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-
Stokesequations for laminar incompressible fluid in terms 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
Variablesdefined 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 thechannel, E
Height thecavity 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 frequentlycauses laminar separation near the leading edge 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 inthe cases offinned surfaces, turbine flow passages , bomb bays, windows , and so on.Cavity flow problem is a special caseof the general problem of separation, having mostof theflow character- isticsof 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 thepresent study is toinvestigate 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- citythus 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. To magnifythe phenomena of
separation and vortex formation the aspect ratios were chosen to give the
reattachment of theflow over the cavity and not inside it.
1.2Phenomenological Discussion
A laminar separated flow can be defined as a separated flow in which
all shearlayers of importance to the problem are completelylaminar. The
separation and the reattachment of the flow over the cavity results in one
or more eddiesin the cavity (Fig. 1). One candecompose theseparated flow
intosix more or less distinctparts: (1) separationpoint region, (2) free
shearlayer, (3) reattachmentpoint, (4) main recirculatingeddy, (5) corner
eddies,and (6) external stream.
Kistler and Tan' definethe separation point as the point where a
streamlinein the neighborhood of the surface breaks abruptly away from
thesurface. The streamlinethat passes through this separation point serves
as a boundarybetween the fluid in the channel and that in the cavity. The
shear layer in the neighborhood of this dividing streamline is called the
freeshear layer. The reattachmentpoint is thestagnation point where part
of the flow is turned back into the separated region and part movesaway
from thisregion. The flowwithin the separated region is made upof one
or more eddieswith streamlines that close on themselves.
1.3 Historical Backqround
During thefifteenth century Leonard0 da Vinci observed and sketched
recirculatingeddies in the flow over various configurations. However,de-
tailedstudies of cavity flow were carriedout only recently. These investi-
gationsgenerally consider steady plane flow of anincompressible Newtonian
2
. .. I "In //////////////////////////////////////////////////////////////////////////////////////~'- ..
- EXTERNALSTREAM -
FREE SHEAR LAYER
S EPA RA TION POINT REATTACHMENT POINT REATTACHMENT POINT SEPARATION (UPSTREAM CORNER) (DOWNSTREAM CORNER)
FIGURE I, PROBLEM DEFINITION OF THE FLOW OVER A CAV ITY fluid in a rectangular cavity boundedby three motionless walls andby a
fourth.movingin its own plane.This is theprototype of practical flow
problems in which the fluid moves over a cavity.
A model proposed by Batchelor2 has been frequently used in analyzing
cavityflow. According to this model, thelimiting (i.e. as l~ tendsto zero)
laminar flow consists of a finite wake embodying a residual recirculating
eddyhaving uniform vorticity. In cavity flow the separation region is
knows'tobe of finite extent. For suchproblems, Batchelor's model appears
to becorrect for the limit N + m This is based on theassumption that Re . viscouseffects are restricted to a thinlayer along the separation stream-
line. Then anexact intergral theorem derived from the Navier-Stokes equa-
tions for steadyflow shows thatthe vorticity is uniform. The specific
va1u.e of the vorticity is obtainedby matching the external boundary conditions
usingthe boundary layer equations. Other theoretical models were alsopro-
posed3 forseparated flows, but with the exception of the complete Navier-
Stokes equations , none hasbeen accepted as having general sound basis. 4 Mills carriedout an analytical study of the prototype problem for a
squarecavity using Batchelor's model. He obtained a solutionof a linearized
form of Von Mises' equationfor steady flow in the boundary layer for con-
stant andvarying pressures around the walls ofthe cavity. From thisanalysis
the vorticity imparted to the core is obtainedand is used to determine the
motion inthe core. Mills alsoperformed experiments toverify his analysis.
The measured velocity profiles were in qualitative agreement with the analysis.
However, the measured vorticity of the inviscid core was about 1/3 thepre-
dictedvalue. This error was attributedto the gap between the fixed walls
and the moving wall. 6 Burggraf3, Mills5 andKawaguti have solved the same problemnumerically
4 using stream function and vorticityequation. Burggraf solved it for a
.squarecavity for a rangeof Reynolds numberfrom 0 to 400. Mills solved
it foraspect ratios of 0.5, 1.0 and 2.0 for = Whereas,Kawaguti N Re 100. consideredaspect ratios of 0.5, 1.0 and 2.0 and a rangeof Reynolds'number
from O to 64.
Theyemployed central differences to formulate the finite difference equations.Burggraf and Kawaguti chose a square mesh for all aspectratios, whereas Mills used it for cavities having aspect ratios of 0.5 and 1.0 only.
Kawaguti first calculated the vorticity at the boundarypoints and then the
streamfunction and vorticity at eachpoint in the interior. Mills utilized
Liebmann's iterativetechnique. He wentover the stream function field twice beforeentering the vorticity field which was alsotraversed twice. The boundaryvalues of vorticity were moved by only one half of the values indi-
cated by theboundary condition, however (as in Kawaguti'streatment) the valuesin the field were(mostly) given their full movement. Mills alsotried
a single iteration per field andfound it better than the above procedure.
On theother hand, Burggraf underrelaxed both stream function and vorticity
inthe interior of the field. During each iteration he considered points in
each row progressively from right to left (the direction opposite to the motion
ofthe moving wall), withthe rows taken in order from top (moving wall) to
bottom.Both Mills andBurggraf employed thecorrected values as soon as
they were available. Kawaguti was unableto obtain convergent solution for
N~~ = 128.
At higherReynolds numbers theviscous layer thickness diminishes.
Burggraf observed that in this case the mesh size mustbe decreased for the
same degree of accuracyand the relaxation (convergence) parameter must then
bedecreased for convergence. (Both of these imply an increased number of
iterations.) He alsofound that for large Reynolds number the mesh size had
5 a strong influence on the location of the vortex center as well as on the
entireflow field. In his study he allowed the maximum machine (IBM 7094)
time for onecase to be 30 minutes.This limitation permitted accurate solu-
tionsfor 5 400. However, approximatesolutions were obtainedfor Rey- N Re nolds numbers ofup to 1000.
Mills andKawaguti reported a circulating flow extending the wholeheight
forshallow andsquare cavities. Similar results were reported byBurggraf
forsquare cavities. Kawagutifound weak secondaryflows in a largeregion
nearthe bottom of a cavityof aspect ratio 2.0. However,he was unableto
determinethe nature of these flows because of a coarsegrid and computational
limitations. Mills claimsto haveobtained this secondary eddy at all Rey-
noldsnumbers. According to Kawaguti thecenter of thevortex moves down-
stream, as theReynolds number increases. However, the more accurateresults byBurggraf, obtained for a largerrange of Reynolds number thanthose by
Kawaguit showed that the vortex center shifted first in the downstreamdirec- tion and thentowards the center of the cavity as the Reynolds numberwas in-
creased.Kawaguti did not observe the corner eddies. Mills reported them
for aspect ratio of 1.0 andBurggraf observes them for all Reynoldsnumbers
for the same aspect ratio.
Mills compared the theoretical results with photographs of flow patterns.
There was a good correspondencebetween theoretical and experimental results foraspect ratio of 0.5. When theaspect ratio was 1.0, theexperiments did not show the two corner eddies whichhe had predictedtheoretically. For the aspect ratio of 2.0, a secondvortex did notappear in the experiment anywhere inthe range 0 5 N 5 100. (However, it appeared at N 1000.) Re Re Thesediscrepancies were attributed to the difficulty of realizing in the ex- periment a.11 the conditions necessary to produce a flowwith such a weak vortex. Burggraf noticed that the vorticity distribution was symmetric at
NRe = 0. At NRe = 100, a very small inviscidcore haddeveloped around the vortexcenter, while at N = 400, theinviscid core had grown to a diameter Re about 1/3 thatof the cavity. The cornereddies were oftriangular-shape and they had a diameterof about ten percent that of cavity at N = 0. Re However , at NRe = 400, the downstreameddy had grown to about 1/3 the dia- meter ofthe cavity, whereas the upstream eddy was relatively unaffected by the Reynoldsnumber. The strengthof the flow field (the maximum valueof thestream function in the cavity) was practicallyindependent of the Rey- nolds number.
Burggrafacquired analytical solution for the high-Reynolds-number
limit (N -f m) withthe stipulation ofBatchelor's uniform vorticity model. Re The problem was solved by theuse of the finite Fourier transcorm. The result brings out the most serious failure of the uniform vorticity model,
namely, thesolution does not show theexistence of secondary corner eddies.
Panand Acrivos7 obtained numerically the creeping flow solutions for
cavitieshaving aspect ratios from 0.25 to 5.0 usingthe relaxation procedure
employedby Burggraf. The flowin the primary vortex (the one nextto the moving wall) remainedunaffected by the location of the bottom wall as long
as theaspect ratio was greaterthan 2.0. For aspectratio of 5.0 the
first threevortices had a lengthto width ratio equal to 1.40. The analy-
tical value of this ratio predicted byMoffatt' was 1.39.
According to Panand Acrivos, for a square cavity, even a grid size as
small as 0.01 was still too coarse to reveal the detailed streamline pattern
insidethe corner eddies, although it was more than adequate for the solu-
tionin the core of the cavity. The solutionin the core remained practi-
callyunaffected by the changes in the structure of the corner vortices. So
7 animproved solution was coaputed after convergence in the core was assured,
by subdividing the region around the corners into finer meshesand iterating
further.This process of subdivision was repeatedseveral times. It dis-
closed a sequenceof eddies which were amazingly similar andsymmetric with
respectto the diagonal of thesquare cavity. The relativesizes and
strengths of these corner vortices were inexcellent agreement with analy- 8 ticalresults obtained byMoffatt . The cornervortices occupied only about
0.5 percent of the total area ofthe square cavity.
An experimentalprogram was undertaken by Panand Acrivos to study the
basicfeatures of the steady flow for different Reynolds numbers. Theycon-
sidered a cavity of a square section in the horizontal plane and withvary-
ingdepth (height). Theywere not ableto avoid the presence of three-
dimensionalfluid motions near the four intersections of the vertical sides.
However, thesemotions did notextend into the mid-section, where to all
appearancesthe flow was indeedtwo-dimensional.
Panand Acrivos found that for a square cavity the downstreamcorner
vortexincreased in size from N = 0 to NRe= 500, inexcellent agreement Re withBurggraf's numerical results. With a furtherincrease in theReynolds
nynber,this vortex began to shrink slowly, until at N = 2700 it retreated Re onceagain into the immediate neighborhood of the cavity corner. Hence,
Panand Acrivos concluded that, for a cavity of a finite ratio, to all in-
tents andpurposes the steady flow in the limit N + m will consistof a Re single inviscid core of uniform vorticity with viscous effects being con-
fined to infinitesimally thin boundary layers along the walls.
A cavity with aspect ratio 10.0 was used as a model for a cavity having
infinitedepth. As withfinite cavities, the primary vortex shrunk at first
as was increasedfrom the creeping flow limit. Beyond NRe= 800 the N Re
8 vortexgrew, with its size becomingproportional to ?5 at Reynoldsnumbers N Re between 1500 and 4000, beyondwhich point instabilities began to set in. It was observed that the core of the primary vortex never attained an inviscid state as N + m. Hence, it was concludedthat for a cavityhaving infinite Re depth, the viscosity and convection play an equally important role in the momentum transfer.
O'Brien' has also tackled the same problem for apsect ratios of 0.5,
1.0 and 2.0, andfor the range of Reynolds number from 0 to 200. She de- termined the numerical solution of a linearized formof the Stokes equation, a fourth order partial differential equation for the two-dimensional stream function. The result was appliedto approximate the Reynolds number solu- tion andapproximations were iterated to a more accurate solution until thefinal answer satisfiedthe complete stream functionequation. This pro- ceduredid not require any iteration for N = 1, however, it demanded an Re increasing number ofiterations as Reynolds number was increased. If the mesh was too large, the iterations did not converge satisfactorily and oscillatedbetween two closesets of values. O'Brien preferred not to solvethe coupled stream function and vorticity equations so as to avoid determiningthe boundary values of vorticity. She explains that in many cases the failure to get convergence has been traced to these boundary values.
Duringthe course of this study the author came across the doctoral thesis of Brandt''. Thisthesis is writtenin Hebrew (unknown tothis author) withan abstract in English. Brandt considered an infinite symmetric channel withfully-developed laminar flow at the entrance as well as exit andhaving recessed walls. He solvedthe fourth-order partial differential equation for thetwo-dimensional stream function. He worked withaspect ratios of 0.25,
0.5, 1.0 and 1.5. He alsotreated aspect ratios much less than 1.0, with
a view to investigate suddenexpansions and contractions in channels.
9 Brandt concludes that the flow in a straight channel produces a vortex
flow in any rectangular recess in the channel walls; if the recess is deep
enough, a main vortex is formed. His numericalprocedure did not give the
cornereddies for aspect ratios of 0.5, 1.0 and 1.5. The constant stream-
line plots for aspect ratio of 1.0 show that the dividing streamline is con-
cave for = 0 (basedon the height of the channel and the average velocity) fRe andconvex for = 100. kRe While theseinvestigations of cavity flow contributed substantially
towards a better understanding of the vortex flow, the phenomenon offlow
separation coupled with the formation of vortex has not been analyzed in
detail. It is well known that problemswherein the vortex motion is generated
by the action of the shear stress ofan outer stream, Jrhich separates and
reattaches itself again, are offrequent practical occurence. The effectof
theprecise nature of the dividing streamline cannot safely be ignored; that
is, the dividing streamline cannot be replaced by a horizontal flat plate
acrossthe top of the cavity in most ofthe problems. At the same time,sharp
cornersin the flow field may besingular points. The present work essentially
deals with numerical investigation of flowseparation and the formation of
vortex driven by an external stream.
10 CHAPTER I1 NUMERICAL FORMULATION
In this chapter the basic equations ofhydrodynamics, suitable for
theproblem of cavity flow are consideredalong with appropriate boundary
and initialconditions. Starting with the equations of mass and momentum
balance for a Newtonian fluid,the assumptions necessary for obtaining usefulforms of these equations are introduced.Later, to get the numeri-
cal solution of the present problem, the flow field was subdividedby a
grid in the horizontal and vertical directions and at eachnodal point the
governing differential equations were represented by difference equations.
2.1 GoverningDifferential Equations
The partial differential equations ofhydrodynmics for constant vis-
cosity may beexpressed as
aP - -at + v.(pu) = 0
(continuity or mass balance)
(momentum balance) where,
t = time
p = density - u = velocity - - F = body force per unit mass
p = pressure
l~ and A = the first andsecond coefficients of viscosity
11 The two coefficients of viscosity are.related by
K is usuallyreferred to as thecoefficient of bulkviscosity. It is I customary’’ to assume that Stokes relation
is at leastapproximately valid. Further, this analysis is restrictedto
incompressiblefluids. These assumptions when appliedto equaticns (2.1)
and (2.2), yieid the followirg equations :
where
v = -u is the kinematic viscosity-. P
The continuity and momentum equations with appropriate boundary condi-
ti’ons form a complete set ofequations for determining the pressure and the
velocityfields. Generally, a pressureequation produced by taking the
divergence of the momentum equation(2.4) is usedinstead of the continuity
equation. The pressureequation is:
Anotherformulation of the equations of hydrodynamics may beobtained
by eliminating pressure from the momentum balance equation and at the Same
time removing thecontinuity equation. This can be affected by intro-
12 duction of the vorticity equation and the stream function concept. Taking the curl of (2.4) and making useof (2.3) we get the vorticity equation:
where the vorticityis defined by
It is not convenientto calculate the components of the velocity vectoru from (2.7). In order to circumvent this difficulty, the conceptof stream function is introduced.
Aziz12 has carried out a numerical study of cellular convection using both of the above approachesto determine the velocity field for a two- dimensional problem. He concludes that the first approach (momentum and pressure equations) yields less accurate results than the latter. The difficulty arises from the highly non-linear nature of the pressure equa- tion and the coupling dueto pressure in the momentum equations. Therefore, in this study the vorticity equation and the stream function concept are used.
The problem was formulated in Cartesian coordinates, x ,y,z , with the corresponding velocity componentsu,v, w and vorticity components<,<,q
(Fig. 2). For two-dimensional flow the vorticity components in this coor- dinate system are
<=" aw ax
13 Fig. 2. Coordinate System and Velocity Notation
and the stream function,$ , is defined by
v= 2ax
Introduction of these definitions of velocities in the expression0 in for
( 2.8) gives
(2.10)
(stream function equation)
In addition to equation (2.10), the z-component of the vorticity equation
14 is used.This equation neglecting body forces is
2 2 at
(unsteady-state vorticity equation)
For steady state the aboveequation reduces to
2 2 0 (2.11b) (v(>+>) - (uk+vk)).=
(steady-state vorticity equation)
The stream function and vorticity equations with appropriate boundary and initial conditions form a complete set ofequations for determining the velocity field.
..2.2 .. Boundary- . . .. ~~ and. " Initial Conditionsfor the Governing Differential Equations The steady state problem is solvedwith the help of equations (2.10) and(2.11b). Since these equations are elliptic, the boundary conditions mustbe specified on allthe boundaries. The entrance and exit of thechannel are at aninfinite distance from the cavity; therefore, it is assumed that at theseboundaries the normal derivatives of all thefunctions are zero.
(Alternately,the analytic values of the functions can be defined. However, past experience indicates that such analytic values may beincompatible with thenumerical solution of the finite difference equations for the interior.)
No slipcondition is assumed on theremaining boundaries. Hence, alongthe moving wall the velocity of the fluid equals the velocity of the.wal1, whereas,on the other wall thefluid velocity is zero.Mathematically these conditions can be expressed as follows:
15 =o (2.12) I.x =.+a
=o pc Ix =?a
$(X, 0) = $m
where Qm is the value of streamf’unction at the moving wall.
L
subject to
whereu is thevelocity of the moving wall m
-- (2.20) (2.21) 2 2 (2.22) 16 Inequations (2.19) to (2.22) the second derivatives are evaluatedsub- ject to the condition that the first derivatives, that is, the velocities arezero. (This corresponds to the no slipassumption.) The numerical procedure was varified by reproducing Burggraf's results for theprototype problem (Fig. 3) with the boundary conditions (2.23) to (2.25) given below and again with the no slip assumption. $(x, 0) = 0 (2.24) (2.25) (2.26) L (2.28) The formationof the vortex was studied by solvingthe stream function equation (2.10) and theunsteady-state vorticity equation (2.11a) for the prototypeproblem. Theboundary conditionsare the same as equations(2.23) to(2.28) for all time t. Inaddition to these, initial conditionsare speci- fied in the interior of the entire field because equation (2.11a) is parabolic with respect to time. 17 Fig. 3. Definitionof the Prototype Problem 2.3 Nondimensionaland Transformed Equations The equations are made dimensionlessusing the length of the cavity, L, and the stream function at the moving wall, as referencedimensions. JI my x *x=- L * * y =x v =-Lv L 'm (2.31) * L2 '*' =- =- rl 'm 'm * JIm t =-t 18 " where thenondimensional quantities are designated by asterisks. The nondimensionalequations are where By definition and Also, for a Couetteflow u = u (1 - $1. m Hence, "InH Gm = - 2 if 19 Note U \ JI E- for the prototype problem m2 Therefore, uH "m Nke - 2v The nondimensional equations may be transformed from the physical ** ** plane(x , y ) to any new plane ( z (x , y*) as follows. and * - Nie (U * -7dz a + v* 2))Q* = 0 (2.38'0) dx az The equations were transformed so that the infinite channel in the physicalplane can be contracted to a finite length in the transformed plane. At the same time thetransformation was so chosenthat the neighborhood of eachof the convex corners was expanded.Thus a uniformgrid in the trans- formedplane is equivalent to a finer grid near the convexcorners in the physicalplane. This facilitates the study of the rapid changes in the flow 20 fieldnear these corners. The transformationchosen is defined by * + tanh a(X + 0.5) z=* 1 1 + tanh(0.5a) * where 'a' is a constant.This transformation maps -m c x < 0.0 into * -x 0-0 < z < 1.0, with a pointof inflection at x = - 0.5which is the up- stream corner. * The functioncan be extended for the domain 0 < x < m by assuming Q that it is antisymmetricabout z = 1.0, that is t * 1 - f(x ) = f(-x ) - 1 (2.40) The mapping functionthus defined (Fig. 4) and its first derivativeare continuous. However, thesecond derivative has a finite jump discontinuity Q withan average value of zero at x = 0. 280- Z* 0,5 I .o X* Fig. 4. The Mapping Function 21 Finally, the boundary conditions listedin the previous section are rewritten in the nondimensional and transformed forms. Equations(2.12) to (2.22) for the channel flow are considered first. ” (2.41) az 1 and I at inlet and exit * 3” *-0 J (2.42) * dz Note, -*-0 - x=?m dx ** ’ (z , 0) = 1.0 subject to ** * x. $(z ,H = 0 o<_z zcl (2.45) * * zc2 5 z 5 2.0 * * where z and z are the values of the transformed coordinates of the up- cl c2 cl stream and downstream corners, respectively. 22 ** * * * * * $(z,H +He)= 0 5 z zc2 zc.1 1. * * * H ** * * * * * * o * * (2.48) z < z < 2.0 c2 Note 2* dz x. 7=o x = 2 0.5 dx In a similar manner the boundaryconditions for theprototype problem are nondirnensionalizedand transformed. The same mapping function(equa- tion(2.39)) was employed.Equations (2.23) to(2.28) became 23 ** * J, bci, y = 0 (2.53a) ** 2* n(z,o)= 224* aY ** * 2* n (z , He ) =- * aY 2.5 Method of Solution Aziz12 has reviewed the numerical methods appliedto viscous hydro- dynamics problems similarto the one considered here. According to him alternating direction implicit methods (A.D.I.) are very effective in solving non-linear parabolic partial differential equations.He also solved a three-dimensional natural convection problem with the A.D.I. method. Hence, in this work the unsteady state vorticity equation is treated similarly. As mentioned earlier, Mills solved the elliptic equations with Liebmann's method. Burggraf and Kawaguti used a slight modificationof Liebmann's technique. In this work the steady state solution follows the treatmentof Lavan and Fejer13, whichis also a modification of Liebmann's method. 24 111 II I I II I I 111 I I I I The flow field is divided in rectangular grid in the transformed plane,Fig. 5, and the finite difference equations are solved for the values of the dependent variables at the nodal points. i-l *L 4 i "AZ*"- i+l j-l j j +I Fig. 5. GridNotation Consider first the numerical solution of the steady-state vorticity and stream functionequations. In the stream function equation the derivatives are substitutedby central differences. In the vorticity equation the second orderderivatives are replaced by central differences; the first orderderiva- tives are approximated by backward or forwarddifferences depending on whether thecoefficients of these derivatives are positive or negative, as explained by Lavanand Fejer. The finitedifference formof equations(2.37) and (2.38b) are : * 2.0 i,j-1 i+i, j 25 {(GAz +( (5* - (2.0 + 2. Az J (2.60) where n = thelevel of iteration fill,* = overrelaxationparameter AY *I z = the first derivative of transf orm ation * '1 z = thesecond derivative of transformation c r = &v*Ay* * "* = 1.0/(2.0 + 2.0 (% Z8j2 + (c + 2.0c2) cr + (2.0 c4 + cs) cz) Az 1 c = -1andc=Oifcr These finitedifference equations ( (2.59)and (2.60)) weresolved along withthe appropriate boundary conditions . Initialvalues were assignedto 26 all nodalpoints and improved values of these functions were obtained by successively scanning over the grid points once with each of the equations. * * The scanningproceeded in thedirection of increasing z anddecreasing y . Thisprocedure was repeated until the maximum error (residue) in the field was smallerthan some prescribed value. Now, considerthe A.D.I. method forequation (2.38a). The procedure used is a perturbation of the Crank-Nicholsonscheme which is defined as 2 (2.61) .where the secondderivative is defined at time level m andwhere This scheme requires an implicit method for the solution of finite differen;:(. !L equations. The perturbedtechnique employed here was formulatedby Douglas- forlinear andmildly nonlinear parabolic equations. He has shown that for a cubicregion his method is stable for anypositive time step. However, anexact stability analysis for noplinear equation is lacking. Beforegoing into details of the numerical method, equation (2.38a) is rewritten in a differentform and some useful quantities are defined * 2* 2* * (2.62) where 27 1 Let z * 2Az Y Y Y *m+ 1 The numericalsolution of equation(2.62) can be obtained at t * from the known solution at t by rn *m+l *m+l where { is the first estimateof vorticity at t . The finalvalue is calculatedfrom (2.64) Rearrangingequation (2.631, 28 Substracting equation (2.63) from equation ('2.64) andrearranging (2.66) Equations (2.65) and (2.66) reduceto 2.0 NAe Az 2.0 Az 2.0 Az i,j+l Az At i ,j C *m 2.0Az i ,j+l 2.0 NAe - i,j (2.67) Az At 29 2.0 Nie .2.0 * ) n*m+l 2. OAy At i ,j Eachof theabove two equationsinvolves the solution of a tridiagonalsys- tem of linear algebraic equations. The timedependent vorticity equations- ((2.67) and (2.68)) and the streamfunction equation (2.59) along with the appropriate boundary condi- tions weresolved to determine the transient solution for theprototype problem. At all nodalpoints initial values were assigned. These values 8 Km were eitherfor t = 0 or for t . Inthe former case they would bezero and in the latter casethey would bethe previously calculated values. The vorticity was first calculated on the boundaries and then in the interior by the A.D.I. method. The streamfunction equation was completelyrelaxed usingthe values of vorticity at the new time.This procedure was repeated for a desired number of time levels. The vorticity at theboundaries and the coefficient of the first order terms in the vorticity equation always lagged one time step behind the rest 30 of the field. However, all thevalues in the field might be iterated at the same time step until the boundary values of vorticity did not change more than a prescribed limit. This would make the computer time and cost prohibitive; as a result, this refinement in the solution was notconsidered. As mentioned before, vorticity at the solid boundaries was calculated from thesecond derivative of stream function. The stream function was ex- pandedabout the boundary under consideration in a Taylor series, and the appropriate value of the velocity was substituted for the first derivatives. The resultingexpression was solved for the second derivative to get its finitedifference form.However, forthe channel flow at theentrance and the exit the normalderivatives of vorticity as well as streamfunction were assumed to bezero. These conditions are enforced by treating the boundarypoints as interior points and reflecting the values of the function at the interior points (the ones next to the boundary) to the points outside theboundary. 31 CHAPTER I11 RESULTS AND DISCUSSION It is generally desirable to check the validityof a.numerica1 pro- cedure by comparison with some known results. Therefore, first the proto- type problem was solved for creeping flowin a square cavity. The results are presented as contoursof constant stream function (i.e. as streamlines) along with Burggraf's solution for thesame case (Fig. 6). (In the present work the contours of constant streamlines were obtained by graphical inter- polation whereas Burggraf employed numerical interpolation.) Good agreement is indicatedin Fig. 6 in the entire flow field with the exceptionof the region near the centerof the vortex, where the maximum values of the stream function in the two cases differ 4.3 by per cent. This variation is probably due to the coarser grid(11 x 13 points) and theuse of one-sided differences for first derivatives as compared with Burggraf's finer grid(50 X 50) and central differences. (The one-sided differences were chosen because the stability analysis by Lavan and Fejer shows the central differencesto be unstable for numerical solutions of channel flow.) After partially confirming the validity of the numerical procedure, it was utilized in the main investigation of the flow in a cavity along the wall * * of a channel. The grid size was0.1 and 0.0625 in y and z -direction, respec- tively. The stream function and vorticity values were relaxed until the -4 residues were smaller than 1.5 x and 2.5 x 10 , respectively. The height of the channel and the length of the cavity were always kept the same but the heightof the cavity was variedto obtain the aspect ratios 0.5 of 1.0 and 2.0. It should be noted that in the prototype problem the boundary layer thickness along the wallsof the cavity becomes thinner as Reynolds number is 32 A BURGGRAF (REF,3) q: =-0,1911 FOR PRESENT WORK t,bi ~-0~2FOR BURGGRAF’S WORK FIGURE 6. COMPARISON WITH BURGGRAF, NRi=O,ASPECT RATIO= 1.0. increased. and the proper representation of the flow field would require finer and finer mesh sizes. In this problemReynolds number is directlyproportional to the velocity of the moving wall, while in the present problem it is pro- portionalto the velocity of upper channel wall. Hence a largeReynolds number (s.ay 1000) in the present problemcorresponds to a much smaller Reynolds number based on the average velocity in the free shear layer on top of the cavity. It is thereforebelieved that the results obtained using 99 grid pointsinside the cavity are accuratefor the entire range of Reynolds numbers investigated (1 - 500). Constantstreamline plots are shown in Figs. 7 to 10 and in Figs. 11 and 12 for theaspect ratios of 1.0 and 0.5, respectively. Onlyone vortex is observedin these cases. An increasein the Reynolds rumber affectsthe vortexflow as follows: (1) thestrength of thevortex first increasesand thendecreases; (2) the vortex center shifts downstreamand in an.upwarddirec- tion; (3) thestreamlines in the free shearlayer cluster together; and (4) the streamlinedividing the cavity flow and the channel flow is concave at low Reynoldsnumbers and convex at highReynolds numbers. Two vortices, one strongerthan the other, are observed(Figs. 13 and 14) for a deepcavity having an aspect ratio of 2.0. The primaryvortex extends to a depth of 75.0 percent and 70.9 percent of thecavity height for Nf = 1.0 Re and N' = 100.0, respectively.This vortex has a heightto legnth ratio of Re 1.464, for NAe = 1.0 and 1.460 for NAe = 100.0. The numericalcreeping flow solutionof Panand Acrivos, for the prototype problem with the same aspect ratio predicts this ratio to be 1.4. The effectof the presence of the cavity on the external stream (the * channelflow) is shown inFig. 15. The variationof velocity u withrespect * to z , at one grfdpoint away from the wall havingthe cavity is given. At 34 -FLOW 1 7-t II/////I///,/I////I/,/////////I/,/,I/////I//,//I////,f 1 A%: FlGURE8, CONSTANTSTREAMLINE CONTOURS, NR:= 10.0,ASPECT RATlO=110. I W co W CD FLOW -0,004 + -0101 09 FIGURE 12, CONSTANT STREAMLINE COfVTOURSl NR~'=10010, ASPECT RATIO=O85, FLOW __gL 41 -5 \', FIGURE 14, CONSTANT STREAMLINE CONTOURSl NRd=100,0, ASPECT RATIO. 2,01 42 ASPECT RATIO = 1.0 0 NR~I =lIO A NR: = 10010 a UPSTREAM4 DOWNSTREAM z* CORNER CORNER FIGURE 15, VARIATION OF u* WITH RESPECT TO Z*ATY*=O.~H#, highReynolds number the velocity in the neighborhood of the cavity deviates less from the corresponding value of the Couette flow at the entrance and theexit.of the channel. Thesudden removal of thepresence of the channel wall results in a smaller acceleration of the particles next to the wall at highReynolds numbers. Inthe numerical procedure followed here, the upstream and downstream corner points (that may besingular) were consideredto be grid points. The behaviorof'the solution at thesepoints was thereforestudied. Figs. 16 * * and 17 show theeffect of changing Ay and Az on thevalues of the vorticity * * at thecorners. These values increase as Ay decreases (Az beingheld constant). However, at one gridpoint upstream of the upstream corner and one grid point downstreamof the downstream corner, values of vorticity were * practicallyunchanged. (Note, for Ay = 0.05, the stream functionand vorticity were relaxed me order of magnitdue lower than those for the other * * * valuesof Ay . WhenAz was decreased,keeping Ay constant,the vorticity values at thecorners were practicallyconstant. Thus, it seems thatthese pointsdo not appreciably influence t.heflow field at a small but finite distance away. Thedevelopment of the vortex in time was studiedusing the- configuration afthe prototype problem for NAe = 10.0. It was decidednot tosolve for the transient solution inside the cavity in a channel wall in order to con- centrate only on the'formation of the vortex andkeep the computer cost low. * The upperhorizontal plate was moved with a constantvelocity at t > 0. The % natureof the vortex is shown inFigs. 18 to 23 fordifferent t . The strengthof the vortex increases uniformly and attains thesteady-state value asymptotically(Fig. 24). Thisvalue is inbetter agreement with the value obtainedby Burggraf than that calculated from steady-state equations. This 44 38 ASPECT RATIO=015 36 3.4 DOWNSTREAM 3,2 A * 31 c F 21% \ 2.6 2.4 212 1 I I 1 1 I 1 1 0,025 0 105 0,125 FIGURE 16.EFFECTOFCHANGING AY*ON VORTICITYATCORNERS, 45 NRl= 10,O ASPECT RAT10=015 0 I 025 0105 0,075 011 01125 A z*- FIGURE 17, EFFECTOF CHANGING Az" ON VORTICITY AT CORNERS, 46 * Urn ISYMBOL~G* 1 . 0 -0,012 0 -0102 V -0103 + -0,0417 * Urn FIGURE 19, CONSTANT STREAMLINE CONTOURS, NR~'=I 010, ASPECT RATIO=I,O, t*=0,06, -0804 -01 IO -01I28 FIGURE 20, CONSTANT STREAMLINE CONTOURS, MRi=I O,O, ASPECTRATIO=I,O, t*=Olll Ln 0 - 0.02 -011 2 + -0,1467 i / $ I ////// I///,, //,/ I /,// / 1 1 I r ///////./...////////////~/ FIGURE 221 CONSTANT STREAMLINE CONTOURS, Nsol=IOtO, ASPECTRATIO=IIO,i*=0,3ll * Urn SYMBOL -01 I2 -0816 + -0,1972 4 FIGURE 23, CONSTANT STREAMLINE CONTOURS, NR~'=10,0, ASPECT RATIO=I10,t*=0,78, i -0.20 -0116 -0.12 3I -0a08 -0004 -0.02 011 002 013 0,4 03 O,G 087 018 t* FIGURE 24, VORTEX DEVELOPMENT, NR~= 10,0, ASPECT RATIO= 1,01 is mainly dueto the fact that the unsteady-state finite difference formula- tion uses central differences whereas the steady-state formulation employs one-sided differencesfor first derivatives in the vorticity equation. Finally, other investigators have reported corner eddies for square cavities. The present study that utilizes a relatively coarse grid does not indicate any corner eddies. 54 CHAPTER IV CONCLUSIONS 1. Steadylaminar incompressible flow in two-dimensional channels with a rectangular cut-out were obtained using an explicit numerical method for solvingthe complete Navier-Stokes equations. Solutions were obtained for aspect ratios of 0.5, 1.0 and 2.0, and for Reynoldsnumbers of 1, 10, 100, and 500. 2. The number ofeddies present in the cavity dependonly on theaspect ratio. For aspectratios of 0.5 and 1.0, onevortex was observed. However, for aspect ratio of 2.0 two vorticeswere present, oneon topof the other. The dividingstreamline was at a cavitydepth of 0.75 at Reynolds number unity and at N' = 100. .TO9 Re 3. The streamlinedividing the external flow and the cavity flow was concave for low Reynoldsnumbers and convex forhigh Reynolds numbers. 4. As theReynolds number was increased,the strength of the vortex increased and thendecreased, and the vortex center moved downstreamand up- ward,creating a thinshear layer. 5. The calculatedvorticity values at the upstream and downstream cornersdid not appear to approach a limit as the grid size was decreased. However, the vorticity values at nodal points next to these corners, leveled off as the number of grid points was increased. 6. Inthe present problem the shear layer on topof the cavity andalong the cavity wall is notvery thin even at largeReynolds number (based on channel flow) hence, results obtained with relatively coarse grid may beaccur- ate even at largeReynolds numbers. 7. Inorder to observe the actual vortex formation the time-dependent solution for the prototype problem was determined using an implicit alter- 55 nating direction method to solve the vorticity equation andan explicit re- laxationprocedure for the solution of thestream function equation. The strength of the vortex increased uniformly and attained the steady state valueasymptotically. 8. The steady-stateresults determined from theunsteady-state equa- tions were in better agreementwith Burggraf's results (for the prototype problem)than those calculated from steady-state equations. This may be duemainly to the fact that in the former study central differences were usedfor first derivatives whereas one-sided differences were used in the latterstudy. 56 r APPENDIX THE FORTRAN PROGRAM In this appendixthe computer program used in this work is briefly described. Theprogram was written in Fortran IV for the IBM 360Computer. The notations used in the program are listed anddefined below. Fortran Symbol ion Explanat * IC Last grid point in the y -direction of the channel * IL Last grid point in the y -direction (channelplus cavity) * JL Last grid point in x -direction B NMAX Maximum number of iterations for $ and steady-state n* equations KL1 Frequency of print-out of maximum residues KL2 Outputoption: 1 - no CHRT output BB*t KL3 Outpu; option: 5 - read $ , rl , u , v and t from magnetictape, 6 - write $*, Q*, us, v* and t* on magnetictape, 7 - do both of above KL4 Problemoption: 1 - prototypeproblem, 2 - channelwith cavity b B Outputoption: 1 - no $ and rl CHRT output KL6 Planeoption: 1 - no transformation, 3 - transformed plane KL9 Frequency of unsteady-state CHRT output KL10 Stateoption: 1 - unsteady state KL13 Initialvalue option: 1 - initialvalues printed u, v, x and y printed(dimensional output) Natureof boundagy condition at J = JL: 1 - functions JI and q* defined,other- wise normal derivatives equal to zero 57 KL18 Must be equal to 1 C AX*/AyX RNN Reynolds number , NAe BPS1 Over-relaxationfactor for streamfunction equation BVRT equalMust be to 1 Y * DPSIM Maximum allowableresidue of 9 w DVRTM Maximum allowableresidue of rl (for steady-state) J2 Value of J at whichcavity the starts JM Value of J aboutwhich the cavity is symmetric J3 Value of J at whichendscavity the LENGTH Dimensionallength cavityof the * DLTT At NTMAX Number of time steps * NMAXl Maximum number of iterations for $ (unsteady state) ZMIN 0.5 Must be KLO, KL7, KLl5, KLl9, KL20, Dummy variables - notused in the program KL21, HL, ZMAX, AA * U Thein velocity y -direction * W Thein velocity x -direction K PSI 9 * VRT rl * DIST x w Y Y 2% DLTZ Az w DLTY AY 58 r Note,during the utilization of the computer program given here, the comments listed in the mainprogram and the ones listed belowshould be kept in mind. 1. Heightof the channel and thelength of the cavity should be the same. 2. Do notuse transformations given by the options KL6 = 2 and KL6 = 4. 3. The prototypeproblem can be solved only for aspect ratio of 1.0. 4. The presentform of the program is notcapable of solving for the transient solution of the flow in the channel having a cavity. 59 L Q, 0 1PFOOR20 1P FOO830 1PFOOR40 1PF00230 1P FOOSOO IFF00510 1PF00870 I 1 1 CASA>> 1 )CASA>> 1 1P~O2000 1PF02010 1PFO2020 1 *I I 1 1 CA SA>> 1 )CASA>> .. -3 -3 11CASA>> 1 1 CASA?? 3 1 CASA>? 11CASAB? 1PF03990 1 PF04000 r REFERENCES 1. Kistler, A.L. andTan, F.C. "Some Propertiesof Turbulent Separated Flows",The Physics of FluidSupplement, (196~)~165-173. 2.Batchelor, G., Journalof Fluid Mechanics, 1 (1956), 388. 3. Burggraf, O.R., "Analytical andNumerical Studies of theStructure of SteadySeparated Flows'', Journal of Fluid Mechanics-,24 (1966), 113-151. 4. Mills, R.D., "On theClosed Motion of a Fluid in a SquareCavity", -Journal of -the Royal Aeronautical Society, 69 (1965) , 116-120. 5. Mills, R.D., "NumericalSolutions of the Viscous Flow Equationsfor a Class ofClosed Flows", Journal of the Royal Aeronautical Society, 69 (1965) , 714-718 6. Kawaguti, M., "NumericalSolution of the Navier-Stokes Equations for the Flow in a Two-Dimensional Cavity",Journal of Physical Society of Japan, 16 (1961), 2307-2315. 7. Pan,F. and Acrivos, A., "SteadyFlows inRectangular Cavities", Journalof Fluid Mechanics, 28 (196~)~643-655. 8. Moffatt, H.K., "Viscousand Resistive Eddies Near a SharpCorner", Journalof Fluid Mechanics, 18 (1964), 1-~8. 9. O'Brien, V., "Periodic BoundaryLayer Flows Over a FlatPlate Part 111: With Disturbance Bars", TG-944, AppliedPhysics Laboratory, The Johns Hopkins University, August 1967. 10. Brandt, A., "Solutionsof Equa.tions in Hydrodynamicsand Magnetohydro- dynamics'' , Weimann Institute of Science, Israel (1964). 11. Bird, R.B., Stewart, W.E. and Lightfoot, E.H., Transport Phenom-, JohnWiley and Sons, Inc., New York, 1960. 13. Lavan, Z. and Fejer, A.A., "Investigationsof Swirling Fiows inDucts'', ARL 66-0083 , Aerospace. Research Laboratories, USAF, Wright-Patterson Air ForceBase , Ohio, May 1966. 14. Douglas (Jr.) , J. , "AlternatingDirection Methods for Three Space Var- iables" , NumerischeMathematik, 4 (February1962) , 41-63. 15. Wilkes, J.O., "The FiniteDifference Computation of Natural Convection in anEnclosed Rectangular Cavity", Ph.D. Thesis,University of Michigan p (1963 1. 79 NATIONAL AERONAUTICSAND SPACEADMINISTRATION POSTAGE AND FEES WASHINGTON,D. C. 20546 NATIONAL AERONAUT1 SPACE ADMINISTRATI OFFICIAL BUSINESS FIRST CLASS MAIL “The aesonautical and space activities of the United States shall be condzlcted so as to contribule . . . to the expansion of hz~nzanknolul- edge of phenomena in the atmosphere aad space. The Administration shall provide for the widest pyacticable md appropriate dissem&zation of informatioft concerning i/s activities and the reszdts thereof.” -NATIONAL AERONAUTICSAND SPACE ACT OF 1958 !\# NASA SCIENTIFIC AND TECHNICAL PUBLICATIONS TECHNICALREPORTS: Scientific andTECHNICAL TRANSLATIONS: Information technical information considered important, published in a foreign languageconsidered complete, anda lasting contributionto existing to meritNASA distribution in English. knowledge. SPECIAL PUBLICATIONS: Information TECHNICAL NOTES: Information less broad derived from or of value to NASA activities. in scope but nevertheless of importance as a Publications include conference proceedings, contribution to existing knowledge. monographs, data compilations, handbooks, sourcebooks, and special bibliographies. TECHNICAL MEMORANDUMS: Informationreceiving limiteddistribution TECHNOLOGY UTILIZATION because of preliminarydata, security classifica- PUBLICATIONS:Information on technology or othertion, or reasons. used by NASA that may be of particular f f interest in commercial and other non-aerospace CONTRACTOR REPORTS: Scientific and 1 npplications. Publications include Tech Briefs, I technical information generatedunder a NASA Tec,~nologyUtilization Reports and Notes, i grant andcontract or grantimportant considered an 1 and Technology Surveys. ! contribr ltion to existing knowledge. Details on theavailability of thesepublications may be obtained from: SCIENTIFIC AND TECHNICAL INFORMATION DIVISION NAT I ONAL AERONAUTICS AND SPACE ADMINISTRATION Washington, D.C. 20546