HEAT AND MOMENTUM TRANSFER IN LAMINAR FLOW PAST A RECTANGULAR CAWTY WITH FLUID SNJECTION Thesis for the Degree of Ph._ D. MICHIGAN STATE UNIVERSITY ROBERT W. JOHNSON 1970 , unm‘ ‘ L I B R A R YI' Michigan Scans University lit-(Paw: This is to certify that the thesis entitled Heat and Momentum Transfer in Laminar Flow Past a Rectangular Cavity With Fluid Injection presented by Robert W. Johnson has been accepted towards fulfillment of the requirements for Ph. D. Mechanical Engineering degree in W}/\/m¢ {z Major professor Date a / 2§‘/ 7‘" / / 0-169 "V amomo av " ' ! "DAB & SUNS' . 800K mm ms. t LIBRARY IINDE as ' i l ) ABSTRACT HEAT AND MOMENTUM TRANSFER IN LAMINAR FLOW PAST A RECTANCUIAR CAVITY WITH FLUID INJECTION By Robert W. Johnson The study described herein included an experimental and analytical investigation of heat and momentum transfer from a rectangular cavity in a wall with a laminar external flow field. Cases considered included a range of Reynolds number from 10 to 1000 with uniform injection of fluid through the back wall of the cavity at velocities ranging from 0 to 5 per-cent of the average free stream velocity. The analytical study consisted of a numerical solution of the full Navier-Stokes and energy equations for two-dimensional steady state incompressible flow over a wall cavity. Flow patterns, velocity profiles, and temperature results obtained were in sub- stantial agreement with the data obtained from experiments which were conducted using air flow over a square cavity situated in a channel wall. Results indicate a significant effect of injection on flow patterns and heat transfer rates. The greatest reduction of heat transfer due to injection is found to occur near the reattachment point on the downstream cavity wall. It was found that the re- duction of heat transfer from the cavity due to injection could Robert W. Johnson be correlated by the equation 3..- 1 - exp(-vi Reo°77) qo for the range of Reynolds numbers and injection rates considered in this study. HEAT AND MDMENTUM TRANSFER IN LAMINAR FIDW PAST A RECTANGULAR CAVITY WITH FLUID INJECTION By ( “« Robert Wkaohnson A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1970 6:6.2 5156 7~/~7o ACKNOWLEDGMENTS The author wishes to express his sincere appreciation and thanks to his major professor, Dr. A.M. Dhanak, for his help and guidance in the work which led to this thesis. Special thanks are also extended to Dr. J.S. Frame, Dr. M.Z. v Krzywoblocki and Dr. T. Triffet for their work and time spent while serving on the guidance committee. Finally, the author wishes to express appreciation to his wife, Susan, for her patience, understanding, and encouragement during the course of this work. ii TABLE OF CONTENTS INTRODUCTION EXPERIMENTAL STUDY ANALYTICAL STUDY _ Momentum.Ana1ySIs Choice of Relaxation Factors Heat Transfer Analysis Flow Geometry Mesh Size Corner Points Boundary Conditions Convergence Rates ANALYTICAL AND EXPERIMENTAL RESUETS CONCLUSIONS BIBLIOGRAPHY APPENDIX I: Boundary Conditions 11: Prediction of Optimum Values for the Factors, a1, dz, 03 III: Tabulated Numerical and Experimental Results iii 15 16 21 23 25 26 26 28 30 32 75 77 79 83 91 10. ll. '12. 13. 14. 15. 16. l7. 18. LIST OF FIGURES Experimental Flow System Cross Section of Cavity Photograph of Test Equipment Traverse Mechanism Mesh System Used Flow Geometry For Analytical Study Isolated Corner Region Analytical Streamline Patterns, Re = 10, No Injection Analytical Streamline Patterns, Re = 100, No Injection Analytical Streamline Patterns, Re = 500, No Injection Analytical Streamline Patterns, Re No Injection Photograph of Flow Pattern, Re 3 10 Photograph of Flow Pattern, Re E 500 Analytical Streamline Patterns, Re = 100, 1% Injection Analytical Streamline Patterns, Re = 100, 5% Injection Analytical Velocity Profiles, Re = 100, No Injection Analytical Velocity Profiles, Re = 100, 2% Injection Analytical Velocity Profiles, Re = 100, 5% Injection iv 1000, Page 11 18 27 29 33 34 35 36 38 39 40 41 42 43 44 Figure 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. Analytical and Experimental Velocity Profiles Showing Reynolds Number Effect, x/B I 0.5, No Injection Analytical and Experimental Velocity Profiles Within Cavity, Re = 1000, No Injection Analytical and Experimental Velocity Profiles Within Cavity Showing Effect of Injection, Re I 500, x/B = 0.50 Analytical and Experimental Velocity at y = 0, x-component, No Injection Analytical Velocity at y = 0, y-component, No Injection Analytical Shear Stress Distribution at y = O, No Injection Analytical Shear Stress Distributions at y = 0, Showing Effect of Injection, Re = 500 Analytical Shear Stress Distribution Near Reattachment Corner y = 0, No Injection Analytical Shear Stress Distribution Near Reattachment Corner Showing Injection Effects, y B O, lie-100 Analytical Isotherm Patterns, Re = 100, No Injection Analytical Isotherm Patterns, Re = 1000, No Injection Analytical Isotherm Patterns, Re = 100, 3% Injection Analytical and Experimental Temperature Profiles Re - 100, No Injection Analytical and Experimental Temperature Profiles Re I 100, 2% Injection Analytical Temperature Profiles Re = 100, 5% Injection Page 45 47 48 49 51 52 53 54 55 57 58 59 60 61 62 Figure Page 34. Analytical and Experimental Temperature Profiles Showing Reynolds Number Effect, x/B = 0.5, No Injection 63 35. Analytical and Experimental Temperature Profiles Showing Effect of Injection, Re = 500, x/B = 0.5 65 36. Analytical Nusselt Number Distribution Along Cavity Walls Showing Effect of Injection Re = 500 66 37. Analytical and Experimental Temperature Gradient at y = 0, No Injection, Comparison with Burggraf's Result 68 38. Analytical and Experimental Temperature Dis- tribution at y = 0 Showing Reynolds Number Effect, No Injection 70 39. Reduction of Heat Transfer due to Presence of Cavity 71 40. Effect of Injection on Heat Transfer, Comparison with Chapman's Prediction 72 41. Effect of Injection on Heat Transfer 73 vi a. n NOMENCLATURE mesh size, x direction, ft cavity width, ft factor defined by equation (8) factor defined by equation (23) specific heat, BTU/1bm.OF factor defined by equation (8) factor defined by equation (23) component of error vector error vector constant BTU hr ft2°F BTU hr ftoF a constant used in calculating the relaxation parameters convection coefficient, thermal conductivity, 2’ 3 a constant used in calculating the relaxation parameters “2’ “3 mass flow rate, lbm/sec number of mesh points in x direction any property number of mesh points in y direction heat transfer rate, BTU/hr 0 temperature, F ''3| 5| has done an analysis for separated laminar flows based on the assumption of a thin mixing layer over a "dead air" region. He predicted the rate of heat transfer to the separated region to be 0.56 of that to a corresponding attached boundary layer. He further predicted that "moderate" amounts of injection into the separated region could reduce the heat transfer (12) rate to zero. Larson experimentally found the ratio of heat transfer rates to separated and correSponding attached regions in laminar supersonic flow to be 0.56 (independent of Reynolds number) as predicted by Chapman. An experimental study by Wolf et.al.(22) using a transient calorimeter technique indicated that Chapman's predictions may overestimate the reduction of heat transfer due to injection for laminar hypersonic flow over a cavity. An extension of Chapman's analysis to the reattachment zone by Chung and Viegas(6) predicted a ratio of heat transfer rates of 4-5 for separated to corresponding attached flows in this region. For a Mach number 0f 11, N16011(14) found the ratio to be half that predicted. (21) Townes and Sabersky conducted experimental studies of laminar flow over cavities as a means of studying the effects of surface roughness. Reihman<15) has conducted an experimental study which was concerned primarily with investigation of the stability of laminar flow in cavities. He found that for a cavity in a wall with a laminar developing boundary layer in low- turbulence external flow, stable laminar flow occurred for values of Reynolds number (based on cavity width) below 2400. For a Reynolds number of approximately 2400 an instability in the shear layer flow was observed with flow velocities fluctuating in a periodic manner. When the Reynolds number based on x (distance measured from the leading edge to the cavity) was less than 30,000 the oscillations were not observed, even for Reynolds numbers greater than 2400. This probably indicates that the oscillations were triggered by the beginning of transition to turbulence in the boundary layer.) Reihman also experimentally obtained velocity pro- files for the region within and above the cavity. Mills(13) solved the Navier-Stokes equations by a numerical procedure for flow in a rectangular cavity with one moving wall. His iterative procedure (referred to as Liebmann's iterative technique) correSponds to the over-relaxation procedure used in the present work.with the "over-relaxation" parameters taken as unity. He presents solutions for flow in cavities with depth/ breadth ratios of 0.5, 1.0, and 2.0 at a Reynolds number of 100. Photographs of flow patterns in a cavity with a moving belt as one wall confirm the validity of his numerical results. Burggraf(2) has published a paper describing a two-part study on laminar separated flows. The first part was an analytical study based on a linearized model for a circular eddy. It pre- dicted a viscous eddy at low Reynolds number which develOped into an inviscid core at higher Reynolds number. The inviscid region becomes significant at about Re = 100. The second part, a numerical solution of the full Navier-Stokes and energy equations, verified to a large extent the validity of his linearized analysis. He pre- sents contour plots of stream function, vorticity and pressure dis- tribution along with isotherms and wall heat flux obtained from the numerical study for a range of Reynolds numbers from 0 to 400. (1) In another study concerning laminar flow at relatively high Reynolds number, Burggraf combines the concept of a thin shear layer as proposed by Chapman with a linearized model for the flow within the cavity. He presents findings which include velocity distribution along the Open-cavity plane y = 0, heat flux dis- tribution, and pressure distribution data. (8) GreenSpan has developed a numerical method of solving the Navier-Stokes equations which converges for all values of Reynolds number, but consumes considerable computer time. This method involved a two-step type iteration in which the Y and w fields were each individually relaxed to the desired accuracy before proceeding to the other. This process was repeated until a con- vergence criteria was satisfied. He studied flows in closed rectangular cavities and gives results for a range of Reynolds number from 200 to 100,000. There have been several studies concerned primarily with (7) turbulent flow over cavities. Fox showed that a shear layer forms over the cavity and borders the external flow with the boundary layer separating at the leading corner of the cavity and reattaching at the recompression corner (or at the base if (4) the cavity is very shallow). Charwat studied effects of boundary layer thickness, depth to width ratio, and free stream Mach number (17) on cavity flow. Roshko measured pressure, velocity, and skin- friction on cavities with varying depth to breadth ratio. Haugen and Dhanak(9’10’11) conducted an analytical and experimental study of turbulent flow over a cavity with controlled boundary layer thickness. They found that the thickness of the oncoming boundary layer has a significant effect on heat transfer from the cavity. A more complete discussion of studies on separated flow in general has been given by Chilcott(5). The present study consists of two parts, one analytical, and the other experimental in nature. The analytical portion is a numerical study which differed from previous studies in that the Navier—Stokes and energy equations are solved for the region above the cavity as well as within it. This eliminates the need to make assumptions as to the nature of the flow at the top of the cavity. In addition, the region near the reattachment corner is solved in detail and injection of fluid through the cavity bottom is con- sidered. The experimental work involved measuring velocity and temperature distributions within and above the cavity and obtain- ing photographs of flow patterns within it. EmeIMENTAL STUDY The air flow system used in the experimental study is illustrated in Figures 1, 2 and 3. It consisted of a flow channel of rectangular cross section with a square wall cavity in one of the walls. The channel width and cavity width were held constant at 0.75 in. The aspect ratio.was 26.7, thus ensuring essentially a two-dimensional flow. The channel walls, constructed of l/4-inch thick aluminum plates and insulated with two inches of styrofoam, were maintained at the desired uniform temperature by seven individually controlled heating circuits. The desired temperatures were obtained by ad- justment of the spacing of the heating wires and of the current in each circuit. To allow uniform injection of air through the back.wa11 of the cavity, this wall was constructed of a 1/4-inch thick plate made of graphite. The material used, Graphitar 2413 furnished by the U.S. Graphite Co., had an overall porosity of 20%. Air was supplied from an available high-pressure air supply to a plenum which was constructed.over the cavity back. Injection rates were controlled by means of a pressure regulator on the supply line and were measured with the same orifice and laminar flow tube used in calibrating the hot-wires (described later in this report). The air supplied to the plenum was heated to the cavity wall tempera- ture by heating wires around the copper supply tube which was 6 lg Ema gm gang ll llll lull, Ill 1. It- VII! lull-l H «Aswan :ma 1. El =.-N\m Kw xeum goo: whoeownwaeuum roam ----1 mua>uo I. / Cannoam coauuaououm haaaam ua< acuuoofieH.\ H 30d” r ( f "4 \ “ é \. f-rjl’ {I/ r . '\ “it" ‘Rl ‘ ~‘/ _ ’ (\ // A 'l . ‘ 1~ / ’ I- ‘ ‘ y, F‘( \ ,+‘{—.’ v , ,j Styrofoam Insulation , ' ,i‘Lf fl VIII." Aka} , \ .:(v' , ‘., v.1, \ \ .74 / /.‘ c ( /JA/// ///////////////////Z/// __———> 3" 'Z ////{/47_ e% I /”'/ \ . .‘o I ' O O . '1 I" :0. H .. 7 / . . I l “A“ h ; 'I I ‘ ./ F I Thermocouple ‘\ //- (J 3" r”) ) / ‘ Porous Locations 1. // -> 1 Plate . ,. 4 / / ’, / , , s ’ ' l i J Heating s4 ,, ’ j ' ‘ ‘ ;‘ Wires I 7 t , - ' ,. a ‘» " L I “f ‘ ’ / ’ " ‘ ‘V A l t 'l k .\ i I 1 / i l \ \ , - ) .5 L Figure 2 CROSS SECTION OF CAVITY HzmzmHzom Emma mo mmam9 q ousmwm 12 the test section. In addition, it was necessary that the tempera- ture remained essentially constant (a variation of 10.050F. was taken as the maximum tolerable) during a velocity traverse. This was achieved by insulating the test section with two inches of styrofoam insulation, and providing a heat sink at the channel inlet. The heat sink consisted of two water tanks 24 inches x 20 inches x 1 1/4 inches spaced 1/4 inch apart. It was found that with the heat sink in place the temperatures in the test section remained very nearly constant (i.e., 10.050F.) for up to an hour. Since a velocity traverse usually was completed in 10-15 minutes, there was no noticeable temperature effect on the hot-wire readings. A Special calibration unit was constructed and used to obtain calibration curves for the wires in the low velocity range. The unit included a smooth hard copper tube with inside diameter of 1.510 inches into which the probe was placed with the wire positioned at the centerline. The unit was deisgned with sufficient entrance length/diameter ratio to assure a fully developed laminar flow at the wire in the velocity range of interest. Thus, the velocity at the wire was twice the mean velocity. The flow rate of air in the calibration unit was measured by two methods. First, the air was passed through a thin plate orifice, built to ASME standards, in the supply line, with the pressure drop across the orifice being measured with a micromanometer accurate to within 10.001 inch of water. The orifice was considered accurate for velocities at the wire of 0.3 ft./sec. to 2.5 ft./sec. For lower velocities a second measuring device was used. It consisted of a brass tube of 0.201 inch ID, between the orifice and the wire, 13 with pressure taps Spaced 10 feet apart. The pressure drop be- tween the two taps was measured with another micromanometer accurate to 10.001 inch of water. For flows in the laminar range, the flow rate could be calculated directly from the usual Poiseuille- flow relation. Using this method, accurate measurements were possible for velocities at the wire of 0.005 to 0.5 ft./sec. Com- parison of velocity readings obtained from the two flow measuring systems showed good agreement in the range of velocities where both were judged accurate to within 15%. The range of Reynolds number (based on cavity width) in- vestigated was limited to 1000 or less since the channel flow tended to become turbulent at higher flow rates. Limitations on accuracy of the hot-wire readings placed a lower limit of approximately 500 on Reynolds number for velocity traverses. Photo- graphs of the flow structures were obtained at lower Reynolds numbers, however, by injection of smoke into the flow channel. Considering the potential errors due to temperature variations within the channel, calibration errors, and limitations on reading accuracy, it is estimated that the velocity measurements were accurate to within.1§% of local velocity except in those regions in the cavity where the air flow direction was oriented to the wire in a manner entirely different than that of the air flow in the calibration unit. These areas are generally near the position y = -0.5. (The coordinates are defined in Fig. 6). Temperature measurements were made with 30-gage copper- constantan thermocouples and a K3 potentiometer. Wall tempera- tures were measured with thermocouples attached to the wall at 13 14 selected locations. Traverses were made with a probe which was positioned by means of the same traverse mechanism which was used in the velocity measurements. The probe was constructed of a 1/8-inch diameter steel tube which was inserted through the cavity back wall and reduced to 1/16-inch diameter tube which was then bent at a 900 angle and extended 1/2 inch toward the measur- ing point. The enameled thermocouple wires, 0.012 inches in dia- meter, extended another 1 1/4 inch from the end of the smaller tube to the junction. A common cold junction for all the thermocouples was pro- vided by making the junction at a copper block which was immersed in an insulated distilled water bath. The temperature of the bath was measured with an Emerson Calorimeter mercury-in-glass thermo- meter readable to 10.0050C. The maximum variation in cold junction temperature during any traverse was found to be 0.020C. Prior to their installation, the thermocouple calibra- tions were checked and compared. The calibrations were checked by immersing the thermocouple junctions in an insulated distilled water bath, the temperature of which was measured with a second Emerson-Calorimeter mercury-in-glass thermometer which had been checked against the thermometer used to measure cold junction temperature and was found to agree within 10.0050F. The tempera- ture of the bath was varied over the range of interest in the experiment. The maximum difference observed between any two thermocouples was 0.020C. Since all thermocouples used had a common cold junction and were found to give a potential difference of 0.0407 millivolt per degree Centigrade, this factor was used 15 in computing temperatures. It is estimated that the temperature measurements were made with an accuracy of 1? per-cent of the total temperature difference RE- Tw which was held at approximately 1.500 to minimize distor- tion of the flow patterns due to free convection effects. ANALYTICAL STUDY Laminar flow over a wall cavity has been successfully studied (See Ref. 3) for high Reynolds numbers using a model which assumes a thin shear layer over the t0p of the cavity so that boundary-layer approximations can be used. The reliability of these studies in predicting over-all heat transfer rates has been verified experimentally (See Ref. 12). However, these studies are not adequate to fully describe the flow near the reattachment corner, within the cavity itself, or flows with low Reynolds number where the dividing streamline may not be straight. Numerical solutions of the full Navier-Stokes and energy equations have the advantage that they can describe virtually the entire flow field. (Although there will be singularities in the solution at the separation and reattachment corners, reductions in mesh size can make the region of uncertainty as small as desired.) A further advantage of numerical solutions is that they can be easily adapted to a variety of boundary conditions. Since the matrices which are to be inverted in solving the Navier-Stokes and energy equations are sparse and usually quite large, probably the best solution technique now available for use 16 is the so-called "over-relaxation" method.* This method takes advantage of the fact that most entries in the matrices are zero and has proven to be faster in solving the Navier-Stokes equations than other matrix inversion techniques. An over-relaxation method was used in this study, with the relaxation parameters taken as a function of local velocity. Momentum AnaIysis The flow field can be described by the full Navier-Stokes equations. In dimensionless form and in terms of streamfunction and vorticity they can be written as %+%=-w (1) 5X BY 2 2 u+a—?=Re[i¥5§-i§9¥] (2) 5&2 a; By ax ax By where Y = streamfunction: §¥'= E-= 3 By U EX.= _ 2.: ;‘ a)? V v _ .. .61 a1 - t t : - _ - - w vor 1c1 y w 5X by Re = Reynolds number = -—- u = component of velocity in x direction (ft./sec.) v 3 component of velocity in y direction (ft./sec.) U 8 average free stream velocity (ft./sec.) v = Kinematic viscosity (ft.2/sec.) B Cavity width (ft.) * Russell (13)gives a review and comparison of the merits of several inversion techniques when applied to the Navier-Stokes and similar equations. 17 For the numerical solution, the region under consideration is covered with a rectangular mesh system similar to that shown in Figure 5. The distances between mesh points in the x and y directions differ by a factor B (B = 0.5 was used in this study). This allows solution with improved accuracy, since changes may take place more rapidly in one direction than in the other. The various derivatives in the equations may then be approximated by the central difference relations Q1: = P1+1,j " Pi-1,j 5:? 2b a}: = P1,1+1 ' Pi,j-l 59 28b 2 - 2P + a P = P1+1,1 1,1 P14,1 (3) -2 2 5x b 52? = Pi,j+l ' 213,1 + P1,j-1 -2 2 2 BY 8 b where Pi j = the value of any prOperty at the i,j th location. 9 In each case the truncation error is of order b2 (or szz). Using these relations, the finite difference forms of equations (1) and (2) become 1+1,1 ' 23,1,1 + Y1-1,1 Yi,j+l " 211,1“ + Yi,j-l = (4) 2 + 2 2 "”m‘ Y u’1+1,L" 1,1 +w1-1,1 +‘”1,1+1 ' i,j + i,j-1 = b szz RerYi,j+l " Y1,1-1;‘”1+1,j ' ”Di-1,1 _ L 23b 2b Y - Y - . i+111 1"];11 . (”i,j-r1 UJILL-1 (5) 2b 25b l8 o i,j-l-l Bb b A f h o o o “1’3 i,j i+l,j o i,j-1 Figure 5 MESH SYSTEM USED 19 These equations apply for all except the boundary points of the region. The governing relations for the boundary points are dis- cussed in another section of this report. For a rectangular region divided into p x q points, there are p-q algebraic equations for the Y field and p-q algebraic equations for the m field. These must be solved simultaneously. The method used in the solution (successive over-relaxation) (13) and Burggraf(2) is somewhat similar to that which was used by Mills in studies of flow in closed cavities with one moving wall. However, in this study the relaxation parameters are taken as functions of local velocity in order to obtain rapid convergence over a greater range of Reynolds numbers. Equations (4) and (5) are solved for Y, and w, res ectivel to ield 1.1 1.1 p y y y -_————1— 820’ +1! +b2 )+v +1 (6) 1,: mfiz) 1+1,j i-I.j “’m 1.1+1 1.1-1 u). . =—--1--- (82+C)w. .+ (Bz-cm. . +(1-D)w. . + 1.1 2(1+Bz) 1-1.J 1+1.J 1.J-1 (1+D)wi,j+1] (7) where =R—95- _ C 4 ”1,3“ Yi,j-l) (8) .M - D a (Yum Yi-1,j) Introducing relaxation parameters 01 and a2, (6) and (7) become 20 a l 2 2 Y = Ln Y +- , ,+Y, ,+b .. + i,j ( l) 1,j 2(l+fi2) [P (Y1+1,J 1-1,j m1,J) Yi,j+l +Yi,j-l] (9) w = (l-a )w +'—::;-- (82+C)w + (Bz-C)w + i,j 2 i,j 2(1+82) i'laj i+lsj (l-D)wi’j_1 + (1+D)wi,j+1] (10) where a1 and dz are multipliers which lie in the range 0 < a1 < 2. (It is seen that the term "over-relaxation" is some- what misleading since when 0 < “i < 1 "under-relaxation" would be more descriptive.) In the iterative procedure the newest available values are th used at each step so that the n values of Y. . and w. . are 1:] 1:] obtained from the relations n n-l (1’1 2 n-l n 2 Y. = 1- Y. +'—‘"—'-‘ +‘Y. . + b . . + i,j ( a1) 1,j 2(1+82) [I (Yi+l,j 1-1,j wl,J) n-l n Yi,j+l +Yi,j-l] (11) i,j 2 i,j 2(11fi2) i-l,j i+l,j (l-D)w:’j_1 + (1w)w‘i‘:j1+l] (12) A crude initial guess is made for the solution (say Y = 1, m = 0 everywhere) and the system of equations is solved first for a low Reynolds number. At each iteration equations (11) and (12) yield improved values at each point of the fields. The process is repeated until the fields (Y and w) stabilize. The fields were considered stable when the maximum residual, R, was less than a 21 predetermined value, where R: -—-1—,—[2 1;; . +1311 . +1.2... .) mil-1+, 2(1+B) 1 s] 1 :J 13.] 19.] (13) n n-l Yisj'l] - Yisj For the present work it was found that a value R = 0.00001 proved to be a satisfactory check for convergence. Solutions at the higher values of Reynolds number are obtained using the solution for a lower Reynolds number as an initial guess. This results in a considerable saving of computer time. Choice of Relaxation Factors: Proper choice of values for al and a2 is necessary for a stable and rapid convergence of the iterative procedure. Choos- ing values of 01 that are too large results in an unstable iterative process which diverges, while choosing values which are too small results in excessively slow convergence. Stability of the solution seemed to be particularly sensitive to the choice of . It was found that by choosing a2 as a function of local velocity in the field at each step it was possible to achieve con- °‘2 vergence of the iterative process for all values of Reynolds number which were attempted during this study while maintaining reasonable solution times. For solution of the Navier-Stokes Equations using a square mesh (5 = 1.0) Russell<18) predicts "optimum" values for the relaxation factors a1 and oz to be 22 01 = .f 2 l +-n -2 a =_L__ 2 1 +f5 where <15 = 15(C2 + D2 + 11263-2 + C1.2)) (15) He bases his predictions partly on empirical findings and partly on an analysis assuming uniform velocity everywhere in the field and neglecting the non-linearity of the equations by assuming that a correct solution is available for one field while analyzing the other. A similar simplified analysis was done for the case where a rectangular mesh is used (see appendix). For relatively low velocities and for large flow fields the analysis yielded the approximate values CI = 2 1 1+; 2 112 TT2 “—57“? 1+6 9 q (16) CI =- 2 2 1 + 2 l 2 —-. 14s 6 However, the square root function requires a relatively large amount of computer time. Therefore, to save computer time and to be assured of convergence the following values were actually used for most of the computations: 23 a = 1.2 K1 (17) 0‘2 = K2 + [Cl + In] Here K1 and K2 were constants which were varied between 1 and 4. These values are somewhat conservative. But, underestima- tion of a2 does not result in too great an increase in the number of iterations required while overestimation of a even slightly 2 leads to divergence (see Ref. 18, p. 12). Therefore, it was found after some experimenting, that better results were obtained by being conservative rather than seeking a value which is truly an optimum for the entire flow field. Heat Transfer Analysis The temperatureafield can be described by writing the energy equation in dimensionless form for two-dimensional steady- state heat transfer in an incompressible, homogeneous, non-gen- erating medium with negligible viscous dissipation. It can be written as — T - T -1 2T" 2T ui:+va==(ere) [L+§—] (18) ax ay -2 '2 BX CY where: _. T ' Tw T =— TE - Iw Pr = Prandtl Number (19) T = Temperature (OF) Wall temperature (OF) I-l ll Temperature at channel center at upstream boundary (CF) «is n 24 Introducing the stream function, Y, this becomes a i at a: -1 a3 a5 a)" an? ' a}? 65" = (”Re) [5132 + a92] (20) Using the relations (3) this can be written in finite-difference form. Y - v _ - ‘ - - 1,1+1 i,j-l . Ti+l,j T14,1 _ Yi+l,j “-141. i,j+l Ti,j-1 = 23h 2h 2h 25h _ F, - 2T. .+T._ . T. . -2T. .+T. ._ = (PrRe) 1[: 1.1,11 1,3 1 1,] + 1,111 1,1, 1,1 1 2 2 h ah Rearranging, we get — = ..._.1_ 2_ t _ 2 I -' Ti,j 2(1+32)[(B C )T1+1,j + (B +C )Ti_1’j + (l-D')Ti,j_1 + (1+D')Ti,j+1] ‘ (22) where: u = EEEEL _ C 4 (Y1,j+1 Yi,j-l) (23) D. =m - 1y ) 4 ai+l,j i-1,j This equation is then solved iteratively using an over-relaxation procedure similar to that used in the solution of the Navier- Stokes equations. Using the newest available values at each step, the nth value for Ti , is obtained from the relation: , CY -n = 1_ fn-l +_ 3 2_ . -n-l + 2 . -n “n 1 '-n (24) I " _ l + (1+1) )Ti,j+l + (1 D )Ti,j_1] 25 where is the relaxation parameter which is calculated for 0’3 each point in the field to ensure stability and rapid convergence. The equation for T. is similar to that for w except for the appearance of PrRe in place of Re. The "optimum" value for 03 was predicted to be (for large p, q and small C', D'): — 2 25 2 2 ‘ 1+3 5 This value differs from that predicted for a in the vorticity 2 equation only by the appearance of Pr in the expressions for C' and D'. To save computer time and to prevent divergence, the value actually used in most of the solutions was: K1 “3 = K2 + 1C1] + lD'] (26) Flow Geometry: Preliminary solutions were obtained for a geometry similar to that considered by Mills (13) and Burggraf (l), viz, a square cavity with one side moving at constant velocity. In addition, the condition with uniform injection through the wall Opposite the moving wall was considered. It was assumed that the injected fluid flowed uniformly out through the moving wall. The region considered was then extended above the cavity and upstream and downstream in the channel. With proper choice of boundary conditions, it was found that "correct" solutions in the area of the cavity could be obtained with one boundary at a distance x/b = .8, measured upstream from the separation corner 26 and the other boundary at a distance x/B = 1.0 downstream from the reattachment corner. This geometry is illustrated in Figure 6. Mesh Size: Since the computer time required for convergence increases greatly with a decrease in mesh size*, it is desirable to keep the nesh as coarse as possible while maintaining a stable and accurate solution. However, a fine mesh is desirable to ensure accuracy of the solution, especially near points of highest shear. In fact, entirely incorrect results can be obtained at high Reynolds numbers if the mesh is not sufficiently fine (See Ref. 2). It was found after some experimentation with mesh size that a mesh Spacing of 1/25 of the cavity width in the direction parallel to the main flow and 1/50 of the cavity width in the direction normal to the main flow gave satisfactory results for the cases considered. Corner Points: The corner points at the top of the cavity require Special attention during the solution of the Navier-Stokes equations. The value for the stream function, Y, is Specified at these points since they lie on a physical boundary. However, the vorticity could be calculated either using the relation given for the points along the * For low-velocity flows, the number of iterations required is pro- portional to the 3rd power of the number of points while for "high"- veolcity flows (large C, D) the number of iterations required is proportional to the l/2 power of the number of points (See Ref.l8) 27 wnbam A¢UHHMH¢Z< mom NMHMZOMU 304m ,\\IW\\ \ m m u \ L.\ \ 6 \\\\\\ il.\\\\\\\ a \\\\\\\\\\\\\\\\\\\N\ \ N 28 channel wall upstream and downstream of the cavity or using the relations for the points on the upstream and downstream walls of the cavity. In this study, both possible values were calculated for each point (as suggested in Ref. 20). The values calculated from the relation for the points along the channel walls were used in the subsequent calculations for neighboring points outside the cavity, and those calculated from the relations for the points along the cavity walls were used in subsequent calculations for neighboring points within the cavity. In order to increase the accuracy of the solution near the reattachment corner at the higher Reynolds numbers this region was isolated and the equations solved for it using a reduced mesh size. The isolated region is shown in Figure 7. Boundaries were taken as five mesh distances upstream and downstream and above and below the corner, with boundary conditions obtained from the pre- vious solution over the larger region. The process was repeated three times, each time reducing the mesh size to half its pre- vious value so that the final mesh size for the region very near the corner was 0.0053 x 0.00253. Boundary Conditions: The boundary conditions at the solid walls are easy to write since Y is known and Specified along these boundaries and rela- tions for m can be easily derived using the condition that there is no "slip" at the wall--i.e., the tangential velocity is zero at the wall. However, at the other boundaries, where there are no physical walls, the problem is not so well defined,since the effects -- -————-—-- I 29 I l I I I I I I I Figure 7 ISOLATED CORNER REGION 30 of the cavity on the external flow may be felt for an indefinite distance upstream and downstream. Therefore, it is necessary to impose some condition which is only an approximation to physical reality at these boundaries. The goal is to find a condition which gives the "correct" solution in the region within and near the cavity. After trying several combinations, it was decided to use, in this study, the conditions: 531:0 BX (27) 552:0 6X at both the upstream and downstream boundaries. The first solu- tions were computed using boundary conditions which correSponded to fully developed flow between parallel plates and actually specifying Y and m at these boundaries. However, at Reynolds numbers of 500 and 1000 slight irregularities (which physically would correSpond to very small oscillations in the flow) were noted in the Y field near these boundaries. Specifying the derivative boundary conditions eliminated these irregularities while still allowing satisfactory convergence rates when using a good initial guess at the solution. The numerical expressions and their derivations are given in the appendix for all boundary conditions finally used in this study. Convergence Rates: Convergence to a value R = .00001 usually occurred after a few hundred iterations. Typical cases were convergence for 31 Re = 500 with no injection in 321 iterations using 7 minutes of CDC 6500 computer time, with the initial guess taken from the solution for Re = 100, and convergence for Re = 500 with 1% injection in 453 iterations using 16 minutes of CDC 3600 computer time,with the initial guess taken as the solution for Re = 500 with no injection. Convergence rates decreased somewhat with in- creasing Reynolds number, but Since reasonably good initial guesses were available the number of iterations required remained about the same. None of the solutions required more than 20 minutes of CDC 3600 computer time. The number of iterations required for convergence of the energy equation was approximately the same as for the Navier- Stokes equations. However, since there was only one field to work.with and since a3 could be calculated for each point in the field and stored (unlike a2 which was calculated for each point at every iteration) the amount of computer time required for convergence was considerably less. Typical convergence times were 5 minutes on the CDC 6500 computer. ANALYTICAL AND EXPERIMENTAL RESULTS Streamline patterns within and above the cavity for a range of Reynolds numbers of 10-1000 are illustrated in Figures 8 through 11. These patterns were obtained by mapping the stream function solutions obtained from the computer programs and linear inter- polation between points when drawing the streamlines. Representative values of stream function are printed at selected points within the field. There appears to be a slight shifting of the center of the vortex from near the center of the cavity at low Reynolds number toward the downstream wall and then back again toward the center with increase in the Reynolds number. This shifting has pre- viously been reported by Burggraf(2). At higher values of Reynolds 13 number small secondary vortices Similar to those shown by Mills( ), Burggraf(2) (8) , GreenSpan , and others appear in the bottom corners of the cavity. A shifting of the position of the dividing streamline with increase in Reynolds number can be observed in these Figures. At very low Reynolds number the zero streamline dips noticeably into the cavity. As Reynolds number is increased, this streamline rises and in fact is seen to be slightly above the y = 0 position for a Reynolds number of 1000. However, it is very nearly straight for the higher values of Reynolds number. 32 33 2282.5 on .3 use .nmzmmueaa magmas. aaoahaaza a seam: O O o 6 0 £9,“ 2 as 3.3. up! wage)“. as». Isa. gage-Mm. : O O 6 O O O o 4 o o o o O I. 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Ema. 38 8.8 8.8 88 8.8. can. 88. 88. nooc. _ anc _ a one.” . _ "68 a 88 .u was. . 008 a 008. n cos . m 008 o. 88.. 008 .u DOB.“ II 35 8.888 oz 68 u 3. £288.... 8.5.28.8 qéafiifi 0H ~91.me 88 .880 .o 88.8 88 .o 88 .o 2.8 .o x \\ .\ 88.808. 2.8.. $8.- 88.. 88.0 88 .88.... .88. .88 .. 8A... .- 88 .o \\ \x... .. / I . . . / . 885... .83 . 88. /86. .88.. 88.0 _. \ . ., . . .2 . . . ._ 88.9.8. 88.... 86.- $8... 88.... E. \ \ 886 88... . 88... 88... 88... 88.0 4.88.1I385I88: I88... 88.8 88.8 88... 88.088... :8. “8.. 89. 2.2. 8.... 3.2. 8.: 3.2. 8.2. 8.2. 8.2. 8.9. 89. 8.0.. 98.. 2.8. .58. 8.8. 88 £8. 1:28...- 98...---28. I 9.8. - 28.-In 8.8. 8.8. 5.8 I- 98.....88 28. 8.8 8.8. 8.8. 88. :8. :88 “.88. 2.8. 8.8. 9.8. 8.8. 8.8. 6.8. 38 -I‘ I ! . III I.OIIIIJ. 88. :8. 6.8. 88... c8... 88. 3.8. 88 8.8 88 88. 8.8 8.8. 88. 88. 88.. 88.. 88.. 88.. 88... 88.. 88.. DE 88.. 88.. 88.. 88.. 88.. 88.88.. 36 28.83% oz .83 u mm .mzmmfifi mzflfimmwm géfififiz HH muswwm Ana .o 098 .o 008 .o unB .o 0000 .o ..o 88.0 88.0 88.0 88$ “8”. mar. 89. 8:. “2... m2... 7.3... 2:. 6:. mg. 29. 39. 5:. 29. :2. - .--..,4 FR...£.W.§§ SKY! 8mm. $-23?! 39.1- «3.94-.- 03.911 .68... 89?- 2am u! 3%.?! as“?! 89.!88; r... , , . . A i- ...,ac!-lf‘|l- ‘ . 4. .. I... 7..'!!..¢ gt-'-L R8. mag. 83. 2mm ammo. Ema. $9. 33 3%. ~13. 2.3. 9%. 5.3. 5...». SS. .§.8®. 89. 8%. 8%. 8%. 88. ~89 8%. 3F. .3... 2E. Aug; g4 88.— 84 g.— 88.‘ g4 05; 3.. g; 88.. 37 Photographs of the actual flow patterns in the cavity were obtained by injection of smoke into the channel for Reynolds numbers of approximately 10 and 500 and are shown in Figures 12 and 13. The shifting of the dividing streamline into the cavity at Re = 10 is as predicted by the analysis and shown in Figure 8. The effect of uniform injection through the cavity bottom wall on the flow pattern is shown in Figures 14 and 15. At low injection rates (IZ), the vortex is slightly distorted but still present. However, as the injection rate is increased the vortex no longer appears. Velocity profiles within and above the cavity at positions x/B = 0.25, 0.50, and 0.75 are shown in Figures 16, 17 and 18 for flow at Reynolds number of 100 with 0, 2, and 5% injection rates. In each case, the velocity in the upper portion of the cavity is seen to be greater at position x/B = 0.75 than at either 0.25 or 0.50, but the magnitude of the x component of the velocity in the lower regions of the cavity is greatest at position x/B = 0.50, as might be expected. The injection appears to reduce the x component of velocity within the cavity, but has little effect at or above the y = 0 position. The effect of a change in Reynolds number on the non- dimensional velocity is illustrated in Figure 19 for flow without injection. Velocities (non-dimensional) within the cavity are greatest at the lower Reynolds numbers, indicating the influence of the relatively greater viscous drag on the fluid near the top of the cavity at these flow rates. 38 Figure 12 PHOTOGRAPH 0F FIDW PATTERN, Re E 10 39 Figure 13 500 no = PHOTOGRAPH OF FIN PATTERN, Re 40 fauna a...» .32 s as .2588 8328.8 afiHflncfi «a «hands 88:89.. 88.- 88.. 88:88... gauge: .28. :8. 8.8. 89. 8.x. 8!. 2.2. 2.2. 3.8. 8.... 8.... 3.x. 3!. .6132. 83. is. =8. 5.3.. 3.8. 2.8. has. 88. 38. 5.8. 8.8. 5.8. 8.5. 8.988. 41 28.8.8.8 s... .8. .. 3. ..Eug 8555...... .5855... n.— 0.53m .21. .8... 9.... 9.3.. I l‘«|ii|A .I 8...... - awn--- - - 88 88. 38. am... : ,.|Ifl-!\.tl.|.l|lpll IKI‘u’I’OI'IIL. .IJDOI: Ill.l! Ci'l" LO’I’il 88.. 88... 88888J .812»... 3.... .8... . it'll. 8.8. 8F. 88. ..8. Ufl< 42 0.1 0 RIB-0.25 . Figure 16 ANALYTICAL VELOCITY PROFILES, Re=100, NO INJECTION cu: 43 x/B-O.25 x/B-O.SO x/B-O.75 -0.4 v -006 J -0.8 r- -1.0 0 Figure 17 ANALYTICAL VELOCITY PROFILES, Re=100, 2% INJECTION cm: Ufl< + 0.4 4’ 0.2 A . ; ,/ 4 : 1 1‘ : : : -0.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 .' - x/B-0.25 x/B-0.50 ’0°2 x/B-0.7S -0.4 4 -0.6 -0.8 v -100 ++ Figure 18 ANALYTICAL VELOCITY PROFILES, Re=100, 5% INJECTION cu: 0.4.. 45 0.2/x’ oia 056 0:8 1:0 liz 1:4 1:6 Re-IOOO Re-SOO Re-IOO -———- ANALYTICAL EXPERIMENTAL DATA I Re-1000 4 Re-SOO Figure 19 ANALYTICAL AND EXPERIMENTAL VELOCITY PROFILES SHOWING REYNOLDS NUMBER EFFECT, x/B=O.5, NO INJECTION 46 In Figures 20 and 21 the velocity distribution within the cavity region only is plotted to show greater detail in the lower regions. Figure 20 shows velocity profiles as a function of position for a Reynolds number of 1000 while Figure 21 shows velocity profiles at a position of xfi; = 0.50 for injection rates of 0, 2, and 5% and Reynolds number of 500. Experimental data points are also presented on these figures for the cases for which they were obtained. Agreement between analytical and experimental results is generally good except in the regions near the center of the cavity where the x component of velocity is predicted to be zero. Since the hot wire anemometer readings also sensed the y-component of velocity, they could not be expected to be accurate in this region where the x and y components are of the same order of magnitude. (In regions where the velocity was very low and its direction was such that it tended to directly oppose the natural convection currents from the wire (y direction) the wire readings actually indicated a less than zero magnitude of velocity, which is meaningless. At such points the velocity was recorded as zero.) The x-component of velocity at y = O is plotted for flow without injection in Figure 22 for Re = 10, 100, 500, and 1000 with experimental values included at the two higher Reynolds numbers. It is evident that an assumption of uniform velocity across the cavity at the dividing streamline would be at best a crude approximation for the lower Reynolds numbers considered in this study. 47 cu: x/B-0.75 x/B-0.25 x/B-0.50 ____. ANALYTICAL EXPERIMENTAL DATA I x/B-0.25 O x/B-0.50 A x/B-O. 75 -1. Figure 20 ANALNTICAL AND EXPERIMENTAL VELOCITY PROFILES WITHIN CAVITY Re-Iooo, NO INJECTION 48 b L F cl: 1X3" No injection -0.2 27. injection A $.41) ‘“ ('IE""““‘ 5% injection A " EXPERIMENTAL DATA 0 No injection -006 ‘ 4’ i 1 A 2% injection -0.8 ANALYTICAL -l.0 Figure 21 ANALYTICAL AND EXPERIMENTAL VEIDCITY PROFILES WITHIN CAVITY SHOWING EFFECT OF INJECTION, Re=500, x/B=0.50 an: 49 EXPERIMENTAL DATA A Re-SOO ANALYTICAL I Rel-1000 0.4 '1 R-- O. 0.3 "‘ ./" . .A 0.2 .' . ‘ Re 1000 //‘ I A .-/ ‘ / ' / 0.1 J o 0.5 0.5 0T6 0‘.8 1. x/B Figure 22 ANALYTICAL AND EXPERIMENTAL VELOCITY AT y=O, x-COMPONENT NO INJECTION 50 The velocity at y = 0 is seen to increase from zero at the separation point and reach a maximum somewhere downstream of x/B = 0.50. As the Reynolds number is increased the position of the maximum velocity shifts downstream and is approximately at x/B = 0.80 for a Reynolds number of 1000. Increasing the Reynolds number also has the effect of flattening the entire profile so that an assumption of uniform velocity across the top of a cavity may be reasonably good at high Reynolds number. In Figure 23 the y-component of velocity at y = 0 is plotted vs. x position for Reynolds numbers of 100, 500, and 1000. The distributions show a similar trend, with an initial negative value and final positive value present in all cases. However, an increase in Reynolds number decreases the magnitude of the y-component, eSpecially near the separation corner and the reattachment corner. This picture is, in general, consistent with the streamline patterns observed in Figures 8 through 13. The non-dimensional shear stress at y = 0 as calculated from the relation ‘38; 2 T ay (8) is plotted in Figure 24 for Re = 100 and 1000 with no injection and in Figure 25 at Re = 500 for injection rates up to 5%. The values near the recompression corner were calculated with a reduced mesh size for cases with Re = 500 and 1000. The region immediately downstream of the recompression corner is expanded in Figure 26 in which the shear stress is plotted for Re = 100, 500, and 1000 with no injection and in Figure 27 in which the shear stress is plotted 0.16 0.12 0.08 0.04 -0.08 51 . Re-1000 X Arx Re-IOO 1 Re-SOO ~ Figure 23 ANALYTICAL VELOCITY AT y=0, y-COMPONENT, No INJECTION IDIX 52 Iona Mpg 20.80.82. oz .ouA a< onesmHmHmHa mmmmem ma... A’ ll O'p- f3.- - (1-01) = where 'F- l (1+fi 2) HE HE ZEKBA'C 2 cos p +./1-D2 cos q‘] (52) (53) (54) 2 For the case where C < 82, D2 < 1, the smallest maximum root,,/i, is obtained when the two roots of (53) are equal. requires (up)2 = 4((1-1) or 2 1 +/’—1_p,2 The value p' is that of u for which yields the maximum A. 80 u'='—1—2-[\/ 4 C2cos-+/_ 1+3 3-1-21) and with large p and q cos 2,; cos 3 l P .nla This (55) (56) (57) (58) (59) 86 so u 2.: ——1 2 [/ 4_C2 +/1_D2:] 1+9 9 (60) 2 L 1 4 2 2 u' =“TT[B - c + l-D + 2/84_Cz /I-D2] (1+6) Now using fl-‘a" “=' 1 - % (small a) (61) we get 2 2 2 “.2; 122[Ba_cz+1_D2+282 _§_§_ E32D2+c13] (1+8) 8 26 (small C, D) and 2 2 2 1-p'22—1—2—5[02+9§+52D2-C—2—] (62) (1+6) 6 28 Discarding higher order terms we arrive at 2 l C2 2 l-p.‘ =—-—i- —2+D (63) “fl 3 Therefore do = 2 (64) 1 :/1 (934432) 2 2 1+8 6 or a = 2 2 2 . 1 + 1 2(9—2- + D2) (65) 1+6 5 87 = 2 (66) (1’ 3 '2 1+ 12c_2_ +132) 1+t=12 Now in the event that both C and D are large, the maximum A is a minimum when 2 a = a0 = (67) 1 +/1_p.2 as before, However u' = i 2 [/C2 2_B4 cos _r +,/D 2 _1 cos 113] (68) (1+6) q- so 2_ ns 2 ' 2 4 -—- + 2 —— 11 —-—2+; L/C -B cos p f:- _1 cos q ] (69) which is equivalent to (54). Since C and D are large (61) no longer applies so we must write “'2 E—lfi[84-C2 +1-D2 +Z/84C2/1DZ] (1+8 ) (70) 2 ~ 1 2 2 ] I = ___.——— 1- 2 4 2 [J- (1+B2)2 [84 "C + D + 2/84‘ C ”B D2+C2D “'2 E —-l-2—-§ [BA-C2 + 1-D2+ 2CD/ 4 ] (71) (1+6 ) 1 4-.§____ §§._.l§ C 22D C D or, for large C, D r“ $4 4 I “'12 E —-12—§- B4-C2 + 1-D2 + 2CD (1 + kLZ 2 ~ % - l§))] (1+3 ) - D C D 4 . “'2 a ——1-2—2- 54-02 + 1-D2 + 2CD(1 - £567 - 15)] (72) c D 88 then 4 1-13 .+2.[2.2 + my + 9C2 + g] (1+6 ) (73) 1 2 2 - - 2 = ‘—2—5 [(C-D) + B /2 +./Q ‘ (1+8 ) c D giving 60 = 9:3 (74) 1 +1—1— [M v » >21 (1+62)2 E 42/6: In the event that either C2 < 82 and D2 > 1 or (75) C2 > 82 and D2 < 1 the expression for u yields a complex number. It is possible to write an expression for ‘1‘ and differentiate it with reSpect to o in an attempt to find the value of for which ‘1‘ is a minimum. However, the resulting relation is an unwieldy poly- nomial which must be solved numerically. Since relations for a2 and 03 were found which gave relatively rapid convergence of the numerical procedure, it was deemed unnecessary to proceed with the solution of the resulting polynomial. For the streamfunction equation the expression for the prOpagation of the error vector, 5, is er} =<1-.1)e. +——-—- n-l “1 2 n-l 2 n 1:.j 1:] 2(1+62) + ei+l,j B ei-1,j n_1 + n J (76) e. . + ei,j+l le-l 89 The analysis is similar to that for the vorticity and energy equations. We again seek solutions of the form 9 q - . e, = 2 2 E. sin I‘LL sin 11-5-1 19.1 r=1 8:]. 15.1 p q Setting 1 - 1— — fl 1-1,j fh Ei+13j \/7\ Eiaj+1 /A E we get the condition 1 - olu /1 - (1-31) = 0 where M ='_"l§‘ [Bzcos E5+ cos E2] (1+8 ) P q The minimum occurs when 2 (01 u) =4(a-1) :0! =6 —— 1 1 C) 1 +-/’ 2 Since p, q are large, and taking r = s = 1 O o m old II H I N n o m A]: H I to so I..._1._ 2 fill—2.4.1 1.13. 1‘ ‘ 2 B ' 2 ‘ 2 1+8 9 q and, dropping higher order terms 9'2 (1+82)2 Thus we get . . = E. . 15.1-]- 1,] (77) (73) (79) (80) (81) (82) (83) 2 42 22 [1+34+2E32-2E32TT “En—2"22n ’Zzn] (84) 90 2 2 2 1 ‘ u' = 22 2 [(1 + 82) (Tr—2 4' LL5)] (85) (1+8 ) P 9 and 2 (10 = / 2 2 2 (86) 1 + ——(LT— + L) 2 2 2 1+6 P 9 APPENDIX III Tabulated Numerical and Experimental Results 91 V ~a¢.H 00¢.H 00¢.H 50¢.a 000.“ A noe.H 50¢.d 00¢.“ 05¢.H 0mm.~ mmn.~ 0am.~ m. 000.“ ~<¢.~ n¢¢.~ nmc.fl 00¢.H N¢¢.~ m0¢.~ mH¢.H 0~¢.H mmN.H 00N.~ ~0n.~ 0. -N.~ 05~.a 05~.~ h0~.~ 05~.H -N.H NMN.H 0¢N.a 00~.~ w-.~ 00~.H ~N~.~ n. 050.0 050.0 050.0 mnm.0 ~00.0 050.0 500.0 mno.0 nu0.0 ~00.0 «00.0 m00.0 N. «50.0 500.0 000.0 000.0 050.0 000.0 000.0 NM0.0 000.0 0m~.0 NNB.0 0N0.0 H. NON.0 Nsa.0 c-.0 0¢N.0 NHN.0 m0~.0 mmm.0 0mm.0 mmm.0 000.0 «00.0 0mm.0 0 000.0 000.0 0H0.0 000.0 000.0 0N0.0 N«(T(I+1.J)-I(1-1.J)))/ES+(T(1-1.J)+T(I+1:J)>*6*'2* 2 TII:J+1)+T(IpJ'111/E2*b(lou)*(1o~GIIaJ))*SA CONTINUE CONTINUE UO 8 J=52:100 U0 9 132070 SA=T(10J) IIloJ1=IE1*((XII*1nJ)‘XI1'1;J1)0(T(1,J*1)'1(l:J-111*(X(IDJ*1) 1 'XIInJ-1))*(TII+1»J1'TI1'1.J)))/F3+(T(1'1.J1*T(1*1:J))*6*'?* 2 T(lpJ+1)+i(InJ'111/t2*t(1:~1*(1.°G(IoJ1)*SA U1F=ABSI1II:J)‘SA) IF (DIP-CHECK) 404,405,405 CONTINUE N1=l N2=J CHECK=DII CONTINUE CONTINUE CONTINUE IF (LI-L11130n31p30 CONTINUE PRINT 24p CHECKINlp N2 LIT=LIT+10 CONTINUE L1=LI*1 IF (CHECK-EPS) 4081408:409 CONTINUE IF (LI-LIN) 25:26:26 CONTINUE COVIINUE PORMAT <2x.10$12.5) U0 20 J=1950 PRINT 210 (TIIIJ)OI:21946) CONTINUE U0 22 J=51p101 112 PRINT 210 (T‘TIJ’DT=1D)1) CONTINUE PRINT 23: LI PRINT 24, CHECK’ DO 80 J=lp50 PUNCH 75: (TTIoJT11321146) 80 CONTINUE DO 81 J=51o101 PUNCH 75. (TII:J):I=1:71) 61 CONTINUE 7b FORMAT (8F10.6) '3 FORMAT (t0*:1110) 24 FORMAT (*0*a1F15.6o2I10) DU 730 I=77n39o6 U0 761 J=2o100 TBAH=TIIoJ)/T(Io76) u=(X(IpJ+1)-X(IoJ'1))/B PRINT 732. J.TBAR.U 731 CONTINUE 730 CONTINUE 73d FORMAT (2x.1I10:2F10.6) 000:0. QCV=Up J=51 479 480 462 451 no 479 1:22.45 TG=TTTI:J+1)'T(Ind-1))/T2.tA*F) PRINT 4881 TDTG UCD=UCD+IG/?5. CONTINUE 1:21 IG=(-I(I.J+2)+4.vTIl.J+1)-3.ai(1.J))/H UCD=UUD+TG/50. PRINT 488) In TG 1:40 IG=(-T(I.J+2)+4.tT(I.J+1)-3.tT(I.J))/a UCO=UCD+TG/50. PRINT 488. TaTG DU 480 1:21.45 UCV=OCV+.5IPR*R*(X(I+1:~)'XTI-J))*(T(I*1:J)+TII,J)) CONTINUE PRINT 481. ncn.OCV FORMAT (¢0*.3510.b) IORMAT (to-.2F10.5) 1:21 u1=0. DO 485 J=2.50 IG=(-|(I+2pJT+4.*T(I+1:.)‘3.*T(IaJ))/(2.*B) PRINT 488. J.TG 453 4.54 465 465 487 113 UT:0T+TG/50. CONTINUE J=51 TG=('TTI+?.J)+4.*TII+1a~)'3.*T(T:J))/(2.*BT PRINT 488. JoTG OT=OT*TG/100. PRINT 484. QT FORMAT (*0'91F15o6) TORMAT (2X.1110.1T10.6) J31 DO 485 1:21.45 TGST'T(I.J*2)*4o*T(T9J*1)‘3.'T(T0J))/B PRINT 488. IoTG UT=QT+TG/Zb. CONTINUE PRINT 484. CT I346 DO 487 J=2.50 TG=(’T(I'Z:J)+40.T(T’1ov)'3..T(TOJ))/(20.B) PRINT 4883 JDTG UT=UT*TG/50. CONTINUE J=51 TG=('T(I‘?.J)*4.*T(I’1.~)'3.*T(T:J)T/(2.*B) PRINT 488. JaTG uI=u1+TG/1UO. PRINT 484 1 0T ENU I H H L T! llil "IL ll 2 4 9 2 2 6 o 3 o TITIIITTIIITIIT 1293 TIHITTIITTITIT