.

-86Then the average convection coefficient hc would be th'e difference hC T h A series of steady heat transfer measurements were made which correspond closely to the free convection correlation given in Reference 22. Figure 32 shows the experimental data obtained for Nusselt number plotted against the product of Grashof and Prandtl numbers. The solid line is a plot of the correlation equation in Reference 22~ The same procedure was used for measuring heat transfer coeffif cient while the cylinder was oscillating. The test section was allowed to come to equilibrium condition under nonoscillatory conditions. The guard heaters were carefully adjusted so as to prevent end losses, At this time the temperature distribution in the cylinder wall was essentially uniform. The test package was then vibrated at a known frequency and with a desired amplitude. The wall temperature difference (T\\. - To) was carefully observed on the precision potentiometer and the guard heaters monitored at all times. For small values of velocity amplitude, the wall temperature difference rose slightly in all the experimental runs. The guard heaters, under these conditions, did not require any further adjustment, At higher velocity amplitudes, however, an abrupt decrease was noticed in the wall temperature and the guard heaters required further adjustment.o This abrupt decrease in wall temperature was attributed to transition to turbulent flow. Figure 33 shows two sets of experimental data obtained at a frequency of 15 cycles per second. The Nusselt number based on total

-87100C Nut 0. 59(Grt Pr)/4 90 ~ —--- Nu =0.56 (Grt ~Pr)'/4 80 70 60 Nut 40 20 107 2 3 4 5 6 7 8 9 108 Gre, Pr Figure 32. Steady Free Convection Data.

-8848 EXPERIMENTAL DATA Ifr - Is CPs 47 _ (Grt Pr = 4.37x107 ~46 ~ I fr' icPs 46!Gr, Pr 3.65 x 107 - SMOOTHED CURVE 45 44 Nut 43 42 41 40 39 38 37 0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ( A/V)xlO 3 Figure 33. Typical Data Showing Nusselt Number Versus Vibration Amplitude Ratio.

-89length of the cylinder, decreases slightly with increasing values of A/g, reaches a minimum and then reversing its trend, increases abruptly over its steady state value. The solid lines in Figure 33 are curves best fitted through the experimental pointso Each of these curves crosses its steady state value at an amplitude ratio whose magnitude is a decreasing function of Grashof number. The curve for Gro Pr = 3565 x 107, crosses at A/e = 8~6 x 10-3 whereas the one for Gro Pr = 4o37 x 107 crosses at A/2 = 7o6 x 10-3, Also the rise in Nu e is sharper for the higher value of Gro Pro Figure 34 contains data obtained for different frequencies and a constant Grashof number. Heat transfer data has been presented in terms of per cent change in Nusselt number caused by oscillations of the test section. Smoke studies revealed that all the experimental points on Figures 33 and 34 that represent a decrease in heat transfer coefficient are in laminar region. The procedure for smoke studies consisted of permitting the test section to achieve different temperatures for different power inputs. Once the test section reached equilibrium, the smoke generator was started. An optimum smoke exit velocity was found by adjusting the compressed air and the various valves. Two Plexiglass windows were provided on two of the enclosure walls, one for lighting purposes and the other for observing the smoke patterns. The criterion for smoke exit velocity adjustment was to obtain the minimum velocity possible for the smoke filaments to be stable and clearly visible. Then the flow patterns were observed for different values of frequency and amplitude.

9 fr 20 CPS8 ~ Gr*Pr=3.6x1o7 7 6 0 5 fr 15 CPS Z Z 4 z 3 o o 2 fr IOCPS 0 -I -21 0 2 3 4 5 7 8 9 10 11 12 B3 14 15 16 17 (A/t )x103 Figure 54. Experimental Data Showing Per Cent Change in Nusselt Number Resulting from Oscillations.

-91For small values of velocity amplitude the smoke filaments retained their streamline shape as in the case of no oscillations. For larger values of velocity amplitude, however, a mixing action was observed on the upper portion of the test cylinder~ The region where the smoke was mixed rather than streamlined, would move downward when the amplitude of oscillation was further increased. It was concluded that this mixing action was due to turbulence. In each run the length of the cylinder in turbulent regime was measured by means of a scale mounted on the test section. Figures 35 through 37 show the results of smoke studies shown schematically for the data presented in Figure 345 Figures 38 through 42 are photographs of the smoke studies. These photographs are representative of what was seen without a camera. Figure 38 represents the smoke pattern in steady state case (A = 0). In Figure 39 the cylinder is oscillating with a small amplitude, however, the smoke patterns are identical to the steady state case. In Figure 40o where the cylinder is oscillating with a higher velocity amplitude, a change is noticed in shape of the smoke filamentso Within the top three inches of the test section the smoke is slightly mixed which signifies transition to turbulence. In Figures 41 and 42 the cylinder is oscillating under more severe vibratory conditionso The mixing action is stronger and it covers a wider region along the length of the cylindero All the photographs were taken by a Graflex f/4o7 camera using Polaroid Land 4 x 5 type 57 filmo This type of film has an ASA equivalent exposure index of 3200. This film was used with a setting of 1/200 second at f/4o7o

o Gr P r =3.6x107 fr 10 CPS 6 5 - LAMINAR LAMINAR LAMINAR LAMINAR TURBULENT 0:3 Qb 8 8 8 8 8 z 0 4 M 7 7 77 21 6 6 6 6 35 C )5 5 5 5 4 1 4 4 4 4 o 2 3 z 3 3 3 2 2 2 2 2 I I I I I I I I I I I 0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 A Figure 35. Smoke Study Data Showing the Nature of Flow Regime Along the Length of the Cylinder for Various Heat Transfer Measurements.

"i "....I" I I i' I:1' 1 I "i.... I...I' i i ii TURBULENT 8 7'7211~ Gr. Pr 3.6x107 6 fr = 15 CP-S 5 4 3 2 TURBULENT 5 8 LAINAR6 Z0 85 1:3 7~~~ / 4 A z ~~ ~~6 3 4 3fl Z w 2 34 o ~ z o z3 3 0, 2 7 2 0 / 0AA 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 Ax 103 Figure 36. Smoke Study Data Showing the Nature of Flow Regime Along the Length of the Cylinder for Various Heat Transfer Measurements.

8 = r20 CPS 7 6r - 5 TURBULE NT 3 8 6 2 I LAMINAR 2 6 1' 6 JI 5 B 0 5. 7 z-' I 4 l l _' 0 00 1 24 5786 3 i 4 - u) 5 2'~Z cW 4 - 2 0 x 2 0 +1 -I 0 I 2 3 4 5 6 7 8 9 0 II 12 13 14 15 16 17 x 103 Figure 37. Smoke Study Data Showing the Nature of Flow Regime Along the Length of the Cylinder for Various Heat Transfer Measurements.

MO Figure 38. Smoke Photograph for Figure 39. Smoke Photograph for T- TO= 139OF, and Tw - T = 139~F, No Oscillation. A = 0.05 inches and fr = 16 cps.

~~L~~~~~~~~ -Z ADZ_ 1..........i _ _ Figure 40. Smoke Photograph for Figure 41. Smoke Photograph for Tw- To = 1760F, Tw - T0= 1760F, A = 0.11 inches and A 0.07 inches and fr = 16 cps. fr 45 cps.

-97-,,, _! - - | | |.; 11111 11111 | | I | |, | gBtygS,.w155oi. 1111|1 l@11 113 | | | | | | BoyeSSe tZm ill S11 S | E E E R 24................................................... as.s.>.Ne.-gB g lillllE | | | B | | g 8,. |E l I! | | E - - * | I | E X | * | l I I | _ _ _ * l | -;>.Ne__:.:;.::s'<v6B::.;.:B5zlg:H..<. s..ysE>i.cl.B..5>B:av; — * | I l | __|_ * | | | I I | _ - _ * | I l I | _ - _ * | l l l | z. B z __ AN __ __!; ______ _.Ny >.B_ t1__,_:St._E j = _ K u% _ 1E _ W_4__ BB.K.s..,Rtsy_ - __'Xt.,f.;.t:g__'>Rht. ypS Sii..E-E:i.,.'-:H.'.'2.,B,,B,,B, g ss ss.- g Bc.l;>>clB N:E i:E <::s;.8cHss -;l_ E sgs.,..set_=z_ ss. E.. __B =_:B{:iNE _ __ | | s s | Uw__.fi-EAc. t gSStSStS-.yg_,L I,, 1.. | 1!1 Si 1ll 1lil l!l l l _ gggySS_ ____ E _ t__ yE.,....S. _____ m -____ S_ __B9'G<ggS' ssBBs-s s --— --iE'i' _E —gf g _ <t.y.<..->f-,.:... ___+ys.E-E-E -.s X — _ g:.:B:.;gt s.sys S=gS*aggugzttzgs i-:'........................................................................ 1. s...E:'>Be>>:225e._.iss s>.>:S _.a ___-_ -_ _ _.s.~.XR.: SW;y,iE__ 0sr _ - Figure 42. Smoke Photograph for Ilv - So _ 176 oF, A = 0.005 inches and fr = 200 cps.

- 98_Another set of heat transfer data has been presented in Figure 43 for a frequency of 12 cycles per second. Each experimental measurement shown in Figure 43 has been accompanied with a photograph indicating the flow regime for exactly the same conditions. The experimental points in Figure 43 which correspond to Figures 44 - 47 are definitely in laminar region and represent decreases in the coefficient of heat transfer. A slight undulation can be seen in Figure 49 on top of the test sectiono This implies that at these conditions a very small portion of the test section is experiencing minute turbulence whose gross effect on the average coefficient of heat transfer is to nullify the decrease which prevails in the portion of the test cylinder still in laminar region. The measurements relating to Figures 50 and 51 show increases in the coefficient of heat transfer. There is a definite disturbance in the smoke filaments near the top of the test cylinder which is interpreted as the start of transition to turbulent flow. The smoke patterns in Figure 48 indicate a slight waviness on the outer filament suggesting that transition to turbulent flow starts from the outside. The over-all study of the smoke photographs brings about a gross indication of transition to turbulence as a consequence of longitudinal oscillation of the test cylinder the mechanism of which is to be determined by further research, Correlation of Experimental Data It was pointed out earlier that for high values of velocity amplitude the coefficient of heat transfer under vibratory conditions experiences an increase over its steady state value. It was also established

EXPERIMENTAL DATA Gr. Pr = 3.56x 0 7 FIG.51 8 U fr = 12 CPS 7 6 5 o: 3 z O 2 - I -2, I I - I -2, I 0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 (A/t)x 103 Figure 45. Per Cent Change in Nusselt Number Resulting from Oscillations as a Function of Amplitude.

-100Figure 44. Smoke Photograph for Gr. Pr = 3.56 x o107, and No Oscillation.

Figure 45. Smoke Photograph for Figure 46. Smoke Photograph for Gr. Pr = 3.56 x 107, Gr. Pr = 53.56 x 107, A/e = 2.9 x 10-3 and A/i = 5.7 x 10-3 and fr = 12 cps. fr = 12 cps.

Figure 47. Smoke Photograph for Figure 48. Smoke Photograph for Gr. Pr = 5.56 x io7, Gr. Pr = 7.56 x 107, A/2 = 8.1 x 10-3 and A/I=9. 5 x 1o-3 and fr 12 cps. fr =12 cps.

Figure 49. Smoke Photograph for Figure 50. Smoke Photograph for Gr. Pr = 3.56 x 107, Gr. Pr = 3.56 x 107, A/l = 11.3 x 10-3 and A/i = 14.7 x 10-3 and fr = 12 cps. fr = 12 cps.

-104Figure 51. Smoke Photograph for Gr. Pr = 3.56 x 107 A/i = 18.1 x 10o3 and fr = 12 cps.

-105that in such cases the flow is in turbulent region. An attempt has been made to correlate this data in terms of a suitable parameter which would take into account both the free convective effect and the influence of oscillations. Figures 33 and 34 indicate that the Nusselt number increases with increasing Gr, A/~ and co. This was the criterion for defining a modified Grashof number Grcv which would have the above-mentioned characteristics. It was found that Nug could be correlated as a function of Grec defined as c-v- DGre (r _ 1/3 _-_* 2 I (Gre) (Pr) Figure 52 is a plot of Nu. versus Grcv ~ Pr where the solid line is the best curve fitted through the data points. The experimental measurements presented in Figure 52 do not test the Prandtl number dependencies since all the data is for air with Pr = 0.72~ The Prandtl number appearing in the definition of Grcv, however, has been extracted from theoretical analysis of steady free convection and it is expected to apply to fluids other than airo No attempt has been made to present a correlation equation because of the limited range of the vibratory as well as thermal and geometrical parameters. More data in a wider range of parameters is necessary if a meaningful correlation equation is to be obtained. Nevertheless, within the limits of this work an equation of the form 1/3 appears to correlate the turbulent datao

60 50 40 Nut 30 20 4 5 6 7 8 9 Grcv' PrxIO7 Figure 32. Correlation of Turbulent, Oscillatory, Free Convection Data.

-107The heat.transfer data obtained in laminar region under vibratory conditions indicate that the average coefficient of heat transfer falls slightly below its steady state value. The analysis of Chapter II shows the same trend. Equation (73) determines the local Nusselt number and is valid as long as w* is greater than 2 co* ~ Since w* is proportional to Wxl/2 then the use of Equation (73), for a fixed value of w, restricts the lower limit of x. In other words, for finite w, x cannot arbitrarily approach zero if Equation (73) is to be used. This implies that the local Nusselt number cannot be integrated over the total length of the heated section to yield an expression for the average Nusselt number. Considering the fact that the experimental measurements include not local coefficients but the average coefficients of heat transfer, a direct comparison between theory and experiment is not possible. It is intended, however, to make use of Equation (73) in order to obtain a semi-empirical expression for per cent change in Nusselt number caused by oscillations. It was pointed out, in Chapter III, that the relative change in Nusselt number is a function only of Pr and A/x, for large values of w* J This suggests that an attempt could be made to correlate the experimental data using A/2 as the main parameter describing relative change in Nusselt number for laminar heat transfer phenomenon. The expression in Equation (73), for Nux, could be integrated, for each frequency a, from a point x = xo where w* = 2 wo* up to the point where x = Yo Then from x = 0 to x = Xo, it is conceivable to assume a distribution for relative change in Nusselt number that is similar in nature to its functionality in the range x = xO to x =.o A two region integration over the length of the heated section will then lead into a

-108semi-empirical equation. This procedure has been carried out for air (Pr = 0.72); the following equation is obtained lo0 Nt{-Nete - -1 33 i T (-) ( ) (77) 0 V Xo _ 3/I In the above equation y, C-.....' (78) Tf Cr and Pi is a constant that should be determined experimentally for each set of conditions excluding the parameter A/o. In Equation (78) xo is in feet, fr in cycles per second and Tf = (Tw + T.) is the film temperature in degrees absolute. The expression in Equation (77) has been worked out for two sets of conditions and the results are presented in Figures 53 and 54 together with the experimental data obtained for the same conditions. The plots in Figures 53 and 54 indicate that as long as the flow is in laminar region Equation (77) agrees with the data as expected, deviations for larger values of A/2 being a consequence of transition to turbulence. Summary of Experimental Results A vertical heated cylinder with a diameter many times greater than the steady-state boundary layer thickness was oscillated vertically

Tw T O 147 IF 3 Nut-Nu0t [2 A9(A A Pi1 { Nuot 09( )+ fr = IOCPS Nu~t SEMI-EMPIRICAL CURVE 2 FOR THE ABOVE CONDITIONS Pi 0.68 0 z Z H 0 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'~ -2 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 7 (Alt) 1030 Figure ~5. Correlation of Laminar Data.

fTwT,0 136~F A fr 3Nu-Nuo A 2 A* tPi fr 15 CPS 100626 +8.65 P Nuo - SEMI- EMPIRICAL CURVE FOR THE ABOVE CONDITIONS P =0.65 0:3 -+1- Z I 8 o, ~_.-__- - -1 -..... 8 A -I - - - - - _ _______ ______ -2 0 1 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 (A/f) x 103 Figure 54. Correlation of Laminar Data.

-111 in atmospheric air and heat transfer measurements were made. The changes in the coefficient of heat transfer under oscillatory conditions were compared to their measured values in the case of no oscillations. It was found that for small values of oscillatory velocity amplitude there was a small decrease in the coefficient of heat transfer. For higher values of velocity amplitude, however, increases were observed in the average coefficients over their measured steady state values. This reversal in the behavior of heat transfer coefficient was attributed to a transition phenomenon from laminar to turbulent flowo Smoke studies were carried out and evidence of a transition to turbulence was observed. Semi-empirical equations were devised for the flow region associated with decreases in the heat transfer coefficient which are of a monotonically increasing character, in absolute value, with respect to the amplitude ratio defined as the ratio of oscillation amplitude to the heated length of the cylinder. The data in the turbulent region was correlated in terms of a modified Grashof number defined such that it is a superposition of the free convective and forced oscillation effects. Furthermore, a correlation equation was suggested that has to be verified by more extensive data.

REFERENCES 1. Schmidt, Eo and Beckmann, Wo "Das Temperature und Geschwindighetsfeld von einer Warme abgebenden Senkrechten platte bei naturlicher Konvektiono" Technische Mechanik und Thermodynamik, (October 1930), 341-349, (November 1930) 391-406o 2. Ostrach, S. "Analysis of Laminar Free-Convection Flow and Heat Transfer about a Flat Plate Parallel to the Direction of the Generating Body Forceo" UoSo NACA Report TN2635, 1952o 30 Sparrow, E. M. and Gregg, J. L. "Laminar Free Convection from a Vertical Plate with Uniform Surface Heat Flux." Trans. ASME, 78 (1956), 43 5 -440~ 4, Sparrow, Eo M. and Gregg, J. L. "Similar Solutions for Free Convection from a Nonisothermal Vertical Plate." Trans. ASME, 80 (1958), 379-3860 5. Finston, M. "Free Convection Past a Vertical Plate." ZAMP, 7 (1956)9 527-529. 6. Millsaps, K. and Pohlhausen, K, "The Laminar Free-Convection Heat Transfer from the Outer Surface of a Vertical Circular Cylinder." Journal of the Aeronautical Science, 25 (1958), 357-360o 7. Yang, Kwang-Tzuo "Possible Similarity Solutions for Laminar Free Convection on Vertical Plates and Cylinders." J. of Applied- Mechanics, (June 1960), 230-236, 8. Sugawara, So and Michiyoski, I. "The Heat Transfer by Natural Convection in the Unsteady State on a Vertical Flat Wallo" Proc. of the First Japan National Congress for Applied Mechanics, 1951. 9. Siegel, Robert. "Transient Free Convection from a Vertical Flat Plate." Transo ASME, 80 (1958), 347-359. 10. Chung, P. M. and Anderson, A. D. "Unsteady Laminar Free Convection." Journal of Heat Transfer (November 1961) 473-478o 11o Lighthill, M. Jo "The Response of Laminar Skin Friction and Heat Transfer to Fluctuations in the Stream Velocity." Proc. Royo Soc., A224 (1954) 1-23. 12. Illingworth, C. R. "The Effect of a Sound Wave on the Compressible Boundary Layer on a Flat Plate " J. of Fluid Mechanics, 3 (1958), 471-4930 -112

-11313. Kestin, J., Maeder, Po F.o and Wang, H. E. "On Boundary Layers Associated with Oscillating Streams." Applo. Sci, Res., Seco A, 10, 1960o 14o Schoenhals, Ro J. and Clark, J. A. "Laminar Free Convection BoundaryLayer Perturbations Due to Transverse Wall Vibration." Journal of Heat Transfer (August 1962), 225-234. 15. Blankenship, V. D. "The Influence of Transverse Harmonic Oscillations on the Heat Transfer from Finite and Infinite Vertical Plates in Free Convection." PhoDo Thesis, The University of Michigan, 1962. 16. Chamberlin, G. B., Jr. "Heat Transfer by Free Convection from a Longitudinally Oscillating Cylinder in Air." MoSo Thesis, Air Force Inst. of Tech., 1957. 17. Sprott, A. L., Holman, JO Po and Durand, F. L. "An Experimental Study of the Interaction of Strong Sound Fields with Free Convection Boundary Layerso" WADC Technical Report 59-717, November 1959. 18. Holman, J. P., Gartrell, H. E. and Soehngen, E. E. "A Study of Free Convection Boundary Layer Oscillations and Their Effects on Heat Transfer." WADC Tech, Report 59-3, December 1959. 19. Soehngen, E. Eo and Holman, Jo P. "Experimental Studies on the Interaction of Strong Sound Fields with Free Convection Boundary Layers." ARL Tech. Report 60-323, October 1960. 20. Goldstein, S. "Modern Developments in Fluid Dynamics." Oxford University Press, Volo II (1938), 641-643. 21. Eckert, E. Ro G. and Drake, Ro M., Jr. Heat and Mass Transfer. McGraw-Hill, (1959) 312-315. 22. McAdams, W. H. Heat Transmission. McGraw-Hill, 1954. 23. Schoenhals, Ro J. "The Response of Laminar Incompressible Fluid Flow and Heat Transfer to Transverse Wall Vibration." Thesis, The University of Michigan, 19600.

APPENDIX I EQUIVALENCE OF THE CASE OF AN OSCILLATING PLATE IN A STAGNANT AMBIENT TO THE CASE OF A FIXED PLATE IN AN OSCILLATING POTENTIAL FLOW The terms in the energy equation will retain the same form whether the system of coordinates is fixed or accelerating. The boundary layer equation, however, being the x-component of momentum equation would undergo some changes. Let us first consider the case where the plate moves in the x-direction with a velocity -U(t) relative to a fixed coordinate axis. In this case, the boundary layer equations become, D aU 1 I- LU t Axt,O, t) - v( So, t) A; a(X, 00 t) - (t) aP f inaly and finally. D - z +-/3(T-Too) + l- -114

-115In the other case where the plate and consequently the system of coordinates are fixed, there are no inertia forces present and the equations take their-classical form of DLU e a.-F = e x~ x,ot) = vCx, o, t) = o; u(x, X9. t) - U(t) p dU' and finally DL-~ L (T —T) L A t Dt _ a Cx, o,t)= vrx, o,t) = O; LCbx, Mt) = U() which are exactly the same as the equations obtained for the first case. In conclusionthe term appears in the first case as an inertia force (body force) and in the second case as a pressure force.

APPENDIX II TABULATION OF THE RESULTS OF THE STEADY STATE INTEGRAL SOLUTION Nux Umax Pr 1A _ D3 = 1/ D3 = ) (Grx) 1(Pggx) 0.005 6 03117 0.03173 1l91931 0.010 4.44177 o0o4695 1.61694 0.02 3.31645 0.07849 1,36287 0.03 2.81689 008476 1.23342 0.60 1.05914 0,32816 0.58206 0.70 1.01935 0. 3 4805 0o 55859 0.72 1.01237 0.35178 0.55437 0,733 1.00799 0o35417 0.55170 25/33 = 07575 o1.00000 0,35859 0,54681 1 0.93727 0.39742 0.50676 2 0.81129 0.50749 0o41478 10 0 64106 0 84461 0.o 23984 50 0.57733 1.32017 0,12104 100 0.56692 1o 58179 o.08744 1000 0.55674 2 83381 O 02825 10000 0o 555674 5.04329 000oo895 00 /5/9 = 0.5555 In Chapter II, the functions H'(O) and F"(O) from Ostrach's numerical solution,(2) have been refered to. These functions are tabulated below for different Prandtl numbers, -116

-117Pr 0 o 01 0 o 72 0.733 1 2 10 100 1000 -HI'() 0o0812 0.5046 0.5080 0.5671 0.7165 1.1694 2.191 3.966 F"(O) 0.9862 0~6760 0.6741 0o6421 0.5713 0o4192 0.2517 0.1450

APPEZNDIX III DEFINITION OF ogs AND f's For A < 1 - 1 )o 3 31 - 231 0(7 c~ - 95 3 3 3s 8!3o.0(7 " 3 3- 5 93 5 -93 310(7 5- 5 C<8 - ~46 ( A + 62 o,) 5-118 -118

-1195g 25Z Pr A (7 S (A) /CZ = 6Z(L) 13 S (A) /3s P- r YP(r) / 6 S(L) A sP r Y(a) 5 - ZZ5 OC7 S(A) 6 S(A),3 /A Pr 6 w4q14 7S (A) 7 s(A),8 = 3 37 X(A) Also values of a5f have been tabulated for some Prandtl numberso Pr 0 757 1 2 10 100 11 1 490 X1o 4068 o0999 0 05088 0o 1225

APPENDIX IV TABULATION OF 7n AND Xn FOR DIFFERENT VALUES OF PRANDTL NUMBER Pr 0.757 1 10 100 n 7n Yn X n 7n Yn n n 0.74794.78715 x lol.73349.69854 x lol.37987.23661 x 101.56817 x 10-1.53633 1.18106.31442.18641.37052.28860.92735.43318.13451 x 10l 2.31654 x 10-2.54153 x 10-1.31562 x 10-2.69534 x 10-1 -.53343 x 10-3.54463 -.21293.43567 x lol 3.78520 x 10-4 -.46146 x 10-2.14668 x 10-3 -.70231 x 10-2.79701 x 10-2 -.22632.61514 -.11422 x 102 4.16995 x 10-4 -.27537 x 10-3.28467 x 10-4 -.54613 x 10-3.34067 x 10-2 -.85837 x 10-1.12067 x lo101 -.30228 x 102 5 -.14461 x 10'5.70913 x 10-5 -.26907 x 10-5.30398 x 10-4 -.10876 x 10-2.31609 x lo-1 -.22374 x lo101.85018 x 102 6 -.72051 x 10-7 -.98765 x 10-6 -.16433 x 10-6.44027 x 10-6 -.28118 x 10-3.11767 x 10-1 -.41383 x l101.26029 x 103 7 -.97839 x 10-9.19626 x 10-6.15736 x 10-8.15484 x 10-6.52779 x 10-4 -.46448 x 10-2.80018 x 101 -.87964 x 103 8 -.74057 x 10-9.19879 x 10'7 -.1276 x x o-8.20331 x 10-7.25515 x 10-5 -.20503 x 10-2.17806 x 102 -.33018 x 104 9.10071 x 10-9 -.10423 x 10-8.20824 x 10-9 -.18604 x 10-9.30956 x 10-5.10462 x 10-2 -.52411 x 102.13755 x 105 10.77962 x 10-11.60197 x 10-10.16748 x 10-10.44023 x 10-9.12459 x 10-5.61173 x 10-3 -.21454 x 103.63254 x 105 1 -.66362 x 10-13 -.23074 x 10-1O.33604 x O-12 -.11193 x 10-9.26228 x 10-6 -.39576 x 10-3.11098 x 104 -.31869 x l06 o 12.94308 x 10-13 -.30780 x 10-11.37768 x 10-12 -.18108 x 10-l1.78923 x 10-6 -.27454 x 10-3.65377 x 104 -.17458 x 107 13 -.18354 x lo-13.17376 x 10-12 -.74089 x lo-13.21235 x 10-11 -.82176 x 10-6.20088 x lo0-3 -.41781 x 105.10329 x 108 14 -.19841 x lO-14 -.26008 x 10-13 -.84330 x 10-14.15532 x 10-12 -.71434 x 10-6.15420 x 10-3 -.28497 x 106.65612 x 108 15.67903 x 10-16.96663 x 10-14.23813 x 10-15.22653 x 10-14.61152 x 10-6 -.12411 x 10-3.20622 x 107 -.44526 x 109 16 -.25376 x 10-16.16774 x l1'1 4 -.15851 x lo-15.27139 x lo114.54876 x 10-6 -.10475 x 10-3.15781 x 108 -.32141 x 101l 17.72365 x 10-17 -.17526 x 10-15.48670 x 10-16 -.35144 x 10-15 -.52409 x 10-6.92687 x 10-4 -.12734 x 109.24586 x 1011 18.11141 x 10-17 -.15447 x 10-17.88915 x 10-17.28028 x 10-16 -.52930 x 10-6.85878 x -4 -.10806 x 1010.19862 x 1012 19 -.90401 x 10-19 -.42466 x l0-17 -.10956 x lo-17 -.24538 x 10-16.55947 x 10-6 -.83206 x 0-4.96200 x 1010 -.16896 x 1013 20.77193 x 10-20 -.11637 x 10-17 -.55454 x 10-19 -.67779 x 10-17.61458 x lO-6 -.84181 x 10-4.89638 x 1011 -.15095 x 1014 21 -.49414 x lo-20.18990 x lo-18 -.14355 x 10-19.11906 x l0-17 -.69915 x 10-6.88811 x 10-4 -.87246 x 1012.14128 x 1015 22 -.12051 x 10-20.16328 x 10'19 -.48769 x 10-20.10418 x lo-18 -.82243 x 10-6.97583 x 10-4 -.88535 x 1013.13822 x 1ol6 23.20156 x 10-21.17528 x 10-20.64467 x 10-21.18712 x 10-19.99970 x 10-6 -.11154 x 10-3.93510 x 114 -.14109 x 1017 24.21571 x 10-22.11509 x 10-20 -.67403 x 10-22.11954 x lO-19.12551 x 10-5 -.13247 x 10-3.10263 x 1016 -.14998 x 18 25.16160 x 0'-24 -.29201 x 10-21.69194 x 10-22 -.35907 x 10-20 -.16267 x 10-5.16331 x 10-3 -.11689 x 1017.16576 x 1019

APPEENDIX V COMPUTATION PROCEDURE All the expressions derived for large values of frequency in Chapter II were simple and explicit in all the parameters involved and therefore, did not require extensive computation. The integral procedures, however, did involve some numerical evaluation. For this part of the analysis, the IBM 7090 Electronic Computer of the University of Michigan Computing Center was utilized. Three sets of computer programs were written for different stages of the computation. The first program was used to obtain a numerical solution to Equation (41) of Chapter IIo This resulted in tabulation of A for different Prandtl numbers (Appendix II). Newton's iteration process was used and the initial values of A, for each Pr, were estimated by a rough plot of Pr versus A. The second program was written for the evaluation of the first three terms of the series solution for B and B1, i.e., yo through Y2 and Xo through X2. This program was simple in nature and involved only evaluation of a set of formulas for a few values of Prandtl number, The third program was set up to calculate coefficients 7n and Xn for the values of n greater than two for which generalized expressions were derived in Equation (58) of Chapter II. These are the coefficients listed in Appendix IV. The series themselves, however, were calculated on a desk calculator. This would allow one to observe the convergence behavior of the -121

-122series and does not.present any problem concerning the order of magnitude of the numbers involved. All the programs were written in MAD (Michigan Algorithm Decoder).

-123 - $ COMPILE MAD,EXECUTE,PRINT OBJECT,PUNCH OBJECT,DUMP 04/18/62 MAD (12 MAR 1962 VERSION) PROGRAM LISTING.......... PRINT FORMAT F1 *001 VECTOR VALUES F=$S///S10,35HFOR PRANDTL NUMBERS LESS THAN 0.7 *002 157///S5 85HSIGMA I DELTA I CONST *002 2 I ERRDEL I UMAX/S2,91H.**.******** I *002 3 **..******* I.*** ******O I *********** I *00. 4* ****** *****$ *002 BASE2 READ FORMAT F2,SIGMA,DELTA.______. __ _*003 __ VECTOR VALUES F2=$2F15.5*$ *004 WHENEVER SIGMA.G.O.757,TRANSFER r[0 BASE5 _____________ _________ ___*005 THROUGH ASE1,FOR I=O 1, (.AHS. (2.52-8.316*DELTA-1./SIGMA+0.15 *006 1/(DELTA.P.3)-0.81/(DELTA.P.2)+0.972/DELTA+6.804,(DELTA.P.2))..006 2L.1E-6).OR. I.G. 15 *006 BASE1 DELTA=DELTA+( 1./S IGMA-2. 52+8. 316*DELTA-6. 804*( DELTA.P.2 )-0.15 *0__07 1/ ( DELTA.P. 3)+0.81/(DELTA.P. 2 ) -0.972/DELTA ) / (-8. 316+13.608*DEL *007 2TA-0.972/(DELTA.P.2) +1.62/(OELTA.P.3)-0.45/(DELTA.P.4)) *007 WHENEVER SIGMA.L.O.757,TRANSFER TO RASE4 *008 BASE5 PRINT FORMAT F4 *009 VECTOR VALUES F4=$///Sl0,35HFOR PRANOTL NUMBERS MORE THAN 0.7 *010 157///S5,85HSIGMA I DELTA I CONST *010 2 I ERRDEL I UMAX/S2,91H*********** I *010 3 ****** **** I *****..*.** I *********** I 010 4-*-* * ** —--------- -- ---- -------------------— __-__ 4 ************$.010 BASE6 READ FORMAT F2,SIGMADELTA __ 011 THROUGH BASE7,FOR J=0 1,.AS. (-1./SIGMA-5.04*(DELTA.P.3)+14. *012 1472*( DELTA.P.4)-12.15*(DELTA.P.5)+4.794*(DELTA.P.6)-0. 7560(DE *012 2LTA.P.7 ).L.E — 7). OR.J. G. 15 *0 12 BASE7 DELTA=DELTA+( 1./SIGMA+5.04* (DELTA.P.3)-14.472*(DELTA.P.4)+12. 013 115*(DFLTA.P.5)-4.794*(DELTA.P.6) +0.756. (DELTA.P.7))/(-15.1? *013 2 (DELTA.P.2)+57.88* ( DELTA.P.3)-60.75* ( DELTA.P.4 ) +28.764*( EL 013 3TA.P. 5)-5.292*(DELTA.P. 6) ) *u13 BASE4 CONST=1./(DELlA*( /5.6*(1.8*DELTA-1.)).P.G.25) 014 )HENEVER SIGMA.LG..757 *015 ERRDEL= 1. E-6/(-8.316+13.608*DELTA-. 972/UELTA.P.?.+1.62/DELTA *016 1.P.3.-0.45/DELTA.P.4. ) * 16 OTHERWISE *. 17 FRRDEL=1.E-7/(- 15.12* DELTA.P.2.+5 7.888*1)ELIA.P.3.-60. 75*DELTA 0 * 8 1.P.4.+28.764*DELTA.P. 5.-5.292*)ELTA.P.6. )' 18 END OF CONDITIONAL 019 UMAX= (9./128. )* ( 75.6* ( t. 8*DEL TA-1. ) ). P..5 *020 PRINht FORMAT F3,SIGMA,DELTA,CONST,ERREL,MAX ----- 021 VECTUR VALUES F3=iS2,F1O.45,S, lHI,2,2FI3.8,S4,lHIS4,tFI.6t,4 *022 1, 1HI S4,E11.4,S 4, lHI, S4,F 1.6,/*D *022 WHENEVER SIGMA.L.0.757,PRINT F(;R'AT F'.,I *023 VECTOR VALUES F5=$,S114, I3* *024 WHENEVER SIGMA.G.O.757,PRINI FORMAT Ff,,J *025 -.-VECTOR VALUES F6=bS:114,1'3*$ -- *026 WHENEVER SIGMA.L.0.75',TRANSFER T1 HASE? *027 WHENEVER SIGMA.G.C.757,TRANSFER TI BASL6 *028 INTEGER I,J -0?9 END OF PROGRAM,030

-124S COMPILE MAD,EXECUTE,PRINT OBJECT,DUMP 000036 MAD (16 APR 1962 VERSION) PROGRAM LISTING......... PRINT FORMAT FZ *001 VECTOR VALUES F2=$//S9,5HSIGMA,S12,5HALPH8,S12,5HALPH9,S12,6H *002 1ALPH14,SI2,6HALPH15,S12,6HALPH16,S12,6HALPH17*$ *002 ALPH3=4.8387097 *003 ------------------- ALPH4=1./93 *004 BASE1 READ FORMAT FL, SIGMA, DELTA *005 VECTOR VALUES -Fl=2F20.5*$ *006 ALPH7=1.8*DELTA-1. *007 ALPHO=1./93./DELTA *008 ALPH1=3.+25./31./ALPH7 *009 ALPH2=5./93./DELTA+5.*DELTA/31./ALPH7 *010 ALPH5=5./93.5./9 31./ALPH7 *011 ALPH6=5./2232./ALPH7 *012 ALPH1O=(5./2232.)*(1./DELTA/ALPH7+SIGMA/ALPH7*DELTA.P.3.) *013 ALPHII145./62.+50./93./ALPH7 *014 ALPH13=5./186./ALPH7 *015 S=(DELTA.P.2.)*(2./21.-6./35.*DELTA+1./9.*DELTA.P.2. *016 1 -8./315.*DELTA.P.3.) *016 W=(DELTA.P.2.)*(I./15.-1./45.*DELTA.P.3.+1./105. *017 1 *DELTA.P.4.) *017 V=DELTA.P.4.*(1./560.-1./540.*DELTA+1./1890.*DELTA.P.2.) *018 X=DELTA.P.3.*(-1./84.+1./120.*DELTA.P.2.-1./315.*DELTA.P.3.) *019 - - Y=DELTA.P.4.*(-./105.-1./56.:*DELTA+1./84.*DELTA.P.2.-1./360.* *020 1DELTA.P.3.) *020 Z (DELTA.P.-5. *(1 /36.-4./315.*DELTA) --- *021 BETAL=3.+(5./252.)/(SIGMA*DELTA*S*ALPH7) *022 BETA2=6.*Z/S *023 BETA3=6.*W/S *024 BETA4=SIGMA*Y/6./S *025 BETA5=DELTA/252./S/ALPH7+5.*SIGMA*Y/6./S *026 BETA6=SIGMA*DELTA.P.3./18144./S/ALPH7 *027 BETA7=6.*V/S *028 BETA8=3.*BETA7 *029 BETA9=18.*X/S *030 ALPH8=(ALPH3*BETA3+ALPHI*BETA2)/(ALPH2*BETA3+ALPHI*BETA1 *031 1-2.*ALPHO*BETA3-2.*ALPH1) *031 ALPH9=(1./ALPHl)*(ALPH3+ALPH8*(2.*ALPHO-ALPH2)) *032 ALPH14=(BETAI*(ALPH1L-ALPHlO*ALPH8-ALPH5*ALPH9)+ALPH2*(ALPH8* *033 1(BETA5-2.*BETA4)-ALPH9*BETA8-BETAA9))/(2.*BETA+ALPH1*BETAl+ *033 23.*ALPH2*BETA3) *033 ALPH15=(ALPH8*(BETAS-BETA5-2.ETA4)-BETA8*ALPH9-BETA9-3.*BETA3* *034 1ALPH14)/BETA1 *034 ALPH16=((4.+ALPHl)*(2.*BETA5*ALPH15-2.*BETA6*ALPH8+ *035 12.*ALPH14*.BE BETA7))58+2.*B*BE7-. TA3*(2.*ALPH6*ALPH9-2.* *035 2ALPH13+2.*ALPH14*(ALP H5+2ALPH4)-2AL1A L PH15))/(5.* *035 3BETA3*(ALPH2+2.*ALPHO)+(4.+ALPH1)*(2.+BETAl)) *035 ALPH17=(2.*BETA5*ALPH15-2.*ALPH8*BETA6+ALPH14*(4.*BETA7+2.*BE *036 ITA8)-ALPH16*(2.+BETA) )/5./BETA3 *036 PRINT FORMAT F3,SIGMA,ALPH8,ALPH9,ALPHl4,ALPH15,ALPH16, ALPH17 *037 VECTOR VALUES F3=$7E17.8*$ *038 TRANSFER TO BASE1 *039 END OF PROGRAM *040

-125$ COMPILE _MAD__EXECUTE,PRINT OBJECT,uUMP 057458 ------ - t_ — r P IN E —---------- -- ---- --'- 057458 MAD (16 APR 1962 VERSION) PROGRAM LISTING-... PRINT FORMAT F2 -*001 VECTOR VALUES F2=$///S10 l1HNSlO,5HSIGMASIO,8HGAMMA(N),S10,9 *002 1HLAMBCA ( N S *002 - -J=1 *003 DIMENSION GAMMA(100),LAMBDA(100) *004 __________________ ALPH41.1_. - ____________________________________ 005 BASEl READ FORMAT Fl,SIGMA',DELTA,N1,N2, GAMMA ( 1 ),GAMMA(2),LAMBDA( *006 ll ),LAMBDA(2) *006 VECTOR VALUES Fl=$2F20.5/2I5/4F15.5*S *007 ALPH7=1.8 CELTA-1!. *008 ALPHO=1./93./DELTA *- 009 __________ ________ALPH1=3.+25./31./ALPH7 —- — _____ *010 ALPH2=5./93./DELTA+5. DELTA/31. /ALPH7 *011 ALPH5=5./93.+5./31./ALPH7 *012 ALPH6=5./2232./ALPH7 *013 ALPHlG=(5./2232.)*(1./DELTA/ALPH7+SIGMA/ALPH7*DELTA.P.3.) *014 S=(OELTA.P.2. )*(2./21.-6./35.*DELTA+1./9.*DELTA.P.2. *015 1_ -8./315.*-IELTA.P.3.) _ 015 W=IDELTA.P.2.)'( 1./15.-1./45.*DELTA.P.2. )( 1./15.-1./45..P. 3.+1./105. 016'1 *DELTA.P.4.) *016 V=DELTA.P.4.*(1./560.-1./54.0.*DELTA+1./1890.*DELTA.P.2. ) -*017 Y=DELTA.P.4.* (1./105-.-1./5'6.*DELTA+1 1/84.*DELTA.P.2.-1. /360.* *018 1DELTA.P.3.) / *018 BETA1=3.+(5./252.)/(SIGMA*DELTA*S*ALPH7) *019 BETA3.=6. *W/S *020 BETA4=S IGMA*Y/6./S *021 BETA5=DELTA/252./S/ALPH7+5.*S IGMA*Y/6. /S *022 BETA6=SIGMA*DELTA.P.3./18144./S/ALPH7 *023 BETA 7=6.'V/S *024 BETA8=3.*BETA7 _*025 THROUGH BASE2, FOR N=N,N2,.ABS. LAMBOA (N-1 ). LE. 1. E-36. OR. *026 1.ABS.GAMMA (N-1).LE. 1. E-36.CR..ABS.LAMBDA (N-1 ).GE. 1. E+36.OR. *026 2.ABS.GAMMA(N-.1),.GE. 1.E+36 *026 A1. —__N___ _ALPH5+~2, m (N- 1. ).*AL PH4) * _(-J). P. N__ ________ _ *027 A2=N* (N-1. )*ALPH6 028 -_ ___________ ALPHO+ALPH2____ ___________N____________A_,_P 0 + AL P H 2____'_______ _ *029 A4=-N*ALPHLO* (-J).P.N *030 A5=2. *N+AL PHI *031 C 1=-N( 2.* (N-2.)*BETA4+BETA5) (-J).P.N *032. —---------------- --- C__ _N- l ~TA6 _-__ _ _A_ *033 C3=(2.*N+1. )*BETA3 *034 4. _,__-_ _______J__-___ C_4-E-N* ( 7N- 1. ETA7+BETA8 ) * (-J }) P. N *035 C5=2.*(N-1. )+BETA1 * —036 LAMBIDA(N)=i(1./(C5*A5+A3-*C3 ))*(A5* (-C1.*LAMBDA(N-1)+C2* LAMBDA(N *037 1-2 )+C4*GAMMA(N- 1 ) )+C3 (A2* GAMMA (N-2 )-A1.*GAMMA (N-1 )-A4*LAMBDA ( *037 -.. —-. — _- __-1' _1 _ }..-I*. _ _ _ _ _ _ __._._._._._.037 GAIMA(N)=(A2*GAMMA(N-2)-A1*GAMMA(N-1)-A4*LAMBDA( N-i )-A3*LAMBD *038 ---------— _ —I_ A_ __ *038 BASE2 PRINT FORMAT F3,N,SIGMAGAMMA(N)tLAMBDA(N) *039 VEC I::R VATI 1 FP__::T= 1 l__../_..:I 7F7?fI - "F040 TRANSFER TO BASE1 *041 I... ____ T _ EGfERJ _N lN, 0N3_042 END OF PROGRAM *043

-126APPENDIX VI EXPERIMENTAL DATA A fr Tw - T. T. hT hr hc BTU BTU BTU in. cps OF OF 2 o hr-ft2- ~F hr-ft2- F hr-ft2- F 0.035 10 147.3 79.1 1.1172 o.ogg998 1.0170 0.082 10 147.5 82.2 1.1160 0.1013 1.0150 0.103 10 147.8 82.4 1.1138 0.1015 1.0120 0.032 10 150.9 82.8 1.0962 0.1127 o. 983 5 0.064 10 151.0 83.4 1.0955 0.1130 O.9825 0.120 10 149.6 83.6 1.1057 0.1127 0.9930 0.0225 15 135.6 76.1 1.0393 0.1026 0.9367 o.o49 15 135.8 76.4 1.0377 0.1029 0.9348 0.074 15 132.2 78.5 1.0660 0.1029 o. 9631 0.107 15 130.3 79.0 1.0815 0.1027 0.9788 0.047 15 150.7 83.6 1.0977 0.1131 0.9846 0.083 15 142.3 84.8 1.1624 0.1113 1.0511 0.026 15 187.3 83.6 1.2433 0.1271 1.116 o.o48 15 187.3 84.0 1.2433 0.1273 1.116 0.081 15 185.1 83.6 1.2581 0.1264 1.132. 108 15 177.6 84.9 1.3112 0.1248 1.186 0.0162 20 150.1 79.6 1.0994 o.0982 1.0012 0.0325 20 150.2 79.8 1.09og86 o.0983 1.0003 0.061 20 142.6 80.0 1.1572 o0.09og64 1.0608 o.o89 20 142.0 82.2 1.1593 0.1175 1.0418 0.120 20 138.1 83.2 1.1920 0.1168 1.0752 0.033 50 139.8 83.1 1.1775 0.0997 1.0778 0.080 50 173.5 86.0 1.3422 0.1191 1.223 0 - 157.1 83.6 1.1526 0.1305 1.0221 0.023 12 157.3 83.0 1.1511 0.1302 1.0209 0.045 12 158.4 81.7 1.1431 0.1298 1.0133 0.064 12 158.2 83.0 1.1446 0.1305 1.0141 0.076 12 157.7 83.5 1.1482 0.1507 1.0175 0.09ogo 12 157.2 83.8 1.1519 0.1307 1.0212 0.118 12 152.1 84.5 1.2040 0.1288 1.0620 0.145 12 147.8 84.6 1.2251 0. 1280 1.0971

UNIVERSITY OF MICHIGAN 3 9015 02826 82851111111 I! 9015 02826 8285