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FLOW in a TWO-DIMENSIONAL CHANNEL with a RECTANGULAR CAVITY by Unmeel B. Merbta and Zalmun Lavan

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FLOW in a TWO-DIMENSIONAL CHANNEL with a RECTANGULAR CAVITY by Unmeel B. Merbta and Zalmun Lavan NASA CONTRACTOR REPORT LOAN COPY: RETURN TO AFWL (WLIL-2) KIRTLAND AFB, N MEX FLOW IN A TWO-DIMENSIONALCHANNEL WITH A RECTANGULARCAVITY by Unmeel B. Merbta and Zalmun Lavan Prepared by ILLINOISINSTITUTE OF TECHNOLOGY Chicago, Ill. for Lewis Research Center NATIONALAERONAUTICS AND SPACE ADMINISTRATION 0 WASHINGTON, D. C. 0 JANUARY 1969 NASA CR-1245 TECH LIBRARY KAFB, NM FLOW IN A TWO-DIMENSIONAL CHANNEL WITH A RECTANGULAR CAVITY By Unmeel B. Mehta and Zalman Lavan Distribution of this report is provided in the interest of informationexchange. Responsibility for the contents resides in the author or organization that prepared it. Prepared under Grants No. NGR 14-004-028 and NsG 694 by ILLINOIS INSTITUTE OF TECHNOLOGY Chicago, Ill. for Lewis Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sole by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTl price $3.00 ... FOREWORD Research related to flow over aerodynamic bodies and to advanced nuclear propulsion is describedherein. This work was performedunder NASA Grants NGR 14-004-028and NsG 694, with Mr. MaynardF. Taylor,Nuclear Systems Division, NASA LewisResearCh Center as Technical Manager. A part of this work was also supported by the U.S. Air Force under contract AF-AFOSR 1081-66. iii ABSTRACT The steady-state flow characteristics in a rectangular cavity located in the lower wall of a two-dimensionalchannel whose upper wall was moved with a uniform velocity, were investigated by solving the complete Navier- Stokes equations for laminar incompressible fluid interms of the stream function and vorticity.Numerical results weredetermined for a range of Reynoldsnumbers from 1 to 500 and for cavity aspect ratios of 0.5, 1.0 and 2.0. A circulatingflow extending the.whole height was observedfor shallowand square cavities. For deep cavity a secondaryvortex near the bottomof the cavity was also noticed. Time-dependent solutions for the vortex flow in a square cavity bounded by three motionless walls and a fourth moving in its planewere obtained for Nie= 10.0 and aspect ratio of 1.0. V TABLE OF CONTENTS FORWAFtD ........................................................... iii ABSTRACT .......................................................... V LIST OF ILLUSTRATIONS ............................................. ix LIST OF SYMBOLS ................................................... xi CHAPTER I. INTRODUCTIOX ............................................... 1 1.1 h-oblen Definition 1.2 PhenoEenologicalDiscussion 1.3 Historical Background 11. NUMERICALFORi4LJLATIOJ ...................................... 11 2.1 GoverningDifferential Equations 2.2Boundzry and Initial Conditionsfor the Governing Differential Equations 2.3 Nondkrnsionaland Transformed Equations 2.4 DifferenceFormulations 2.5 Method of Solution 111. RESULT'S A!JD DISCUSSIONS .................................... 32 IV. COIiCLUSlOIIS ................................................ 55 APPENDIX .......................................................... 57 REFERENCES ........................................................ 79 vi i I . LIST OFILLUSTRATIONS Figure Page 1 ProblemDefinition of the Flow Over a Cavity ............. 3 2 CoordinateSystem and Velocity Notation .................. 14 D efinition of Prototype ProblemPrototype 3 of Definition .......................... 18 4 TheMapping Function ..................................... 21 G rid Notation 5 Grid ............................................ 25 6 Comparisonwith Burggraf . NAe = 0. AspectRatio = 1.0 .. 33 = 1.0, Aspect 7 ConstantStreamline Contours . N;e Ratio = 1.0 .............................................. 35 8 ConstantStreamline Contours . Nie = 10.0, Aspect Ratio = 1.0 .............................................. 36 9 ConstantStreamline Contours . NAe = 100.0, Aspect Ratio = 1.0 .............................................. 37 10 ConstantStreamline Contours . Nie = 500.0,Aspect Ratio = 1.0 .............................................. 38 11 ConstantStreamline Contours . Nie = 1.0, Aspect Ratio = 0.5 .............................................. 39 12 ConstantStreamline Contours . NAe = 100.0, Aspect Ratio = 0.5 .............................................. 40 13Constant Streamline Contours . NAe = 1.0, Aspect Ratio = 2.0 .............................................. 41 14 ConstantStreamline Contours . NAe = 100.0, Aspect Ratio .2.0 .............................................. 42 * * * * 15 Variation of u withrespect to z at y = 0.9H .......... 43 16 EffectofChanging Ay" on Vorticity at Corners ........... 45 17 Effect of Changing Az* on Vorticity at Corners ........... 46 18 Constant Streqine Contours . NAe = 10.0, Aspect Ratio = 1.0, t = 0.02 .................................... 47 = 10.0, Aspect 19 ConstantStreamline Contours . N;e Ratio = 1.0, t* = 0.06 .................................... 48 20 ConstantStreamline Contours . NAe = 10.0, Aspect Ratio = 1.0,.. t* = 0.10 .................................... 49 ix Figure Page 21Constant Streamline Contours. N;Ce = 10.0, Aspect Ratio = 1.0, t" = 0.14 ................................... 50 22 ConstantStreamline Contours. NAe = 10.0, Aspect Ratio = 1.0, t" = 0.31 .................................. 51 23 ConstantStreamline Contours. Nie = 10.0 , Aspect Ratio = 1.0, t" = 0.78 ................................... 52 24 VortexDevelopment. NAe = 10.0, AspectRatio = 1.0 ..... 53 X LIST OF SYMBOLS Symbol Description A Variabledefined after equation(2.62) * C -Az * AY Constants defined after equation ( 2.60) '1-'6 Variables defined after equation (2.60) C Variabledefined after equation (2.62) D Variabledefined after equat,ion (2.62) - F Body force vector H Height of thechannel 8 H H Nondimensionalheight of the channel, E Height the cavity He of * He Nondimensionalheight of the cavity, He -L L Lengthof the cavity Reynoldsnumber, - V Reynoldsnumber, -'In V P Pressure t Time * %l t Nondimensional time, - t L2 - U Velocity vector - U ,v ,W Components of velocity vector, u in x, y, and z directions * * * U Nondimensionalvelocity in x or z direction, u 'm * * V NondimensLonal velocity in y -direction, -Lv JIm xi Symbol Description U Velocity of the moving wall m* U Nondimensional velocityof the moving wall,-Lu m "m X Distance along the lengthof the cavity Y X X Nondimensional distance along the lengthof the cavity,- L Y Distance along the heightof the cavity B Y Nondimensional distance along the height of the cavity,xL * z Transformed coordinate defined by equation(2.39) z *, First derivative of transformation *" Z Second derivativeof transformation t * z z -coordinate of the upstreamcorner cl * * z z -coordinate of the downstream corner c2 * %* Overrelaxation parameter for $ B Relaxation coefficient of rl , defined after equation (2.60) 5l* 6 Operator, defined on page28 - 5 Y5Yll Components of vorticity vector,LU * L2 11 Nondimensional component of vorticity in z-direction,- ll 'm 'm -* * 11 First approximation of Q at new time step Coefficients of viscosity V Kinematic coefficient of viscosity P Density Stokes stream function Nondimensional stream function,L 'm 'm xi i Symbol Description Maximum value of stream function %l - w Vorticityvector, VXU * AY* Grid size in y -direction * * Az Grid size in z -direction V Del operator V2 operatorLaplacian vx operator Curl A Operator,defined page on 28 A2 Operator,defined page on 28 Subscripts * i Grid point number in y -direction R Ll Grid point number in z -direction R Y Independentvariable of theoperator 6 or A or A2 )i 2 Independentvariable of theoperator 6 or A or A2 Superscripts n Iteration number m Time level xiii CHAPTER I INTRODUCTION On airfoils at largeangles of attack, the adverse pressure gradient frequently causes laminar separation near the leadingedge resulting in a severe stall condition. If suchearly separation does not occur, the flow invariablyseparates near the trailing edge causing much milder stall. Flow separation onaerodynamic surfaces can also be due to the presence of per- turbancesand cavities, as in the cases offinned surfaces, turbine flow passages , bomb bays, windows , and so on.Cavity flow problem is a special case of the general problem of separation, having mostof the flow character- istics of the latter. Hence, inthis study cavity flow has been investiga- ted with the motivation of obtaining a better understanding of the phenomena of flow separation and vortex formation. 1.1 Problem Definition The purpose of the present study is to investigate the flow character- istics in a rectangular cavity, located in the lower wall of a two-dimensional channel. The natureof the vortex formed in the cavity will depend on the Reynolds number and the height to length ratio (the aspect ratio) of the cavity.This ratio together with the channel height and lengthdefines the geometry. The natureof the flow approaching the cavity would alsoinfluence thevortex. However, for simplification,the length of the channel was takento be infinite and the upper wall of the channel was moved with a constant velc- city thus keeping the flow approaching the cavity identical in all cases. This also facilitates defining the conditions at theupstream and the down- stream boundaries of the channel. The problem thus relates %o the flow over a rectangular cavity in the lower wall of a two-dimensional infinite channel where the upper wall is moving with a uniformvelocity (Fig. 1). The flow is assumed to be laminar,incompressible and Newtonian. The results are obtainedfor dif- ferentaspect ratios and Reynolds numbers.
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