lAN IMPLICIT NUMRICAL SOLUTION FOR THE LAMINAR AND TURBULENT FLOWOF AN INCOMPRESSIBLE FLUID ALONG THE AXIS OF A 90—DEGREE CORNER

_ by .David Tillman Klinksiek

Thesis submitted to the Graduate Faculty of theUVirginia Polytechnic Institute and State University

„ 4 in partial fulfillment of the requirements for the degree of7

DOCTOR OF PHILOSOPHY’

in 7 7. A —

4 ' Mechanical Engineering

APPROVED: ,

H. L. Moses, Chairman .

gg _ ( n.CR.

A. Co pÄ}1¤ E. F. Brownl

es@/3/eeC„__,„,<lC_. . /74 ,R. K. Will g C. B. Ling

7August, 1972

_ ·l

Blacksburg, Virginia

L Ll ii „

ITABLE OF CONTENTS

EssACKNOWLEDGMENTS ......... . .......l....... iiLIST OF FIGURES .......... . . ............ v

l LIST OF TABLES . ........................ viiTABLE OF NOMENCLATURE viii

I. INTRODUCTION ...........7 ........ 1 tII. REVIEW OF LITERATURE .................. 4‘ lA. Investigations of Laminat Corner Flow 41 B.· Investigations of Turbulent Corner Flow . . . . . . . 6

~ III. ANALYSIS 10A. Governing Equations for Laminat Flow ........ 10l

1. Corner Region Equations ............. ll ·2. Two-Dimensional Boundaty Layer Equations ....12B.Governing Equations for Turbulent Flow . .-..... 13n

C. Eddy—Viscosity Model . . . . ............ 15D. Finite—Diffetence Formulations g........... 17

l. Finite—Difference Approximations for the Two-Dimensional Laminat and Turbulent Boundaty Layer. 17

y 2. Finite-Difference Approximations for the CornerRegion Laminat and Turbulent Boundaty Layer . . . 18

E. Boundaty and Initial Conditions ........... 20

iii

Aiv

TABLE OF coutsuws - continued O

EssF. Solution Technique . . ................ 26

1. Laminat Case ......... ·.......... 262. Tutbulent Case .................. 273. Step Size and Grid Spacing 28O

IV. RESULTS 30. A. Laminat Flow 30

B. Tutbulent Flow . . . . ...........A..... 42V. CONCLUSIONS AND RECOMMENDATIONS.............. 55O

VI. BIBLIOGRAPHY ......l. 0......... l....... 57VII. APPENDIX A . .....................W. . 60VIII. APPENDIX B .„....................... 90

vtm............................. 135

LIST OF FIGURES

Figure 4 ggg3.1 Two—Dimensional Finite—Difference Grid ......... 19M 3.2 Corner Region Finite—Difference Grid .....p..... 213.3 General Geometry of Corner ......... _...... 2223.4 Enlargement of Plane 1—2—3—4-5-1 from Figure 3.3 .... 234.1 U-Velocity Profile Compared with Blasius Solution . . . 31

4

4.2 V—Velocity Profile Compared with Blasius Solution . . . 324.3 6, 6*, and 0 Compared with Blasius Solution ...... 334.4 Skin Friction Compared with Blasius Solution ...... 354.5 U—Velocity Profile on Bisector Compared with Dowdell,Carrier, and Rubin and Grossman Solutions . ...... 364.6 U—Ve1ocity Profile on Bisector for Various Values ofReynolds Number ....... . ............ 384.7 „Average Continuity Equation Residual for VariousV Values of Reynolds Number, Laminar Flow. . . . ..... 39 °

. 4.8 Skin Friction Along Y = 0 Wall in Corner Region .... 404.9 Streamwise Turbulent Velocity Profile Development'Compared with the Law of the Wall ........... 434.10 6* and 0 Compared with a Momentum Integral Analysis . . 44

- 4.11 Skin Friction Compared with Ludwieg and Tillmann.0 0 0 0 0 0 0 I 0 I 0 0 0 I I 0 0 0 0 0 0

04.1210-Velocity Profile Development Compared with Bragg'sData in Two-Dimensional Region . ............ 474.13 Ü¥Ve1ocity Profile on Bisector Compared with Bragg's _• • • • • • b • • • • • • • • • • • • • • • • • ••v

2_ vi

LIST OF FIGURES - continued

Figure V lEäää

4.14 ÜÄVe1ocity Profile on Bisector Compared with Bragg'sI I I I I I I I I I I I I I I I I I I I I I I I Q

I4.15ÜÄVelocity Profile on Bisector Compared with Bragg'sI I I I I I I I I I I I I I I I I I I I I • • •

•4.16U¥Velocity Profile on Bisector Compared with Bragg'sQ Q I I I I I I I I I I I I I I I I I IV I I I I•l

4.17 Average Continuity Equation Residual for Various 'Values of Reynolds Number, Turbulent Flow ....... 52A.l Grid Used for Representation of g'(x) and g"(x) .... 62A.2 Variable. Grid Spacing Used for uy, uyy, uz, and uzz. . 64A.3 Grid Used for Finite—Difference Approximations forContinuity Equation in the Corner Region ....... 85A.4 Corner Grid ................ .2

..... 87 .

LIST OF TABLES ß

A Page· Table l4.1 u, v, and w on and near the Biaector, Laminar Flow . . 414.2 u, v, and w on and near the Bisector, Turbulent Flow . 53·

vii 1

l, TABLE OF NOMENCLATURE

A · van Driest's damping factor

Matrix coefficient e

Bn(i),Bn(j) Matrix coefficient ‘

. Cn(i), Cn(j) Matrix coefficient_ Cf Skin friction

1 1

D¤(i),Dn(j) Matrix coefficientDij Rate—of-strain tensor for laminar flowEli Time average rate—of-strain tensor for-turbulent flowFi Body force

3‘

H Dummy variable ·HS

1Boundary layer shape parameter, 6*/6

J Defined by equation 3.16l

—L Nondimensionalized mixing length for two—dimensionalboundary layerLC Nondimensionalized mixing length for corner region ”

Lp Plate length and reference length .V P.Time average pressure

Qo Reference velocityQi Velocity vector for laminar flow with componentslU, V, and Wöl Time averagg_vg1oc1ty;yector for turbulent flow withcomponents U, V, and WRex n

Reynolds number based on length from leading edgeRee Reynolds number based on momentum thickness

viii l

ix

ÜTABLE OF NOMENCLATURE — continued _

U,V,W * nVelocity components for laminar flow-U,-V-{W- Time average velocity components ÜUm,V„,W; External velocity components for laminar flowÜ;,V;,W; Time average external velocity components for _turbulentflowXi

Displacement vector with components X, Y, and ZX,Y,Z

ÜDisplacement vector components _ _Ü

i Index for y directionj

ÜIndex for z direction

n Ü Index for x direction Ü Ük Upper bound on i index within the corner region ·m

ÜUpper bound on j index within the corner region

2 Mixing length: two—dimensional boundary layer Ü .Ü

RC Mixing length: corner regionr1,r2,r3,r4 Defined by equations B.2a—b and C.2a-b in Appendix A

81,82,83,Sk, ·S5 S6

S B Defined by equations B.2c—d, C.2c-d, D.2a—d, E.3a-d,9° 10° 11° and E.7a-b in Appéndix A ‘s ,s ,s12 12- lk

u,v,wÜ

Nongdmgnsionaldzed velocity components correspondingto U, V, and W and U, V, and WÜ,v,w Nondimensionalized space average velocity componentsV in finite—difference grid¤*„v*„w* Half-step nondimensionalized velocity components in· finite—difference grid -

TABLE OF NOMENCLATURE — continued

4us Shear velocity, (Tw/p)1/2

U+ .., .u U/us

. x,y,z Nondimensionalized displacements corresponding to„X, Y, and Z ‘

Ax.Ay,P_

A Az Finite—difference increments. At a given node, the -‘y+’ +’increment with a + subscript is always in the positiveAz_ · direction to the next node and greater than or equalto the negatively subscripted increment.

Y+ Yus/V · 4 46 — Bpundary layer thickness defined as positive where· U/Um -4.99 in the boundary layer · ._6l Two-dimensional boundary layer thickness parameter

4. used ig_mixing length model and defined as position —where U/Um - .999 in the boundary layer

62 Boundary layer thickness parameter used in cornerc region mixing length model6* Displacement thickness defined by equation 4.l _ '6 Eddy-viscosity

En Nondimensionalized eddy·visco6ity8 Momentum thickness defined by equation 4.2p Dynamicviscosityv

Kinematic viscosity .4p Density „Tw _Wall shear stress

~ ·V

I. INTRODUCTTON

The growth and general characteristics of two-dimensional laminarand turbulent boundary layers have been analyzed for many years by ethe momentum integral method.- With the advent of the digital com-

l

puter, numerical techniques for the analysis of two—dimensionalboundary layers have become possible.

The standard momentum integral technique used to evaluate laminarboundary layers is the Karman-Pohlhausen method. If the pressuregradient is too severe, Launder's method of dividing the boundary-layer into strips can be applied with success. For either of the

’methods mentioned no empirical input is necessary other than thevelocity profile assumption for each method. If a numerical technique ~ _is used, the governing partial differential equations aresolveddirectlywith the application of the appropriate boundary conditions.No assumption about the Velocity profile is necessary since the pro-file will be predicted by the solution technique.

A momentum integral method for the evaluation of turbulentboundary layers requires a shear stress correlation in addition to aVelocity profile model. When a numerical technique is used toanalyze a turbulent boundary layer, a model for the shear stress,or turbulence mixing, across the boundary layer is required.· Thus,unlike the analysis of laminar boundary layers by a numerical method,the evaluation of turbulent boundary layers by a numerical methoddoes require an empirical input for the shear stress across theboundary layer. . _ V

A1

2

Extending this discussion to three-dimensional laminar and turbu-lent boundary layers, additional empirical information is required to .solve the three-dimensional flow field by a momentum integral method. VFor either flow classification, a model for the transverse velocityprofile is necessary and, in addition, for turbulent flow, a shearstress correlation is required. To solve the three-dimensional flow lfield by a numerical method, no empirical input is required forlaminar flow. However, analysis of turbulent flow by a numerical ‘method again requires a model for the shear stress within the flowregion.

The flow of a fluid along the interior axis of a corner formed bytwo planes intersecting at ninety degrees is a three—dimensiona1 flowphenomenon which results from both walls influencing the flow fieldnear the intersection of the planes. Far removed from either of theplanes, but close to the remaining plane, the flow field is two- l U

- dimensional.I

l — nThe analysis of the flow field close to the corner region by a .

momentum integral method can be accomplished if a suitable model forh

the velocity profile can be established. For the laminar case, trans-_* verse velocities exist, but according to the argumcuts found in the

l1

literature, a closed vorticity pattern is not established or has notfbeen experimentally observed. If the flow is turbulent, closed vor-Tticity patterns have been observed for non-circular ducts. The fluidflows along the bisector toward the corner and then out along thewalls to the mid—polnt of the duct and back into the free stream. _

Thus a secondary flow is superimposed on the free stream.This investigation has adopted a numerical method which was used

to predict the three components of velocity in the corner region for_ either laminar or turbulent f1ow.p A model for the shear stress

across a two-dimensional turbulent boundary layer was modified forthe corner geometry when turbulent flow was evaluated. This numericalmethod gave good results for laminar flow when comparisons were madewith other analytical solutions which were obtained by different lnumerical techniques. This numerical method also produced good re-sults when a comparison was made with the one experimental work foundl ’ in the literature for turbulent flow.

The numerical method developed in this investigation is sufficient-ly general so that adverse pressure gradient flows and developingflows in enclosing ducts can be analyzed. However, to perform theanalysis, some minor revisions must be made in the numerical method.

_ V II. REVIEW OF LITERATURE

A. Investigations of Laminar Corner Flow '~

The first attempt at an analytical solution to the problem ofl

flow in the corner region was by Carrier (1). His solution was ob-tained by solving onlytthe streamwise component of the Navier-Stokesequations in conjunction with the continuity equation. Asymptotic

· series were used to represent the solutions for u, v, w, and p. Thesolution for the u profile was taken to be a combination of the Blasiusprofiles, which should exist on the two surfaces at points far removedfrom the corner, plus a function which has influence only in the

‘ corner. Results are given only for the streamwise isovels in the qcorner region. _

Due to criticism by Kemp (2), Dowdell (3) investigated the effectthat the neglect of the crosswise components of the Navier—Stokes ‘°

equations would have on the streamwise velocity component in thecorner. Further, Dowdell (4) also obtained a system of equationsamenable to the solution of the flow field along the intersection ofthe planes at any angle. His analysis indicated that a 4 percentdifference occurred between his analysis and that of Carrier (1) inthe case of planes intersecting at ninety degrees for the mainstreamvelocity component in the corner region. Dowdell's (4) analysisfurther indicated that a flow reversal occurred deep in the cornerregion for planes intersecting at small angles.

‘*

5

Rubin (5) discussed the above references and a method of solvingthe flow field in the corner. He divided the flow into three regions:an irrotational flow region, a two-dimensional boundary layer on eachsurface far removed from the other surface, and the corner region _where both surfaces influence the flow. Solutions for the potentialflow and the two-dimensional boundary layer were obtained as asymptotic

A

expansions which utiliaed the small perturbation parameter (v/2Uax).These solutions were then matched at the intersecting boundaries ofthe various regions.

H·, 4

_‘”

T Pal and Rubin (6) presented the asymptotic solution to the cornerregion equations. Again, the small perturbation parameter was(v/2UwX) and stretched similarity variables were used for the cross-_flow directions. This led to a system of four partial differentialequations which were then solved by the asymptotic expansion method. ‘_

The expansions were matched to the asymptotic expansions of the two-dimensional boundary layer and the irrotational potential core pre-sented by Rubin (5).

’Rubin and Grossman (7) reduced the governing equations of the

flow in the corner region to four Poisson-like partial differentialS

equations which were solved numerically by the Gauss-Seidel method ofl

successive iterations. The asymptotic expansion of Pal and Rubin (6)was used to establish the boundary values for the numerical analysis.In this series, a single arbitrary constant of unknown value was ·

’ present. This constant was varied until interior mass sources wereA

no longer generated by the numerical method.

4 ASignificant differences were obtained when a comparison was made

- Z ‘between Rubin and Grossman (7) and 0arrier (1). The results of Rubinand Grossman (7) were in some cases 25 percent greater than the resultsof Carrier (l) for the streamwise velocity component along the corner

. bisector. Also, the isovels of Rubin and Grossman (7) did not have thesharp curvature that Carrier (1) isovels indicated. The skin frictionfrom the corner to the two—dimensional boundary layer region wasfound to increase monotonically from zero at the corner to itsasymptotic values at the two-dimensional boundary layer.

4

° B. Investigatlons of Turbulent Corner Flow

g Most of the work attempted in the turbulent flow along a corner _has been in the fully developed region of square and rectangular ducts.Nikuradse (8) was the first to study this type of flow experimentally Rand observed that the mainstream isovels were displaced toward the

_ walls at the corner and away from the walls at the mid-point betweenthe corners. Gessner (9) has presented detailed experimental resultsand analysis of the turbulence and mean flow characteristics of fullydeveloped flow in a rectangular channel. —

g Bragg (10) presented experimental results and attempted to1analyze the turbulent boundary layer along the corner. He used amomentum integral technique in which a streamwise simplified Navier-Stokes equation was integrated across the corner layer in conjunctionwith the continuity equation. . q

7In

order to integrate the momentum equation across the corner_.j af

layer, the streamwise velocity profile must be specified as a functionl f:of both y and a, which are the cross—flow coordinates. In the laminar

sub-layer, Bragg defines a velocity scale UC = (v ägäg] so that

z=0the resultant form of the velocity profile in this region is U/UC =(Ucy/v)(Ucz/v). This form is very similar to the u+ ¤ y+ currently inluse to describe the laminar sub-layer U-profile in two-dimensionalboundary layer theory. The outer U-profile was assumed to be similarto the classic law of the wall correlation using a shear velocitybased on properties of the flow along the bisector and the cross—flowcoordinates. » 7

The wall shear stress was assumed to be correlated as a function _of the wall shear velocity far removed from the corner (two—dimensionalVflow) and the distance from the corner. This correlation seemed tohold until a position very deep in the corner region was reached.Another velocity scale determining shear stress in this region wasAdeveloped and used in the correlation for wall shear stress deep inthe corner region.

The comparison of experimental results with the various models ‘

for mean streamwise velocity profiles and wall shear stress indicatedthat fairly good agreement was obtained with no pressure gradient and

I

poor agreement was obtained with an adverse pressure gradient. How-ever, this comparison was only made for the corner bisector.

I

rl

Toan (ll).also attempted an analytical solution to the turbulent 1

·flow along a ninety-degree corner. The mainstream momentum equation,csgecuer with the continuity equation, was solved assuming that a

_ modified Ludwieg-Tillmann shear stress equation was valid and that the aa 'T- 'mainstream velocity p§¤£11e could bg represented by a power law.

Results were unsatisfactory since the influence of the secondary flow·on the mainstream velocity was not taken into account. An improvementin the mainstream velocity profile representation was attempted byintroducing a new parameter which wa: an indication of the intensity

V .of the secondary flow. Also, it was assumed that the shear stressalong the wall in the corner was constant from the work of Gersten

" (12). These assumptions resulted in better predictions of the main-stream isovels but still were not satisfactory since the curvature of jthe isovels in the vicinity of the bisctor was not duplicated. V

_ Another method was presented using the isovels as one coordinate andtheir orthogonal velocity gradient lines as the other coordinate.This method was presented only for a qualitative study of the second-

Vary flow because of the difficulty in predicting the isovels. Toanalso mentioned Bragg's (10) work and stated that Bragg's hypothesis»(for the velocity scale required that the isovels be hyperbolas.

Veenhuizen and Meroney (13) presented measurements of the primaryand secondary flows in the developing section of a square duct whichwas designed so that the pressure gradient was zero. Also, the turb-lence distribution was measured at various points in the duct. Re-sults indlcate that the primary and secondary flow patterns and theq

.

turbulence distribution are similar to that found in fully developed ‘duct flow investigated by Gessner (9). No attempt was madeto solve the momentum, vorticity, and continuity equations.

Launder and Ying (28) presented experimental and analytical re-sults for fully developed turbulent flow in square ducts. Their _ ‘analysis indicated closed vorticity patterns that agreed reasonablywell with the experimental results. However, a coarse grid spacing(ll x ll) was used which made it necessary to fit the mainstreamprofile with a semi-log correlation.

III. ANALYSIS .

Before the problem of turbulent flow along the intersection oftwo right angle planes was solved, it was advantageous to solve the

laminar flow case and then add an appropriate approximation for theReynolds stresses. The corner flow problem was solved using theAlternating Direction Implicit (ADI) f1nite—difference technique. Useof this method resulted in the generation of many tridiagonal matrixeswhich were solved for a half step in the x—direction, stepping implic—itly in the y—d1rection and explicitly in the z—d1rection. Then, forthe second half step in the x—d1rect1on, stepping implicitly in thez—direct1on and explicitly in the y—direct1on.

13 '

A. Governing Eguations for Laminar Flow

The equations of motion and continuity for the laminar unsteadyflow of an imcompressible fluid are given by Schlichting (14):

1 BP B¤6r"äa*+ä<“ °j1> +F1·1 1 A 1 ·= 1,2,3j = 1,2,3 (3.1)

where D/Dt is the substantial derivative, and „ .

Bil = O. j =192931

1 3 · A 10

_ 1. Corner Region Eguations ’ .For the case under investigation, the flow was steady and there

were no body forces. It was further assumed that the second deriva-tive of any velocity component with respect to the x—direction wasmany times less than with respect to the y and z-directions. And,since the flow was over a surface similar to a flat plate, thepressure gradient in all three directions was also negligible.

Thus, the equationsof motion 3.1 were reduced as follows withU

—the continuity equation 3.2 remaining unchanged: A2 28U 8U 8U 8 U 8 Uv( 2 + 2) , V (3.3a)8Y 8Z Y .V av av fav azv azv YU-—+v—+w—=v[—-—+—-—) ,a¤d (3.31:)Y8X8Y az 2 2. BY az

Y aw aw law azw azwu—+v——+w——=v(——+—-) . (3.3c)ax BY az 2 28Y 82

The equations 3.2 and 3.3a-c were then non-dimensionalized by—the following substitutions: —

kU = uUw , '

(3.4a)-1 2 1

V · v(LP/Umy) / , V (3.4b) VU A * -1 2 » ·Vw . W ¤ w(LP/Umy) / , (3.4c)X = xLp , ‘

(3.4d)U Y · 2 , (6.6a)and Z = z(vLp/U°°)l/2 . Y (3.4f)

12 _

· Substitution of the non—d1mensional quantities of equations 3.4a-f into equations 3.2 and 3.3a-c yielded the following results:

uux + vuy + wuz = uyy + uzz , _ _ (3.5a)

uvx + vvy + wvz = vyy + vzz , ~ (3.Sb) .uwk + vw& + wwé = wyy + wéz , and (3.5c)

. ux +2vy + wi = O , (3.5d)

where the subscripts x, y, and z denote partial derivatives of thequantities u, v, and w with respect to the subscripts, i.e, Bu/Gx =· ·u O I A I

'xAfter the equations 3.5a-d were cast into finite-difference form,

there resulted three unknowns and four equations because of theassumptions applied to equations 3.1 and 3.2 that all the pressuregradients were zero. Thus, it was decided to use the continuity .equation as a check of the solution technique.

A brief review of equations 3.Sa—c revealed that these equations‘were in essence an initial value problem for the xédirection and aboundary value problem for the y gnd z—directions. This type ofpartial differential equation can be readily solved using the ADInumerical method. ‘

2. Laminar Two-Dtmensional Boundary Layer 2 2A

For the analysis of the two—dimensional flow along the Y=0 andZ-O walls far removed from the corner region, the standard boundarylayer assumptions were applied to equations 3.5a and 3.5d. The

C U13

result of these assumptions yielded the following governing equationsfor the flow in the two—d1mens1onal boundary layer along the Y¤0 wall:

uux + vuy ¤ uyy _ (3.6a)' and u + v ¤ 0 . (3.6b)X Y

B. Governing Eguations for Turbulent Flow ·

The equations of motion and continuity for the turbulent flow of l

an incompressible fluid are very stmilar to the laminar case as givenby equations 3Ll and 3.2 except that the Reynolds stresses due tovelocity fluctuations must be included. In the following equations,‘time averaging has been used and the body forces neglected:

D6 —1 SP S · —- - ————-Dm “1axi+°äi'; (D D11 ' °D1D1)·1-1,2,3 _ D

__ - j=l,2,3 (3.7)SQIand -——- = 0 . 1=1,2,3 (3.8)SX1 _In equation 3.7, D/Dt 1s the substantial derivative. D

The difference between equation 3.1 and equation 3.7 was theaddition of terms representative of the Reynolds stresses. These _additional terms rendered the equation 3.7 unsolvable without further-information. However, this problem was circumvented by use of thehypothesis advanced by Boussinesq (19) for these stresses, which re-sulted inaareduction 1n the complexity of the equation 3.7. His

1 hypothesis 1s given by the following equation:

— • éön. 3.9

14

Substitution of equation 3.9 into eguation 3.7 yielded the following; results: ’ -

I { ^E-

_D6 ——. 36 o1

j · .ß _i j-1,2,3 (3.10)‘Assuming that the flow is steady, the pressure gradients are zero

or negligible, and the second derivative of any variable with respect· to the x axis is many times smaller than derivatives with respect to

the other coordinates, the following three equations evolved fromlequation 3.10 and were used for the flow field in the corner region:

-8-Ü -633 -65 6 BU 6 GF (3.11a)9

.-63 —8”V—~—8_V-_ 6 Q;7 _6·UV + w az [gv+s)8J + az [Ev-I-e-)8;| , and (3.11b)-63 -6:3 +6°:3_ 6 BW 6 6:3 (3.11::)

With the addition of the definition,

y(

s = v sn , (3.12)to equations 3.a-f (the various velocity components are time averaged),the turbulent momentum equations were non-dimensionalized into theAfollowing form:

I”

Z uux + Vuy + WUZ ‘= [(1+€n)l1y]y + [(1+€¤)t1z]z ,41

Wy + WZ .. [(1+gn)vy]y + [(14-g¤)Vz]z _„ land (3.131:) Vuwk + vw& + wwz ¤ . (3.13c)The non+dimensionalized continuity equation, 3.5d, remained the

same and is repeated here for clarityz

15 A ’

ux + vy + wi = 0 (3.5d)

For the two—dimensional boundary layers which were allowed to _l

develop along the walls far removed from.the corner region, the y andhz momentum equations (equations 3.13b and 3.13c respectively) werezero and the cross flow components, either v or w, depending on which

l

wall the flow was developing, were zero. Also the second derivativeIwith respect to the coordinate parallel to the wall was zero. Thus,to solve for the developing two—dimensiona1 flow along the Y=0 wall,

l

the following equations were used to calculate the flow field:uux + vuy ¤ [(l+s¤)uy]y (3.14a)

and Uk + vy = 0 •l (3.14b)

C. Eddy-Viscosity Model T .

A suitable eddy—viscosity, €, model was required for the solutionof the equations 3.13a-c and 3.14a-b. A literature survey revealedlmany eddy—viscosity models for two-dimensional flow. The one model

„ which has gained widespread acceptance was that proposed by Prandtl(20). This model in which the eddy-viscosity is formulated as afunction of a mixing length and a velocity gradient is given by

2d.Ü (s Z . (3.15)

For three-dimensional flows, Prandtl (21) suggests the followingreplacement for the term Iggl in equation 3.15:

16—- 2 —— 2 - 2 - -V 2 ( —- —- 2 ·V J VZ) + (UZ + W;)__ __ 2 1/2

V + (Vx + ny) 1 . (3.16)If the usual two-dimensional boundary layer approximations are applied .6 aüto equation 3.16, J reduces to E? .

V When the usual boundary layer order of magnitude approximationsare applied to the various derivatives in equation 3.16, the express-ion for the eddy-viscosity model in the corner region becomes1 1/2 (-· 2 -— 2 ‘6 2c2[(Uy) + (UZ) ] . (3.17) V V

The eddyzviscosity can be written for the two-dimensional regionnon-dimensionally as "2 16u - L uy 1 (3.18a) ·and for the corner region as

6 - L zu >2 +< >21l/2 <6 16L>n C uy uz , .1 where V2 22 Ü·wL 3/2 VQVCZ ÜLOL 3/2L L = —— —-E 1 and L = ——— _——P— (3.18c) ·. 2 v c 2 vL L -P P 2 1

From the recent work of Pletcher (22), the following mixing length Vmodel was used for the two—dimensional flow.

ß ·= 0.41 [1.0 - exp (-y+/A)] -1- E- <0.1 (3.l9a)6_ 6 1 62 2 2 _

[17

·X&-= 0.41 [1.0 — exp (—y+/A) EX·‘ ‘2 2

Y 2 3— 1.53506 (E- — 0.1) + 2.75625 (Y/62 — 0.1)· 2A A - 1.88425 (Y/62 - 0.1)4 0.1 j_Y/6l j_0.6 (3.19b)

X&·= 0.089 ' Y/6ß > 0.6 (3.19c)2

6ß is a normal distance from the wall to a value of ÜYÜ; that is 'within the boundary layer. However, the value of ÜYÜQ that should _be used was not explicity stated by Pletcher (22). A parametric studywhich solved for Ülprofileson flat plates was made to determine thebest value of Ü7U„ to use for determining the distance, 62. This .study indicated that the position in the boundary layer where ÜYÜQ= .999 gave the best distance for 62 to use in equations3.19a-c.

The value of the constant A was taken to be equal to 25 since _' Klinksiek (18) has demonstrated that this value gives the best

correlation with the law of the wall constants suggested by Coles (26).For the corner region, the distance 6ß and Y were modified to

conform with the corner geometry. The details of this modification„ . can be found in Appendix A. · [ ‘

— D. Finite-DifferenceFormulations1.

Finite—Difference Approximations for the Two—DimensionalLaminar and Turbulent Boundary—Layer

The Crank—Nico1son (17) implicit finite—difference method was

18 .V

used to analyze the flow developing in the tw¤*di¤¢¤8i¤¤al flat Plata 4region for the following reasons: '

(a) Klinhsiek (18) has shown that the resultant finite-difference equations are stable and convergent. . _ _

.l

(b) Truncation errors are of 0(Ax)2, 0Ay+, and OAz+.(c) Since the finite-difference approximations were implicit,

A.

the resultant matrix was tridiagonal, which allowed the4

use of the Thomas algorithm presented by Von Rosenburg (16)Afor the solution of the unknowns.

_ Figure 3.1 shows the grid that was used to generate the finite—-»

”'ldifference approximations for the two-dimensional boundary layer flow.,

_ The equations were written about the point (i,m+l,n+1/2) for anyvariable H with a variable grid spacing in the y-direction for flow on

V

the Y-O wall. The second subscript, m+l, is the index notation forthe z-position on the Y-0 wall where the flow is assumed to be two-dimensional. The variable grid spacing was used in the z—directionfor flow on the Z=0 wall. Both of the variable grid spacings were l

the same. The variable grid spacing was used to ensure a high con-centration of nodal positions in the region of large velocity gradi-ents. The detailed development of the finite-difference forms ofequations 3.6a-b and 3.14a-b can be found in Appendix A.

2. Corner Region Finite·Difference Approximationsfor Laminar and TurbulentFlowThe

Alternating Direction Implicit (ADI) method was selectedover the Gauss-Siedel method of solving the set of partial differencud

19

1+1 ,m+1 ,11 1+1 ,m+l , 11+1

_ Ay+ ·

r AY_

i__l,m+l, l~

‘ ··1,l11+1,Il+1Ax '*'***·ä

Figure 3.1. Grid for Two-Dimensioual Bouudary Layer ou _Y = O wall. I 1

20

equations 3.5a-c and 3.l3a—c. The reasons for this selection were

l as follows:* (i) The ADI method has been shown to be stable and convergent by

Peaceman. and Rachford·(15). (However, stability and convergence were —

proven only for equal Ay and Az spacings and linear equations).(ii) The truncation errors are of order (Ax)2, Ay+, and Az+.

‘ U

This allows the use of larger Ax lncrements as the solution continues.‘ A

(iii) The finite—difference equations for this type of method are ·· written such that the matrixes formed are tridiagonal, thus allowing

l the use of the Thomas algorithm for the solution of the unkncwns.Figure 3.2 illustrates the grid used to generate the finite·

difference approximations for the corner region. Variable grid spacing ° A

was used in the y and z—directions to ensure a high concentration ofnodal locations close to the wall where the largest gradients were

„ ‘ Nexpected and also to conserve digital computer storage space since .Cthe values of u, v, and w were saved at the initial, intermediate and

·’ the new x—position for each nodal point. The variable grid spacing

was the same for both the y and z—directions. .The detailed development of the finite-difference form of

equations 3.5a-c and 3.13a—c can be found in Appendix A.

E. pßoundary and Initial Conditions

The figure 3.3 illustrates the geometric shape (area 1-2-3-4+5-1)V ”

in which the finite-difference equations presented in the Appendix Awere assumed to approximate the flow.’ Figure 3.4 is an enlargement of

21

T~Y Ei2 I

1+1,5, +1 „ 1 ”

1 aj a¤+l 2_ '

’ ‘ Az_ _ .~\ ' ._ ea ,n+1/2

R G 2 AL Y 2. Y ·

A g -1,j,u+1/2 VX

k 2Y']-aj au 1

Figure 3.2. Corner Region Finite-Difference Grid.l l

· 22

MainstreamOW

5

,«,z1 Corner_ Region „ ·1 Y- Two—Dimensional

2 Boundary Layer

Figure 3Q3. General Geometry of Corner.A

.

- F 23

Y »_. _

l

Two-l

DimensionalBoundaryLayer

_ 2 3 ·Biaeetor

4 VTwo-Dimensional ‘

Corner ‘ Boundary LayerRegion _

5 Z

Figure 3.4. Enlargemenc of Plane 1-2-3-4-5-1 from Figure 3.3.

· „ 24

the plane l-2-3-4-5-l to facilitate the following discussion of thecorner region and the boundary conditions associated_with this region.

Along the Y=O wall from l to S, the boundary conditions imposedwere that all velocity components, including the half step components,were zero, i.e., no slip along the wall and no flow into or out of

l

the wall. A similar condition for the velocity components was imposedalong the Z=O wall from l to 2. The line 4-5 was assumed to be the °boundary between the corner region and the two-dimensional boundaryllayer which was developing along the Y•O wall. Thus, this line 4-5

° was’assumed to be one of the boundaries for the corner region. Thevelocity components along this boundary were further assumed to followthe two—dimensional flat plate case and these velocity componentswere calculated numerically at each x-step. Thus, the u and v _velocity components were specified by the development of the boundarylayer along a flat plate and the w-velocity component was assumed toAbe zero along this boundary, i.e., no flow from or into the cornerregion across this boundary. Similar boundary conditions were estab-lished along line 2-3. On this boundary, v-O, and the w component4was equal to the v component along the line 4-5. The u components on

. line 2-3 were equal to the u coponents on line 4-5. . _Along the lines 2-3 and 4-5, the boundary values for the half '

step velocity components were determined as follows. At each nodal.2

position, the u, v, or w velocity component at the old x-position was4

summed with the respective u, v, or w component at the new x-positionU

and divided by two.S

. —

_ 25

Along line 3-4 it was assumed that the corner region was mergingsmoothly into the irrotational free stream. Boundary conditions wereset along this line as u„= uw, w - ww above the bisector, w°= 0 belowthe bisector, v = vw below the bisector and v = O above the bisector,Uwhere vw and ww were obtained from the numerical two-dimensionalboundary layer calculations.

Another very important consideration was the position of thelines 2-3 and 4-5 with respect to the corner. The location of theselines had a definite influence on the results obtained for the flowfield in the corner region. These lines were arbitrarily fixed at Athe beginning of calculations a given position from the corner. Aftera certain number of steps in the x-direction, the slope of the u-iso-vels at lines 2-3 and 4-5 at each nodal position with respect to thecoordinate perpendicular to these lines was determined. If theresults were greater than the slope of the u-profile in the two-dimensional boundary layer at the position where U/Uw ¤ .99, thelines 2-3 and 4-5 were re-positioned further from the corner and the

l

corner region flow field recomputed. ” A

The initial conditions were applied at the leading edge of theplates which formed the ninety-degree corner. For the case underinvestigation, the leading edge was a singularity at which point uwas one and v and w were infinite as the boundary layer approximationsindicated. However, for this investigation, v and w were assignedvalues of zero and u was set equal to one since in reality the v and 4

lw velocity components at the leading edge should be small if not,indeed„ zero.

· 26

F. Solution Technigue

1. Laminar CaseV

Because the equations 3.5a-c and 3.6a were non—linear, it wasnecessary to develop a method of linearization which is presented inAppendix A. The method required that the values for the quantities u,v, and w be assumed known at the new x—position. Since the abovementioned quantities were the variables to be determined at the newVx—location, an iterative procedure was used to ensure that the assumedlvalue and the calculated value of the u, v, and w quantities at thenew x-position would agree.

The following is an outline of the procedure that was used tosolve equations 3.5a-c and 3.6a—b:

V ‘·4

(a) Project all values of the variables u, v, and w from theold x—position (i,j,n) to the new x-position (i,j,n+l) forall values of i and j.

(b) Determine the values of u, v, and w in the two—dimensionalboundary layers by solving the finite—difference equations

Ü

presented in Appendix A for equations 3.6a—b.l

(c) Determine the values of u, v, and w in the corner regionby solving simultaneously the finite—differenceequationspresentedin Appendix A for equations 3.5a—c. AThe solutionwas performed three times, each time using the newly cal- _Vculated values to linearize the equations.

(d) After a certain number of steps were taken in the x—direction,V

the slope of the u-isovels at each nodal position along

27l U

6lines 2-3 and 4-5 were moved one nodal position furtherfrom the corner and steps (c) and (d) were then repeated.

(e) Repeat steps (a) through (d) for each new position in the6x-direction.

Due to the numerical methods used, the resultant form of the -finite-difference representations of the equations 3.5a-c and 3.6a ” ’

always produced matrixes of unknowns in a tridiagonal arrangement.The solution for the unknowns was accomplished by the Thomas algorithmpresented by Rosenburg (16). A description of this technique can be”found in Appendix A.

.2. Turbulent Case6

The method used to linearize the turbulent flow equations andthe reason for iterating these equations are the same as given in‘ section III. F.1.

1 _·— . ‘¥ · .In addition, the eddy-viscoslty had to be determined at the new .

x-position. The eddy-vlscosity is dependen:<x1the u-profile at the V ·new x-position since the position of ÜYÜQ ¤ 0.999 in the boundarylayer determines the value of 62. Therefore, a check was introducedto ensure that the assumed value of öß at the new x-position was

_ _ within j;l percent of the resultant 62 when the iteration of the two- ·dimensional flow equations was completed.

l 6l

The following is an outline of the procedure that was used to _solve equations 3.14a-b and 3.13a-c: · ·

l

(a) Project all values of the variables u, v, and w from . ,the old x-position (i,j,n) to the new x—position (i,j,n+l)

28

for all values of 1 and j. _(

(b) Determine the values of u, v, and w 1n the two—dimens1onalU

bpundary layer by solving the finite-difference equationsa presented in Appendix A for equations 3.14a-b. A check was

performed to ensure that the assumed value of ÖL at the newx—pos1t1on was within jll percent of the calculated öl valuewhen the iteration of equations 3.14a-b was completed.

(c) Same as step (c) of the laminar case. ·l(d) Same as step (d) of the laminar case.(e) Same as step (e) of the laminar case. -

- The resultant form cf the f1nite—d1fference representation ofequations 3.13a-c and 3.14a always produced matrixes of unknowns in _ äsa tridiagonal pattern. Again the Thomas algorithm was used to solvel

_ for the unknowns. g V

3. Step Size and Grid Spacingl

_ For bnth turbulent and laminar flow, the calculations werestarted at the leading edge and Ü;/V was approximately 6.25 x 10-7

sec/ft. The x—1ncrement size was assigned the physical d1mens1on of0.001 inches and was held at this value for the first twenty steps. ”·1After the twentieth step was calculated, the x-increment was allowed Vto increase at a rate of 10 percent of the previous x-increment size.IHowever, the x—1ncrement was never allowed to exceed the boundary „·layer thickness.

1The same spacing for the grid increments was used for the y and

z-directions. The first five increments from the wall were assigned ·

29 ~

~ equal values of 0.001 inches. The size of each of the next 15 incre-ments was increased in succession by 5 percent of the previous incre-ment size. For the next 20 increments, the percent increase was 10percent. All the remaining increments were increased by 15 percent.

l _ IV. RESULTSU

A. Laminar Flowl

'

The two—dimensional u and v—profi1es predicuaiby this method arecompared in Figs. 4.1 and 4.2 with the Blasius solution. As thesefigures indicate, the method was accurate in predicting the velocityprofiles but the prediction of the v-profile was not accurate untillReynolds numbers of over 10,000 were reached. This discrepancy wasattributed to the truncation error of the finite-difference equation

_ that was used to represent the continuity equation 3.6b.

ll Figure 4.3 illustrates the boundary layer thickness predictioncapability of this method for the tw¤~dimensional case. Inaccuracies _occurred until the Reynolds number was greater than 1000. The in-

1

accuracies of the magnitude depicted by Fig. 4.3 near the leading edgewere expected because of the initial conditions applied.

Figure 4.3 also demonstrates the accuracy with which the methoddcan predict the displacement and momentum thicknesses. These thick-nesses were obtained by integration of the u—velocity profile by thetrapezoidal rule in accordance with equations 4.1 and 4.2, which ‘l · define the displacement and momentum thicknesses respectively. _

5 .6* = [ (1 - dy _ (4.1)

· ‘ 9·= [ ü·(l - U/U“Qdy (4.2)0

' . 30 —

31

<rO kho ~¤V . •-I «-1 E3 Q Q

_ * u u u ~¤ C:N N N 1 OGI 0 CD •r·I

V M MMII I 1-I

° <1 0 ‘gA Ü Ö U3<r • :1

GII ·¤I °U r-I — PI _G tu MQ

•-G‘ Üw 54 uOM-• q; ••-IQIQI°•¤I-• Q.

.[-gs.! Z , •·g‘* 3

° ‘ Hä“~. U

{gg• >-•>¢U1 _ Q G

·_:

Q 6 Ԥl

E•U2‘ C: ev· 4 *7 {

. D

* 4U <.·

U‘ . o 1-•U :'”' 2PI g ;:.,g

I •Q

Q c~ 00 r~ OO 0 O U

H Ö•-1 c o c ce ca o c> o o 6:8\D

‘32

I V "4

» Q In 0 O _O O O • _~

I •-I •-—I •—-I 0‘ • ' II II VII4N N NU U G) •V M M M G ·

‘ ;*°O

‘ · ’I VI I 1-IJ,„ <I III O C? 3I - Q ‘ In ON rn0 U1U ¤-I 5G W! ••-IIII U mQ ‘ >~• I-I •¤-I

I6In GI I—• •-IO II-I GJ M —¤.• aa E c>EE M :3 • J3v ZQ44 pe 3· H• 2 v 4

"°‘ 3.4

E° 8I M Q- ¤·-·I

·r-I. '+-I —I .‘‘‘8 Ä

54 o · -84

U. >

' , A. Q •Q N

4 . 4¤-4GIH

äh4 ·•-I

| OV OO GS Q IN O In Q Q N ¤-—I O•-1 o o c c o o o o V c: c:X oo

SEA ll 4

33 A -

°' •-•. H

¤I 5•-J g _ · .I >·• 0 E ‘H U 54 GI R! uu"U vg. · 5O ••-I O.4 an cn :2

· I I I IA , 4. . O H

°¤~ EG) U QIJ C°• r-I In · ugE-! 0 GI CD

H U GI Q0 ·•-4 ct A 5IH HM ·•-1ID gt) QIvzZ E-• · pn

I 4 Q "'¤ A -5•-I ~«·-I$ .

. •·¤ IQ)äChnig § I

— N<D"U

I { O Qy-| 6

-8IO

6«§QGIHN 5O 00— •·-I ••-IIn•-I

O · .O

<¤¤¤1¤¤r> 6 v¤¤ ‘«=6 ‘s>A

344

. As is evident in Fig. 4.3, the method accurately predicted thedisplacement and momentum thicknesses after the Reynolds numberreached a value of 1000.

Figure 4.4 compares the predicted skin friction coefficient with‘ theoretical, and again excellent agreement was achieved when the

Reynolds number was greater than 1000. The skin friction was cal-_

culated by equation 4.3. 4 4

cf ·· 21:w/pU„2 7 (4.3)

The shear stress at the wall, tw, was obtained by using the valueof u at the first station away from the wall, dividing this value by ‘ -the normal distance from the wall to this station, and multiplying bythe dynamic viscosity. This calculation is given by equation 4.4.

lltß Ä uU(2,m+1,n+1)Aywa11 (4.4).

._Figure 4.5 illustrates the results of this numerical technique

_ for the corner region when compared to the techniques advanced byCarrier (1), Dowdell (4), and Rubin and Grossman (7). Only the numer-‘ical results at a position sufficiently removed from the leading edgeare used for comparison. This method compares favorably with the re-4sults of the other investigators of the laminar corner flow problem.4Deep in the corner, all of the methods seem to agree with discrepanciesdeveloping as the free stream is approached. This method predicts alarger increase in the u-velocity as the free stream is approached

‘ than e1:her of the previous methods. ’

Q 35

' O„ O:-4

<1'N

8ÜA

1 tn 6::ct u -4 ¤>~•w -1-4 °¤·4$-44-4 $-4 ·UOw·w 5wß-Jg :-4gw O

M Z V)O

VJD-:-4

E92' <' •-G

4 O 4-•:-4 •:·-4

. 3 ·QU .wH .2'L

7 E. \ O ·. __ I8 uE ß'Q ,‘ -:-4JJ

M U1 _ *g

:--4 lr-:G•:-4MU)

J<" .wH:3N OD

O -:-4 .. -4 Lu

•—• :-:¤oOQJ

c ' QO ¤¤'F1¤I1„:I UPIS

‘ _ 36

O_ N‘ rsN

. . 1 " gQ „„ cs -5c ä ·¤,-4 5

ac °°MI 3 · O ·¤$@0 Q ~o 81 •·-|•\/ sxeuoo '° H1 U Hä u °’

·•-lll •-·n eu ‘•'*M 0 ¤ ·-• QÜ Nt °H H {US0 3 .0 u O¤c¤ 0 :s cuZ OZ U' Qu n n • ' „§ Z}1 0 [340 gg. U

g 8 Q·•-I4 ¤ Q ¤G)

. g g' x fg0 .Q|·~· Q>-•OG |>< HEJ <= Q, ° UM U9 ~In

OGOOH0 ,,3Ü N mm- O’ ‘ H<*0 “‘”’

_ U1 4 6 Q · '5*°°

OS'% O -10‘ ·” DU

1 AO U

° H. O goC2 °Ö de 4:- cn os --1 g E-—e o o o e Q 6 6 C; _C; C5

::>|::>8 _

. 37

A Figure 4.6 demonstrates the change with Reynolds number of theu—velocity profile along the bisector. This Variation was most pro-

* bably caused by the initial conditions applied at the leading edgefor the Velocity components. The numerical method predicted a singlecurve for the u-Velocity profile after the Reynolds number was great- her than 100,000. A similar trend was observed for the two-dimensionalcase but the u-Velocity profile reached its limiting value at a

4

smaller Reynolds number. None of the previous investigators reported·

whether or not any change in the u—Velocity profile on the bisectoroccurred with Reynolds number. _

4· „ ·

Because of the assumption that the pressure gradient was zero 'and the final form of the finite—difference equations, four equationsinvolving only three unknowns resulted. Therefore, the continuityequation was used to independently verify that the Velocity components

4

predicted by the solution of the three momentum equations werereasonable. .Figure 4.7 illustrates that the continuity equationresidual approaches zero as the calculations advanced in the x- ~direction. The high Value for the continuity equation residual atthe leading edge was expected.

· ~Figure 4.8 is provided to illustrate the value of the skin 'friction from the corner to the two—dimensional boundary layer. A ‘ 4

monotonic increase in skin friction was predicted.

Table 4.1 presents typical results from the corner regionsolution to illustrate that the numerical method predicted a

l

symetric flow field. The values of u at the same positions on

38 4

l•0 IQ Q Lj h

O O

0.86y O

E140.6C) ÄÄ 66 ‘

O 2.·:>I::»8 6

0.4 66 UAO O Rex = 13,336O

R = 52 3700.2 , ex ’’

O [1 Rex = 410,673E) R = 819 497O O EX ,

0 •!§0. 1.0 2.0 — 3.0 ‘··° 5·°ReX I/...6. 6 R ·X 2 _

Figure 4.6. U-Velocity Profile on Bisector for Various Values -of Reynolds Number.

J I 0V O•-4 .

. J2 2El0g '¤v-I2h0M .

*4-I{ OE _

.tn g”

O. •·-I'>

. · Iß:3O‘ ·•-I

. ~ · H '09-I

N •-IQ Q .nd és _

- Q0· MGOä' ·x-IO u•'* Q· 5¤‘

ISI. ‘ >·•

uS'' 2 2O e-I QU In

. .- 3) H2 E2 ” ·<: .3J(WQ •

H ~1'

° ° O2

c 3. 3 2. <= 3 2I | T f

40

IhQ _ . .v-IN —.

Q\ •

•-·l _ Q _I 2 20O ox pg•'-|JJ

~ _ g···* ¤*·· s-•FH¤

° .2 ‘° =U} ’ H

SI<¤ as2 I ..><I 2H N2 I °. II‘”

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S)2 ‘ EI["<ü

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GO·•-IJJ

·«-•I 0H. pg, UI 2* 2A4_ U)6 II

<t0_H2 go

Q ~«·-IG In I<|’ N . G2 2 2 2 2 2• G G G G GG G G; G G G

O UPIS · -

41 l

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'J>3 ¤>3 ¤>3 ¤P3 ¤>3U0 ~Q N <r M C C GMOCNN C¢C CCP-! CCC CCC {Q) CNN CCC CC:-4 CNC C«—|I\ J3_;

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CCQ CCr·| •—iC<‘|’ CCC <I'C•··ICCC CCC CCC ¢CC C•··ir·I' CMN:-1 •—|CN CNC NCC CCC C"NCC NNC •-INN r··|CC ¤—|CC U')CCC CCC CCC CCC CCC ä'ICC OCC ICC CCC COC' Q

N <I' •··I •-I ‘ ¢ Cab r~ v-I an us _ M .M so »o m <r <·. • • • • • ‘

>•><

_42

opposite sides of the bisector were within 0.1 percent of each other.AThe value ofuv

atuagiven position on one side of the bisector was ' „

within 0.1 percent of the value of w at the symmetrical position on _the other side of the bisector. The same result was observed forsimilar comparisons between w and v. On the bisector, v and w were A

within 0.1 percent of the same magnitude. A ‘e

A B. Turbulent FlowAA

Because turbulent flows cannot be described completely byl

mathematical statements which can be solved analytically, solutionsfor even the simplest flow geometries do not exist. Therefore, thereliability of the numerical method to predict two-dimensionalturbulent boundary-layer development was ascertained by comparisons

_ with empirical correlations, exparlmental data, and a simple momentum_integral solution.

A

The prediction by the numerical method of the u-velocity pro- A

file development in the two-dimensional region was compared to Coles'‘ (26) wall-wake model. This model was considered the best of the

empirical correlations for turbulent flows. Figure 4.9 illustrates_ that excellent agreement was obtained since the u-velocity profile

developed as expected. aThe development of the wake function wasl

‘ also predicted but was not as great as that given by Coles (26).

The method was further checked by comparing the predicted 6* and6 development against a simple momentum integral technique presentedby Schlichting (14) which assumed a one-seventh power law velocity

43 .

? o8 G ·G’«-J °

4-•*4-Ju-I - <*O °“ "83 m mn ~¤O|¤ , 3 G G G 3<¤¤ G ·G· ·—· ~2 ·¤Q *-‘ ‘ V >< >¢ :-2 2 § •"

· UO *2 3 3 °' G3¢ <T <I' wc" u u u .‘,‘.}

>¢ >< >< ° 3° 2 S2 2 8 ·¤• "' ä. ¤•, 4 EI O g4 U

JJO GO o cu‘* E'.C] • ,9,„G Q!JJ l '2 Ü 5gu. •

~ _ Q2 ES • N ,9,Nm I ·•-•

ädl Gm'?Hé' 3 ~ °*• ,,

. TQ Q.

* 2 §’

•‘ c -2I " "3Q ._ . g 3

Q.‘

2 °‘ ä'1 ah-;· -_

2 V ¤Z~ F9 **7 N é oo —=· °N N N. _ N ,..; 3 °°<"

V _ - . 4) .. sn go

. °'— •r-I

44 6

0.35 ° {

· {

0.30 6 O

6* -———-ä0.25 V I 4O .

Oé0.20 ~-.-1. CD V G 4 4s-• Ü ·o

12 -<—————-—- 00.15‘ O e

0.1 Q __ 0 .———— Momentum

, Ü Numerical0.05

1..106 2..106 3:1:106 ~ 4xl06 6..106 6..106Re {

V _ xFigure 4.10. 6* and 0 Compared with a Momentum Integral Analysis.

‘ 45

profile in conjunction with the Blasius law of friction. Figure 4.10illustrates the methods capability of predicting 6* and 6 comparedg .with the momentum integral technique. 6* and 0 were obtained byintegrating the predicted u-velocity profile by the trapezoidal rulein accordance with equations 4.1 and 4.2. ‘

VThe wall shear stress estimated by equation 4.4 is only valid for

the turbulent flow conditions if the first nodal position in the normaldirection to the wall is located within the laminar sublayer. Theresultant skin friction obtained from equation 4.3 is compared inFig. 4.11 with the Ludwieg and Tillmann (27) empirical shear stress

· correlation. The local value of 9 and HS necessary in the correlationwere obtained from the momentum integral method.- Although the agree-ment is poor, the general slope and shape of the skin friction curveis predicted by the present method.

ll

Figure 4.12 demonatrates the ability of the numerical method topredict the two—dimensional turbulent boundary layer u-velocity

(profile as reported by Bragg (10). Very favorable results are indi-cated by the comparison.

Figures 4.13, 4.14, 4.15 and 4.16 demonstrate that the method wasV

4_ capable of predicting the u-velocity profile for turbulent flow along

the bisector of a corner. The method tends to underpredict the_ _experimental results as the corner is approached, which is most

probably due to the eddy-viscosity model used with this method.The method preddcted symmetry about the bisector reasonably well

l but the symetry was not as good as in the laminar case. Table 4.2

4_ 46

~ .004 l

.Cf

.0033 O

«,

6. „ » l

Ludwieg and· Tillmann

C) Nuerical.002

lxlO6 2xl06. 3xl06 4xl06 . 5xlO6 6xl06Re ”

.x

Figuré 4•ll- Skin Friction Compared with Ludwieg and TillmannCorrelation.

47

1.0 • •"o 'Ö

U

.6* ’ °

• A . .

.|¤|D8 p _ E U. Reg = 2740U4 ’ ——-—-—- Numerical

O Experimental ‘

.2 I

0 _ lp

l

O U2 O4 U6 A U8 llo

S _ Y, in. _

Figure 4.12. U—Velocity Profile Development Compared with Bragg'S_ Data in Two—Dimensional Region. ”

w

' <I’.[g, .

3 Ng N IIV °J: N

. <r§ u,,

M Q„ >‘ ta, nG•-I Q . •

3 3° vi *3, 8'

6 eu OF2 *1 2 <=2 ¤ ·‘%° -3 .··=· O ° c- é ¤¤. O—m Q w• 2 E',·‘*ü* Ö· ‘°°

6 - . H‘ °? s‘ 6 3 O ·¤Q) .3 ZS•¤«ä[UH

- QQQU. -6* gw

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g pn

oo[1

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49

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°·•·*

1 1 T 1* "·• .V '[¤¤p1:Sa*5 '[¤uo]:suam1:puoNV

V

— 53

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‘UN<t‘r-I I~¢*'1O <I'r-II~ •—|GN NMQ G1U U1QQ <I'0G ¤'1<|'<f N:-4N‘ NOONI-OU1 NU1U1 Nlhlh NU1U1 N!-F1!-¢'1 O ‘IOI IOC 000 I00 **0 Q'v-l

•-OH _ · 2 ·DIJ

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v-IU V

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G V VuQ)QO

-¤« . ‘ F1E <I'¤1U1 U1v—·|0 UNQG QQQ LONG O .OInO <'•1<"1<I' ¤·—|OO OGG OwU1l-O ¢1U1U1 •-I>‘ NU1U1 NLOU1 N<I'<f‘ r-4<f<f •"‘|~T<I O- 4-J·r—|

UOQ) .P '.

' G QQQ ¤1ON <"1<|'U1 Owl-NGOQQ O1<'<' ‘Q¢'*1€'1 OQQ •·—I ·_ _ NU1U1 N<I‘<t‘

•-·|<I'<I' •·-|<t<t •-¢<!'<!' O>. 1DV

’\ VVU}°fw<r‘-¤U~ N «—• ·oo :~·w vw4 al Uv cw N vw o vw .. •-•·

3 wow .· oo ~¤ _.vw vwN O O O O ·O ·

l4 is included to give a representative result for the u, v, and w

components on and near the bisector. 4Figure 4.17 presents the continuity equation residual as a . _

function of the Reynolds number based on distance from the leading ”

edge. As before in the laminar case, the residual tended to zero asthe solution proceeded in the x—direction. This again was taken asan indication that the predictions of the velocity components in thecorner region were reasonable. r

The method did not predict the negative secondary velocities onthe bisector as shown by published experimental work. However, a

‘

small amount of u-isovel distortion was predicted in the region ofthe bisector, but the magnitude was not as pronounced as that indi-

i

cated by experiments. Neither was a closed vorticity pattern pre-dicted. lThe failure of the numerical method to predict the secondary

flow pattern is probably due to the simple eddy—viscosity model used __ for the corner region. _ _

_ AV. CONCLUSIONS AND RECOMMENDATIONS

A technique is presented to solve the equations of motion forboth the laminar and turbulent flow of an incompressible fluid along

A

the axis of a ninety-degree corner. The Alternating Direction Imp1icit„ A

n (ADI) finite—difference method was used to cast the governing partial·differential equations into a finite-difference form.

An eddy—viscosity model expressed in terms of a mixing length wasA

used to represent the turbulent shear stresses in the_corner region.A ’ -The mixing length model given by Pletcher (22) was adapted for use inAthe corner region.

e. _ ~ „

‘The final form of the finite-difference·equationsAwas such that ·. _the corresponding coefficient matrixes were always tridiagonal.

A 1 ··Therefore, the Thomas algorithm (16) applied.

This method was used to solve the laminar and turbulent flowalong the axis of a ninety-degree corner from which the followingobservations and conclusions were obtainedzv

1. The ADI finite—difference scheme resulted in‘a stableA

l_ and convergent method for the corner region.

· 2. For the laminar corner flow case, good agreement with _A AA A

the solutions of Carrier·(l), Dowdell (4), and Rubin —A

and Grossman (7) was obtained.A

A A

0. For the turbulent corner flow case, good agreement ·with the results of Bragg‘s (10) experiments was

·obtained.

A _ 55 ·

lV 56 l '

U4. The computatioual time required to solve the Bragg (10)

geometry was 69 minutes on an IBM 370, model 165.5. This method should be extended so that adverse pressure

A

_ gradient flows can be studied. The procedure is sufficiently{general that these studies can be conducted without a majoralteration.

6 U

- 6. Additional experiments should be conducted to ascertainthe true nature of the turbulent shear stresses in the ·6corner region.

V · VI. BIBLIOGRAPHY _ V

' 1. CARRIER, G. F., "The Boundary Layer in a Corner," Qparterly ofApplied Mathematics, Vol. 4, No. 4, pp. 367-370, 1947.2. KEMP, N. H., "The Laminar Three-Dimensional Boundary Layer and aStudy of Flow Past a Side Edge," Master's Thesis, CornellUniversity, June, 1951. ·3. DOWDELL, R. B., "Corner Boundary Layer," Sc.M. Thesis, BrownUniversity, May, 1952. ~ ·4. DOWDELL, R. B., "Incompressible Boundary Layer Flow Along InteriorCorners," Proceedin s of 10th Midwestern Mechanics Conference,1 Developments in Mecäanics, Vol. 4, pp. 1355-1378, Äigust, 1967.5. RUBIN, S. G.,

"Incompressible Flow Along a Corner," Journal of_ Fluid Mechanics, Vol. 26, No. 1, pp. 97-110, 1966.6. PAL, A. and S. G. RUBIN, "Asymptotic Features of Viscous FlowbAlong a Corner," Quarterly of Applied Mathematics, Vol. 24, No. 1,pp. 91-108, April, 1971.

· _U

7. RUBIN, S. G. and B. GROSSMAN, "Viscous Flow Along a Corner: _Numerical Solution of the Corner Layer Equations," Qparterly ofApplied Mathematics, Vol. 24, No. 2, pp. 169-186, July, 1971.8. NIKURADSE, J., "Untersuchungen uber die Geschwindigkeitsverteilungin Turbulenten Stromungen," VDI—Forschungsheft 281, Berlin, 1926. V9. GESSNER, F. B., "Turbulence and Mean Flow Characteristics ofFul1y—Developed Flow in Rectangular Channels," Ph.D. Disserta-· tion, Purdue University, January, 1964.

10. BRAGG, G. M., "The Turbulent Boundary Layer in a Corner," Journal„ of Fluid Mechanics, Vol. 36, Part 3, pp. 485-503, 1969. Vu11. TOAN, N. K., "Applications of Integral Methods to the Turbulent _Corner Flow Problems," ASME Paper No. 71-WA/FE-36, Presented at ‘

Winter Annual Meeting, November 28-December 2, 1971.12. GERSTEN, K., "Corner Interference Effects," AGARD Report No.299, March, 1959.

13. VEENHUIZEN, S. D. and R. N. MERONEY, "Secondary Flow in a Boundary lLayer," Project THEMIS, T.R. No. 3, ONR Contract No. N00014—68-A- -0493-0001, June, 1968.

V S7 V

, 58 V V

14. SCHLICHTING, H., Boundary-Layer Theory, Sixth Edition,‘ McGraw-Hill Book Company, p. 62, 1968.

15. PEACEMAN, D. W. and H. H. RACHFORD, JR.,"The Numerical Solutionof Parabolic and Elliptic Differential Equations}'Journa1 of the_ _ Society of Industrial Applied Mathematics, Vol. 3, No. 1, pp. 28-41, March, 1955. V Ä

16. VON ROSENBURG, D. U., Methods for the Numerical Solution ofV Partial Differential Equations, American Elsevier Publishing. Company, Inc., New York, p. 113, 1969.V17. CRANK, J. and P. NICHOLSON, "A Practical Method for NumericalEvaluation of Solutions of Partial Differential Equations of theHeat Conduction Type," Proceedings of Cambridge PhilosophigalSociety, Vol. 43, pp. 50-67, 1947.

- 18. KLINKSIEK, W. F., "An Implicit Numerical Solution of the Turbulent„ Three—Dimensiona1 Incompressible Boundary-Layer Equations," Ph.D.VDissertation, Virginia Polytechnic Institute and State University,June, 1971. „

19. HINZE, J. O., Turbulence, an Introduction to Its Mechanism andTheory, McGraw-Hill Book Company, Chapter 1, p. 20, 1959.20. SCHLICHTING, H., Boundary-Layer Theory, Sixth Edition, Mc-GrawHill Book Company, p. 548, 1968. ‘

21. GOLDSTEIN, S., Modern Developments in Fluid Dynamics, Vol. 1,Dover Pub1icationg,Inc., pp. 206-208, 1965.

22. PLETCHER, R. H., "On a Finite-Difference Solution for the Con-stant-Property Turbulent Boundary-Layer," AIAA Journal, Vol. 7,. No. 2, pp. 305-334, February, 1969.

23. EAST, J. L., "An Exact Numerical Solution of the Three DimensionalIncompressible Turbulent Boundary-Layer Equations," Ph.D.‘ Dissertation, Virginia Polytechnic Institute and State University,1970. V ‘ · #. [“

24. SCHLICHTING, H., Boggdary-Layer Theory, Sixth Edition, McGraw-HillBook Company, p. 129, 1968.

25. KELLY, L. G., Handbook of Ngggriggl Hgthgds ggg Applicgtigns, First _Edition, Addison-Wesley Publishing Company, Inc., pp. 224-225, 1967.26. VCOLES, D., "The Law of the Wake in the Turbulent Boundary-Layer,"Journal of Fluid Mechanics, Vol. 1, Part 2, pp. 191-226, 1956. Ä.27. LUDWIEG, H. and W. TILLMANN,"Investigations of the Wall ShearingStress in Turbulent Boundary Layers,' NACA TM 1285, 1950.

„ S 59

28. LAUNDER, B. E. and W. M. YINC, "Fully-Developed Turbulent Flow in‘ Ducts of Square Cross Section," Report No. TM/TN/A/11., Imperial _College of Science and Technology, Department of MechanicalEngineering, July, 1971.

~S

· VII. APPENDIX A 6

Appendix A contains the development of the finite—difference

equations which were used to approximate the equations 3.5a-d, 3.6a-b,3.13a-c, and 3.14a-b. Also, the Thomas algorithm and the model usedfor the corner region mixing length are presented.

60 l

_ 61 _V z

— A. General Discussion of Finite-Differences

It is possible to represent the value of a function removed a dis-tance from a position where the value of the function is known by a ”

Taylor series. l.

(Ax)2(Ax)3‘

„ Using the figure A.l as a reference, the central difference repre-sentation of the second derivative at x can be approximated to a trun- .cation error of by writing g(x+Ax) and g(x-Ax) in two Taylorseries and then adding these two series together. This equation is as ‘A follows: '

(Ax) g . gThe equation A.2 assumes that the function is continuous through- V ·

out the region of applicability, is many times differentiable and theneglected terms are many times less than the first term retained in the ·resultant series. 6

U _Again using two Taylor series for g(x+Ax) and g(x-Ax) and sub- ‘

zstracting these series from one another, the central differencerepresentation for the first derivative can be obtained and will have atruncativn OYYOY of The result is as follows:

, (x+Ax) - (x—Ax) (Ax)2 zg (x)

· Fig. A.l. Grid Used for Representation of g'(x) and g'Ü(x). „

~ ’ _ 63 . .

_ The same assumptions applied to equatiou A„2 must also be applied‘ _ to equation A.3 in order to ensure a valid approximation of the first

derivative. ‘.

V If variable grid spacing between nodes is desirable, the ·l

following equations are representative of u , u , u Q and u when3

y yy z _ Vzzapplied to figure A.2:

. Ay_ . Ay+———- d — —-— - b ”22 /1},+ (u( ) u(¤)) + Ay- (u(e) ¤( ))

(A}; + A}·_>6.(A.4a) -

—¥Ay+Ay_ 4 Ay+“’.

(Ay+) ..-—-_-·4+12 ¤(€)yyyy 4 V ( )

· Az_ V Az+°E; (¤(c) - ¤(¢==)) + E: (¤(e) - ¤(a)) (Az-112

. “z(°) = (Az+ + Az_) + 6 u(e)zzz3(A.4c)

36 M-}-<¤<¤> — ¤<e>> +-J- <¤<e> - ¤<a>> 3 6

V .AZ+Az-.uzz(e)= (Az+ + Az_)/2 +. 3. u(e)zzz

6 A 4 64 4 . 4.

Az Az _- u d) + ‘-

u(a) ¤(@) „ ¤( ) ~

Ay_

u(b) V .

Z · 4 U

. · Fig. A.2. Variable Grid Spacing used for u , u , 4 A · ·4 ·uz, and uzz. V VV . F

._65_

Thus, it is obvious that in order to keep the truncation errorsmall, equalgrid spacing should be used since both of the finite—

4

difference equations A.2 and A.3 are of approximately the same orderof truncation error. However, in order to conserve memory

l-

8tOT88€ and to keep computer calculation times to a minimum, the '

variable grid size was used in this investigation.4

Examination of equations A.4a—d indicates that the first and „second partial derivatives of a function can be determined if the

Vfunction is known at various points. By replacing the derivativeslin partial differential equations with their finite-difference '

V approximations,aaset of equations can be generated where the onlyunknowns are the values of the function itself at various grid‘ ~

~ locations. And, if the boundary conditions imposed on the region arel ‘

known or specified 1in‘ which the partial difference equation is“

assumed to be valid, uns set of equations can be solved for the value '_“

of the function at the various grid locations within the region. ‘

B. 4Finite—Difference*Approximations for the Two—Dimensional Laminar= Boundary Layer l

The Crank—Nicolson (l7) implicit finite-difference method was » h

~ — used to represent the x-momentum equation, 3.6a, in the two-dimension—„5

al flat plate region. Using the dummy variable H, the variousV

derivatives in equation 3.6a were written about the point i,m+l,n+l/2.

The position specified by the index m+1 is the line 4·5 as shown in4

Fig. 3.4. · · _

‘ A HX - (H(i,m+1,n+1)_; H(l,mH1,n))/Ax „n.

1 (B.la)

-————-————- -—- + -

.’1-1 1 {Ay" u(1+1 1 11

y 2(Ay++Ay—) Ay+ [ ,m' ,n) H( ,m+l,n) + H(i+1,m+1,n+1)

p ~ H(i,m+l,n+1)] +·K;- [H(i,m+1,n+l) - H(i-l,m+l,n+1)

V·

+ H(i,m+1,n) - H(i-l,m+1,n)]} · (B.1b)

l . 1 ‘ 1 ’ l ‘ °..·Hyy ?Ä;;;Z;i7-{Ei; [H(i+l,m+l,n+1) + H(i+l,m+l,n) - H(i,m+1,n+l)

I - H(i,m+l,n)] -·K%— [H(i,m+1,n+1) + H(i,m+1,n) » _

— H(1—1,m+1,n+1) - H(i—1,m¥1,n]} l V.(B.lc)

·U

The non-linear coefficients in equations 3.6a were assumed to beh lthe space average of the previous nodal location and the new nodal „location in the x-direction at each y node in order to linearize theequation. Thus, the non—1inear coefficients were calculated asfollows:

The above equation assumes that the value of—H at the new nodal . ‘

location in the x—direction was known but the entire problem was to A· determine what the value of H was at this nodal location. There-V

fore, as a first appronimation, H(i,m+1,n+1) was assigned the value of

67 '

H(i,m&1,n) and an iterative procedure was used to determine succes-ively better approximations to H(i,m+1,n+1). 7

In order to present the finite-difference equations in a more‘ more concise form, the~fo1lowingAdefinitions were introduced for the

e

coefficients which make up the matrix. ” _'

r ,11E 2 Ay_ (Ay+ + A>'_) . ' °

. - Ay8 S X . E . ..; . ...L...... S6 rz6 2 15}*+ (Ay++AY_) (Bib)

l O O 1

7 Vl .

815 Ay_ (Ay), + AY) V (B°2°) -1 Ax _ 1 l

S2E Ay+ (Ay_,_ + AY_) (B°2d) .

Using the equations B.1a-d, the finite-difference approximationl

of equation 3.6a. is as fo11ows: A·

(rl-s1)u(i-1,m+1,n+l) + (1.-r1—r2+s1+s2)u(i,m+1,n+1)

+ (r2—s2)u(i+1,m+l,n+1) = (r1+s1)u(i-1,m+1,n) ·

· + (1.-r1+r2-s1—s2)u(i,m+1,n) + (sz-r2)u(i+1,m+1,n) (B.3)Klinksiek (18) has shown that the finite-difference approximation _

used for the continuity equation 3.6b was important_from a stabilityconsideration. The finite—difference form of the continuity·equation recommended by Klinksiek (18) was written. about the point(1-1/2,m+1,n+1/2).

4 lAy_ ' _

v(i,m+l,n+1) = v(i—1,m+1,n+1) - iK;· {u(i-1,m+1,n&1) V“

+ u(i,m#1,n+1) - u(i-1,m+1,n) - u(i,m+1)n)} (B.4)I ‘

' ~ 68 ‘

The equation B.4 contains a backward difference approximation for

g the ux derivative and therefore has a truncation error of 0(Ax). _To further condense equation B.3, the following notation was

employed: A · .‘ _ ·

A1(i) = rl — sl (B.Sa)° B1(i) = 1. + rl - r2 + sl + sz - (B.5b)

sz (B.5c)

g D1(i) = (rz + s2)u(i—l,m+l,n) + (l.—r1+r2—s1—sé) • ·

· Using the notation above, the following algebraicpl

_equation resulted and was solved in conjunction with * ' l

the finite—difference representation of the continuity equation toyield the u and v-profiles for the two—dimensional boundary layerdeveloping on the Y = 0 wall. l

x—momentum ·

A1(i)u(i-1,m+1,n+1) + B1(i)u(i,m+1,n+l) + C1(i)u(i+l,m+l,n+1)

i = 2,3..... 2 (B.6)

Inspection of equation B.6 reveals that a tridiagonal·matrix wasl 8

generated. The solution technique used to solve this type of matrixde

is discussed in the last portion of this Appendlx. l

C. FiniteeDifference Approximations for the Laminar Corner Region ‘‘

The Alternating Direction Implicit (ADI) finite·difference methodwas used to solve the x, y, and z momentum equations 3.5a-c in the

U 69 ‘

corner region. In accordance with the ADI method,·the derivatives in’

U one of the transverse directions were written about the point ’ - . U4i,j,n+l/4 and the derivatives of the second transverse direction about °

the point i,j,n. Then, the derivatives in the second transverse UU direction were written about the point i,j,n%3/4 and the derivatives

V I-

of the first transverse direction about the point i,j,n+l/2 using thel

half-step values. Because of the method used to write the

derivatives with the ADI method, half-gtgp values of the desired Uvariable are generated at the point i,j,n+1/2. The values generatedat this point, however, are not the true representation of the

variable at this point since to arrive at this point, the calculations1 l

are executed explicitly in one of the coordinate directions. ‘

U These half·—step values are only used for calculations in the nextV l

half step for the desired value of the variable at the end of the h'

U full step. Explicit stepping continuously in one of the coordinatedirections can cause instabilities to develop which are circumvented

el

in the ADI method by stepping implicitly in the next half step in the Uecoordinate direction which was done explicitly in the previous half U

l”

step. For a detailed study and analysis of the ADI method, consult' .Peaceman and Rachford (15). ‘

U · U Vl U

(Using the dummy variable H, the various terms of_eduations

3.5a-c were written in the following finite-difference form for the· ”first half—step in the x-direction using the equations A.4a-d as

1 .·

models.U

·l

4. 70

At l

H = (H*(i,j) - H(i,j,¤))/(Ax/2)7

(C-la)~ x ^?_ y H A?+_ [B7 (H*(1+1,;I) — H*(1.;I)) + E-_(H*(i.:l) - H*(1-1.j))2 +

_'

l _ Az_ Az+ l

z iAz+ + Az_

(H*(i+l.j) - H*(i.j)) — K;7 (H*(i.:l) — H*(i—l„j)) l g- + H ·· - „Hy? _

p „ (A?+ + Ay_)/2 - - n ·(C•ld) n

+ -Hzz =

(Az+And,for the seeond half step in the x—direction, the following

finite—difference equations were used.l

HX = (H<1,j.¤+l) · H*(i,j))/(Ax/2) . (C.lf)

- E7 (H*(i+1,j) - H*(1.j)) + E,-(H*(1,j) — H*(i—1.j)) ‘H AY+ +A?_Az_

0 Az+Kg- (H(i,j+l,n+l) — H(1,j,n+1) + E- (H(i,j,n-I-1). + — p. _ Hz = , q _ Az++ Az_ I- ‘ l °

fÄ-ä- (H*(i+l,j) — H*(i,j)) — K5- (H*(i„j) -H*(1—1,j)) ‘

H - <c 11>yy (Ay+ + Ay_)/2 °

V71

U'

KE;-(H(i,j+l,u+1) 5 H(1,j,n+1)) - K;:·(H(i,j,n+l) .

‘5 H(1,j—1,n+1)) 5 1 · (C.15)

‘ The symbol (*) denotes the intermediate value of the variable at the __ point 1, 5, n+l/2. I

The non-linear coefficients in equations 3.5a-c were assumed tobe the space average of the previous nodal location and the new nodal

U l_ location in the x—direction at each nodal location in order to lineariae

·

gl B

the equations., The ha1£¥-step values of u, v, and w were not used

for the value of these non-linear variables since the half-step V _ ”

values of u, v, and w were to be used only in the calculation of u, v,

and w at the new x-location. Thus, the non—linear coefficients were_A-calculated as follows:

, if =B

(6.110_ The value of H(i,j,n+l) for the above equation was obtained in

I

the same manner as the value for H(i,j,n+l) was obtained for equation ·

5 B.ld. 1 . 1 ' ( _°A ”

In order to present the finite—difference equations in a moreA ” B

concise form, the following definitions of the coefficients whichmake up the matrixes for the first half step were introduced in 1

addition to the definitions introduced in section B.

72

f- Az- xr. , 4; , ..1 ,......L......tg”E 2 Az_ (Az+ + Az_) (C‘2a)

— Az ß- Y. , 4.; , ..1 , ......J........ ·rg -

E 2 Az+ (Az+ + Az_) <C'2b)

.. L . 4;,SaAz_ (Az+ + Az_) (C°2°) ‘

- L , Q;. ,.....L.....Su Az+ (Az+ + Az_) l - (C°2d)

UUsing the above notation and the notation given by equations _

f ;B.2a-d, the fin1te—difference approximations for equations 3.5a-c for” the first half x step were as followsz

x—momentum(-1:1

— s1)Vu*(i-1,j) + (1.+1:1-1-2 + 61+62) u*(i,j) + (rz-sz) u*(i-!:l,j)

= (r$+s3)u(i,j—1,n) + (1.—r3+r“—s3-s“)u(i,j,n) + (su-1k>u(1,j+1,¤)· (C.3a)

- y~momentum

(-1:1-s1)v*(i-1,j) + (1.+rl-r2+s1+s2)v*(i,j) + (1:2—s2)v*(1+1,j)I

= ([3-{-g3)V(j_,j-]_,u)·+ (1,·—I3+I“-S3-Sl})V(i,j ,II) + (8“'Y“)V(i;j+]-gn)

(C.3b) ·

z-momentum(—x1—s1)w*(1-1,j) +(]„+p1—r2+s1+s2)w*(i,j) + (r2—s2)w*(i=1,j)

= (r3+s3)w(1,j-1,n) + (11-r3+r“-sa-s“)w(i,j,n) + (64-r“)w(i,j+1,n%CBC)

To further condense the equations, the following notation wasintroduceds ~ - _

A (1) F — r - s (C.4a). 2 * 1 1V B (1) = 1. + r — r + s + s (C.4b)· 2 · 1 2 1 2C 1 = . - .2( ) rz sz (C 4c)VD (1) F (r + s )u(i,j-l,n) + (1. - r + r - s - s ).2 3 3 3 u 3 u• u(1,j,n) + (su - r“)u(i,j+l,n) (C.4d)D3(1) = (ra + s3)v(i,j-1,n) + (1. —r3_+ ru — sa — sk)

• v(1,j,n) + (su — r“)v(i,j+l,n) (C.4e)‘·

gD“(1) = (ra + s3)w(i,j—1,n) + (1. - rs + ru — sa — s“)

‘ Using the above notation, the following algebraic equations resulted.V

They were solved simultaneously for the first half x step and yieldedthe intermediate values of u, v, and w, 1.e., u*, v*, and w*.

x—momentum _V

A2(1)¤*(1—l,;l) + B2(1)¤*(i.j) + C2(1)¤*(i+1„j)V = D2(1)1 = 2,3,....k

g j = 2,3,....mV

(C.5a)y—momentum. 1 ‘_··._ .

A2(1)v*(1—l»:I) + B2(1)v*(1„;|) + C2(1)v*(1+1„;l)· = D3 (1) y „ V1 = 2,3,....k ‘

j =

O . V l

ZV 74 1

·- V z¥mbmentum ‘ ' . ¤A2(i)w*<i—l.;\)+B2(1>w*(1.j) + C.2(i)w*(i+l,j> · D“(i)

· 1 = 2’3,0Irlk

lA

V J = 2,3,0Il-Om. V-

4Inspection of equations C.5a—c revealg that all three equations

lgenerate matrixes of coefficients which are tridiagonal. The solution”‘of this type of equations is presented later in this Appendix.

·‘Similar to the first half x step, the second half x step matrix

coefficients generated by the finite-difference approximations were

VA

_ condensed for ease of notation. Tha the same notation for the _ a- coefficients of the finite—difference approximations given by equations

x B.2a—d and C.2a-d yields the following for the three momentum equationsfor the second half x step. I '

x—momentum

V (-ra—s3)u(i,j}l,n+l) + (l.+r3—r“+s3+s~)u(i,j,n+ll‘ V V+ (r“—s4)u(i,j+l,n+l) = (r1+s1)u*(i—l,j) p+ (l.—r1+r2-sl-s2)u*(i,j) + (sz-r2)u*(i+l,jl

V (C.6a) V

y-momentum l VV

A r 4“(-r3—s3)v(i,j-l,n+1) + (l.+r3-r“+s3+s“)y(i,j,nÄl)U .+ (ru-s„)v(1,g+1,¤+1) = (r1+s1)v*(i-l,j)

l + (l.—r1+r2+S1-é2}V*(i,j)·+ (S2-!-'2)V*(i+l,j) Ü (C·6b) °

I 75 I

_I

z—momentunI

WI _ '

I (-r3+s5)w(i,j-l,n+l) + (l.+r3-r~+s3+s„)w(i,j,n+l)_ + (rg-s~)w(i,j+l,n+l) ¤ (r1+s1)w*(i—l,j)

I'

- + (l.—r1fr2—s1—s2)w*(i,j) + (s2—r2)w*(i+l,j)I

(C.6c)W

Again', to further cgndgnga the equatlons, the following notation

W was employed:

= r“—a# _ (C.7c) _

W1 · D5(j) = (g1+s1)u*(i-1,j) + (l•'Y1+I2°8,'S2) . W

_ WI u*(i,j) + (s2—r2)u*(i+l,]) x I (C.7d)

UW D6(j) = (r1+s1)v*(i-1,j) + (1.-r1+r2-sl-sz)

v*(i.;l) + (S2-r£)v*(i+l.:l) ’ (C•7e>D7(j) = (r1+s1)w*(i-1,j) + (1.-r1+r2—s1-sz)

· w*(i,j) + (sz-r2)w*(i+l,j) I (C.7f)I

Using the notation above, the following algebraic equationsV Yéßultéd, which were solved simultaneously to yield the values of V .

u, v, and w at the next full x step.I I V I

„I

x-momentum ‘ .W I _ 1 _I,

_. ‘

A3(j)u(i,j-l,n+l) + B3(j)u(i,j ,11+1) + C3(j)u(i•j+l;¤+l)V = DSÜ) I I

W

I

U

I

-:

II

J{ O O lk

W i = 2,3,....mI

(C.8a)

76 4 _y—momentum

4 4 4 4

A3(j)v(i,j—l,n+l) + B3(j)v(i,j,n—I-1}+4

C3(j)v(i,j+1,n+1) = D.6(j)

_ j = 2,3,....k _l

4i= 2,3,,...m (C.8b)

z—momentum4

A3(j)Vw(i,j-l,n+1) + B3(j)w(i,j,n+l) + C3(j)w(i,j+l,n+l) = D7(j)

j = 2,3,....k4

‘ i = 2,3,.....m4 (C.8c)

Again, as for, the first half x step, the equations C.8a-c generateds matrixes of coefficients which were tridiagonal. The solution for

4.

equation of this type is presented later in this Appendix.

D. Finite-Difference Approximations for the Two-Dimensional TurbulentBoundagy Layer _

The Crank-Nicolson (l7) implicit finite-difference method was used .‘ to represent the x-momentum equation, 3.14a, in the two-dimensional

_ flat plate region. Using the dummy variable H, the representation ofthe HX and Hy derivatives are the same as equations B.la-b. The termon the right hand side of equation 3,14; was written about the point(i, m+l, n+l/2) where the position specified by the index m+l is the ’

.4

line 4-5 shown in Figure 3.4. 4V s ‘

• - [1.+4:5 (1-1/2,m+1,¤+1)]4

y+ n „_+

' ' G 77

• m+l 2) _HH m+l n)

-5 [Lil-; (i-1/2,m+l,n)‘]· Y 7_ n ._ Hgi„m+l„nQ — Hgi—1zm+lzn) GA Ay- } (D.1l

where: G l__ · - €n(i+1,m+l,n+1) + €n(i,m+l,n+l)eu(i+1/2,m+l,n+1) = —————————————-———G——————-———————- (D.2)

Similar space averages were used for evaluating the other values of suin the above equations. ”

The non-linear coefficients in equations 3.14a were determined byequation B.ld.

A The following definitions for the coefficients which make up theA

4*matrix were used to present the finite—difference equations in a moreconcise form.

Ax7” = ——-——-—-——-———-—-————— ———————————————— D.2S6

G(l.+EG(i-1/2,m+1,n+1)) UAx -S6 S

G (Ay__)<Ay+ + Ay_> A(D°2b). (l.+Eh(i+1/2,m+1,n))Ax.S = ""—'*—·——·—————— · (D,2g)7 G ({\y_) (A>·+ + Ay__)A 7

B = (1.+EG(i-1/2,m+1,n)) Ax 6 G (D Gd)6 G (Ay_)(Ay+ + Ay_) G °

Using the equations B.la, b and d, B.2a and b and D.2a-d, thefinite-difference approäämation of equation 3.l4a is glven by:

(-r -6 )u(i-l,m+l,n+l) + (1.-r +r +6 +6 )u(i,m+l,n+l)+ (r -6 ) ·2 6 1 2 s 6 2· su(i+l,m+l,n+1) = (r2+6a)u(i-l,m+l,n) + (l.+r1-r2—s7-68)u(i,m+l,n) 6+ (67-r1)u(i+l,m+1,n) A

(D.3)

·· To further qqudenge the equation D.3, the following notation was— _ 6 employed: A

·A (i) = - r -6 . · (D.4a) -u 2 6AB (i) = l.-r +r +6 +6 (D.4b)7 ·6* 1 2 s 6C (i) = r -6 ’ (D.4c)u 1 s _

· ’ D8(i) = (r2+s8)u(:L-l,m+l,n) _ _° + (1.+r -r -6 -6 )u(i,m+l,n)· I ‘ _ ‘ 1 2 7 6A

+ (6%-r1)u(i+l,m+l,n) · U (D.4d)

The above notation results in the following algebraic equation,This equation was solved in conjunction with the continuity equationB.4 to predict the u and v profiles for the two-dimensional boundarylayer developing on the Y=0 wall. A

x-momentumA _A A ·

A4(1)u(1-1,m+1,n+1) + B~(1)u(1,¤r+1,n+1) + C~(i)u(i+l,m+l,n+l)

. Again a tridiagonal matrix was generated and its solution .A

technique is discussed in the last portion of this Appendix.

E. Finite-Difference Approximations for the Turbulent Corner RegionAAgain the ADI method was used to solve the x, y and z momentum

equations 3.13a-c in the corner region.

9 79

.,Usingche dummy variable H, the derivatives on the left hand side.and nonlinear coefficients can be expressed by equations C.1a—c and ==

· C.lk for the first half step in the x—d1rection. The.dgp1va;1vgg gnthe right hand side were expressed in finite-difference form as follows; ‘

2 — -9

H*(1+1,j) — H*g1,j)H = ~————————- 1.+ 1+1 2 +1/2)) _[(1+611) yly [Ah + Ay-] {( €n( _- Ay+

(1.+E“H Y- „

_ 2 · —- H(izj+1,n) — H(i,j,n) ·_ [(l+€u)Hz]z — {(l,+€n(i,j+]./2,u))Az-,.

—-zjz - i,j—1, - .- (1.+c—:n(1,j—1/2,n))H(i I1) AzH( Io} „ (E•lb)

For the second half step, equations C.2a-c and C.1k were used forthe left hand side and equation E.la was used for the expression ofHyy in the momentum equations. The following finite-difference V

‘ expressions were used for the right hand terms,Hzz.

_ . 2 - —-[(1+en)HZ]Z - [Az+ + AZ-] {{l+en(1,j+1/2,n+1))

· 2+ n _

_ H 1 +1 — H 1 '—1 +1(,j,¤ ) Az( ,3 ,¤ )} (EJ)

_ U 80 U

To place the finite-difference equations in a more work- VV

able form, the following definitions of the coefficients which make upthe matrixes for the first half step were introduced in addition to

V

equatious B.2a-b and C.2a—b. _ _ '

U 8=(l.+€;(i+l/2,j,n+l/2))

{U Ax

U_U

U (E ga)9 U E (^Y+)(AY+ + AY_) 0 ' °

U S =(l.+€;(i-1/2,j,n+l/2))

UAx

U U (E 3b)10 ÄU E (AY_)(AY+ + AY_) °(1.+E;(1„i+l/2„¤))

S11 “ "°‘°°‘°°“—‘i;1 {'°""‘<A=„,><°A‘z‘°;‘+"'A'z°_'>" 3 ‘E·3°’(l.+EV(i j—l/2 n)) _ — · ·S = } (E•3d)12 . ;~ U ( z_)(Az+ + AZ_)

Using equations C.la—c, C.1k, B.2a-b, C.2a-b, E.la-b and E.3a—d,the finite—difference approximations of equations 3.13a-c can be

V„

written 8SVf0110WB·fOI the first half x—step: V 1 .

x—momentumV

1 U+1 + (rg-8é)¤*(i+1,j) = (r3+812)¤(1„j—1„¤) 0 i VU

1 ”+(l.—r +r -6 -6 )u(i,j,n) + (s -r )u(i,j+l,u) (E.4a)U9U u 11 12 11 u 1 U

U Z-momentumV

U‘

UV

V (r -6 )v*(i;l,f) + (1.+r -r +6 +6 )v*(i,j) .1 10 1 2 9 10+ (r -8 1)v*(i+l,j) = (r +8 >v(1,;I—1,¤) ~ ‘2 9 1 3 12 U ·

1

· . _ 81 V ‘ l

z-momentum l

1(rl-S10)w*(i—l,j)+ (l.+t1—I2+89+S1°)W*(i•j) 9 ·+ (r -S )w*(;|_+]_,j) = (r +6 )w(i,j·l,¤) l

2 9 3 12

+ (1.-1: +1: -6 -6 )w(i,j,¤) + (S -r )W(i•j+l•¤) (E·‘*t) V‘ 3 u 11 12 11 u _. lWA6 before, to furth@I‘cond@¤tS the eq¤eti°¤S• the fS11¤w1¤6

additional notation was introduced: ' ‘

5 1 10B (1) = 1.+r -r +6 +6 V (E-5b)5 1 2 9 10 .~ 1C (1) = 1 ,8 l (E.5c)‘ s _ 2 9D (1) = (r +6 )u(1,j-1,n) + (l.-r +r -6 -6 ) VV9 3 12 3 .3 11 12

· • u(i,j,n) + (611-r“)u(i,j+l,n) (E.5d)D 1 = - .— -- —l 10( ) (r3+s12)v(1,j l,n) +(l•

v(1,j,n) + (611-rk)v(1,j+l,n) ’ _ V (E.5e)

V D (1) = (r +6 )w(1,j-l,n) + (I.—r +r -6 -6 ) ”11 3 12 · 3 0 11 12 .

° W(i•j•¤) + (S11*rk)W(i,j+l,¤) ‘ (E.5f)

With the above notation the following algebraic equations resultedwhich, when solved, yielded values for u*, v*, and w* at the first

· half step.

x-momentum

A5(1)u*(1-1,j) + B5(i)u*(i,j) + C5(i)¤*(i+l,j) = D9(i) .4 5 -· 1- 2,3,....1;I

.1 j- 2,3,;...111 (E.6a)

82 „ ·

X—momentumn

A5(i)v*(i—l.j) + B5(1)v*<i.j> + C5(1>v*(i+l,;i) = Dl0(i>· » i = 2,3,....k

_ j = 2,3,....mu ·(E.6b)

z-momentum _ «»w A5(i)w*(i-1,j) + B5(i)w*(i,j) + C5(1)W?(i+l,j) = D11(1)

'= 2,3,••••k

j = 2,3,....Im, (E.6c)

« Again, the above three equations yield tridiagonal matrixes. The~ _ solution.techniqme is presented in the last portion of this Appendix.

‘ Similar to the first half x—step, the second half x—step matrix· . coefficients were condensed for ease of notation. With the addition of

two more equations, ‘

6 = (1 +E (1 j+l/2 n+1)) {—————£———-—-}·W

(E 7a)13 ' n’ ’

(Az+)(Az+ + Az_) ° „

_ - Axsin —+ Az-)} • (E.7b)

the following equations were generated for the three momentum equations _for the second half

step.x-momentum 42 3

(-r3—s1&)n(i,j-1,n+l) + (1.+r3—r“+s13+s1“)u(i,j,n+1)_

+ (r“—s13)u(i,j+l,n+1) = (r1+s1°)u*(i—l,j)A•

( + (1.-r +r -6 -6 )u*(1,j) + (6 -r )u*(i+1,j) 3(E.8a)1 2 10 11 sg 2 ~

83 ‘ 3

X-momentuml

. A(-r -6 )v(i,j-l,n+l) + (1.+r -r +6 +6 )v(i.j,n+l)9 lk A 9 u 13 lk ·+‘(r -6 )v(i,j+l,n+l) = (r +6 )v*(i—l,j) 3

u 13 1 10+ (1.-r +r -6 -6 )v*(i,j) + (85-I )v*(i+l,j)· „ (E.8b)_ 6 Ä n 1 2 10 11 9 2 .

z-momentum

(-rg-6i#)w(i,j-1,n+1) + (l.+r3-r“+613+61“)w(i,j,n+1)

A— + (rk-613)w(i,j+l,n+1) = (r1+61°)w*(i-1,j) ‘

. + (l.-r +r -6 -6 )w*(i,j) + (6 -r )w*(i+l,j) (E.8c)1 2 10 11 9 2

Again the equations were further condensed by employing the followingnotationsz .

A (j) =·-r -6.·A ‘ — (E.9a) A6 „ 9 lk „

B (j) =A1.+x -r +6 +6 (E-9b)6 9 u 13 lßC (j) = r -6 · . (E.9c)A 6 u 13D (j) = (r +6 )u*(i-l,j) + (1.-r +r -6 -6 )u*(i,j)12 1 10 1 2 10 11‘

+ (6 -r )u*(i+l,j) 1 (E.9d)9 2D (j) = (r +6 ‘)v*(i-l,j) + (1.-r +r -6 -6 )v*(i,j)_ 13 1 10 1 2 10 11 —

+ (6 -r )v*(i+l,j) A (E.9e)9 2D (j) = (r +6 )w*(i-1,j) + (1.-r +r -6 -6 )w*(i,j) ‘

14 1 10 1 2 10 11V. = + (69—r2)w*(i+l,j) ,(E.9f) — ‘

‘ With the above notation, the following algebraic equations re- Agultgd whigh were solved for u, v, and w at the next full step in thex direction. *3

I84 ‘ 2 . . .

TA6(j)ut(1,j-1) + B‘(j)u*(i,j) +·C6(j)u*(i,j+l?.€ DIZÄJ) .

V

3 3 3 3r

j = 2,3,....k . ‘

1 = 2,3,....m (E.10a)A6(j)v*(i,j}l) + B6(j)v*(i,j) + C‘(j)v*(i•J+l) *.@13(J) r

L 2 j- 2,2,....1. .n . A 1 = 2,3,.°..·.m (E.iob>

~ A6(j)w*(i,j-1) + B6(j)w*(i.j) + Ö6(j)W*(i.J+l) ' D1“(J)· ur 343 __ . j ·= 2,3,....k 3 3" r 1 = 2,2,....m (1:.10.:) _

_ Again these equations also yielded tridiagonal matrixes and .were solved bj the Thomas Algorithm which is discussed last in theAppendix.F 1 ' Ü r

3 · r.--

r.» <

F. Finite—Difference Aggroximation of Continuitj3Eguation in the r· 1 Corner Region 3The continuity equation for the corner region was written

lin

finite-difference”form about the point (1+1/2,1+1/2,n+l/2) so thatrl

3l

truncation error would be of O( x}2, 0( j)2, and 0( 3z)2 for theN V

3V

truncation error. Figur.e A.3 illustrates the grid used for the 3 3equationwhich can be written as follows: 2 3

3 {[u(:l+l,j+l,n+l) + u(i+1,j,n+l) + u(:i.,j+l,n+l) + u(i,j,,n+l) ‘3

— u(1+l,j+1,n) — u(1+1,j,n) — u(1,j+1,¤) - u(1,j»,-11)]/dxI.

3T+ [v(i+l,j+l,¤+1) + v(i+1,j,n+1) + v(i+1,j+1,n)‘ 3+ v(1+1,j,n) 3— v(1,j+1,n+1) - v(1,j,n+1) — v(i,j+1,n) - v(i,j,n)]/dy

r ‘A+ [w(i+l,·j+l,n+1) + w(i+1,j+1,3n) + w(i,j+1,n+1) + w(i,j+1,n) 34‘—rw(i+l,_1,n+l,) ·#w(1+1,j,n) - w(i,j,n+l) — w(i,j,n)3j/dz}/.4. = 0 (F.1)

l,

r

_ ' 85

1+1,;+1,11+1

1 Z V · V‘ — 1dxV V

i+l,j ,n 1

V V_isj•n i,j+].,1’1V 1 · dz 1

Fig. A.3. Grid used for finite-difference approximation for ·1 _ continuity equation in the corner reg1on. 1

·1» 1 1 1 1 8

.~ 86 '

° The above equation was used to obtain the residual of the conti-4

A „ . nuity equation at the points (1+1/2,j+l/2,n+l/2) throughout the corner A‘ region. ' 4U

G. Mixing Length for the Corner hegion 4 _

Pletcher's (22) mixing length model for the two-dimensional_flow.4

was adapted for use in the corner region in the following manner.4

Figure A.3 illustrates the corner region and the grid lines.

When attempting to estimate the eddy~viscosity at each nodal· position, a modified mixing length was used. For each nodal position,

_ the line A-A was passed through the node and the grid position where4

. the two—dimensiona1 boundary lines intersected, point—C.. It was fur-4

ther assumed that the isovel 0.999 continued in a circular arc B-hfrom one of the two—dimensional boundaries to the other. The distancealong the line A+A from the wall to where the circular arc B—B interj ‘

' sected the line4A-A was considered the corner boundary thickness for' the mixing length model. The distadhe along the line A-A from the

intersection with the wall to the nodal position in question was con-sidered the Y distance. Together, the two positions on the line A-A

· iwere used in Pletcher's mixing length model to generate a modified _4 4

mixing length at each nodal position. The shear velocity, us, which ·4‘h

4was necessary for Pletcher's formulation, was taken as equal to the

4

two-dimensional region shear velocity since experiments by Gersten4

„ (12) have shown little or no change in total skin friction for planes V4

at ninety4degrees to a flat plate with an equivalent4wetted area. E.

VAÜIIIIl—1

V · ’ Fig. A.4. Corner Grid.

88d 8

„ W“

H. Thomas A1gorithmV·d

·_

T

The following describes the Thomas algorithm as applied to a

general tridiagonal matrix as the one given below for n equations for

the dummy variable h. · ”

blhl + clhz 4 _ V W W - dl_ Wazhl + bzhz + c2h3 V

• dz Jd · aghz + b3h3 + c3h4 = d3

‘W an—lhn-2 +‘bn-lhn-l + cn-lhn S dn-l

fx lrah

+bu“·=la• 8n n-1 n n n

W Where: al = 0 or ho“¥ O and dä = du - cnhu+lThe matrix is then transformed into upper bidiagonal matrix by a

”Gaussian elemination generating the following coefficients:

l

el = bl 'f1·

el -'bi°— aiCi_l/eiW

i = 2,3,....n U

The transformed matrix is as follows Wl

hl + elhz W = fln H h2 + ezhg 8 y y ¤ fz

hn-1._y 4

lv-, d r hn A ¤ fu . e

·l Thua, once fu is determined, hh is known and the remaining values

~

of the dummy variable h can be obtained from the following equation.|A

dV hi · fi e.ciui+1/ei i - n—l,n—2,,..«l

· The above method was considered most appropriate for the ~

solution of the system of equations generated by this finite-differencel

technique} The method greatly reduced the number of calculations

necessary to solve the ever increasing large system ef equationsl

· generated as the solution proceeded in the x-direction. Without this _* method, matrix inversion methods would have been necessary which

generally produce an even larger number of equations for solution —· and thus an even larger number of calculations for each x·step.

l.

-· VIII. APPENDIX B _

U. hp'The following contains

laligting of the computer programs. The

first program is for the laminar case and the second is for the _ _

jTg·’ turbulent case. If desired, different initial u, v, and w profiles ' V.“;,· V}

qéä be uéed by adding the appropriate data cards to the BLOCK DATA\x~

subprogram.f ' I · [ l _Any time the subprogram OUTPDT is called, the matrixes containing

the u, v, and w components are printed. Thus, the programer can have

these matrixes printed by controlling with·one or more parameters ~and obtain information only when necessary or desired by calling the

subprogram OUTPUT. pQ*

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· AN IMPLICIT NUMRICAL SOLUTION FOR THE LAMINAR AND TURBULENT FLOW · — l

OF AN INCOMPRESSIBLE FLUID ALONG THE AXIS OF A 90-DEGREE CORNER

David Tillman Klinksiek

_ · (ABSTRACT)Ii

I4

A method of solving the equations for the three—dimensi¤nal, in-compressible laminar and turbulent flow along the intersection oftwo planes at ninety—degrees has been developed. The Alternating _

”Direction Implicit (ADI) finite-difference method was applied for both)

types of flow. The turbulent stresses in the cdrner region·weremodeled with an eddy—viscosity model which was obtained from mixinglength theory. The method was compared with other types of solutionsfor the laminar case and good agreement was achieved. For the tur-bulent case, the method was compared with experimental data and good

agreement was obtained. ” ‘ i '

The three momentum equations were solved simultaneously and thel

gcontinuity equation was used to verify the method. The method appeared ' l

to predict the velocity components correctly since the continuityA ' equation residual approached zero as the solution prdceeded from the

leading edge in the mainstream flow direction.

No analysis was presented for the convergence or stability of the ‘’ finite—difference equations and no convergence or stability problems

. were encountered when the finite—difference equations weresolved.~The method predicted svmmetry about the corner bisector in all cases·and gave the expected u-velocity profile along the blsector for both

l the laminar and turbulent cases. ,