\” THE INFLUENCE OF HEAT TRANSFER ON THE ’ ’ STABILITY OF PARALLEL FLOW BETWEEN , ~ CONCENTRIC CYLINDERS Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY YUNG KOOK CHOO 1975 This is to certify that the thesis entitled The Influence of Heat Transfer on the Stability of Parallel Flow Between Omentrie Cylinders presented by Yung Kook Choc has been accepted towards fulfillment of the requirements for 31.1). “ms-cal W Jegree in >wa (4.6% Major professor Date my 2' 1975 0-7639 I .1 if] E" . I i. V ' ABSTRACT f 11%;} ‘-._ ’ I’i THE INFLUENCE OF HEAT TRANSFER ON THE STABILITY L) 0F PARALLEL ELow BETWEEN CONCENTRIC CYLINDERS By Yung Kook Choo The stability of parallel flow between concentric cylinders at different temperatures is investigated for infinitesimal velocity and pressure disturbances. Primary interest is in the effect of heat transfer and the radius ratio of the inner to outer cylinder on the critical point of the neutral stability curve. The modified Orr-Sommerfeld equation includes perturbations for the velocity and viscosity as well as its gradient terms, and results in a fourth order ordinary differential equation. This equation and boundary conditions represents an eigenvalue problem that is transformed to an equivalent initial value problem and solved numerically using a fourth order Runge-Kutta integration scheme. The results indicate a strong dependence of the critical eigenvalues on both the heat transfer and the radius ratio a/b of inner to outer cylinder. Heating the inner cylinder destabilizes the flow. The critical Reynolds number of an isothermal flow monotonically decreases supporting the accepted conclusion that the critical Reynolds number becomes infinitely large as a/b Yung Kook Choo approaches zero. The critical Reynolds number of the non-isothermal flow seems to approach a finite value by showing an inflection point on the curve R vs a/b near the radius ratio a/b=0.4. For the crit radius ratio a/b=0.4 the upper branch of the neutral stability curve (a vs. Reynolds number) of the non-isothermal flow appears to have a vertical, or near vertical, asymptote rather than the horizontal asymptote of the isothermal flow. THE INFLUENCE OF HEAT TRANSFER ON THE STABILITY OF PARALLEL FLOH BETWEEN CONCENTRIC CYLINDERS BY Yung Kook Choo A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1975 To Sung, Kenneth, Esther ii ACKNOWLEDGEMENTS The author wishes to express his gratitude to his Major Professor, Dr. Merle C. Potter, Professor of Mechanical Engineering for his helpful comments and guidance and for his encouragement throughout this study. The author also wishes to express his gratitude to Dr. J. F. Frame, Dr. M. C. Smith, and Dr. C. R. St. Clair for serving as members of the doctoral guidance committee. Appriciation is also expressed to Dr. R. w. Little and the Department of Mechanical Engineering for supporting the computer use. Appreciation is also expressed to the Board of Water and Light for providing educational assistance. The author also wishes to express his thanks to his wife Sung and his parents for their constant encouragement and support throughout the author's graduate program. iii LIST OF TABLES ....................... LIST OF FIGURES NOMENCLATURE ........................ TABLE OF CONTENTS Chapter I. II. III. IV. INTRODUCTION .................... l.l Review of Literature .............. 1.2 Purpose of the Present Study .......... FORMULATION OF THE PROBLEM ............. 2.l Governing Equation ............... 2.2 The Governing Equations of the Main Flow . 2.3 The Linearized Equation for Small Disturbances 2.4 Eigenvalue Problem ............... NUMERICAL METHODS . ................. 3.l Numerical Solution of the Governing Equations of the Main Flow ........... 3.2 Numerical Solution of the Stability Equation . . 3.3 Iteration Scheme for the Eigenvalues ...... RESULTS, CONCLUSIONS, AND RECOMMENDATIONS FOR FURTHER STUDY .................... 4.l Numerical Results ................ (a) Governing Equations of the Main Flow (b) Stability Equation . . . . . . . . . . . . I 4.2 Conclusions ................... 4.3 Recommendations for Further Study ........ iv Page vi ix xi Chapter Page ILLUSTRATIONS (Tables and Figures) .............. 35 BIBLIOGRAPHY ......................... 86 APPENDICES .......................... 90 A. RUNGE-KUTTA SCHEME FOR THE FOURTH ORDER DIFFERENTIAL EQUATION .............. 90 B. COMPUTER PROGRAM B-l Description of the Computer Program ........ 94 B-2 Listing of the Computer Program .......... 96 Tables 1. 10. 11. 12. 13. LIST OF TABLES Velocity and Temperature Distribution of Main a/b = 0.99, Ti = 20°C, T0 = 0°C, R = 3900 ....... Velocity and Temperature Distribution of Main a/b = 0.99, Ti = 40°C, T0 = 0°C, R = 6390 ....... Velocity and Temperature Distribution of Main a/b = 0.99, Ti = 60°C, To = 0°C, R = 8760 . . Velocity and Temperature Distribution of Main a/b = 0.99, Ti = 80°C, To = 0°C, R = 11050 . Velocity and Temperature Distribution of Main a/b = 0.99, Ti = 100°C, To = 0°C, R = 13370 . Velocity and Temperature Distribution of Main a/b = 0.1, Ti = 20°C, To = 0°C, R = 401460 . Velocity and Temperature Distribution of Main a/b = 0.1, Ti = 40°C, T0 = 0°C, R = 626120 . Velocity and Temperature Distribution of Main a/b = 0.1, Ti = 60°C, To = 0°C, R = 826850 . Velocity and Temperature Distribution of Main a/b = 0.1, Ti = 80°C, To = 0°C, R = 1013510 . Velcoity and Temperature Distribution of Main a/b = 0.1, T.i = 100°C, To = 0°C, R = 1190270 Significance of dk/dr and Viscous Dissipation a/b = 0.99, Ti = 20°C, To = 0°C, R = 3900 ....... Significance of dk/dr and Viscous Dissipation a/b = 0.99, Ti = 40°C, To = 0°C, R = 6390 . . Significance of dk/dr and Viscous Dissipation a/b = 0.99, Ti = 60°C, To = 0°C, R = 8760 . . vi Flow Flow Flow Flow Flow Terms Terms Terms Page 36 37 38 39 4O 41 42 43 44 45 46 47 48 Tables Page 14. Significance of dk/dr and Viscous Dissipation Terms a/b = 0.99, Ti = 80°C, To = 0°C, R = 11050 ....... 49 15. Significance of dk/dr and Viscous Dissipation Terms a/b = 0.99, T1 = 100°C, To = 0°C, R = 13370 ....... 50 16. Significance of dk/dr and Viscous Dissipation Terms a/b = 0.1, Ti = 20°C, To = 0°C, R = 401460 ....... 51 17. Significance of dk/dr and Viscous Dissipation Terms a/b = 0.1, Ti = 40°C, To = 0°C, R = 626120 ....... 52 18. Significance of dk/dr and Viscous Dissipation Terms a/b = 0.1, Ti = 60°C, To = 0°C, R = 826850 ....... 53 19. Significance of dk/dr and Viscous Dissipation Terms a/b = 0.1, Ti = 80°C, To = 0°C, R = 1013510 ...... 54 20. Significance of dk/dr and Viscous Dissipation Terms a/b = 0.1, Ti = 100°C, T0 = 0°C, R = 1190270 ...... 55 21. Critical Eigenvalues for Various Radius Ratios of Inner to Outer Cylinders ................ 56 22. Eigenvalues for Ci = 0, a/b = 0.99, Ti = To = 20°C . . . 57 23. Eigenvalues for Ci = 0, a/b = 0.99, Ti = 60°C, T0 = 20°C ....................... 57 24. Eigenvalues for c1 = O, a/b = 0.99, Ti = 100°C, To = 20°C ....................... 58 25. Eigenvalues for Ci = 0, a/b = 0.8, Ti = T0 = 20°C . . . 58 26. Eigenvalues for c1. = 0, a/b = 0.8, Ti = 60°C, T = 20°C ....................... 59 o 27. Eigenvalues for c1 = 0, a/b = 0.8, Ti = 100°C, T = 20°C ....................... 59 o 28. Eigenvalues for Ci = 0, a/b = 0.6, Ti = To = 20°C . . . 60 29. Eigenvalues for c1 = 0, a/b = 0.6, Ti = 60°C, To = 20°C ....................... 60 30. Eigenvalues for c. = 0, a/b = 0.6, T. = 100°C, T0 = 20°C ....................... 61 Tables 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. Eigenvalues for c1 0, a/b = 0 4, Ti = TO - 20°C Eigenvalues for c, O, a/b = 0.4, Ti - 40°C T = 20°C ....................... o Eigenvalues for c1 0, a/b = 0.4, Ti = 60°C, T = 20°C ....................... o Eigenvalues for Ci 0, a/b = 0 4, Ti = 100°C, T = 20°C ....................... o Eigenvalues for c1 0, a/b = 0 3, Ti = To - 20°C Eigenvalues for c1 0, a/b = 0 3, Ti - 60°C, T = 20°C ....................... o Eigenvalues for Ci 0, a/b = 0 3, Ti = 100°C, T = 20°C ....................... o Eigenvalues for Ci 0, a/b = 0.25, T1 ' To - 20°C Eigenvalues for Ci 0, a/b = 0.25, Ti = 60°C, T = 20°C ....................... o Eigenvalues for Ci 0, a/b = 0 25, Ti = 100°C, T = 20°C ....................... o Eigenvalues for Ci 0, a/b = 0.4, Ti = 30°C, T = 20°C ....................... o Viscosity and conductivity of water .......... viii Page 62 62 63 64 65 65 66 67 69 69 Figure 10. 11. 12. 13. 14. LIST OF FIGURES A Neutral Stability Curve ................. Main Flow Configuration .................. Dimensionless Main Flow Velocity Profiles for a/b = 0.99 and Various Boundary Temperatures (Ti’ To) ........ Dimensionless Main Flow Velocity Profiles for a/b = 0.1 and Various Boundary Temperatures (Ti’ To) ........ Dimensionless Main Flow Velocity Profiles for (Ti’ To) (100°C, 0°C), and a/b = 0.1 and 0.99 ........... Significance of dk/dr Term for a/b = 0.1 and (Ti’ To) = (100°C, 0°C) ....................... Significance of dk/dr Term for a/b = 0.99 and (Ti’ To) = (100°C, 0°C) ....................... Nave Number a Versus Reynolds Number R for c. = 0, a/b = 0.99 and Various Boundary Temperatures ....... Nave Number a Versus Reynolds Number R for c. = 0, a/b = 0.8 and 0.6 and Various Boundary TempeIatures . . . . Nave Number a Versus Reynolds Number R for c = O, a/b = 0.4 and Various Boundary Temperatures ....... Have Number a Versus Reynolds Number R for c- = 0, a/b = 0.3 and Various Boundary Temperatures ....... Have Number a Versus Reynolds Number R for Ci = 0, a/b = 0.25 and Various Boundary Temperatures ....... Critical Reynolds Number Rcrit Versus Radius Ratio a/b Effect of Heat Transfer on the Neutral Stability Curve for a/b = 0.4, T0 = 20 and Ti = 20°C, 30°C, 40°C, 60°C, 100°C .......................... ix Page 17 70 71 72 73 74 75 76 77 78 79 80 81 82 Figure 15. 16. 17. Page Eigenfunction at the Critical Point for a/b = 0.99, To = 20°C .................... 83 Eigenfunction at the Critical Points for a/b = 0.4, To = 20°C .................... 84 Eigenfunction at a = 3.2915, R = 29613, Cr = 0.258 for a/b = 0.4, Ti = 100°C, T0 = 20°C ......... 85 a1’°2 = +1 C CY‘ C NOMENCLATURE xi Radius of inner cylinder, cm Coefficients used to obtain combined eigenfunctions Radius of outer cylinder, cm Complex propagation speed of wave Have speed Amplification rate Constant coefficients used in the theoretical expression of viscosity where i = 1, 2, . . ., 10 Specific heat at constant pressure, cal/g degC Constant coefficients used in the theoretical expression of conduc- tivity where i = 1, 2, . . ., 10 NmZ Eckert number = CplTi ' TOI Test function F Thermal conductivity (Dimensionless) Thermal conductivity, cal/s cm degC Thermal conductivity at a selected base temperature, cal/s cm degC Pressure Prandtl number = no cp/ko AR t* T* T(I,1) T(I.2) T(I,3) T(I.4) N(I,l) N(I,2) Radius (Dimensionless) Radius, cm Reynolds number = p(b-a)Nm/uo Finite change in R Time (Dimensionless) Time, 5 Temperature (Dimensionless) Temperature, °C Temperature at node I when both dk/dr and o are neglected Temperature at node I when dk/dr term is kept, but 6 is neglected Temperature at node I when dk/dr term is neglected, but A is kept Temperature at node I when both dk/dr and t are kept Radial velocity Circumferential velocity Dimensional radial velocity, cm/s Dimensional axial velocity, cm/s Dimensional circumferential, velocity, cm/s Axial velocity Velocity of main flow Velocity of main flow at node I when both dk/dr and o are neglected Velocity of main flow at node I when both dk/dr is kept, but t is neglected xii “(1,3) ”(1.4) <9 ‘1’ Superscripts ()' Subscripts (I,- I)o () crit xiii Velocity of main flow at node I when dk/dr is neglected, but a is kept Velocity of main flow at node I when both dk/dr and o are kept Mean velocity, cm/s Coordinate in axial direction (Dimensionless) Coordinate in axial direction, cm Nave number in flow direction Finite change in A constant representing R g;- Viscosity (Dimensionless) Viscosity, g/cm 5 Viscosity at a selected base tempera- ture, g/cm 5 Density, g/cm3 Complex eigenfunction of velocity perturbations Viscous dissipation function Stream function Differentiation with respect to r or perturbation quantities Quantities on inner cylinder Quantities on outer cylinder Critical eigenvalues CHAPTER I INTRODUCTION 1.1 Review of Literature The phenomenon of transition from laminar to turbulent flow which is fundamental for the science of fluid mechanics was first investigated experimentally by Reynolds (1883). He discovered in his pipe flow experiments that transition from laminar to turbulent flow always occured at nearly the same Reynolds number (critical Reynolds number). Efforts to clarify and to explain theoretically the process of transition were done notably by Lord Rayleigh (1880, 1887). Based upon this work, independent studies by Orr (1907) and Sommerfeld (1908) in which disturbances of the form ¢(y)exp[iq(x-ct)] were imposed on the main flow led to the now well known Orr-Sommerfeld stability equation for two-dimensional plane flow, which is given by (w-c) (¢"-a2 I - H". = - 5},- II‘V - 201%" + 0.4.. I where N Velocity of the main flow 0 II +. 0 Cl" 1C1 Complex wave propagation speed (Cr is the velocity of propagation of the imposed disturbance and Ci is the amplification rate of the disturbance) 1 a = Have number of the imposed disturbance ¢ = Eigenfuntion (amplitude funtion) R = Reynolds number. Theoretical investigations are based on the assumption that laminar flows may be unstable to certain small disturbances. The behavior of such disturbances is followed in time after they are superimposed on the main flow which remains to be determined in particular cases. If the disturbances decay with time (°i<0)’ the main flow is considered stable. 0n the other hand, if the distur- bances increase with time (°i>0) the flow is considered unstable, and there exists the possibility of transition to either a secondary laminar flow or a turbulent flow. Neglecting the effects of viscosity, Lord Rayleigh (1914) was able to show that any velocity profile that possesses an inflec- tion point is unstable. Much later, Tollmein (1935) proved that this was not only a necessary but sufficient condition for the occurrence of instability. Prandtl (1914) postulated the existence of a viscous bound- ary layer and was able to define transition, separation and drag coefficients on bodies. Incorporating the viscous boundary layer into stability theory, Prandtl (1912) considered flow over a flat plate and included the effects of the largest viscous terms near the wall. This work along with calculations performed by Tietjens (1925) gave the startling result that the introduction of viscosity into the equations did not produce damping as was presumed but amplifica- tion for sufficiently large Reynolds numbers for particular wavelengths of the disturbances. This result was obtained not only for unstable velocity profiles but also the profiles which are stable when vis- cosity was neglected. Tollmein (1929) demonstrated that the effect of viscosity must be taken into account not only near the wall but also in the critical layer, a narrow region surrounding the critical point at which the main flow velocity and the wave propagation velocity are equal, that is, w = Cr' In addition, he also showed that the influence of vis- cosity leads to instability only if the main flow velocity profile is other than a straight line. The method developed by Tollmein, based on asymptotic theory, provided the mathematical basis for later progress in the stability area. Lin (1945, 1946, 1955) was able to provide a firm mathematical basis for the asymptotic expansion theory and was able to explain the nature of the functions near the critical point. He discussed what he called the inner viscous layer which includes the critical point, and the outer viscous layer, a wall layer. The asymtotic expansion method was used almost exclusively until the advent of modern high-speed digital computers which permit the use of more accurate numerical techniques. The stability of plane Poiseuille flow was investigated by Thomas (1953) with a numerical scheme. He obtained a value for the minimum Reynolds number, Rcrit’ for neutral stability of 5780, which is based on maximum channel velocity and the half-width. This value has been shown to be more accurate than Lin's (1945) value of 5300 or Stuart's (1954) value of 5100 based on asymptotic techniques. Potter (1965) studied the stability of plane Couette-Poiseuille flow by asymptotic expansions and later (1967) performed numerical calculations for symmetrical parabolic flows. The values obtained it were in close agreement with those of Thomas. The point of instability as determined theoretically and the for Rcr point of transition, as observed in experiment, to turbulent flow often differ considerably. An explanation for these differences was thought by some to be due to the fact that the derivation of the Orr- Sommerfeld equation is based on the assumption of two-dimensional disturbances only. Squire (1933) showed that if three-dimensional disturbances are considered the flow is more stable, thus attention is typically limited to two-dimensional disturbances. The distance between the point of instability and the point of actual transition depends upon the degree of amplification present and the intensity of fluctuations present in the main flow. But the actual mechanism of amplification can be.obtained from the.study of the magnitudes of the parameters in the interior of the curve of neutral stability on which c1 = 0. Calculations of this kind were first performed by Schlichting (1933) for the flat plate. More recently, Shen (1954) repeated Schichting's calculations, and Stuart (1956) investigated the amplification of unstable disturbances by accounting for the first order effect of the non-linear terms in the equations. Emmons (1951) observed that any disturbance which triggers transition may be "local in time" and once initiated, the turbulent spot moves downstream growing steadily in all directions. This phenomenon was studied by Schubauer' and Klebanoff (1955, 1956). They indicated that there is no well defined point of transition but that the process of transition from laminar to fully developed turbulent flow extends over a finite distance. Transition from laminar to turbulent flow in a boundary layer is now believed to take place within 4 stages. At the first stage, infinitesimal two-dimensional waves called Tollmein-Schlichting waves, begin to amplify and become unstable. The two-dimensional waves become three dimensional and result in hairpin eddies at the second stage. In the third stage low Speed turbulent streaks or bursts (Emmons' spots) originate near the wall, and finally in the fourth stage the burst rate becomes constant and the transition to fully turbulent motion is completed. Morkovin (1958) reviews some of the recent advances in the study of transition and discussed the mechanisms involved in the above mentioned stages. Stability theory yields a critical Reynolds number that cor- responds to stage one. Since the third stage is the first point at which large scale variations take place, this is often considered to be transition by engineers. These differences along with the slower response times of earlier instrumentation serve to explain some of the discrepancies between theory and experiment. Stability predictions in channel flow yield critical Reynolds numbers that also correspond to infinitesimal disturbances but the stages of transition are not as apparent as in boundary layer flow. Free stream disturbances or disturbances which result from wall rough- ness amplify and lead to the transition described above but the effect is propagated throughout the flow and the entire channel becomes turbulent. Barnes and Coker (1905) confirmed experimentally that the critical Reynolds number for a pipe flow increases as the disturbances in the flow before the pipe are decreased. Ekman (1910) succeeded in maintaining laminar pipe flow up to a critical Reynolds number of 40000 by providing an inlet which was made exceptionally free from disturbances. The upper limit to which the critical Reynolds number can be driven with extreme care is not known at present. Transition process of a pipe flow was investigated in detail by Rotta (1956). His measurements reveal that in a certain range of Reynolds numbers around the critical the flow becomes intermittent which means that it alternates in time between being laminar-and turbulent. Recently Gill (1965) performed an exhaustive analysis of the stability problem of a pipe flow and found the flow to be always stable to axisymmetric disturbances. Davey and Drazin (1969) solved the equations for the axisymmetric case and concluded that the flow is stable to all such disturbances. The stability of a flow over heated and cooled flat plates were investigated by Mazzan, Okamura, and Smith (1960). They found that the stability of water flows to Tollmien-Schlichting waves is extremely sensitive to initial heating or cooling. They also indicated that in the case of a heated wall, terms involving the first and second derivatives of viscosity in the modified Orr-Sommerfeld equa- tion are found to have a considerable destabilizing effect. Mott and Joseph (1968) investigated the linear stability of parallel flow in concentric cylinders by solving the stability equa- tion in which the viscous gradient terms were neglected. The critical Reynolds number of an isothermal flow was found to be a monotone function of the radius ratio of the outer to inner cylinder, increas- ing from the plane Poiseuille flow limit to the Hagen-Poiseuille flow limit. The neutral curves for skewed profiles caused by heating the inner cylinder were found to have a second minimum, which for sufficiently skewed profiles, gave the lowest value of the Reynolds number. They also found that heating resulted in a more stable flow. Potter and Graber (1972) investigated the stability of plane Poiseuille flow with heat transfer. The study indicated that inclusion of viscosity gradient terms, even though they are small, is very important. It also indicated that a double critical point does not appear in the neutral stability curve when the viscosity gradient terms are included in the equation in contrast with the result of Mott and Joseph (1968). Finally, their results indicate that heating one of the plates results in a more unstable flow. 1.2 Purpose of the Present Study The purpose of the present study is to investigate the sta- bility of a parallel flow of water between concentric cylinders to infinitesimal axially symmetric disturbances under the influence of heat transfer. The main flow is affected by the heat transfer between cylinders of different temperatures through the variation in viscosity and conductivity with temperature since the viscosity and conductivity of water are quite sensitive to temperature. The viscosity gradient subsequently causes additional terms to appear in the stability equation whose solution will be sought in the present study. The effect of radius ratios will also be investigated by changing the inner radius while the outer radius is fixed. The main flow (fully developed laminar flow) is examined, using the energy and momentum equations with the values for water given by Poots and Rogers (1965) for the viscosity and conductivity. They are taken from the experimental data in tables compiled by Mayhew and Rogers (1964). The differences between this study and previous work done by Mott and Joseph (1968) are that in this investigation all the viscous gradient terms which appear due to the heat transfer are included in the modified Orr-Sommerfeld equation and the complete energy and momentum equations are solved to determine the main flow. Non-linear stability theories typically involve methods which use the eigenvalues from the linear theory as an imput. Such theories have not been applied to pipe flow since for the isothermal case the flow is always stable. Heat transfer has been shown to destabilize the flow. Thus, it is of interest to investigate the flow between concentric cylinders as the inner cylinder is heated and reduced in diameter. As the diameter of the inner cylinder is decreased to zero, the flow may be unstable at a finite Reynolds number if the heating is sufficient. If it is, then eigenvalues can be found for a pipe flow. CHAPTER II FORMULATION OF THE PROBLEM 2.l Governing Equations For incompressible axially symmetrical flow (i.e. all variables are independent of e and v6 = 0) with variable properties the equations of motion in cylindrical coordinates, using dimensional velocity com- ponents v and vZ and other dimensional variables indicated by r astericks, are Dv r _ 89* 8 p613?" ' ar* + 3r* [1‘ ”an” 8v 3v 2 + 33.,- Eu* (3,73% + ——,,..I ] 2E? avr Vr (2 1 1) + r* 15??“ - FT') ° ' DY — - 92:. _§_. 0* Dt* av 8v 1...?— ~k 1: _C _z_ + r* 3r* [ H r (32* + 3P*) 1 (2-1-2) D ——° + v ——3 +v ___8 . 10 The equation of continuity for the same flow is %;-5%;- (Y, r*) + 5§;-(vz) = 0. (2.1.3) For incompressible, steady, parallel flow (v2 = vz(r*), Vr = v6 = 0), and constant boundary temperature the energy equation can be written as dv * (r*k* §%;l + u* (37%)2 = 0. (2.1.4) l_._£L_ r* dr* To write the equations in non-dimensional form, choose the average velocity Hm, the difference between outer and inner radii (b-a), the difference between the temperatures on the inner and outer cylinders (To'Ti)’ and viscosity (no) and conductivity (k0) at a constant temperature, as reference quantities. The dimensionless variables will then be "' r*-a =z_:__ r ' b-a z b-a T*-T. , T = T -T) p = 5—97- (2.1.5) o 1 Mm _ u* _ k* u-—- k-—- 110 ko v v - _E. = .2. u - N w N1 t =tflfl 11 Introducing these quantities in the governing equation yields Bu au Bu ——-+ + a "a? "32’ = - 92. l. 9.. 3” 9.. .99 .9! 3r + R [ (2p——) + 32 {0 (DZ + 3r)} Br Br +£(9Jl--u__)] (216) a1+r 3r a1+r ' ‘ 8w 3w 3w 7+”§F+”§z‘ 32 az az a1+r 3r 1 DZ 3r (2.1.7) .1 a. 91!. . 3??- Br I: U {a]+r} ] + 32 0 (2..18) 1 d -—- dT dw 2 _ a1+r dr [ {a]+r} kHF'] + sgn (To-T1)PrEcku(a;fl - 0 (2.1.9) Reynolds number = p(b-a)Hm where R = 11o “o C P = Prandtl number = ————E- r ko m I H 2 = m ck — Eckert number EETTT:T;1 lm 12 For the variable viscosity 0 and conductivity k in the equations the theoretical expressions of Poots and Rogers (1965) are employed: lO ,- u = H exp (C10 ) (2.1.20) i=0 lO 1. = n exp (oio ) (2.1.11) 1:0 where 0 = (Ti-50) + (To"Ti)T 50 and Ci and Di are constants displayed in Table 42. These values agree with experimental data within 0.2%. 2.2 The GoverninggEquations of the Main Flow The problem shall be simplified by stipulating that the main flow velocity N depends only on r, whereas the remaining two components are assumed to be zero everywhere. The equation of motion for such a parallel flow is l d an = gg_ 31+? '6? [11 {3177'} Eff] RdZ (2.2.1) where R%§~= A is a constant, P being the pressure in the main flow. The boundary conditions for equation (2.2.1) are N = 0 @ r = 0 and r = 1. (2.2.2) 13 The energy equation for the main flow with constant tempera- ture on the boundary is as given by equation (2.1.9). The boundary conditions are T = 1 @ r = 1. (2.2.3) The mean velocity, by which the velocities are non-dimensiona- lized, of the main flow is defined as b I v22flr*dr* w E a m n(b2-a2) (2.2.4) Expressing v2 and r* in terms of the non-dimensional variables defined in equation (2.1.5) and then substituting them in equation (2.2.4) yields the following constraining equation: 1 f w[a1+rldr=a1+-)- (2.2.5) 0 These equations with boundary conditions will be numerically solved by iteration. Iteration is necessary since the viscosity in equation (2.2.1) is not known until the temperature is known. Conversely, the temperature cannot be determined from equation (2.1.9) until the velocity N(r) is known. A brief description of the iterative process is presented in Section 3.1. 14 2.3 The Linearized Equations for Small Disturbances It was assumed that the main flow is a parallel flow. Hence, the perturbed dependent variables can be written as u = u' v = v' (2.3.1) w = N + w' p=P+p' neglecting temperature, conductivity and viscosity perturbation. N and P are the velocity and pressure of the undisturbed main flow, respectively. Substitution of (2.3.1) in the equations of motion yields the first order equations for the small disturbances (u', v', w', and p' are assumed small in the sense that all the quadratic terms in the fluctuating components may be neglected with respect to the linear terms): a). at '9P.'_ 1.2. 39.11. at + N 32 Dr R { ar( “ 8r ) II. .219. aw 2131.29-11; 32 E (az 3r ) J + r ( r r )} (2.3.2) aw dII. _3_1N_'_____3'_1_ a_ w SE-+a?“+wz 32+ 32(23z) +-}-3— L ur(—§-§-'—+-§~Iii)]1 (2.3.3) 15 9.. 3,. (ru') + 3‘1.” 0 (2.3.4) 1. r 82 -_r:_ where r - b-a . Hith axisymmetric disturbances the motion may be represented in terms of a stream function such that u' = -%-%2(r9) (2.3.5) . _ 1 8 W - T‘- 70“?) with which the equation of continuity is automatically satisfied. Consider further that the stream function takes the separated form 9 = -i¢(r) exp [ ia (z-ct)] (2.3.6) With this expression we assume the disturbance to be a sinusoidal function of time and position. The quantity a is a dimensionless wave number and is a real. The complex wave speed c is given by where c1 is the amplification rate. A positive or negative c1 implies growth or decay respectively of the perturbation. This study is concerned with neutral stability, that is, c, = 0. The function ¢(r) may be complex. Eliminating the terms containing p' in equations (2.3.2) and (2.3.3) by cross differentiation, then introducing the assumed 16 form of the disturbances into equations (2.3.2) - (2.3.4) to separate the variables, yields the following modified Orr-Sommerfeld equation, and ordinary differential equation, and N is 2.2. (W-C)L¢ - r 3,7- (% g—I'II 3 2 1. 2 d d 3 d 3 2 d - [u(L¢) +—‘i{2—i+— -(——+20I i 01R d dr3 r dr r2 )dr 2 2 3 + (‘3‘ '- %—)¢} + 7d 1‘ (Lch + 201°C) 1 (2.3.8) F dr 2 d 1 d 1 2 ._..—+ -—— - - a drz r dr ';2 4 3 2 d 2 d 3 2 d --+ --§-+ ("TZ' - 2a 1 "‘3' dr4 5 dr r dr* + (3 _ gg_( §L_ (_ 3 + 2&2 + 4) :3' r dr :4' r2 a the solution of the governing equations presented in Section The necessary boundary conditions result from the no-slip velocity conditions on the inner and outer cylinders and from con- tinuity. They are a = o' = 0 @ r = a1 and a1+l (2.3.9) 17 2.4 Eigenvalue Problem The differential equation and boundary conditions derived in the previous section represent an eigenvalue problem with the eigen- values a, c, and R. The wave propagation speed, c as stated earlier is complex, with the imaginary part being the exponential growth or decay rate of the assumed disturbances. Since we are interested in studying, neutral stability, °i is set to zero. To solve for the eigenvalues it is necessary to specify one of them, say Cr and solve for the remaining two, namely a and R from the complex equation (2.3.8). Setting c1 = 0, the eigenvalues are found and then plotted as shown in Fig. l. The resulting curve is called a neutral stability curve. _——- crit Unstable Stable Rcrit Figure l.--A Neutral Stability Curve 18 The neutral stability curve will have a minimum R, called the critical Reynolds number R for which a flow is neutrally stable. crit’ Reynolds numbers greater than Rcrit result in growth for that particular flow and Reynolds numbers smaller than Rcrit result in decay. It is assumed that all wave numbers are present in any real flow. Thus, it is assumed that a stable flow at point A or point B is not possible, It is only possible to have a stable (laminar) flow below Rcrit' , and Associated with R . there is also a critical wave speed c °r1t crit Y‘ o o wa b . . a cr1t1cal ve num er “crit CHAPTER III NUMERICAL METHODS 3.1 Numerical Solution of the Governing Equations of the Main Flfiw The energy equation (2.1.9) and the equation of motion (2.2.1), which is subject to the constraint (2.2.5), with boundary conditions (2.2.2) and (2.2.3) are solved by an iterative method. The numerical integration of the above equations are performed by using the fourth order Runge-Kutta method. The iteration procedure which is used to solve the governing equations of the main flow through the concentric cylinders with variable u(T) and k(T) is as follow: (1) Find an initial guess of the temperature distribution from T(r) = 751%757- 1n [ l + gTI] which is the solution of the energy equation when the conductivity is constant and the viscous dissipation is negligible. (2) Find an approximation to viscosity u and conductivity k using the following expression given by Poots and Rogers (1965): 19 20 10 i u(r) = H exp (Cie ) i=0 k(r) = H exp (01° ) i=0 where T(r) is included in the expression otr) = (T1430) + (To-Ti) T(r) so and Ci and Di are given in Table 42. (3) Find an approximation to the velocity of the main flow H by solving the equation of motion and the constraining equation: (3a) Pick a value for A and find w by solving the equation (3b) of motion 1 d dH - ——,]., agony) 3,1- A with boundary conditions, N = O @ r = 0 and r = 1 Check whether N satisfies the following constraint: 1 l f o N [a]+r]dr - (a1 + 2) < e wheres: is a preselected small number. If H does not satisfy the constraint, then pick a new value for A and repeat the steps (3a), and (3b). 21 (4) Find a new approximation to the temperature, T(r), by solving the energy equation, 1 T ] a1+r dr [ {a]+r} k— d . 2 + sgn (TO-Ti) Pr Eck u(°" = with boundary conditions, T = 0 @ r = 0 T = l @ r = l (5) Repeat steps (2) and (3) using the new T(r) of step (4). (6) Check whether the velocity and temperature satisfy prescribed convergent criteria (e.g., max ITnew - Toldl _5 10's). If they do they are the desired solutions. If they do not, then repeat the steps (2) through (6) until convergence is obtained. 3.2 Numerical Solution of the Stability Equation The numerical solution to equations (2.3.8) will generate two independent solutions because the solutions are started at the inner cylinder with two boundary conditions. These solutions will not each satisfy the remaining boundary conditions at the outer cylinder. How- ever, a proper linear combination of these functions will yield the eigenfunction which must satisfy the two boundary conditions at the outer cylinder. 22 The integration of the equations is begun at the inner cylinder and proceeds step by step across the annulus to the outer cylinder. A fourth order Runge-Kutta technique, detailed in Appendix A, is used to solve the equation. The two independent solutions are each initialized at the inner cylinder and integrated simultaneously step by step across the annulus. To use the Runge-Kutta integration scheme to solve a differen- tial equation of order n the problem must be transformed to an equiva- lent initial value problem where the function and its n-l derivatives are initially specified. For equation (2.3.8), the boundary conditions (2.3.9) provide starting values for ¢i and ¢% (i = l and 2 represent the independent solutions) with values for the remaining derivatives ¢¥ and o? arbitrarily chosen. The highest order derivatives, namely ¢iv, does not require initialization since they are determined in terms of the lower order derivatives from the describing differential equation. The arbitrary starting conditions mentioned above must be chosen so as to insure independent functions, at least at the start of the integration. Assigning a non-zero value to one of the two unspeci- fied derivatives for each of the two solutions generally helps keep the solutions linearly independent. The coefficient of the highest order derivative of the stability equation (2.3.8) is very small since it contains (GR)-] which is of the order of 10'4 in the present problem. Hence, the equation is highly singular. There will be two linearly independent solutions satisfying the two boundary conditions on the inner cylinder, and an appropriate linear combination of these solutions will produce a 23 solution satisfying the boundary conditions on both inner and outer cylinders. It is well known that during the numerical integration one of the two independent functions for the fourth order problem grows much more rapidly than the other solution away from the starting boundary. Kaplan (1964) referred to the growing function as the "growing solution" and called the other the "well-behaved solution." The growing solution stems from the singular portion of the stability equations and the well behaved function originates from the inviscid portion. The governing stability equations can be written as (W-Cll-(t - eta—1:1. = - 37,; [110240 +73% {21" +-3—} dzu 2 +;;§-(L¢ + 2a ¢)] (3.2.1) where L is a second-order operator and L2 is a fourth-order operator defined in Section 2.3. The left hand side represents the inviscid part and the right hand side represents the viscous part which accounts for the singular behavior. In this particular problem, the two solutions exhibited two distinct growth rates, the larger one corresponding to the growing solution, and the smaller one corresponding to the well-behaved solution. The fourth order differential equation results in only one growing solution which exhibited a growth on the order of 1018 across the annulus, which is in agreement with the observations reported by Reynolds and Potter (1967). 24 It is clear that generation of a second solution which is independent from the rapidly growing solution will be difficult. Any slight round-off error will in effect throw in some small multiple of the growing solution, which will likely dominate the second solution by the time the outer cylinder is reached. The problem is to generate numerically a second solution which is not a multiple of the growing solution. Two approaches may be considered. First, one might use double precision, extending the accuracy 0f the digital computations (to say, 24 digits). To reduce the truncation error associated with any numerical scheme, an obvious answer is to choose a step size that is as small as possible, consistent with the particular limitations of machine storage and speed. Performing all arithmetic operations in double precision should greatly reduce the error. This is in fact, found to be true. However, since all the functions are complex, adding double precision to the program significantly increases machine computation time and storage require- ments. Therefore, this approach was not used for the present study. A second approach, first used by Kaplan (1964), involves suppression of the growing solution during the calculation of the well-behaved solution. His method, that does not require double precision, consists of subtracting from the well-behaved solution a portion of the growing solution at every step of the integration. This procedure called "filtering" prevents the growing solution from ever dominating the well-behaved solution and thus maintains the needed functional independence. 25 Kaplan‘s scheme was implemented in the computer program for this problem. The CDC 6500 computer nominally carries 14 significant digits in single precision, which is equivalent to double precision on IBM equipment and in particular the IBM 7090 used by Kaplan. These computations were in effect then performed in "double precision." The filter used consisted of a ratio of the inviscid solutions, the left hand side of equation (3.2.1), namely, Inviscid part of the well-behaved solution (3 2 2) Filter = Inviscid part of the growing soluthn ‘ ° . . _ . d l dH where, Inv1$c1d part — 1aR [(N-c)L¢ - r aF-(F-EF)¢] (3.2.3) The above ratio determines the fraction of the growing solution to be "extracted" from the behaved solution, so the amount subtracted off is the product of this ratio and the value of the growing solution at the particular integration step. 3.3 Iteration Scheme for the Eigenvalues Integrating across the annulies with assmumed eigenvalues, two independent solutions are generated at each step. Upon reaching the opposite cylinder, the functions are linearly combined to form the total eigenfunction that must satisfy the boundary conditions. The two conditions that must be satisfied at the outer cylinder are ¢ = o' = 0. Consider the following set of combined functions at the outer cylinder: 26 °l¢lo + °2°20 = to (3.3.1) °1°lo + °2¢l20 = 1’6 (3.3.2) a1¢10 + azo 20 = ¢b (3.3.3) Equations (3.3.1-3) are used to solve for the coefficients a1. Note that there is no specific boundary condition for as, hence the choice here is arbitrary. This system can now be written as ¢h> ¢2J °l 0 a2 1 I. II. For functions that are linearly independent the determinant of the matrix containing known values of °io and °io will be non-zero. When the suppression scheme was not used this determinant was effectively zero and indicated the functions to be linearly dependent, that is, one column is a multiple of the other. For independent functions, the °i can be determined by finding the inverse of this matrix or as was done for the computer program by simply writing out the solution. The iteration scheme for eigenvalues has several options (i.e., (1) find R, a for cr fixed, (2) find R, Cr for a fixed, (3) find a, Cr for R fixed). The first option of finding R, a for cr fixed is further explained below. The a., once determined can now be substituted into equation 1 (3.2.4). With the correct eigenvalues ¢0 will be zero; hence o0 will 27 serve as the test function (F). If FF, where F is the conjugate of F, is less than 10"12 the convergence criteria is satisfied and the eigenvalues used to generate the functions are assumed to be the correct ones. If convergence is not attained let F1 = F. Increase a by 1%, recalculate F and let F2 = F. After setting a to its original value, increase R by 1%, recalculate F and let F3 = F. The finite difference approximations for the change in F with respect to a and R are F F '%E : :0 1 (3.3.5) F F F ~ - %R~'%R—] (33:5) These are substituted into the complex equation 3F 8F - fiAa + a—RAR + F1“ 0 (3.3.7) from which Aa and AR can be calculated. The new values for the eigenvalues are a = a + Ad new old (3.3.8) Rnew = Rold + AR It was found that if the initial guesses are relatively good, convergence is obtained in about three iterations. 28 The initial estimates for the eigenvalues of the near parallel plate case (i.e. when the radius ratio is near unity) were obtained from the data presented by Potter and Graber, (1972). The initial estimates of eigenvalues for the isothermal case and radius ratios a/b with 0.8 or less were obtained from the data presented by Mott and Joseph (1968). For nonisothermal case with radius ratios of less than 0.99 guesses had to be made referring to both results of Potter and Graber, and Joseph and Mott. After a few critical points for different radius ratios and temperatures on the boundary were obtained and plotted, the initial point on the next stability curve for smaller radius ratio could be made by extrapolation. Completing a neutral stability curve for a fixed radius ratio, a/b, and boundary temperatures (Ti, 16) was also done by extrapolation after a few points were obtained and plotted. CHAPTER IV RESULTS, CONCLUSIONS, AND RECOMMENDATIONS FOR FURTHER STUDY 4.1 Numerical Results (a) Governing Equations of the Main Flow The velocity and temperature profiles of the main flow were calculated using the iteration technique discussed in Section 3.1. For the heated inner cylinder several cases were considered. The tempera- ture of the outer cylinder was taken at 0°C with the inner cylinder temperature at 20°C, 40°C, 60°C, 80°C, and 100°C. The radius ratio a/b, inner cylinder to outer cylinder, was varied between 0.99 and 0.1. The energy equation (2.1.9) can be written as dT dk dT k d _ [{a]+r}fi]+a;; Fur-0 a1+r ‘HF where p = sgn (To-Ti) Pr Eck u(%¥)2. To determine the significance of the dk/dr term and viscous dissipation p the governing equations of the main flow were solved for the following four cases: (1) Neglect both dk/dr and o. (2) Keep dk/dr, but neglect p. (3) Neglect dk/dr, but keep o. (4) Keep both dk/dr and a. 29 30 The velocity and temperature profiles of these cases for the selected boundary temperatures and radius ratios are presented in Tables 1 through 10. The significance of the dk/dr term and viscous dissipation are presented in Tables 11 through 20. For the near parallel plate case (a/b = 0.99) with Ti = 20°C, T0 = 0°C and R = 3900, the following indicates the maximum deviation between the temperature T(I,1) of case (1) and the temperatures T(I,k) of cases (2), (3), and (4): max T 1’2 ' T 1’] x 100 = 3.9% T I, 103) - T(I,1) = max T( ’1) x 100 1.0% max T(I'fi) ' T(I") x 100 = 4.9% Since the temperature T(I,1) of case (1) contains the highest error, the case with the smallest error will have the largest ratio. That is, the T(I,4) values of case (4) are the most accurate. The value 3.9% of case (2) is larger than the 1.0% value of case (3). This indicates that the dk/dr term has a greater effect on temperature than t has. For a/b = 0.99, Ti = 100°C, and R = 13370 the deviation was max T [’ZT E’I I 1 x 100 = 7.03% max T I’3T f’I 1" x 100 = 1.0% max T(I’4I(;’I§I") x 100 = 7.84% 31 This indicates that the dk/dr term has a significantly greater effect on the temperature distribution than the o term, with the effect increasing with increasing R and temperature difference. Similar results were found for a/b = 0.1. The significance of the dk/dr term when Ti = 100°C and To = 0°C is shown graphically on Figure 6 for a/b = 0.1 and on Figure 7 for a/b = 0.99. The effect of variable properties on temperature is relatively small, but may be important in certain situations. The temperature T with the dk/dr term is always lower than that without the dk/dr term at any position r except on the boundary. Some representa- tive velocity profiles are presented in Figures 2, 3, 4, and 5. They clearly show the significant effect of temperature, acting through the viscosity, on the velocity profile. Based on the result of this analysis the dk/dr term has always been kept in the energy equation for the present study. But, based on the small error introduced, the viscous dissipation has been neglected, as is customary. (b) Stability Equation Numerical calculations of the eigenvalues were performed using the technique discussed in Sections 3.2 and 3.3 for Ti = 20°C, 60°C, and 100°C with To = 20°C, a/b = 0.99, 0.8, 0.6, 0.4, 0.3 and 0.25. The resulting eigenvalues for these cases are presented in Tables 22 through 41. and the associated neutral stability curves are presented in Figures 8 through 13. Critical eigenvalues are summarized in 32 Table 21 and are plotted in Figure 13 to demonstrate the effect of heat transfer and radius ratio a/b on the critical eigenvalues. From Figure 13 there are three observations that are of particular interest: (1) For any fixed value of a/b between the values of 0.99 and 0.25, with T0 = 20°C, the critical Reynolds number decreases with increased temperature of the inner cylinder. (2) For the isothermal flow with Ti = T0 = 20°C the critical Reynolds number increases with decreasing radius ratio. For a/b = 0.99, R = 7687. The crit value of R increases to the value of 105370 crit for a/b = 0.25. The isothermal curve supports the conclusion that the critical Reynolds number of an annulus approaches infinity as a/b approaches 0. (3) The critical Reynolds number Rcrit(°/b) of non- isothermal flow also increases as a/b decreases for a/b between 0.99 and 0.25. But, non- isothermal R -curves in this case show an crit. inflection point near a/b = 0.4 giving the possibility of 11m Rcrit = a finite value. a/b+ O The effect of heat transfer on the stability was further inves- tigated for a/b = 0.4 with To = 20°C, Ti = 20°C, 30°C, 40°C, 60°C and 100°C. The resulting neutral stability curves for these cases are 33 shown in Figure 14. When Ti = 20°C (isothermal case) the neutral stability curve (a vs. R) has the shape of a typical open loop which is shown in Figure 1. However, the neutral stability curve of the non-isothermal flow for a/b_§ 0.4 exhibits a different shape as shown in Figure 14. 4 and a of the In case of a/b §_0.4, R is of the order of 10 upper branch keeps increasing for non-isothermal flow. This results in small (01R)'1 which is the coefficient of the highest order deriva- tive of the stability equation. The differential equation for these small (aR)" is highly singular and its solution is very difficult to obtain. Some eigenfunctions are plotted in Figures 15, 16, 17. 4.2.Conclusions Based on the results obtained and discussion of the previous section it can be concluded that: (l) Neglecting dk/dr results in up to 6.8% error in the temperature distribution and up to 2.7% error in the velocity distribution for high R and T = 100°C. The effect of neglecting o influences the temperature and velocity distributions by less than 1%. (2) Neglecting dk/dr results in an error of up to 13.3% on the heat transfer and up to 3% on the shear stress. The effect of neglecting o influences the heat trans- fer and shear stress by less than 1%. (3) Heating the inner cylinder destabilizes the parallel water flow through the concentric cylinders. 34 (4) The critical Reynolds number of an isothermal flow monotonically decreases supporting the accepted conclusion that R becomes infinitely large as crit a/b approaches zero. (5) The critical Reynolds number of a non-isothermal flow increases as a/b decreases between the values of 0.99 and 0.25. The results indicate that Rcrit may indeed approach a finite value when a/b + 0. (6) For a/b = 0.4 the upper branch of the neutral stability curve of the non-isothermal flow appears to have a vertical, or near vertical, asymptote rather than the horizontal asymptote of the isothermal flow. But, convergence of the iteration process toward an acceptable set of eigenvalues was extremely difficult at the high.(aR) values. Hence, the existance of the upper vertical asymptote, especially for small (To'Ti)’ is not certain. 4.3 Recommendations for Further Study Finite-amplitude instability of parallel flow of a liquid between concentric cylinders would be investigated using the eigenvalues found in this study as an input. 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A ....03 . - ....00 0.00 0 - ...00: - 00.003 .xwz 0000 0 - .0....0 - 00.00. .xwz 0 0.000 5000 00.000 .x02 .00... 00.0.0- 0000.. 0000.0- 000... 0000.0- 0000.0 0000.0- 00.. .0. 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.. 00.0.0 00.0 . 000.0...00 000.0..030 000.0..000 00000..030 000.0...00 000.0...30 000......0 0000....30 ...0 . 00000000500 0 00.00.0> 00 00>.u0>.00o u 00.0 000.. A 0 .000 A 0. .0000 A .0 .00.0 05000 00000000000 000000> 0:0 00\¥0 00 00000.0 A 00 .00.0--.0. 0.00. 50 . - 00.00: 0000.0 A 00.000 0000 0 - 00.003 -,00.003 .xwz 00.000 - 00.000 .xwz . - 00.003 . - 00.000 0000 0 - .00.003 - 00.000 .xwz 0000 0 - 00.000 - 00.000 .0”: . - 00.003 . - 00.000 0000 0 - 00.003 - 00.003 .xwz 0000 0 - 00.000 - 00.000 .xw: m 00000 5000 000000 .xmz 0000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 00.0 000 0000.0 0000.0 0000.0 0000.00 0000.0 0000.0 0000.0 0000.00 00.0 0 00000.0000 00000.0030 00000.0000 00000.0030 00000.0000 00000.0030 00000.0000 00000.0030 0000 0 00:0000Qaw0 0 0000000> 0o 00>000>000o 00000 00000 A 0 .000 A 00 .00000 A 00 .00 0 A 000 05000 :o0pma0000o 030000> 0:0 00000 00 00:00000cm0m-.00 00000 00.003 00.000 51 00000000500 0 0000000> 00 00>000>0000 00000 0000 0 A .00.00: - 00.003 .0“: 0000 0 A 00.000 - 00.000 .xwz 0000.0 A .00.003 . 00.000 0 I a H OOFOOH 0 I 0 l H 00 003 00 003 .000 00 000 00 000 .000 . - . 00.003 . - 00.000 0000 0 - 00.003 - 00.000 .xwz 0000 0 - 00.000 - 00.000 .xwz 0 00000 5000 000000 .x0z 0000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 00.0 000 0000.0 0000.00 0000.0 0000.00 0000.0 0000.00 0000.0 0000.00 00.0 0 00000.0000 00000.0030 00000.0000 00000.0030 00000.0000 00000.0030 00000.0000 00000.0020 0000 0 05000 00000000000 one 0o0 0 . 00 A 00 .0000 A 00 .0.0 A 000 000000> 000 00000 00 000000000000-.00 00000 52 . - 2.33 . - 2.30 000 0 - 0300 - 0.03 .0.0.0 0000 0 - 00.00 - 00.00 0.00 . - 00.000 . - 00.000 0000 0 - 00.003 -00.00: .xwz 0000 0 - .000000 - 00.000 .xwz . - 00.003 . - 00.000 0000 0 - .00.003 - 00.000 .0“: 0000 0 - 00.000 - 00.000 .xwz 0 00000 5000 000000 .x0z 00000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 00.0 000 0000.0 0000.0_ 0000.0 0000.0. 0000.0 0000.00 0000.0 0000.00 00.0 0 00000.0000 00000.0000 00000.0000 00000.0020 00000.0000 00000.0020 00000.0000 00000.0030 0000 0 00000000500 0 0000000> 00 00>000>0000 00000 000000 05000 0000000000 9 a 0 . 00 A 00 .0000 A 00 .0.0 A 000 00000> 000 00000 00 000000000000-.00 00000 53 . - 00.003 . - 00.000 0000 0 - .00.003 - 00.000 .0“: 0000 0 - 00.000 - 00.000 .00: . - 00.000 . - 00.000 0000 0 - ‘ 00.003 - 000003 .xwz 0000 0 - . 00.000 - 00.000 .xw: . - 00.000 . - 00.000 0 0000 0 - 00.003 - 00.003 .xwz 0000 0 - 00.000 - 00.000 .000 0 00000 5000 000000 .x0z 0000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 00.0 000 0000.0 0000.00 0000.0 0000.00 0000.0 0000.00 0000.0 0000.00 00.0 0 00000.0000 00000.0030 0000.0000 00000.0000 00000.0000 00000.0030 00000.0000 00000.0030 0000 0 00000000500 0 0000000> 00 00>000>0000 00000 omwmmm A 0 . 00 A 00 .0000 A 00 .0.0 A 000 05000 00000000000 0 000000 000 00000 00 000000000000-.0_ 00000 54 . - 00.003 . - 00.000 @NNO O I AFQHVZ. I AGaHy oxwz NWNO O I ApaHvH I fi¢nmnm .XWE 0000.0 A 00.00: . - 00.000 00.000 - 00.000 .0“: 0000 0 - .00.000 - 00.000 .xwg 0000.0 A 00.00: . - 00.000 00.003 - 00.00: .0“: 0000 0 - 000000 - 00.000 .0“: a 00000 5000 000000 .x0z 0000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 00.0 000 0000.0 0000.00 0000.0 0000.00 0000.0 0000.00 0000.0 0000.00 00.0 0 00000.0000 00000.0030 00000.0000 00000.0030 00000.0000 00000.0030 00000.0000 00000.0020 0000 0 00000000500 0.0000000> 00 00>000>000o 00000 0000000 05000 0000000000 a 0 0 .000 A 00 .0000 A .0 .0.0 A 000 000000> 0:0 00000 00 00000000000m-A.00 00000 55 00 00: 0 0000.0 00 000 . 0000 0 A 00.00: -.00.000 .000 A 00.000 - 00.000 .xwz 0000.0 A 00.00: . - 00.000 2.00. - 0.00 .0.0.0 020 0 - 3.9 - 00.? .000.. . - 00.003 . A 00.000 0000 0 - _00.003 0-00.000 .0“: 0000 0 00.000 - 00.000 .xwz 00 00000 0000 000000 .002 0000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 0000.0 0000.0- 00.. 000 0000.0 0000.00 0000.0 0000.00 0000.0 0000.00 0000.0 0000.00 00.0 0 00000.0000 00000.0000 00000.0000 00000.0020 00000.0000 00000.0030 00000.0000 00000.0030 0000 0 00300000500 0.0000000> 00 00>000>000o 00000 0000000 A 0 .000 A 00 .00000 A 00 .0.0 A 000 05000 00000000000 030000> 0:0 00000 00 00000000cm0mA-.om 00000 56 Tab1e 21.--Critica1 Eigenva1ues for Various Radius Ratios of Inner to Outer Cylinders a a/bl/ T.Z! c a R 1 r 9.9 Cm 0.99 20°C 0.395 2.0292 7,687 60°C 0.406 2.0914 4,707 100°C 0.422 2.1618 3,041 8.0 Cm 0.80 20°C 0.384 1.9976 8,540 60°C 0.378 2.0316 6,428 100°C 0.386 2.0913 4,562 6.0 Cm 0.60 20°C 0.348 1.9465 12,978 60°C 0.333 1.9931 11,341 100°C 0.340 2.0805 8,313 4.0 Cm 0.40 20°C 0.273 1.8453 34,509 60°C 0.274- 2.0366 26,625 100°C 0.293 2.2553 17,087 3.0 Cm 0.30 20°C 0.227 1.8613 73,321 60°C 0.251 2.2906 40,564 100°C 0.279 2.6563 22,467 2.5 Cm' 0.25 20°C 0.210 2.0143 105,370 60°C 0.2465 2.75847 45,847 100°C 0.27873 3.5593 23,154 1] b = 10(M1 g/ To = 20°C 57 Tab1e 22.--Eigenva1ues for c = 0, a/b = 0.99, T. = = 20°C 1 o a R Cr Remarks 1.7936 9366 0.36 1.9118 8188 0.36 1.9849 7800 0.39 2.0197 7701 0.394 2.0292 7687 0.395 Critica1 Point 2.0420 7696 0.396 2.1043 7843 0.400 2.1780 9140 0.395 2.1949 11513 0.38 2 1796 15227 0.36 Tab1e 23.--Eigenva1ues for = 0, a/b = 0.99, T. = 60°C, T = 2 °C 1 o a R Cr Remarks 1.8918 5286 0.380 2.0311 4757 0.400 2.0689 4712 0.404 2.0914 4707 0.406 Critica1 Point 2.1437 4800 0.409 2.2415 6023 0.900 2.2488 6500 0.395 58 Tab1e 24.--Eigenva1ues for c. = 0, a/b = 0.99, Ti = 100°C, To = 20°C a R Cr Remarks 1.9801 3305 0.400 2.0987 3067 0.416 2.1618 3041 0.422 Critica1 Point 2.1923 3062 0.424 2.2657 3262 0.425 2.3244 3886 0.416 Tab1e 25.--Eigenva1ues for c. = 0, a/b = 0 8, T. = T = 20°C 1 o a R Cr Remarks 1.8895 9115 0.37 1.9631 8666 0.38 1.9898 8582 0.383 1.9976 8540 0.384 Critica1 Point 2.0101 8550 0.385 2.0338 8553 0.387 2.0679 8657 0.389 2.1367 9493 0.388 2.1675 10994 0.383 59 Tab1e 26.--Eigenva1ues for = 0, a/b = 0.8, T. = 60°C, T = °C 1 o a R Cr Remarks 1.9542 6589 0.37 1.9986 6459 0.375 2.0201 6436 0.377 2.0316 6428 0.378 Critica1 Point 2.0449 6432 0.379 2.0601 6450 0.38 2.0766 6474 0.381 2.1636 7.87 0.38 2.2036 8605 0.37 Tab1e 27.--Eigenva1ues for c. = 0, a/b = 0.8, T1 = 100°C, To = 20°C a R Cr Remarks 1.9479 4877 0.37 2.0102 4673 0.378 2.0279 4633 0.38 2.0677 4574 0.384 2.0790 4566 0.385 2.0913 4562 0.386 Critica1 Point 2.1680 4665 0.39 2.2402 5127 0.388 2.2828 5932 0.38 2.3004 6910 0.37 60 Tab1e 28.--Eigenva1ues for c = O, a/b = 0.6, Ti = T = 20°C 0 a R Cr Remarks 1.8696 13376 0.34 1.9046 13126 0.344 1.9152 13094 0.345 1.9257 13055 0.346 1.9361 13018 0.347 1.9465 12978 0.348 Critica1 Point 1.9639 13045 0.349 1.9821 13120 0.350 2.0553 14208 0.35 2.0997 17301 0.34 Tab1e 29.--Eigenva1ues for = 0, a/b = 0 6, Ti = 60°C, T0 = °c a R Cr Remarks 1.9519 11385 0.33 1.9647 11363 0.331 1.9781 11344 0.332 1.9931 11341 0.333 Critica1 Point 2.0126 11391 0.334 2.1447 13664 0.33 Tab1e 30.--Eigenva1ues for c- = O, a/b = 0.6, 61 .Ti - 100 c, To - 20 c a R Cr Remarks 1.9608 8620 0.33 2.0497 8326 0.338 2.0641 8313 0.339 2.0805 8312.8 0.34 Critica1 Point 2.0996 8332 0.341 2.1244 8389 0.342 2.2425 9479 0.34 2.2737 10205 0.3364 2.2956 10965 0.3324 2.3157 11999 0.327 2.3298 13074 0.3216 2.3421 14423 0.3152 2.3466 15069 0.3123 2.353 16161 0.3076 2.3585 17293 0.303 62 Tab1e 31.--Eigenva1ues for c. = 0, a/b = 0.4, _ ___ o 1 a R c Remarks 1.5597 46188 0.24 1.6268 41294 0.25 1.7039 37452 0.26 1.7484 35934 0.265 1.7788 35171 0.268 1.8020 34785 0.270 1.8303 34590 0.272 1.8453 34509 0.273 Critica1 Point 1.8664 34646 0.274 1.8931 34953 0.275 2.0365 52082 0.26 2.0508 63245 0.25 2.0563 76399 0.24 2.0593 9211] 0.23 Tab1e 32.--Eigenva1ues for c. = 0, a/b = 0.4, T1 = 40°C, To = 2020 a R Cr Remarks 1.85 32844 0.26336 1.90 32343 0.26683 1.925 32321 0.26814 Critica1 Point 1.95 32414 0.26922 1.9782 32689 0.27009 2.05 34700 0.26992 2.20 50298 0.25354 2.30 73576 0.23197 2.40 90442 0.21875 2.50 99862 0.21134 63 Tab1e 33.--Eigenva1ues for c. = 0, a/b = 0.4, 1. = 60°C, 1 = 20°C 1 o a R cr Remarks 1.9095 27601 0.266 1.9635 26933 0.27 1.9959 26702 0.272 2.0151 26645 0.273 2.0366 26625 0.274 Critica1 Point 2.0637 26705 0.275 2.1048 27021 0.276 2.15 27708 0.27625 2.2366 30162 0.274 2.3 33220 0.26971 2.4 40249 0.25926 2.5 48107 0.24813 2.6 54347 0.23956 2.7 58605 0.23349 2.8 61338 0.22913 64 Tab1e 34.--Eigenva1ues for c. = 0, a/b = 0.4, Ti - 100 c, To - 20 c a R Cr Remarks 1.7249 25950 0.245 1.8945 20706 0.266 1.9821 19063 0.275 2.0627 18027 0.282 2.1175 17549 0.286 2.1661 17265 0.289 2.1849 17193 0.290 2.2052 17133 0.291 2.2285 17096 0.292 2.2553 17087 0.293 Critica1 Point 2.2897 17133 0.294 2.3513 17382 0.295 2.4169 17891 0.29487 2.6842 22326 0.2835 2.7741 24151 0.278 2.8825 26068 0 272 3.0451 28101 0.265 3.1039 28607 0.263 3.2081 29264 0.26 3.2915 29613 0.258 65 Tab1e 35.--Eigenva1ues for c. = 0, a/b = 0.3, Ti = T = 20°C a R Cr Remarks 1.8232 73900 0.225 1.8415 73574 0.226 1.8613 73321 0.227 Critica1 Point 1.8901 73649 0.228 1.9293 74475 0.229 Tab1e 36.--Eigenva1ues for = 0, a/b = 0 3, T. = 60°C, T = °C 1 0 a R Cr Remarks 1.8884 44727 0.2275 1.9169 48369 0.23 2.0493 43717 0.24 2.1999 41085 0.248 2.2553 40678 0.25 2.2906 40564 0.251 Critica1 Point 2.3365 40565 0.252 2.4675 41431 0.253 66 Tab1e 37.--Eigenva1ues for c. = 0, a/b = 0.3, Ti = 100°C, To = 20°C a R r Remarks 1.8744 36401 0.23 2.0823 28777 0.25 2.1437 27324 0.255 2.2111 25996 0.26 2.2405 25504 0.262 2.2714 25035 0.264 2.2794 24921 0.2645 2.3785 23770 0.270 2.4980 22912 0.275 2.6039 22537 0.278 2.6563 22467 0.279 Critica1 Point 2.7360 22472 0.280 2.7492 22485 0.2801 67 Tab1e 38.--Eigenva1ues for c1 = 0, a/b = 0.25, T. = T = 20°C 1 0 a R Cr Remarks 1.8089 113580 0.2 1.9565 106150 0.208 1.9831 105660 0.209 2.0143 105370 0.21 Critica1 Point 2.0594 105740 0.211 2.1588 108180 0.212 Tab1e 39.--Eigenva1ues for c. = 0, a/b = 0.25, T. = 60°C, To = 20°C a R Cr Remarks 2.4094 47947 0.24 2.4736 47134 0.242 2.5565 46416 0.244 2.6920 45897 0.246 . 2.75847 45847 0.2465 Critica1 Po1nt 2.8711 45971 0.24678 68 Tab1e 40.--Eigenva1ues for c. = 0, a/b = 0.25, T1 = 100°C, To = 20°C a R Cr Remarks 2.853 24548 0.273 2.94492 24122 0.275 3.0105 23916 0.276 3.0887 23710 0.277 3.1505 23584 0.2776 3.2039 23495 0.278 3.2374 23448 0.2782 3.2786 23396 0.2784 3.3363 23334 0.2786 3.3806 23292 0.2787 3.5593 23154 0.27873 Critica1 Point 3.5501 23161 0.27874 3.5371 23170 0.278752 69 Tab1e 41.--Eigenva1ues for c. = 0, a/b = 0 4, 1. = 30°C, 1 = 20°C 1 o a R cr Remarks 1.86 34348 0.26736 1.885 34288 0.2688- Critica1 Point 1.9 34385 0.26943 2.0 37315 0.26983 2.1 50543 0 25687 2.15 72537 0.23782 2.17 85069 0.22895 2.185 93088 0.22382 2.195 97876 0.22092 Tab1e 42.--Viscosity and Conductivity of Water 10 i 10 . u* = H exp (C10 ), k* = H exp (0101) i=0 i=0 1 Ci Di 0 -5.208294 -6.470986 1 -0.836625 0.095773 2 0.228220 -0.069060 3 -0.072689 0.027692 4 0.036071 0.204396 5 -0.025843 -0.145549 6 -0.041654 -0.793563 7 0.018216 0.309234 8 0.072355 1.126945 9 -0.006439 -0.179732 10 -0.037178 -0.523533 0 = (T* - 50)/50 70 :o_um;:mwmcou zopm cwcz--.~ mgzmwm <-< =o_uumw <15] H 11114111 71 Isotherma1 (20.0) \ (40,0) (80,0) \(60,0) (100,0) \ .\\ Figure 3.--Dimension1ess Main Ve1ocity Profi1es for a/b = 0.99 and Various Boundary Temperatures (Ti, To) 72 .0 - .6 - '2 ’ (20°C, 0°C) (60°C, 0°C) .8 L 4 . (100°C, 0°C) 0 l l L 1 I 1 1 l l 0.0 0.2 0.4 0.6 0.8 1.0 Figure 4.--Dimension1ess Main Ve1ocity Profi1es for a/b = 0.1 and Various Boundary Temperatures (Ti’ To) 73 2.0 P 1.6 1.2 0.8 0.4 0.0 l 1 l l 1 1 1 1 1 0.0 0.2 0.4 0.6 0.8 1.0 Figure 5. --Dimension1ess Main Ve1ocity Profi1es for (T To) (100°C, 0°C), and a/b= 0.1 and 0. 99 1.0 0.8 0.6 0.4 0.2 - Neg1ect both 74 dk/dr and 0 Keep dk/dr Neg1ect 9 1.0 0.8 3 0.4 Figure 6.--Significance of dk/dr term for a/b = 0.1 and (Ti’ To) = (100°C, 0°C) 1.0 75 1.0 _ 0.8 _ Neg1ect both . dk/dr and 0 . 0.6 _ 3 0.4 . . Keep dk/dr Neg1ect 9 0.2 4 0.0 0.2 0.4 0.6 0.8 1.0 r Figure 7.--Significance of dk/dr term for a/b = 0.99 and (Ti’ To) = (100°C, 0°C) 1.0 0.8 0.6 0.4 0.2 76 up up _. Aop .vpv mmgauogmasm» xgmuczom msowgm> uca mm.o u n\m .ou.u no» m 5695:: mupocxmm mzmgm> 8 gmnsaz m>mz--.m mg=m_m op w muop x m m Aom .omv T d Aom .omv V/SN .8: l o.F w.~ o.N N.N v.m 77 Ac» .P+v mugsuusmasmh xgaucsom maovgm> can .o.o ccm w.o u n\w .o u 0 Low 1 LQAESZ mvpochmm msmsm> 6 Lwnsaz m>m311.G 0.2.2...— m-o_ x a m_ o_ e_ N_ o_ m 8 5 1 1 d d u 1 fi - 8.0 u n\8 1111111 m.o u 8\8 / Aom .ONV / v AON .oov Aom .oo_v \ \ \\\ ’ \\\\\ \\ _ SN .2: 1\\ Aom .omv \\ \ \\ \ ‘ \ “ lll“ ‘ flow .oo_v m... m._ o.~ N.N ed 78 Aop .thpmmsaumgmasm» agmuczom msowgu> use .e.o u a\m .o n .u so» a Lane:z mvpocawm msmgm> 8 smasaz m>m:--.o_ mgzmwm ¢-op x a N_ o. m 8 e N flow .omv AON .omv X AON .oo_v th .PHVmegzuagmasmp xgmucsom mzowgm> 8:8 .m.o u n\8 .o n .8 68$ a gmnEzz mupocxmm mzmgm> 8 gmnssz 8>83--._P wgsmwu euop x 1 NF o— w o v N d d d u ‘ - 79 m.— o.~ N.N ¢.N SN .83 8m .8314 3.5 3.3 2.6 2.4 2.2 2.0 1.8 80 -——-(100, 20) l 5 (60, 20) 4 l 7 (20, 20) R X 10- Figure 12.-~Wave Number 0 Versus Reyno1ds Number R for . = O, a/b = 0.25, and Various Boundary c Témperatures (Ti’ To) n\8 ovumm 838881 mamgm> upgum 888532 mupocxmm _8uwuwguu-.mp 88:88; 81 88888 88_ 88. 88.xm —q-q~— - —q- — —d--u - Noo A808N .80888 n - . 584 ._88 8 8 Auoou .8588_8 1 . 8.8 1 8.8 - 8._ A808N .80888 82 T1 = 100 C 3.0 - 2.8 - Ti = 60°C 2.2 ’ 2.0 ' 0 2 4 _4 6 8 10 RX10 Figure 14.--Effect of Heat Transfer on the Neutra1 Stabi1ity Curve for a/b = 0.4, To = 20°C, and Various Ti 83 1¢ ‘uoinounguafiga JO 4JPd KJPUIBPWI .808N u 88 .88.8 n 8\8 888 8=_88 _8888_88 888 88 8888883888888--.8p 88=8_8 L «swumm mmm_:opmcws_o o.p w.o o.o ¢.o ~.o c no.0: 8o.on No.01 oo.o No.o 8o.o mo.o . 1 J0 ‘uoiqounguafiig 19313133 803 go 449d 1083 84 8.88 n 88 .8.8 u 8\8 888 888888 _8888_88 888 88 8888888888888--.8_ 888888 8.. 8.8 8.8 8.8 8.8 8 d u q q q u - d u .I We 11 1 1. l 1 .8 88. . . . 8 88. 4 8.88 4 88 . \\\..IIIV: ) {xxx /fl”I/:N _: ””3” n, 0! .5 o // __ .27 8 r ///// x L / II 11 I, II..\\ co. . \ . 8.888 n 88 N; m; 2.4 - 2.0 . 1.6- 1.2. Figure 17.-~Eigenfunction at a C r To 85 0.258 for a/b 20°C BIBLIOGRAPHY 86 BIBLIOGRAPHY Barnes, H. T. and Coker, E. 6., "The F1ow of Water Through Pipes," Proc. Roy. Soc. London, A 74, 341 (1905). Co11atz, L., "The Numerica1 Treatment of Differentia1 Equations," 3rd ed. Ber1in, Springer, (1960). Crowder, H. J. and Da1ton, C., "0n the Stabi1ity of Poiseui11e F1ow in a Pipe," J. Comp. Phys., 7 (1971). Davey, A. and Drazin, P. G., "The Stabi1ity of Poiseui11e F1ow in a Pipe," J. F1uid Mech., 36 (1969). Ekman, V. H., "Archiv fur Math," Astr. Och Fys. VI, No. 12 (1910). Ehnnns, H. w., "The Laminar-Turbu1ent Transition in a Boundary Layer," Part I, Journa1 of Aeronautica1 Sciences, 18 (1951). Gi11, A. E., "On the Behavior of a Sma11 Disturbances to Poiseui11e F1ow in a Circu1ar Pipe," J. F1uid Mech., 21 (1965). Kap1an, R. E., "So1ution of 0rr-Sommerfe1d Equation for Laminar Boundary Layer F10w over Comp1iant Boundaries,“ ASRL-TR- 116-1, Cambridge Aeroe1astic and Structures Research Lab. Report, M.I.T., (1964). Ke11er, H. B., "Numerica1 Methods for Two-Point Boundary-va1ue Prob1ems," B1aisde11, (1968). Kutchey, J. A., “The Stabi1ity of P1ane Poiseui11e F1ow Subject to a Transverse Magnetic Fie1d," Doctora1 Dissertation, Michigan State University (1970). Lin, C. C., "0n the Stabi1ity of Two-Dimensiona1 Paral1e1 F1ows," Part I, II, III, Quart. App1. Math. 3, (1945, 1946). Lin, C. C., "The Theory of Hydrodynamic Stabi1ity," Cambridge University Press, London (1955). Morkovin, M. V., "Transition from Laminar to Turbu1ent Shear F1ow - A Review of Some Recent Advances in its Under- standing," Trans. ASME, (1958). 87 88 Mott, J. E. and Joseph, D. D., "Stabi1ity of Para11e1 F1ow Between Concentric Cy1inders," The Phy. of F1uids (10), 11 (1968). Orr, H. M. F., "The Stabi1ity or Instabi1ity of the Steady Motions of a Perfect Liquid and of a Viscous Liquid," Proc. Roy. Irish Academy, (A) 27 (1907-09). Pai, S. I., "Viscous F1ow Theory I - Laminar F1ow," D. Van Nostrand, 1956. Poots, G. and Rogers, M. H., "Laminar F1ow Between Para11e1 P1ates, with Heat Transfer, Water with Variab1e Physica1 Properties," Int. J. Heat Mass Transfer, 8 (1965). Potter, M. C., "Linear Stabi1ity of P1ane Couette-Poiseui11e F1ows,‘l Doctora1 Dissertation, University of Michigan (1965). Potter, M. C., "Linear Stabi1ity of Parabo1ic F1ows," The Phy. of F1uids (3), 10 (1967). Potter, M. C. and Graber, E., "Stabi1ity of P1ane Poiseui11e F1ow with Heat Transfer," The Phy. of Fluids (3), 15 (1972). Prandt1, L. P., "Bemerkungen Uber die Entstehung der Turbu1enz," ZAMM, 1 (1921), and Phys. 2., 23 (1922). Prandt1, L. P., "Uber den Luftwiderstand von Kuge1n," Gottinger Nachrichter (1914). Ray1eigh, Lord, "0n the Stabi1ity or Instabi1ity of Certain F1uid Motions," Proc. Lond. Math. Soc., 11, 57 (1880) and 19, 67 1887 . Ray1eigh, Lord, "Further Remarks on the Stabi1ity of Viscous F1uid Motions," Phi1. Mag, and Journa1 of Science, 6 Series, 28 1914 . Reyno1ds, 0., "An Experimenta1 Investigation of the Circumstances which determine whether the Motion of Water sha11 be Direct or Sinous, and of the Law of Resistance in Para11e1 Channe1s," Phi1. Trans. Roy. Soc. (1883). Reyno1ds, w. C. and Potter, M. C., "Finite - amp1itude Instabi1ity of Para11e1 Shear F1ows," J. F1uid Mach. (3), 24 (1967). Rotta, 0.. 3Experimente11er Beitrag zur Entstehung turbu1enter Stromung im Rohr," Ing. - Arch, 24 (1956). 89 Sch1ichting, H., "Zur Entstehung der Trubu1enz bei der P1attenstro- mung," flaghr. Ges. Miss. Gott., Math. Phys. K1asse, (1933). A1so, ZAMM, 13 (1933). Schubauer, G. B. and K1ebanoff, P. S., “Contributions on the Mechanics of Boundary Layer Transition," NACA TN 3489 (1955) and NACA Rep. No. 1289 (1956). Shen, S. F., "Ca1cu1ated Amp1ified Osci11ations in the Plane Po1seu111e and B1asius F1ows," J. Aero. Sci., 21 (1954). Sommerfe1d, A., “Ein Beitrag Zur Hydrodynamischen Erk1aerung Der Turbu1enten F1uessigkeitsbewegungen," Atti de1 Congr. Internat. dei Mat., Rome (1908). Squire, H. B., "On the Stabi1ity for Three-Dimensiona1 Disturbance of Viscous F1uid F1ow between para11e1 Ma11s," Proc. Roy. Soc. London, A 142, (1933). Stuart, J. T., "On the Stabi1ity of Viscous F1ow Between Para11e1 P1anes in the Presence of a Co-P1anar Magnetic Fie1d," Proc. Roy. Soc. A, 221 (1954). Stuart, J. T., "On the Effects of the Reyno1ds Stress on Hydro- dynamic Stabi1ity," ZAMM Sonderheft (Specia1 issue) 32-38 (1956). See a1so, J. F1uid Mech., 4 (1958) and J. F1uid Mech., 9 (1960). Thomas, L. H., "The Stabi1ity of P1ane Poiseui11e F1ow," Physica1 Review (4), 91 (1953). TietjenS. 0., "Beitrage zur Entstenhung d9? Turbu1enz," Zflflfln 5 (1925). To11mein, H., "The Production of Turbu1ence," NACA Tech. Memo 609 (1931), originai1y pub1ished in Nachr. Ges. Wiss. Gott., Math. Phys. K1asses (1929). To11mein, H., "Genera1 Instabi1ity Criterion of Laminar Ve1ocity Distributions," NACA Tech. Memo 792 (1936), origina11y pub1ished in German in (1935). Hazzan, A. R., Okamura, T. and Smith A. M. 0., "The Stabi1ity of Water F1ow Over Heated and Coo1ed F1at P1ates," J. Heat Transfer (Trans. ASME), 90 (1960). APPENDIX A RUNGE-KUTTA SCHEME FOR THE FOURTH ORDER DIFFERENTIAL EQUATION 90 APPENDIX A Runge-Kutta Scheme for the Fourth Order Differentia1 Equation The Runge-Kutta scheme for differentia] equations of the fourth order which is i11ustrated by Co11atz (1960) is used to integrate the stabi1ity equation. It is out1ined be1ow: At r = r0 V00 8 ¢(r0) V10 = Ar ¢'(r0) 1 12 V20 2 8 (”0) 3 = SA?) m v30 6 ¢ ('0) where Ar = step size. Using the equation (2.3.8) 0 __, M4]: 6"30 2‘,20 [Lo v r) 1 2 (Ar)3 (Ar)2 Ar 00 4 =%—§-’— ‘°‘R[(w-c)(¢"+ — 6-1754-0124) r 2 d N 1 dN - (3:2" ;' 3;) ¢] -‘—d2 [4"+1—¢'-(] 8214] “ dr r IF: _1_d_g m _3__ .._ 3 2. U dr [ 2¢ + P ¢ (;2'+ 2a )¢ 2 + [ - 24'“ M324 2 42) 4“ - (%-%—)4' r r 3 2012 4 - ( - '1' + “‘2' + a 1¢1 r r _ AY‘ Va1ues at r - ro + —§- _ 1 1 1 1 %V" +fl +fl +fl +‘G v11 ‘ V = 3 3 v V*2*’*'2‘G 21 20 3o 1 v31 ’ V3o + 2G1 As before a = (A§)fi_1, (6V31 2v21 v11 v r) 2 24 (Ar)3 (Ar)2 Ar 11 - ._E Va1ues at r — r0 + 2 v =v +14! +lv +111 +l—G 02 00 2 10 4 20 8 30 16 1 - 3 1 Vm‘vm+vm+4%0+?1 _ 3 3 V32‘ v20"2"30+2‘51 v32 = v30 + 262 Again as before 4 6V 2v v = (Ar) 32 22 12 G3 24 f ( ’ ’ V02’ r) (Ar)3 (Ar)2’ Ar Va1ues a Again 93 t r = r0 + Ar v03 = v00 + v10 + v30 + G3 v13 = V10 + 2v20 I 3V30 + 463 v23 ' V20 + 3v30 + 663 V33 ' v30 + 463 04 = (%%1f_f 6v33 2v23 V13, v03, r ) EX;S3’ (Ar)2’ Ar The eigenfuction ¢ and its derivatives at r = r0 + Ar are ¢(’0 + A”) = V00 + V10 + v20 I V30 + G . =.l_ . ¢ (r0 + Ar) Ar (V10 + 2V20 + 3V30 + G ) II = 2 u 4 (r0 + 4r) Z;;;2-(v20 + 3v3O + 0 ) 6 m ¢"'(r + Ar) = -—-—-(V + G 1 0 (Ar)3 30 -.l_ _ where G - 15 (881 + 4G2 + 463 G4) . - J. - G - 5 (961 + 682 + 663 G4) G" = 2 (G1 + G2 + G3) «0 _ 2 APPENDIX B COMPUTER PROGRAM 94 APPENDIX B COMPUTER PROGRAM 8 - 1 Description of the Computer Program The main program initia11y reads the required input data such as contro1 codes, number of integration steps across the annu1us, radii and temperatures of inner and outer cy1inders, and initia1 conditions to so1ve the governing equations of the main f1ow. It finds an initia1 guess of the temperature distribution assuming constant conductivity and neg1igib1e viscous dissipation. First approximation to viscosity and conductivity is found using this approximate temperature distribution. Pick a va1ue for X, which is an unknown constant in the equation of motion, and then ca11 sub- routine RKUTTA1. Subroutine RKUTTA1 integrates the equation of motion for two 1inear1y independent initia1 conditions using the fourth order Runge- Kutta integration scheme. Returning to the main program the two so1utions are combined in order to have a new so1ution which satisfies the boundary conditions at both cylinders. The resu1ting ve1ocity N is checked whether it satisfies the continuous constraint. If it does not, then subroutine PLAM is ca11ed to find a new va1ue for A. 95 96 Subroutine RKUTTAZ is ca11ed from the main program to inte- grate theicomp1ete and/or simp1ified energy equation as desired. Again, two independent so1utions are produced and combined so as to satisfy the boundary'conditions on both inner and outer cy1inders. Using this new temperature distribution repeat some of the above steps to find new main ve1ocity distribution. Check whether the main ve1ocity and temperature satisfy a prescribed convergent criteria. If they do not, then repeat the required previous steps unti1 convergence is obtained. Subroutine YUNG ana1yzes the effect of dk/dr and 0 terms on the ve1ocity and temperature distributions of the main f1ow. Once the temperature distribution and the main ve1ocity distribution across the annu1us is found, the subroutine EIGENVP is ca11ed. The fi1tering scheme and the interation scheme for the eigenva1ues discussed in Section 3.2 and 3.3 are imp1emented in this subprogram. The integration of the stabi1ity equation is done by sub- routine RK4 which uses the scheme i11ustrated in Appendix A. B - 2 Listing of the Computer Program The program deve1oped to compute the ve1ocity and temperature distributions of the main f1ow, and the eigenva1ues of the stabi1ity equation is 1isted be1ow. 10 15 29 25 30 35 40 45 50 55 6” PFOGRAH GODS annononncnon 97 YCHOO TRACE PPOFRAM YCHnC(IMPUT.OUTPUT365) nlnlnslnv R(2011.H1(201). H2t201). ”0(201). T1<201). T2I201).T0(2011 11).F(11).D(11). H3(201).VIS(201). Ktznl). DKDRI2011.T3(201). 1 CODE(10) cOnrou /A/ 31. FRP. H. LAMBDA. 03L. IPOS CUHFOW/B/L. DR. DT. SUH. RI COMPON/C/ PP,ECK, T150.T1050. VISZO. K20 chFON/n/ NONLIM.RO COMFON/E/ SIGN. TI. TO. N. NONI. HONZ CUHION/F/ REY. MONITOR. 166 . ICJ crayon 9.41.H2.T1.T2.c.n . VIS. <5 DKDR TYPF REAL LAHBDA .K20. K TVPI INTEGER CODE MODIFIED 0N 9-27'19/4. READ AND PRINT DATA VISBVISICnT) KiKID-T) HIIVELOCITY T1ITEHPERATURE REAP 175. (CCDEII). 1:131") 175 FURVAT(IOI5) ' 1r CODEI1):1, THEN 90TH Ok/nw AND Viscous DISSIPATION TERMS HILL HE NEGLECTFD. Ir 0005(2)=1. DK/DR TERM Is KEPT AND Viscous plssxp. Tennso. I? 0005131=1. THEM OK/UPSO. ANDVISCOJvDISSIP. TERM HILL BE KEPT. Ir 0005(41=1. THEN ROTH DK/DR AND VISCOUS 0155!“. TERMS HILL BE KEPT. IF CUDF(1)=CODF(2)=CODE(3)=CODE(4)IU. THEN ALL THE ABOVE CASES HILL BE CONS]: Eatn. 1: covecs)=i. rnen suaRoUTIHE YUNG HILL NOT as CALLED. CODE(?1=0 FOR DETAILED PRINTOUTS. 81 FOR LIHITEn PRINTOUTS. :2 FOR "IN. PRINTC 3RI'IT'UTS. 300F(7)=1F0" 131.2.3....... I10 F04 [81.6.11....... IN YONG. Ir norE(81=n TO CALL supnourrwe EIGEHVP, :1 TO SKIP SJBROUTINE EIGENVP. AND :2 r06 IEY=°EY(CRITTCAL1 AND THEN RESALCULATE THE HHOLE c THING. IC°:O 11(r005411 .En.1 .00, cnuF421.80.1 .on. CODE(3).EO.1 .on. CODE(41.EO.1) 00 150.1) 109:1 1C6: COOE(7) RrAD 150, HEY 130 FCRVAT( 710.0) CCCOI‘. RFAP 3. (C(I). 1:1.11) REAP 1, (9(1): 1:1.11) RfAP 2, 01(1). SLOPEI. SLOPE? RtAP 3. 1. PO FORIAT("F10.7) FURFAT(TF10.5) 3 FPRIATIIS. F10.5) vt—D DEFINITICN AND CALCULATION PR-PRANDTL NUMBER CP:SPECIFIC HEAT H: STEP SIZE EP5 3 MEASURE 0F CONVERGENCE USP CUUITS NUMBER OF CALCULATIONS HIrH REYNOLDS NUMBERS HHEN THE VISCDUS DISSIPATION TER“ IS KEPT. NCR=1 H=1.D/N CF:1.9 H 2 N + 1 PH: .0040H I13.“: 0.0004 ' U RH0¢1.0 HJHS : 1.0 3160 :1, RATIO = 1.0 TVO = (20.0 - 50.0) / 50. V1870: “.0100? K?O= 0.0014307 7O PEUGRAH JO 9O 1)” 135 119 .3 11 121 98 PF = V1528 t CP / K20 OOIIO READ ADDITIONAL DATA RFA' ?, RI. TI. TO IFITI-TDI 9100.9200.°100 CROP FOPE(2)=O M”111=C IFICODEII).AE.1) HON2=1 IIIVODE(1).”E.1) C0”E(3)=1 II(r0nE(1, .50. 1) CODEI3I=O C ECK : ECKERT NUMBER 910'1 I)" = R0 .31] 07 = TO - TI YCHOO TRACE L=@186724U.71 VV = ( (V15?0'RFY )/DR )092 AvT 3 ARSIDT) ECK 8 VV/(CDoAUT) / G C . C C REA" TNF FIRST TRIAL VALUE O’iLAHRDA RIAF 7. LAAREA. UEL C C PRINT DATA AND OTHER CONSTANTS DEFINED ABOVE I'( CODF(6)-1) 6210.6220.6230 6210 PVIIT 109". KEY PFI'T 6: 5(1). [31:11) PVIVT 7. (0(1). I=1.11) PVI'T 8. 41(1). SLUPEI. HIII). SLOPE? 6920 P”IIT 9. 51. PO. TI. To P”I'T 19. N. H. CP. RHU PLIIT 130 VISQUO K2": PR: ECK 5 FORIATI- 0 RF12.6) 7 FURCATI' CONSTAMTS.D. I" K‘in) ARE'a/o JXORFlng’ 3 r”RtAT(o01~11IAL "U”DITIONS. VELJCITY Avo 11s rrnsr DERIVATIVE-.l. 1 10X. 2F1".?. 1CX. ”F10.2 ) 9 FLRIAT(.UPI 3 'r602 .SX.0RO . .;60?JSX.TI' .F60205xD.Tn 3 .F6.7) 10 FIKPATtfioM : 916,6x,aH=oF7.3,5X. wC°8oF5.105X.!PHO=oF5.O) 11 rvniaT(.q VIS2U=* F12,7 .BX. 0K20=o F12.7 .Ax. 0PR30F12.7.8X. 1 ’EIK=' F12.7) 1‘90 F“RVA1(~-DEVAOLDS HUHPED : o F15.0) C c 6930 rlsr = (rt-501/50 71090 = (Ta-111/sn C C C HU“LIN COUNTS NUMBER JF ITFRATIONS. C DVL ITgraTIOT IN THIS CASF INVOLVES THE SOLUTION OF BOTH F0. OF MOTION AND C EvEQGY IQ. C IT COUNTS “UNdER OF TRIALS OF LAMBOA VALUES TO GET THE SOLUTION C OF THE E0. OF MOTION. C HUNIT0R=1 S HON1=D S MON2=O ln'r utulIN=0 II05:U [T s I H \3 125 13" 145 150 155 PEUGHAM 160 165 170 175 139 000°C 0 COCO If 17 7Dn 7O 99 111:0 CrhTI‘IUE ICOLNT COU”TS THF NUMBEf OF IVP SOLVEU FOR A CHOSEN LANBDA. 110131T : O CEHTIWUE III. IFU‘IUT ’ 1 7 15016016 N?(3) 3 SLODEI S“ TU 17 H7(1) 3 SLOPE? ICOINT = ICOLNT + 1 SULVF EU. OF MOTION WITH THE ASSUHPTION THAT UK/DRzPHlso. FIND HI(L¢1) FPO" HI(L) BY RUNOE ' KUTTA. I)" 15 L 3 10V- IF(MONITOR .GT. 1) CD To 20 1+< HONLIv - 1 1 19.20.20 1*(Icnu”r.GT.1) so 10 20 R‘LI = u. (L-1) Ab = t ”I . tR-9(L) 1 / RI 11(1) = 1.0/ ALOGIRO/RI) . ALOGIAG) AR3an. D“ 70" 1:1.11 JJ:J-1 9&3: A03 ‘ D(J)-(T150oT1050«T1(-0)ooJJ CONTINUE 0(1): Fx° fi (PI0JR*R(I¢2)) 135 27 CFNTIHUE sinfoh = H/3.0 . SIHSON E' = SIWSON - (PO+RI) l’.n EVR = ADSIEP) IFIFOBEIOI’l) 6610066100062” ”)9 661T PPIIT 53.FR.LAHCDA 53 FlRPAT. a1t1.3).11(1.3)- 1 '1II.4),T1(1.4) 1‘1 C"HTIHu: 11 FURPAT(~ «F7.3. 4(3!.2F10.61) PVIFT 301 301 ‘U'H‘T(O' H2‘InK): 12(InK) FOR 1310101 AND K3104 ., D" 300 1:1,“ , 1C6 P0111 11. 5(1): ”2‘101,0T?(101)0d2(109)012‘102)0u2(103,0T2(!.3)0 1H211p4111?(104) 300 (01T1NHE C CALCLLATE DIFFERENCE AND PATIO 0F VELOCITIES. pm): 1'2 .2 F(RrAT(o. u(1,2)-N(1,1105K0H(1.31-H(1.1105Xou(I.41-H(I.11'10x-RATIO?1'SX'F 1021.9x.nA11031-OXoRAT1041.) vv2=VP4=0. s NP?=PP4=0 VVEAK=0. t ”FEAK=0 D“ 13 1:1.n er91=w1(1.?1-H1(1.11 th31=u1(1.3)-u111.1) VIL41=H1(I.41-H1(1.‘) A1 = Aaclu1(1.111 17(11(I.11.FC.U. .00. A” .LT. 1.JE-101 GO TO 15 R4191=V9L71Ih1(1.1) 30 9'1 10” 1)“ 12" 125 130 YJNG 1) ‘0 2n 41 41 42 302 360 107 HAT31= "EL31/w1rt.1) APAT31:AQG(PAT31) 19(ARAT51.3T,VPFAK) VDEAK=APAT31 17(ARAT31.E‘,VP'AK) HPEaxai RAT41=VCL41IL1(I.1) AV2=AfiS(?AT?1) T”ACE AP4=AQS(RATA1) IF(AR?.GT.v02)VP2:Av2 I'(AR?.GT.VPZ)MR2:1 1*(hn4.fir.v04) VP4=AR4 1*(IR4.fir.qu) "P4=I GD 10 30 RAT?1=0. * PAT31=0. S HAT4180. CUNTXNUF I*(IC6.1E.1) GU In 13 P’IIT 14. VFL21.VEL31.V¢L41.RAT21.RAT31.RAT41 CONTINUE FDRVAT(O ~3t£11.4.7X).5x.3(E11.4.SX)) PVIPT 310. VPEAK, StHPEAK) F‘RVAY<~- HAx. RAT31 = tEtS.50 a? p s .r7.3) CALCLLATE DIFFERENCE AND RATIO OF TWO DIFFERENT TFHPERATURE§. PPIIT 350. ”P2: 5(NPZ)a V94. 5(M‘4) FDRFAT(¢ “AX. HAT21=0515.‘*AT RI'F7.3.510'HAX RT4180515.50AT RIO 1F7.3) PVIIT 50 IOIMAT(t- T(I:?)-T([.1)05X'T(I.5)-T(I.1)0520T(I.4)-T¢Io1)010X'RT2108X0RT3 11‘17X'R731t12x'9T4l*) TP2=TP4=0. t IP2=IP4=0 TVEAK=0. t IPEAK:n D“ 40 1:1." T?1=T1(I.2)-T1(I.1) 761=Ti<1.1)-11 s ART4: ABS(°f41) I’(AHT2.3T.*F2) TP23ART? I’(ART2.3T.TP?) IP88! IF(AHT4.GT.TP4) TP4=ART4 1I(ART4.aT.Yp4) 194:! G“ To 41 HT?1=o. % 913180. % RT41=0. cflnTINUE 1*(106.ME.1) Go T0 40 PPIIT 4?. T21.T31.T41. RT21.RT31.RT41 CUNTI‘I'JF. ‘0'4A7<- ~3 GO TO 6 VI$1(l):t./(2cH)t(-‘tVIS(I)+4*VI$(I+1)- VIS(I*2)) G“ To 4 v131zl)=1./(2*H)o(39v18(I)-4~VI<(I-1>o VISIT-2)) CPNTINUF VIS?:DV131/RR 4380d2/PR 00 6 I=1.H [T(I .69. 4-1) no Th 7 VIS?II)=1./“a02 «(ch15(l)- 5cVI$¢I+1)+ 4cv13(1‘2) -v15(1o3)) 45(1)=1./H~«2'(°ofll(II-SOH1(101).4cu1(1#2)swl(l¢3)) G" T] 6 v15?(l)=1./u..e ~<2oVIS(I) -5ons(I-1)o 4-v15(I-2)-v15(!-3)) u3(1)=1./H.q20(?aul(II-59N1(I-1) o4~H1(I-2)-H1(I-3)) CUNTINUE - pwxrr a, PAL, PREV. PCH. ntLTA PPIIT 9. IcnnF. HAXTT 61 70 75 30 85 90 as 100 115 11n 109 NIHIAR:0 230 [VPtn [I':n NFaPaR:H§4PAR¢1 [1(NEHPA%.GT. M951) 60 To 300 RIAP 3, 34. REY. ALFNA C”: RFAL(CN) PRINT 10. ca. REY. ALPHA SDUHDARY COWJITTCNS... P1=P28(0.0.0.J) 351 An INITIAL CONDITIUWS FQR THE RF.ATEO INITIAL VALUE PRURLEH AND THEN gflLVE TUE IVA HY 4TH anER R-K HETHOJ. P1(1p1)=31(132):(n.000-n) p?(1.1):P7(132):(0|n0n00) 95(’.2)=(1.n;0.01‘P3(101)8(nonlnoo) R4(1,1) =(n.0.1,F-B) S P“102"‘°.000ou, [C81 PPIIT 901.IC.P3(1.1).P3(1.2) .‘4(1.1).P4(1.2) 961 FORLAT(0 015.4(7615.5)) 100 CDNIINUC El: Ui'ALpHAOREY PHAIALPWAtt? 4on n0 500 111:1.HH CALL 9K4(H2.REY.61.CR.PHA.IIlplgUG) CALCLLATE THE SOLUTION TJ INVISID ECUATION TO BE USED 00. FILTERIK EIGENVP TRACE Ciflfl C -n AAD DETERHINE GROWING SOLUTION. JJ=IIII*2)¢1 [1811101 ”0 71" I311? Un(I)3(I41(Ju)-CH)'(P3(IIoIIOIIRIIII'PZIIIoI)'(1/(R(II)O'2 1T ‘ PHA)-°1(II.I) I -(H3(JJ)-1/<(II)0H2(JJI)0P1(IInI)I 1 'OIOAL91AcOEY AA=1EALIP?IIIoII'CUWJG(PZIIIn n“: REAL(°1(Il.l)ocnnJG(P1(II A‘BI: AA/RB 710 n(II-IIS‘IOPTIAABBI GisAnAXII fiIII.1).G(II.?)) C EXTRACT THE PORTION OF GIOHING SDLUTION(U0(1)) FRO" UOIZI. 7‘0 U” = ABS(REAL(U0(1)I) 750 RRIII) =U0(2)/U0(1) 1)): .IT)) 78n P1(II.2):P1(II.2) - RRIII) 0P1(II.1) P?(II.2)8P2(II.2) - RRIII) ~Pd(IIol) P3III02,393(II02) ' HR(II’ 'PSIII.1) P‘(II.2)=P4(II.2) - KRIII) tP4(II.1) .snu CbNTIHUE IF/P1(J.1) 530 DD 940 KJ=1.HH Pl(I.1)=PI(I.2)-A6-P1(I.1) p2(l.1)=p?(1.?)-A6-°2(I.1) P$(I.1):PT(I.2)-A6-P5(l.1I P‘II.1)=P4(I.?)-A6'P4(I.1) ado I=I-1 1*(ICOOF.EO.1) Co Tn 950 DC 65” I=lnv pwlrr 16. 1. P1(1.1).P2(I.1).PS(1.1).P4(I.1) 650 CUNTIVUC son IOIHATIt ~15. 4(2612.4)) c SJPFRPOSITICV or THF Tun SOLUTIONS. 950 Oh: P2(J.1)tP3(J.2) - P?{J.2)-P3IJo1) 6?: P?(J.1)/ RE cit-92(J.2)/ BE 0 cutCK TVE BOOHOARY CONDITIONS fiC(])8 CL'P9(J01) ’ CZ‘OZ'JOZI Ur(?)=01o"3(4.1) ¢ C20P3IJ.2) -(L..o.0) . THC s AnS(RFAL(RC(1))) + ABSIAIVAGIRCIIIII .ABS(REAL(8C(2))) 1 +Al$tA14AGTOC(2))I IP=IP+1 TI(IP)=C1'PI(JoII‘CQ'pIIJOZI TEST¢IP)=TF(IP)tCOHJG(TF(IP)) PVIVT 2a. TF(IP). T=5T(IP). ALPHA. DEY. CR IFI TESTIIP) - DELTA) 11.11.1? 19 COITINUE 60 TO (13.14.15IIP 1‘ DAL=PAL0ALPHA * ALPHAzALPHAoDAJ 6“ T0 100 14 ALPFA=AL°HA-CAL s UREYIPRFYfiRFY s RFYSREY¢DREY 8 Go To 100 15 0T1:(TF(?)- TFI1))/DAL x REY: REY-GREY EIGENVP TRACE DI2=ITF(3)-TI(1))/UREY s DEN=AIMAGI(CONJGIDT1II-DT2) DAL s AIflAGITF(1)0C0NJUIUTZIIIUFV DVEY= A!1AG( CUNJG