NASA CONTRACTOR REPORT NASA CR-178818 A COMPUTER … · 2013-08-30 · NASA CONTRACTOR REPORT NASA CR-178818 A COMPUTER CODE FOR THREE-DIMENSIONAL INCOMPRESSIBLE FLOWS USING NONORTHOGONAL

NASA CONTRACTOR

REPORT

NASA CR-178818

A COMPUTER CODE FOR THREE-DIMENSIONAL INCOMPRESSIBLE

FLOWS USING NONORTHOGONAL BODY-FITTED COORDINATE SYSTEMS

By Y. S. Chen

Universities Space Research Association

Science and Engineering Directorate

Atmospheric Sciences Division

Systems Dynamics Laboratory

Final Report

March 1986

[NASA-CR-178818) A C£_POTEE CODE FORTHREE-DIMENSICNAL INCCMPRESSIEL£ FLOWS OSING

NONOR_HOGONAL BCD¥-FI_ED COC_DINATE SYZI_MS

Final Report [Uni.ersities 5_ac£ Research

Association) 86 _ HC A05/M_ A01 CSCL 20D G3/34

N86-26544

Unclas

42915

Prepared for

NASA-Marshall Space Flight Center

Marshall Space Flight Center, Alabama 35812

https://ntrs.nasa.gov/search.jsp?R=19860017072 2020-04-12T10:09:31+00:00Z

TECHNICAL REPORT STANDARD TITLE PAGE

I. REPORT NO. 12. GOVERNMENT ACCESSION NO. 3, RECIPIENT'S CATALOG NO.

NASA CR-178818 14. TITLE AND SUBTITLE 5, REPORT DATE

March 1986A Computer Code for Three-Dimensional Incompressible 6. PERFORMINGORGANIZATIONCODEFlows Using Nonorthogonal Body-Fitted Coordinate Systems

7. AUTHOR(S) 8. PERFORMING ORGANIZATION REPOR r #

*Y. S. Chen

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10, WORK UNIT. NO.

George C. Marshall Space Flight Center 1. CONTRACTORGRANTNO.Marshall Space Flight Center, Alabama 35812 NAS8-35918

13. TYPE OF REPORS & PERIOD COVERED

2. SPONSORING AGENCY NAME AND ADDRESS Final

Contractor Report

National Aeronautics and Space Administration

Washington, D.C. 20546 ,._. SPONSORINCAGENCYCODE

15. SUPPLEMENTARYNOTES *Universities Space Research Association

Prepared by Atmospheric Sciences Division, Systems Dynamics Laboratory, Science

and Engineering" Directorate.

6. ABSTRACT

In this report, a numerical method for solving the equations of motion of

three- dimensional incompressible flows in nonorthogonal body- fitted coordinate (BFC)systems has been developed. The equations of motion are transformed to ageneralized curvilinear coordinate system from which the transformed equations arediscretized using finite difference approximations in the transformed domain. Thehybrid scheme is used to approximate the convection terms in the governing equa-tions. Solutions of the finite difference equations are obtained iteratively by usinga pressure-velocity correction algorithm (SIMPLE-C). Numerical examples of two-and three-dimensional, laminar and turbulent flow problems are employed to evaluatethe accuracy and efficiency of the present computer code. The user's guide andcomputer program listing of the present code are also included.

7. KE'/ WORDS

N avieI - Stoke s Solver

IncompressibleC urvilinear Coordinate

Computer Code

19. SECURITY CLASSIF. (o¢ thl= report_,

18. DISTRIBUTION STATEMENT

Unclassified-Unlimited

SECURITY CLASSIF. (of rid= I_ie)

Unclassified Unclassified NO. OF PAGES ]22. PRICE87 NTIS

MSFC - Form 3292 (May 1969)

For sale by National Technical Information Service, Springfield, Virginia 22151

TABLE OF CONTENTS

INTRODUCTION .............................................................

TRANSFORMATION OF THE EQUATIONS OF MOTION ........................

DISCRETIZATION OF THE EQUATIONS OF MOTION ..........................

SOLUTION PROCEDURES ....................................................

NUMERICAL EXAMPLES ......................................................

A. 2-D Laminar Driven Square-Cavity Flows .........................B. 2-D Laminar Flows Over a Backward-Facing Step .................C. 2-D Turbulent Flows Over a Backward-Facing Step ...............D. Developing Laminar Flow Inside a 90-Deg-Bend Square Duct ......

CONCLUSIONS ..............................................................

REFERENCES ................................................................

APPENDIX A.

APPENDIX B.

APPENDIX C.

COMPUTER CODE STRUCTURES AND USER'S GUIDE .........

LIST OF FORTRAN SYMBOLS ................................

PROGRAM LISTING ..........................................

Page

1

2

5

7

12

12151818

23

24

27

31

41

,oo

III

LIST OF ILLUSTRATIONS

Figure

I.

1

3.

,

t

6.

7.

1

9.

i0.

13.

14.

15.

16.

17.

Title Page

Three-dimensional grid structure and labeling around a gridnode P ............................................................. 6

Locations where the variables are stored ............................ 8

Control volumes where the mass conservation is evaluated

for solving the pressure correction equation ........................ 11

Physical geometry and wall boundary conditions for laminarflows inside a wall-driven square cavity ............................ 12

Mesh systems used for driven cavity problem ....................... 13

Velocity vector plots ............................................... 14

Comparisons of velocity profiles along the mid-section ofthe square cavity .................................................. 14

Convergence history for the driven cavity problem, Re = 400 ....... 15

Physical geometry and boundary conditions of laminar flowsover a backward-facing step ........................................ 16

Reattachment length versus Reynolds number for laminarflows over a backward-facing step ................................. 17

Streamline plots for laminar flow over a backward-facing step ....... 17

Physical geometry and boundary conditions of turbulent flowsover a backward-facing step ........................................ 18

Locus of flow reversal inside the recirculation region forturbulent flow over a backward-facing step ......................... 19

Stream line pattern of turbulent flow over a backward-facingstep with 2:3 expansion ratio ....................................... 19

Contours of turbulent kinetic energy of turbulent flow overa backward-facing step with 2:3 expansion ratio .................... 19

Geometry and mesh system of a 90-deg-bend square ductdeveloping laminar flow problem .................................... 20

Primary velocity patterns of laminar flow inside a 90-deg-bendsquare duct ........................................................ 21

Secondary velocity patterns of laminar flow inside a 90-deg-bend ... 22

Primary velocity profiles for a 3-D 90-deg-bend square duct ........ 23

iv

NOMENCLATURE

Symbol

A

AP

A u

A v

A w

A o

C v

C 1

C 2

CB

D

e

J

k

k'

P

Pr

Q

S

S u

S v

S w

T

t

U

Definition

link coefficient of finite difference equation

link coefficient of the node at the center of a control volume

link coefficient for the u-equation

link coefficient for the v-equation

link coefficient for the w-equation

link coefficient for time marching scheme

specific heat constant at constant volume

turbulence model constant, = 1.44



diffusion coefficient for the pressure correction equation

internal energy per unit mass (jour/kg)

Jacobian of the metric transformation

turbulence kinetic energy (m2/S 2)

thermal conductivity of the fluid

pressure in the fluid (N/m 2)

production term for the turbulent kinetic energy

energy added per unit volume (jour/m 3)

source term

source term of the u-equation

source term of the v-equation

source term of the w-equation

temperature (°K)

time (sec)

velocity in x direction

V

V

W

X

Y

Z

Greek

I"

£

A

¢

_o

°k

gE

;t

;eff

P

Z

i

n

Subscript

i

ref

max

velocity in y direction

velocity in z direction

X-coordinate (m)

Y-coordinate (m)

Z-coordinate (m)

diffusion coefficient

turbulent kinetic energy dissipation rate (m2/s 3)

difference operator

vsriable of general transport equation

solution at the previous time level



curvilinear coordinate


molecular viscosity (N-S/m 2)

turbulent eddy viscosity (N-S/m 2)

effective viscosity (N - S/m 2)

density (kg/m 3)

summation over all values around a grid node P


index of all possible values

reference value

maximum quantity

vi

0

!

Superscript

previous time level solution

current solution

correction quantity

vii

CONTRACTOR REPORT

A COMPUTERCODE FOR THREE-DIMENSIONAL INCOMPRESSIBLE FLOWSUSING NONORTHOGONALBODY-FITTED COORDINATE SYSTEMS

INTRODUCTION

With the currently increasing computer capability and various flow solversdeveloped, numerical simulations of three-dimensional incompressible flow problemsusing Reynolds-average Navier-Stokes equations are now becoming more feasible inmany engineering design and analysis applications. In many real world flow problems,the boundary geometries are complex such that it is more accurate to describe thegeometries using body-fitted coordinate (BFC) systems. Especially for internal flowproblems with complex geometries such as those of the hot gas manifold (HGM) of theSpace Shuttle Main Engine (SSME), the use of nonorthogonal BFC systems for numeri-cal solutions can be beneficial in many aspects. It is not only the boundary geomet-ries that can be represented more closely using BFC systems, but also grid-refinedsolutions can be obtained without increasing an excessive amount in computer memory.In addition, once a particular flow problem has been set up, the redesign or opti-mization process of the boundary shapes can be performed very easily using BFCsystems.

Several numerical methods [1,2, 3,4,5,6] has been developed for solving theincompressible Navier-Stokes equations in 3-D BFC systems. The main differencebetween these methods lies in the way of finding a pressure field such that theflowfield can be as close to divergence-free as possible (i.e. to satisfy the mass con-servation equation). This is the main feature and difficulty of solving the incom-pressible flow problems. Numerical methods of References 1, 2 and 3, for instance,have employed the pseudocompressibility approach and time-iterative scheme to gen-erate the pressure field so that the continuity equation is satisfied when a steadystate solution is reached. In these methods, artificial smoothing techniques must beused to obtain a strong coupling between the velocity and pressure fields. Methodsof References 4, 5 and 6, on the other hand, have utilized a successive pressure-velocity correction scheme by using a Poisson's equation for pressure correctionderived approximately form the continuity and momentum equations. For these lattermethods, grid staggering between the v_locity vectors and the pressure nodes mustbe used to ensure stability of the numerical solutions.

There are several possible methods of grid staggering associated with differentfeatures in solving the pressure correction equation. These grid staggering methodswere discussed in Reference 6, from which one of the methods was shown to be themost promising arrangement (i.e. with the velocity vectors located at the faces of avolume which contains the pressure and other scalars at its center). But, thismethod has one drawback, that the velocity components are solved using differentcontrol volumes. It is for this reason that a grid staggering system similar to theone used by Vanka et al. [4] is developed in the present study. The presentmethod of grid staggering and pressure correction equation that was described byVanka [4[ and Maliska [6]. Also, using the present method, the same control volumeis used for the velocity components and scalar quantities.

In the following sections, basic elements for establishing the present computer

code for solving the curvilinear Navier-Stokes equations in three-dimensional space

(CNS3D) will be described. These are followed by a series of standard numerical

examples used to evaluate the accuracy and efficiency of the present numerical

method. The numerical exampres include laminar flow driven-cavity problem, cases

of laminar and turbulent flows over backward-facing steps, and 3-D laminar flows

inside a 90-deg-bend square duct. Applications of the present code to the internal

flow problems of SSME will be included in future publications.

A user's guide to the present CNS3D code is provided in Appendix A.

Appendix B contains a list and definitions of all the major fortran symbols used in

the computer program which is listed in Appendix C.

TRANSFORMATION OF THE EQUATIONS OF MOTION

For incompressible Newtonian fluid, the continuity, momentum and energy

equations can be written as:

U t + E x + Fy + G z = S(1)

where (x,y,z) represent the Cartesian coordinates, and

U=_

ep

pu

pv

pw

pe-Q

E

"pu

p uu - _/u x

puv - ]_v x

puw - ]JwX

pue - k'TX

pvpvu- _Uy

F =_Rvu _Vy

Ovw p.Wy

PVe k' Ty

G

P

pw

pwu- puZ

pWV- _WZ

p WW - ]JW z

pwe - k' TZ

S

(PUx)x + (PVx)y + (PWx)z - Px

(UUy) x + (_Vy)y + (u Wy) z - Py

+(_ -P(_Uz)x + (;Vz)y Wz)z z

2 + w 2) + + Uy) 2 + (wu[2(Ux 2 + Vy z (Vx y )2 + (u + Wx )2+ VZ Z

_ 2 (u x + v + Wz )2]3 y

e = the internal energy per unit mass = CvT for perfect gas

Q = energy added per unit volume

k' = thermal conductivity of the fluid

Equation (1) is transformed to a general curvilinear coordinate system (_, n,

_), which results in equation (2).

U t + E_ _x + En nx + E_ _x + F_ _y + Fn ny + F_ _y

+ G_ _z + GT] _z + G_ _z = S(2)

where

_x = J(Yn z - y_ z n)

_y = -J(x n z - x z n)

_z = J(x y - x yn)

nx = -J(y_ z - y_ z_)

my = J(x_ z - x z_)

n z = -J(x_ y_ - x y_)

_x = J(Y_ Zn Yn z_)

5y =-J(x_ zn Xn z_)

_z = J(x_ Yn - Xn Y_)

J = i/[x_(yq z - y5 Zrl) Xn(Y _ z - y_ z_) + x (y_ zn - Yn z¢)]

The transformation coefficients, _x' _y' _z' nx' ny, nz, _x' _y' and _z' are com-puted numerically using second order central differencing. In the transformed domain,the grid sizes (i.e., A_, An, and A_) are set to be unity. This simplifies the cal-culation of the transformation coefficients.

For turbulent flow computations, the present code has employed the standardk-¢ turbulence model [7] to provide the turbulent eddy viscosity ut" The standardk-¢ turbulence model (which consists of a turbulent kinetic energy equation, k-equation, and a turbulent kinetic energy dissipation rate equation, E-equation) isgiven as:

p _eff k ) = _(Pk)t + uik ok x i x i P(Pr_) (3)

P eff )E

(PE)t + Puik o x i- C ¢) (4)x. = P k (CIPr 2

1

where the effective viscosity Ueff is calculated from:

Peff = p + ut = _ + p C k2/Et.l

and the turbulent kinetic energy production term, Pr' is defined as:

k 2 2 2

Pr = Cu --_ [(Uy + vx) + (v z + Wy.) + (w x + u z)

2 2+ 2(u x

2+ V

Y+ w 2)]

Z

The turbulence model constants are:

C = 0.09 , ok = 1.0 , o = 1.3H

C = 1.44 C = 1.921 ' 2

Also, the molecular viscosity u in equation (i) is replaced by the effective viscosity

Ueff for turbulent flow cases.

In order to save the computational efforts, the widely used wall function

approach [8] is employed to provide the near wall boundary conditions for the momen-

tum and energy equations and the k-_ turbulence model. This approach avoids the

requirement of integrating the governing equations up to the wall which requires alarge number of additional grid points near the wall.

Equations (2), (3), and (4) form a closed set of nonlinear partial differentialequations governing the fluid motion. This set of equations are to be solved bymeans of finite difference approximations which are performed in the transformeddomain. For treating the convection terms, the hybrid scheme [9] is employed forsimplicity (although other more elaborate schemes such as central differencing plusartificial dissipation scheme, QUICK scheme, or skew upwind differencing scheme, etc.can be implemented [10]). These are described in the following sections.

DISCRETIZATION OF THE EQUATIONS OF MOTION

In this section, finite difference approximations are used to discretize thegoverning equations, equations (2), (3), and (4). Second-order central differencingis used for the diffusion terms and the source terms. The hybrid differencing scheme[9] is employed to approximate the convection terms in the governing equations. Thefinite difference discretizations are performed in the transformed domain. The solu-tion procedure for the discretized equations using a velocity-pressure correctionalgorithm (SIMPLE-C) of References II and 12 will be described in the next section.

The governing equations of motion can be represented by the following modeltransport equation in which ¢ denotes all the dependent variables respectively and ris the diffusion coefficient..

(P¢)t + [pu¢ - I'(¢_ _x + Cn nx + _r _x)]_ _x

+ [pu¢ - r(_¢ _x + Cn nx + ¢_ _x)]n nx

+ [pu, - r(¢g gx + Cn nx + ¢_ _x)]¢ Cx

- + #_ _y) ] _y+ [pvqb F(¢_ _y + q_n ny _

+ [pvq_ - F(¢_ _y + Cn ny + _ _Y)]n ny

+ [pvq_ - F(qb _Y + _n ny + _ qy)]_ _y

+ Cn n + ¢_ _z )] _z+ [pw¢ r(¢_ _z z

- + Cn n + ¢ _z)] nz+ [pw¢ r(¢¢ Cz z _ n

+ Cn n + ¢ _z)] _z = S (5)+ [pw¢ r(¢_ _z z _

5

Discretization of equation (5) is performed using finite difference approximations inthe transformed domain. The second order central differencing is used for approxi-mating the diffusion terms. For the convection terms, the hybrid differencing scheme[9] is employed (i.e., using central differencing for cell Peclet number less than orequal to 2 and switching to upwind differencing when the cell Peclet number isgreater than 2). The finite difference equation is arranged by collecting termsaccording to the grid nodes around a control volume as shown in Figure i. The finalexpression is given by equation (6) in which A represents the link coefficientsbetween grid nodes P, E, W, N, S, T, B, NE, NW, NT, NB, SE, SW, ST, SB, ET,EB, WT, and WB as shown in Figure i.

Ap _p = A E _E + AW CW + AN _N + AS ¢S + AT CT + AB CB + Sl (6)

where

S I = S + Ap ° _pO + AN E ¢NE + ANW _NW + ANT _NT + ANB _NB

+ ASE q_SE + ASW _SW + AST ¢ST + ASB ¢SB

+ AET _ET + AEB _EB + AWT _WT + AWB _WB

Ap = A E + A W + A N + A S + A T + A B + Ap °

Ap ° = pp°/At

The subscript o denotes the solution at the previous time level. A fully implicitformulation is employed for solving the time dependent transient problems.

Figure 1.

ET

TJ

WB

Three-dimensional grid structure and labeling around a grid node P.

Thus, the nonlinear equations of motion are approximated by a system of linearalgebraic equations which have the form of equation (6). Only one program subrou-tine is designed to calculate the link coefficients and the source terms. The numberof algebraic equations depends on the number of interior grid points. For a gridsize of 10 x 10 x 10 the number of algebraic equations to be solved would be around512. This large system of equations are preferred to be solved by some iterativemethods, such as Gauss-Seidel iteration, line-underrelaxation method [13] or Stone'smethod [14], etc., rather than using direct methods such as Gaussian eliminationmethod. Only a few (6 to 10) iterations through the whole computational domain areneeded and a complete convergence of the system of algebraic equations is notrequired. Since equation (6) is only a linearized version of the governing equationswhich are nonlinear and coupled in nature, solutions of the equations of motion mustbe obtained through global iterations among the equations. A tentative solution toequation (6) will not affect the final results significantly. On the other hand, if toomany iterations are used to get a better solution of equation (6), then a great dealof computing time would be virtually wasted. However, the above argument can notbe applied when the pressure correction equation (which will be derived in the nextsection) is solved. Since during each global iteration it is desirable to retain adivergence-free velocity field, better solution of the pressure correction equationwould in effect promote the convergence of the whole numerical scheme. Therefore,more iterations are usually used to solve the pressure correction equation.

SOLUTION PROCEDURES

The governing equations used in the present analysis are nonlinear and stronglycoupled. Iterative procedures are employed to drive the equations to a convergedsolution. It is particularly important for incompressible flow to make the flow fieldsatisfy the continuity equation and the momentum equations at the same time. Thisrequires a correct pressure field associated with a divergence-free velocity field.A velocity-pressure correction procedure is developed in the present study to drivethe pressure field and the velocity field to be divergence free. This kind of pro-cedure requires grid staggering between the velocity components and the locationswhere the pressure is estimated and stored such that the velocity field and thepressure field will not be uncoupled.

In the present study, staggering grid systems as shown in Figure 2 (for 2-Dcase) are used. The velocity components, u and v, are solved and stored at thegrid nodes and the pressure, p, is located at the corners of the control volume ofu and v. In this way, solutions of u and v can be solved using the same controlvolume and coupling between u, v and p can also be enforced. To estimate thepressure field, a pressure correction equation is derived approximately from the dis-cretized momentum and continuity equations. The velocity and pressure fields arethen corrected using the solutions of the pressure correction equation.

as.First, the finite difference momentum equations (for u, v and w) can be written

ApUup* = _ A.u u.* = P * + S1 1 X U

i

(8a)

X

X

X X

X

X

O

X

u, v AND SCALARS

PRESSURE

Figure 2. Locations where the variables are stored(staggering grids are used).

Ap v Vp* = E A'v vi* = P * + S (8b)I y vi

Ap w Wp* = Z A'w w.* P * + S (8c)I I Z W

i

where u*, v*, w*, and p* represent the solutions of equations (8a) and (Sb). Tosatisfy the continuity equation the velocities and pressure are corrected according tothe following relations:

u = u* + u' (9a)

v = v* + v' (9b)

w = w* + w' (9c)

P = P* + P' (9d)

A new set of momentum equations can be constructed approximately using thedivergence- free flow field, u, v, w, and p:

Ap u Up = E A'u ui - P + S1 X U

i

(10a)

Ap v Vp = __ A. v v. - P + S1 1 y vi

(lOb)

Ap w Wp = E A'wl w.1 - Pz + Sw

i

(10c)

By subtracting equations (8a) through (8c) from equations (10a) through (10c),

respectively, the following equations result:

Ap u Up' = Z A'ul U-'l - Px'

i

(lla)

Ap v Vp' = E A'vl v.'l- Py'

i

(llb)

Ap w = E A'w 'Wp' I wi' - Pz

i

(llc)

According to SIMPLE-C algorithm [11], equations (11a) through (11c) are rearrangedto be:

(Ap u - E AiU) Up' = E A'ul (ui' - Up') - Px'

i i

(12a)

(Ap v -_ AiV) Vp' = .v (vi' - Vp') - P 'i y

i

(12b)

(Ap w E Aiw) Wp' = E Ai w (w i' - Wp')

i i

pz

(12c)

The first terms on the right-hand side of equations (12a) through (12c) are neglectedto simplify the formulation. Thus,

Up' = - ( 1 ) P ' = - DApU _ u x u- A i

i

P ' (13a)X

Vp, = _(ApV _I A'V)l PY' =- Dv Py' (13b)

WP' = - IA _ ) P v = - D P'pW A.Wl z w z " (13c)

i

Using the decompositions of equations (9a) through (9c), the continuity equation canbe written as:

u x + Vy + w z = (Ux* + Vy* + Wz *) + (Ux' + Vy' + Wz ') = 0 . (14)

Substituting equations (13a) through (13c) into equation (14), the following pressurecorrection equation can be obtained:

-[(DuPx') x + (DvPy')y + (DwPz') z] = -(Ux* + Vy* + Wz *) (15)

Equation (15) is a Poisson's equation with the source term equal to the local diver-

gence of the flow field. To enforce the coupling between the velocity and pressurefields, the source term of equation (15) is first evaluated at the control volumes

centered between the velocity nodes as shown in Figure 3. An averaged source termis then calculated at the cell center of p node for solving equation (15). In this way,the difficulties in solving the pressure correction equation, as described by Vanka [4]and Maliska [ 6], are eliminated. Coupling between the velocity and the pressure fieldis also assured.

According to the above analyses, the present numerical method contains thefollowing solution steps:

1) Guess initial velocity and pressure field.

2) Solve for the velocity field using equations (8a) through (8c).

3) Solve for other scalar transport equations.

4) Solve the pressure correction equation, equation (15).

i0

"ROLVOLUME

Figure 3. Control volumes where the mass conservation is evaluatedfor solving the pressure correction equation.

5) Correct the velocity and pressure fields using equations (13a) through

(13c) and equation (9d).

6) Go back to step (2) until solution converges.

A converged solution is obtained when the following criterion is met:

2Error = (IAU]max + lAY[max + [AWlmax) / Ure f + ]P'[max/PUre f

-4_ 3x i0

where Au, Av, and Aw represent velocity changes during each iteration due to the

solutions of the momentum equations.

In solving the momentum equations in step (b) above, underrelaxation factor

of about 0.6 is recommended. With this, Ap's in equations (8a) through (8c) are

modified according to the underrelaxation factor. For the correction of velocity field,no underrelaxation is required. But the correction of pressure field should be under-relaxed slightly (around 0.9) when the grid nonorthogonality is strong. This isdifferent from that suggested by References 11 and 12 (which recommend no under-relaxation for pressure correction).

11

NUMERICAL EXAMPLES

In this section, several numerical examples are employed to demonstrate theefficiency and accuracy of the present numerical method. To serve this purpose, 2-Dand 3-D, laminar and turbulent flow cases are included. These cases are: (a) 2-Dlaminar driven square-cavity flows; (b) 2-D laminar flows over a backward-facingstep; (c) 2-D turbulent flows over a backward-facing step; (d) 3-D developinglaminar flow inside a 90-deg-bend square duct. Detailed descriptions and results ofthe computation of the above cases are included as follows.

A. 2-D Laminar Driven Square-Cavity Flows

The first test case is concerning laminar recirculating flows inside a squarecavity. Only one side of the walls is moving at a constant speed tangent to thatwall. This case has been studied extensively by Burggraf [15] and has often beenused as one of the standard testing cases for numerical methods in solving theincompressible Navier-Stokes equations. Physical geometry and wall boundary condi-tions are illustrated in Figure 4. Reynolds number of the flow (based on the cavitysize and the moving wall velocity) studied in the present analysis is 400. Twodifferent mesh systems, as shown in Figure 5, are used to study the effect of gridnon-orthogonality on the accuracy of the present method. The grid system of Figure5(a) is uniform and orthogonal while the grid system of Figure 5(b) is non-uniformand non-orthogonal.

Y

i

W

UPPERWALL, v = oII

__ "/f////////////1111/I///I/1111//I/I/111////I//111/11/1111,lj

lj

lj

cj

rA

vA

vA

v_

vj

IA

,/vj

rA

tA

v_

IA

U=O_V = 0'/

tj

//

/,f,

/j

r,

/A

g*

c 0"illllllllllll/illllillllillilllillllilllllll////lllliJ

U=O V=O

//

U=O

V=O

Figure 4. Physical geometry and wall boundary conditions forlaminar flows inside a wall-driven square cavity.

12

_ I

c_

E

o°r-i •

_'_

o

o

_z

m

_ om bl_

• o

b_

13

Results of the computations are shown in Figures 6 and 7. Velocity vectorplots of the predicted flow fields are compared in Figure 6 for the mesh systemsshown in Figure 5. Detailed comparisons of the predicted velocity profiles along themid-section of the cavity are illustrated in Figure 7. Predicted results of Burggraf[15] are also included. Good agreements between the present calculations and those

(a)

,T.tfc_7..-_,_- - ---- ----' '- ,.'_\\ t . . _--r"..... -_

i I I I t t # # Ili't.,r_ ----0_ _\ I

,_rrtttt'_','.zl/.. ---..'/ltt, ,1 r I i i_;llt¢_,'--,, .' _1] 1_' ",, :,,

J ) t \ \ \\\\\- "--..'."/l"ll/li'l,

, , _ '_ t \ \ ,,,..._--_ _//'J/Vl6 '/

(b)

Figure 6.

ylw

1.0

0.8

0.6 --

0.4

0-0.4

0.2

Figure 7.

Velocity vector plots. (a) Orthogonal grid.(b) Nonorthogonal grid.

I I I I

PRESENT METHOD

(ORTHOGONAL GRID)

------ PRESENT METHOD(NONORTHOGONAL GRID)

BURGGRAF [15]

Re = 400

I I I I I

-0.2 0 0.2 0.4 0.6 0.8 1.0

U/Uupperwall

Comparisons of velocity profiles along the mid-sectionof the square cavity.

14

of Burggraf [15] are also included. Good agreements between the present calcula-tions and those of Burggraf are shown in Figure 7. Discrepancies between the pre-sent predictions and Burggraf's results are mainly due to the hybrid differencingscheme used in the present method. The upwind part of the hybrid scheme produceslarge numerical diffusion which tends to reduce the strength of the vorticity insidethe cavity. Effects of differencing schemes in approximating the convection terms onthe predicted results will be studied in the next test case.

Convergence history of the computation of the present case using uniform gridsis given in Figure 8 which shows that the present numerical method is quite different.Almost identical convergence rates were found for the non-orthogonal case.

0

-1

LOG (AUmax_\ Uref /,

(AVmax._LOG \ Uref / -2OR

LoG/APmax_

-3

Figure 8.

I I I

U

V

P

100 200--4

0 300 400

ITERATION

Convergence history for the driven cavity problem, Re = 400.

B. 2-D Laminar Flows Over a Backward-facing Step

This test case concerns 2-D laminar recirculating flows over a backward-facing

step with 1:2 expansion ratio. The dependence of the size of the recirculationregion (characterized by the reattachment length) on the Reynolds number (basedon the inlet bulk velocity and twice of the inlet channel width) of the flow is of

major concern. The physical domain and boundary conditions are illustrated inFigure 9 in which a fully developed laminar flow velocity profile is imposed at theflow entrance. A non-uniform grid of 45 x 45 was used for numerical computations.Several cases with different Reynolds numbers from i00 to 800 have been studied.An experimental and theoretical study about this problem, which results will be usedas the basis of data comparisons, has been provided by Amaly et al. [16].

15

h

hU=O/

y=O//

U = O. V = 0

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII,

,/1/1/1/1//////////////////////////////////////////////////////////////////,

Xr "I u = o, v = oJ

30 h

du

_'x=O

dv_--_= o

Figure 9. Physical geometry and boundary conditions of laminar flowsover a backward-facing step (1:2 expansion).

To save computational efforts, the solution of one case with Reynolds number

i00 is obtained in the first run. Then, a series of cases with increasing Reynoldsnumbers (i.e., 100, 200, 300, 400, 600, and 800) are calculated using the precedingresults of lower Reynolds number as the initial guesses of the flow field. In this

way, an average of 500 iterations for each case were needed to obtain convergedsolutions.

Two different differencing schemes in approximating the convection terms are

employed to demonstrate the effects of the differencing schemes on the predictions.One of the schemes is the widely used hybrid scheme [9]. The other scheme employsthe central differencing scheme plus an artificial dissipation term used to stabilize

the solution which is similar to the one used by Rhie [ 17]. The artificial dissipationterm becomes effective only when the cell Peclet number (or cell Reynolds number)exceeds 10.

Results of the present predictions using two different differencing schemes arecompared with the experimental measurements [16] and other predictions as shown in

Figure 10. It can be seen clearly from Figure i0 that the present method with hybrid

scheme gives results similar to those predicted by TEACH code [16] while the present

method with central differencing and artificialdissipation reveals predictions close tothose predicted by INS3D [18] and the method of Kim and Moin [19]. This is reason-

able since the TEACH code and the present method (with the first scheme) use the

hybrid scheme which introduces large numerical dissipation by its upwind part (for

cell Peclet number greater than 2). This tends to reduce the reattachment lengthfor Reynolds number greater than 400. The second scheme, which is similar to the

ones used in INS3D and the method of Kim and Moin, has the numerical accuracy

close to second order by setting the artificialdissipation to be as small as the solu-

tion stability permits such that better accuracy of the predictions is expected.

16

16

14

12

10

e-

s

6

4

2

0

I I IPRESENT CODE WITH CENTRAL

DIFFERENCING SCHEME

PRESENT CODE WITH HYBRID

SCHEME • _**"

INS3D [18] __e ,/_

......... K,M MO,N[181TEACH CODE [16] _y -I

e EXP. [16] .,c_t_,_ I

/ _I 1 I

0 200 400 600 800

REYNOLDS NUMBER

Figure 10. Reattachment length versus Reynolds number for laminarflows over a backward-facing step (1:2 expansion).

Stream function plots of the predictions using the two differencing schemes for

Reynolds number 600 are compared in Figure 11. It is shown in Figure 11 that the

second scheme gives a smooth shape of the recirculation zone while the hybrid scheme

gives a sudden change in the shape of the recirculation region upstream of thereattachment point. Also, larger sizes of the separation regions on the step side

wall and along the upper wall are predicted using the second scheme.

(a)

I23h

IX 23h

(b)

Figure 11. Streamline plots for laminar flow over a backward-facing step

(l: 2 expansion). (a) Hybrid Scheme. (b) Central differencing

plus artificial dissipation scheme.

17

C. 2-D Turbulent Flows Over a Backward-Facing Step

In order to demonstrate the applicability of the present method to turbulentflow case, one of the standard test cases presented in the Stanford Conference [20]is selected here (i.e., turbulent flow over a 2:3 expansion backward-facing step).The standard k-c turbulence model was used to provide the eddy viscosity for thetransport equations. The physical geometry and boundary conditions imposed areshown in Figure 12. The calculation domain extends upstream of the expansion planeby 4 step heights and downstream of the expansion plane by 30 step heights toassure a fully developed velocity profile at the exit. A uniform velocity profile islocated at the inlet plane. A 45 x 42 grid was used in the computation. 300 itera-tions were required to obtain converged solutions. Only hybrid differencing schemeswere used in this case.

U=O,V--O

T'

_-4h_ u = o, v = o24h m

du

_'=o

dv_-_'=o

dk

de

Figure 12. Physical geometry and boundary conditions of turbulent flows

over a backward-facing step (2:3 expansion).

Results of the computation are shown in Figures 13, 14, and 15. These resultsare compared with the experimental measurements [20]. The under-prediction of the

reattachment length is mainly due to the fast development of the mixing layer down-stream of the expansion plane which is the characteristics of the standard k-_ tur-

bulence model. Numerical diffusion provided by the hybrid scheme also contributessome part to the discrepancies between the predictions and measurements.

D. Developing Laminar Flow Inside a 90-Deg-Bend Square Duct

This test ease simulates a three-dimensional developing laminar flow inside a90-deg-bend square duct as illustrated in Figure 16(a). The symmetry plane islocated at z = 0 where the symmetric boundary conditions are imposed. A fullydeveloped velocity profile of laminar flow inside a straight square duct is prescribedat the entrance which is 2.8 duct widths upstream of the bend, A zero pressure

18

PRESENT METHOD (k-e MODEL)

o EXP. [20]

y/h

1.0

.8

.6

.4

Q

® ®

2 4

O

l I0

0 6 8

x/h

Figure 13. Locus of flow reversal inside the recirculation region forturbulent flow over a backward-facing step (2:3 expansion).

5.4h

Figure 14. Stream line pattern of turbulent flow over a backward-facing

step with 2:3 expansion ratio.

i_=--------_---= - ......... 02 .01

Figure 15. Contours of turbulent kinetic energy (k/Uo 2) of turbulent flow

over a backward-facing step with 2:3 expansion ratio.

19

z

(a)

(b)

Figure 16. Geometry and mesh system of a 90-deg-bend square ductdeveloping laminar flow problem.

gradient exit (which is 4.5 duct widths downstream of the bend) boundary condition

is imposed. The Reynolds number of the flow (based on the duct hydrolic diameterand the inlet bulk velocity) is 790. A 21 x 18 x 10 grid was used for numerical

computations. The front view and side view of the mesh system are illustrated in

Figure 16(b). Experimental measurements of Humphrey et al. [21] are used for datacomparisons.

Velocity vector plots on three sections along the main flow directions (i.e., on

x-y plane) are shown in Figure 17. Secondary flow patterns at several stations

across the bend are illustrated in Figure 18. These results are very similar to thoseobtained by Vanka [22] and Rhie [23]. Grid sizes of 50 x 22 x 15 and 58 x 15 x ll

were used by Rhie and Vanka, respectively. The present investigation, using onlyless than half of their grid numbers, gives highly encouraging results. Detailed

comparisons between the measured and the predicted main ve]oeity profiles are givenin Figure 19.

With the above successful numerical simulations, it is believed that the present

numerical method can be applied to general fluid dynamics problems with goodnumerical accuracy and efficiency.

2O

o

r/l

I

b_ •

II

0.-I

r/l ,.-..,

0

_ °

bl

0_,

II

0

%

b_°r-I

21

.... , r t t/e

...,11111

\\\....._._..__.....-- ...--....-....-....-.till A

v

/, . , , , , _ _111

1 I ' tf\\...,...____.__._..-.-,,..If/ill

A

1.1

l,,.-'l.,-'-'--"-- "" "" "" "" \ %,,

11 . , , r llll

I \ \ "" "" """ -- "3

L

A

v

N

L

.Q

I

ob_

coo

o o

N

_m• ,"_ II

o

r_tl

N_0_-,I

22

(a) 0

r/d 0.5

1.00

(b) 0

r/d 0.5

1.00

o= ooW,,o°o=,oo, .......0.5 1.0 1.5 PRESENT RESULT(21x18x 10 GRID)

U/Uref .... RHIE (22 x 15 x 50 GRID) [23]

® EXP., HUMPHREY ET. AL. [21]

I I

O0 t

= 0

I_ I _

0.5 1.0 1.5

U/Uref

Figure 19. Primary velocity profiles for a 3-D 90-deg-bendsquare duct. (a) z/d = 0.0. (b) z/d = 0.25.

CONCLUSIONS

A numerical method for solving the steady or transient incompressible Navier-

Stokes equations in three-dimensional body-fitted coordinate systems has beendeveloped. In the present paper, the basic numerical algorithms and grid arrange-ments have been described in detail. A brief user's guide to the present computer

code (CNS3D) has been included in Appendix A. A program listing has also beenattached in Appendix C.

Several numerical testing examples of 2-D and 3-D, laminar and turbulent flow

problems included in the present work have demonstrated that the present computercode is efficient and robust, and can be used as a reliable tool for engineering

design and analysis applications. Applications of the present code to the internalturbulent flow problems of the SSME will be presented in the future publications.

23

REFERENCES

.

*

.

.

.

0

.

.

.

10.

Ii.

12.

13.

14.

15.

Chorin, A. J. : A Numerical Method for Solving Incompressible Viscous FlowProblems. Journal of Comp. Physics, Vol. 2, 1967, pp. 12-26.

Steger, J. L. and Kutler, P.: Implicit Finite-Difference Procedures for theComputation of Vortex Wakes. AIAA J., Vol. 15, No. 4, April 1977, pp.58 I- 590.

Kwak, D., Chang, J. L. C., Shanks, S. P., and Chakravarthy, S.: AnIncompressible Navier-Stokes Flow Solver in Three-Dimensional CurvilinearCoordinate Systems Using Primitive Variables. AIAA Paper 80-0253, Reno,Nev., Jan. 1984.

Vanka, S. P., Chen, B. C.-J., and Sha, W. T.: A Semi-lmplicit CalculationProcedure for Flows Described in Boundary-Fitted Coordinates Systems.Numerical Heat Transfer, Vol. 3, 1980, pp. 1-19.

Shyy, W., Tong, S. S., and Correa, S. M.: Numerical Reeirculating FlowCalculation Using a Body-Fitted Coordinate System. Numerical Heat Transfer,Vol. 8, 1985, pp. 99-113.

Maliska, C. R., and Raithby, G. D.: A Method for Computing Three Dimen-sional Flows Using Non-Orthogonal Boundary-Fitted Coordinates. InternationalJ. for Numerical Methods in Fluids, Vol. 4, 1984, pp. 519-537.

Launder, B. E., and Spalding, D. B.: Mathematical Models of Turbulence.Academic Press, London, 1972.

Launder, B. E., and Spalding, D. B. : The Numerical Computation of Turbu-lent Flows. Comp. Meth. Appl. Mech. Engr., Vol. 3, 1974, pp. 269-289.

Patankar, S. V.: Numerical Heat Transfer and Fluid Flow. McGraw-Hill Book

Company, 1980.

Syed, Saadat A., Chiappetta, L. M., and Gosman, A. D.: Error ReductionProgram - Final Report. NASA CR-174776, January 11, 1985.

Van Doormal, J. P., and Raithby, G. D.: Enhancements of the SIMPLE Methodfor Predicting Incompressible Fluid Flows. Numerical Heat Transfer, Vol. 7,1984, pp. 147- 163.

Latimer, B. R. and Pollard, A. : Comparison of Pressure-Velocity CouplingSolution Algorithms. Numerical Heat Transfer, Vol. 8, 1985, pp. 635-652.

Anderson, D. A., Tannehill, J. C., and Pletcher, R. H.: Computational FluidMechanics and Heat Transfer. McGraw-Hill Book Company, 1984.

Stone, H. L. : Iterative Solution of Implicit Approximations of MultidimensionalPartial Differential Equations. SIAM J. Numer. Anal., Vol. 5, pp. 530-558.

Burggraf, O. R.: Analytical and Numerical Studies of the Structure of SteadySeparated Flows. J. Fluid Mech., Vol. 24, Part l, 1966, pp. 113-151.

24

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

Armaly, B. F., Durst, F., Pereira, J. C. F., and Sch}_nung, B.:mental and Theoretical Investigation of Backward-Facing Step Flow.Mech., Vol. 127, !983, pp. 473-496.

Experi-J. Fluid

Rhie, C. M. : A Pressure Based Navier-Stokes Solver Using the MultigridMethod. AIAA Paper 86-0207, Reno, Nev., January 1986.

Rogers, S. E., Kwak, D., and Kaul, U.: On the Accuracy of the Pseudo-compressibility Method in Solving the Incompressible Navier-Stokes Equations.AIAA Paper 85-1689, AIAA 18th Fluid Dynamics and Plasmadynamics and LaserConference, July 16-18, 1985, Cincinnati, Ohio.

Kim. J. and Moin, P.: Application of a Fractional-Step Method to IncompressibleNavier-Stokes Equations. J. of Comp. Phy., Vol. 59, 1985, pp. 308-323.

Kline, S. J., Cantwell, B. J., and Lilley, G. M. (Ed.): The 1980-1981AFOSR-HTTM-Stanford Conference on Complex Turbulent Flows. StanfordUniversity, I, II and III.

Humphrey, J. A. C., Taylor, A. M. K., and Whitelaw, J. H.: Laminar Flowin a Square Duct of Strong Curvature. J. Fluid Mech., Vol. 83, Part 3, 1977,p. 509.

Vanka, S. P. : Block-lmplicit Calculations of Three-Dimensional Laminar Flow

in Strongly Curved Ducts. AIAA J., Vol. 23, No. 12, December 1985, p. 1989.

Rhie, C. M.: A Three-Dimensional Passage Flow Analysis Method Aimed at

Centrifugal Impellers. Computers and Fluids, Vol. 13, No. 4, 1985, pp. 443-460.

Chieng, C. C. and Launder, B. E.: On the Calculation of Turbulent Heat

Transport Downstream from an Abrupt Pipe Expansion. Numerical Heat Trans-fer, Vol. 3, 1980, pp. 189-207.

Amano, R. S.: Development of a Turbulence Near-Wall Model and Its Applica-

tion to Separated and Reattached Flows. Numerical Heat Transfer, Vol. 7,

1984, pp. 59- 75.

Chen, Y. S. : Applications of a New Wall Function to Turbulent Flow Compu-

tations. AIAA Paper 86-0438, Reno, Nev., January 1986.

25

APPENDIX A

COMPUTER CODE STRUCTURES AND USER'S GUIDE

The global structure of the present computer code (CNS3D) can be representedby a flow chart, shown in Figure A-1. The user is referred to Appendix C fordetailed information. First, the program requests inputs, from logic unit 5 (LU = 5),

of program control parameters that specify the maximum number of iterations, thetype of flow (i.e., laminar or turbulent), number of iterations for solving the pres-sure correction equation (typically 10), and underrelaxation factors for solving thetransport equations, etc. This is followed by the definitions of all the program con-stants including turbulence model constants (these constants are subject to changeaccording to the user's specific flow problem). Next, the program asks for inputs ofthe initial flow field guess from a restart file (LU = 8) which contains the grid

system coordinates and flow field data that may be created by the user (includinggrid generation) or obtained from the previous solutions. Format of this data file isalso subject to change according to the user's preference. Next, wall boundary con-

trol parameters, boundary grid normal distance to the wall, and wall boundary direc-tion cosine are calculated in subroutine DIRCOS. Subroutine TRANF is then invoked

to obtain the grid transformation coefficients. Before the solution procedure starts,the inlet mass flow rate is calculated which will be used to control the outlet massflow rate to enhance mass conservation. The solution procedures consist of a series

of subroutine calls to SOLVEQ starting from the solutions of the velocity vectors, u,v, and w, and then the solutions of scalar quantities (including the energy equationand the turbulence model equations) and finally the solution of the pressure correc-tion equation to update the velocity and pressure field such that a divergence-freeflow field can be retained.

After each global iteration of the solution procedures, the numerical of itera-tions and the maximum flow field corrections are checked with the initial settings.

If the convergence criterion is satisfied or the number of iterations reaches theprescribed value then the solution procedures stop and the flow field solutions willbe written on the pre-assigned disc file (LU = 7).

For instance, if a steady-state laminar flow problem (Reynolds number of 600)

is of interest and a converged solution is expected within 300 iterations and thenumber of iterations for solving the pressure correction equation is I0 and theunderrelaxation factors are 0.5 and 0.95 for transport equations and pressure correc-

tion equation, respectively, the first inputs from LU = 5 would be:

Line

1. 300 1 10 1

2. 0.5 0.5 0.5 0.95 0.5 0.5 0.5 0.5

3. 600. 0.0

In the second input sequence (i.e., from restart file), the program reads inL x M x N lines of data records. See Figure A-2 for grid structures. Notice that

the program requires variable dimensions of (L+I, M+I, N+I) for solving the pressurecorrection equation. It is important to check the COMMON table for proper variabledimensions.

u., _:

_o_

_Z

0

Z

_z

d _a_

m

(,/)

re_

\_z_- /

I

a

ee--

z__aa

I

_z_0w_

ez_

_o_g

O0

E

bno

o

[]oo

otoo

o

o

o

o

o

o

N_0_,-¢

28

(I = 1, J = 1, K = N),\ = = =

\

(I=L,J=I,K=I)(I=I,J=M,K=I)

(I=1,J=1, K=1)

Figure A-2. Grid mesh structures for 3-D calculations.

If the flow problem involves symmetric or cyclic boundary conditions, then theuser can look into the subroutine SYMOUT to specify the appropriate boundary con-ditions (the conditions shown in the program listing of Appendix C are for symmetric

boundary conditions at K = 1). For cyclic boundary conditions at K = 1 and K = N,data at K = 2 and K = N-1 can be used to obtain boundary conditions at K = 1 and

K = N by requiring same gradients across K = 1 and K = N. This method is simplebut will lag the boundary conditions by one iteration. A direct method withoutlagging the boundary conditions can also be employed by modifying the subroutineof linear algebra solver LINERX such that the boundary conditions can be part of thesolution of the TDMA (tridiagonal matrix) solver.

In case of incorporating different wall functions for turbulent flow problems(e.g., References 24, 25, and 26), subroutine BOUNC and WALFN can be modifiedaccording to the user's method of wall treatments. The set of wall functions givenin the program listing of Appendix C are derived from the conventional wall law andthe equilibrium turbulent kinetic energy relations [8].

When additional source terms are to be added to the transport equations due to

flow problem requirements, modifications to the source term calculation section in thesubroutine SOLVEQ can be carried out. Notice that in the subroutine SOLVEQ sourceterms for the velocities v and w are included in the u-source section. Purpose of

this is to save some computing time since these source terms use similar calculationroutines.

Some times it is required to solve more transport equations other than the basic

ones included in Appendix C. To modify the program to incorporate more equations,

29

several changes are necessary. First, new variables must be added to the COMMONtable (this can be easily done through the computer editor session). Then, newsource term sections are added in the SOLVEQ subroutine. Finally, subroutinesWALFN and SYMOUT are modified to incorporate the new variables into the boundarycondition setting routines.

3O

A(K)

AB(I,J,K)

AE(I ,J,K)

ALC

ALE

ALK

ALP

ALU

ALV

ALVIS

ALW

AN(I ,J,K)

ANAB

ANVI(I)

ANWl(I)

AP(I ,J,K)

APO(I,J,K)

ARDEN

AREA

AS(I ,J,K)

AT(I ,J,K)

AW(I ,J,K)

B(K)

BB(I ,J,K)

BOUNC

APPENDIX B

LIST OF FORTRAN SYMBOLS

= Matrix elements of a tridiagonal matrix

= Link coefficients through the bottom face of a control volume

= Link coefficients through the east face of a control volume

= Underrelaxation factor for symmetry or cyclic boundary conditions

= Underrelaxation factor for the c-equation

= Underrelaxation factor for the k-equation

= Underrelaxation factor for the pressure correction equation

= Underrelaxation factor for the u-equation

= Underrelaxation factor for the v-equation

= Underrelaxation factor for the effective viscosity

= Underrelaxation factor for the w-equation

= Link coefficient through the north face of a control volume

= Sum of the link coefficients at all faces

= Modified wall boundary link coefficient for v-equation

= Modified wall boundary link coefficient for w-equation

= Sum of the link coefficients around a control volume

= Link coefficients in time marching direction

= Area times density across a section in physical domain

= Area of a section in physical domain

= Link coefficients through the south face of a control volume

= Link coefficients through the top face of a control volume

= Link coefficients through the west face of a control volume


= Coefficients in Stone's partial faetorization technique

= Subroutine for getting turbulent wall boundary conditions throughwall functions

I 31

C(K)

C1

C2

CB

CE

CK

CMU

CMU1

CMU2

CN

CS

CT

CW

CX(I ,J,K)

CY(I,J,K)

CZ(I ,J,K)

D(K)

DDB

DDE

DDN

DDS

DDT

DDW

DE(I ,J ,K)

DEO(I,J,K)

DEN(I,J,K)

DENO(I,J,K)

DENC


= Turbulence model constant, = 1.44


= Convective flux through the bottom face of a control volume

= Convective flux through the east face of a control volume

= Von Karman constant, = 0.4


= CMU**0.25

= CMU**0.75

= Convective flux through the north face of a control volume

= Convective flux through the south face of a control volume

= Convective flux through the top face of a control volume

= Convective flux through the west face of a control volume

= Grid transformation coefficient, _x

= Grid transformation coefficient, _y

= Grid transformation coefficient, _z


= Diffusive flux through the bottom face of a control volume

= Diffusive flux through the east face of a control volume

= Diffusive flux through the north face of a control volume

= Diffusive flux through the south face of a control volume

= Diffusive flux through the top face of a control volume

= Diffusive flux through the west face of a control volume

= Turbulent kinetic energy dissipation rate,

= DE at the previous time level

= Density of the fluid

= DEN at the previous time level

= Density at the center of a surface

32

DENIN

DIRCOS

DITM

DK(I ,J ,K)

DKO(I,J,K)

DTT

DU(I,J,K)

DV(I ,J ,K)

DW(I ,J ,K)

E

EREXT

ERRE

ERRF

ERRK

ERRM

ERRU

ERRV

ERRW

EX(I ,J ,K)

EY(I,J,K)

EZ(I,J,K)

F(I ,J,K)

FO(I ,J,K)

FI(I ,J,K)

FLOW

FLOWIN

GEN(I,J,K)

= Initial value of density of the fluid

= Subroutine for calculating the boundary grid sizes and directioncosines

= Wall boundary average value of dissipation rate

= Turbulent kinetic energy, k

= DK at the previous time level

= Time step size, 5t

= Diffusive coefficient for the p'-equation

= Diffusive coefficient for the p '- equation

= Diffusive coefficient for the p'-equation

= Wall law constant, = 9.01069

= Convergence criterion tolerance

= Maximum correction in E

= Maxlmum correction of a variable

= Maximum correction in k

= Maximum correction in p

= Maxlmum correctlon in u

= Maximum correctlon in v

= Maximum correctlon in w

= Grid transformation coefficient, n x

= Grid transformation coefficient, ny

= Grid transformation coefficient, n z

= Tentative variable of the transport equations

= F at the previous time level

= Variable quantity at the previous iteration step

= Outlet mass flow rate

= Inlet mass flow rate

= Turbulent kinetic energy production rate

33

HINUM

I

IBC(I)

IE

IG

IITO

IITY

IJLO(I,J,K)

INIT

INPRO

INSOE

INSOK

INSOP

INSOT

INSOU

INSOV

INSOW

IS

ISWE

ISWK

ISWP

ISWU

ISWV

ISWW

IT

ITT

J

= Large number, = 1.E30

= Index along the _ grid lines

= Boundary grid index

= Index assigned for the transport equations

= Problem control parameter, =1 for laminar flow and =2 for turbulentflow

= Total number of wall boundary grids

= Boundary grid face type

= Boundary grid sequential order

= Subroutine for initializingvariables

= Logical parameter for updating the effective viscosity

= Logical parameter for solving the E-equation

= Logical parameter for solving the k-equation

= Logical parameter for solving the p'-equation

= Logical parameter for solving the T-equation

= Logical parameter for solving the u-equation

= Logical parameter for solving the v-equation

= Logical parameter for solving the w-equation

= Starting value of I of the solution domain

= Number of sweeps for solving the E-equation

= Number of sweeps for solving the k-equation

= Number of sweeps for solving the p'-equation

= Number of sweeps for solving the u-equation

= Number of sweeps for solving the v-equation

= Number of sweeps for solving the w-equation

= Last value of I of the solution domain

= Number of time steps

= Index along the n grid lines

34

JBC(I)

JS

JT

K

KBC(I)

KS

KT

L

LO

L1

L2

LINERX

LT

M

MO

M1

M2

MC(I,J,K)

MT

N

NO

N1

N2

NEWVIS

NLIMT

NT

P

PCXI


= Starting value of J of the solution domain

= Last value of J of the solution domain

= Index along the _ grid lines


= Starting value of K of the solution domain

= Last value of K of the solution domain

= Maximum dimension of grid system in I direction

=L+I

= Starting point of blockage region in I direction

= Last point of blockage region in I direction

= Subroutine for solving algebraic equations

=L- 1

= Maximum dimension of grid system in J direction

=M+I

= Starting point of blockage region in J direction

= Last point of blockage region in J direction

= Wall blockage region control parameter

=M- 1

= Maximum dimension of grid system in K direction

=N+I

= Starting point of blockage region in K direction

= Last point of blockage region in K direction

= Subroutine for updating the effective viscosity

= Limit of maximum number of iterations

=N- 1

= Static pressure (relative)

= Pressure gradient, P

35

PDUV

PEDA

PP

PPBLK

PSCI

PTA

PW

RENL

SIGE

SIGK

SINX(I)

SINY(I)

SINZ(I)

SMNUM

SOC1

SOC2

SOC3

SOLVEQ

SP(I ,J ,K)

SPK(I,J,K)

SU(I ,J ,K)

SUK(I,J,K)

SX(I ,J,K)

SY(I ,J,K)

SYMOUT

SZ(I ,J,K)

TAUN(I)

= Blockage control parameter for link coefficients

= Pressure gradient, prl

= Pressure correction, p_

= Global pressure correction

= Pressure gradient, P

= Wall boundary source term for the momentum equations

= Wall value control parameter

= Reynolds number of the fluid



= Wall boundary direction cosine



= Small number, 1.E-30

= Source term due to shear stress



= Subroutine for solving general transport equation

= Linear part of the source term

= Secondary linear part of the source term

= Constant part of the source term

= Secondary constant part of the source term

= Grid transformation coefficient, _x

= Grid transformation coefficient, _y

= Subroutine for setting flow boundary conditions

= Grid transformation coefficient, _z

= Wall shear stress

36

TIMT

TJO(I,J,K)

TM(I ,J,K)

TMO(I,J,K)

TMULT

TRANF

TXXE(I,J,K)

TXXW(I,J,K)

TXYN(I,J,K)

TXYS(I,J,K)

TXZT(I,J,K)

TXZB(I,J,K)

TYYN(I,J,K)

TYYS(I,J,K)

TYXE(I,J,K)

TYXW(I,J,K)

TYZT(I,J,K)

TYZB(I,J,K)

TZZT(I,J,K)

TZZB(I,J,K)

TZXE(I,J,K)

TZXW(I,J,K)

TZYN(I,J,K)

TZYS(I,J,K)

U(I ,J,K)

UO(I ,J,K)

UC

UCXI

= Total time

= Jacobian of metric transformation

= Temperature

= TM at the previous time level

= Wall shear stress

= Subroutine for calculating the grid transformation coefficients

= Metric coefficient for east face diffusive flux

= Metrlc coefficient for west face diffusive flux

= Metric coefficient for north face diffusive flux

= Metric coefficient for south face diffusive flux

= Metric coefficient for top face diffusive flux

= Metrlc coefficient for bottom face diffusive flux

= Metrle coefficient for north face diffusive flux

= Metrlc coefficient for south face diffusive flux


= Metric coefficient for west face diffusive flux

= Metric coefficient for top face diffusive flux

= Metrlc coefficient for bottom face diffusive flux

= Metrlc coefficient for top face diffusive flux

= Metric coefficient for bottom face diffusive flux


= Metric coefficient for west face diffusive flux

= Metric coefficient for north face diffusive flux

= Metric coefficient for south face diffusive flux

= U-velocity

= U at the previous time level

= Velocity at the center of a surface

= U-velocity gradient, u_

37

UEDA

UINC

USCI

UX

UY

UZ

V(I ,J ,K)

VO(I,J,K)

VISC

VISE(I,J,K)

VCXI

VEDA

VSCI

VX

VY

VZ

W(I ,J,K)

WO(I,J,K)

WALLFN

WALVAL

WCXI

WEDA

WSCI

WX

WY

WZ

X(I ,J ,K)

= U-velocity gradient, un

= Velocity correction at outlet plane

= U-velocity gradient, u

= U-velocity gradient, ux

= U-velocity gradient, Uy

= U-velocity gradient, uZ

= V-velocity

= V at the previous time level

= Molecular viscosity,

= Effective viscosity, Ueff

= V-velocity gradient v_

= V-velocity gradient v

= V-velocity gradient v

= V-velocity gradient v X

= V-velocity gradient vY

= V-velocity gradient vZ

= W-velocity

= W at the previous time level

= Subroutine for calculating the wall functions

= Subroutine for assigning wall values

= W-velocity gradient






= X-coordinate

w_

W

W

WX

W

Y

WZ

38

Y(I ,J,K)

YN(I)

YNI(1)

YPLN(I)

Z(I ,J,K)

= Y-coordinate

= Wall normal distance from the last grid

= Wall grid volume size

+

= Nondimensionalized YN, y = u y/v

= Z-coordinate

39

APPENDIX C

PROGRAMLISTING

_ p_ _ N_'_

41

J;IF..

,le-

0

,11

42

0

r'_

--)

II •

_, _,_J •

un -1_ _

I[ •

U_ • •

_ ,-,_; .... ., ,, ,,,_o..) T-..'_ I-._ _ cO oEI "-_

I

I I

!

UJ_MMM_NII_H ._Z_ZZ_ZIL_Z_

_ H _ H H H 0 } _ _ _ U _ uJ _ _ _ _ _t H _4 H H H H H _ H U u

_.v

Z3) :1 IIUJ I1 II II

>OOO

OOOOCJO 0 _ 0 O L3(3 Q _10000 r30000 O O 0 U 0000000 O 00000

(3U

OC) O0

0000

0000

43

ORIGINAL PAGE ISOF POOR QUALITY

in i¢_ i¢_ In,o,o,.o_,o ,or,- _ r,.- _ r.,.e,,0o0oooooo0o0_ ooco _ o, o.,_o,_ oooo

-1 -) b_• •v

_,_ ui rJ

rJ(3 IIii ii

• • L_

vvu_00_

OflOU

N

: 8 ooo8oo_Ii 4_ _ _ Ii Ii 4_ II 41 {i O ::)

. .+ ......... ;;_ "_ .

., ._. .... . ...........• -; ,-_

v _vvvvvv I_

v I _ ,,I v_ v I v) v1 ,_ _ uJ

: T_ _...... _ ..... . . __

! .... ,u _÷ '_,. ._ .............. _H _ .... i__ _ _ " ::) _ 7_ ......

_ _,) _ ........ _ _ _ _ ....

: .... ooo. o .......... h_1 ooy ......... 0 o _ . ..I I I I I I I I I

U U U U U _-) _ _ U

0000

U r_) _l" _) ULJ U n:l -at 0 0 _ UO0 000000 (1[ N_I -_._tr3Mf_OUO U _1

O 0 IE} 0 L_ 0 e'30 O O _J rJ ('J 0 ('-I 0 0 r?J O t,_ r.3 f_O 000 r_J

oo oooo ........... _o o oo _ooooooooooooo_ ooo ooo ooooggOOn 1) OOI) tTIC_O (_ O r Joo Or_ O I-Jr_ O O ¢')O r J I_ I_ OO Or l (10 O¢T1 i}O

44

i_.. i_. r_ f,,- i_. r.- r _ . no 0o o_ 0o flu 0o o_ o, ok o., o, o_ o_ o_ LgOOO

-1

• Z

") Iu

O_ v

(_ _ "1

C),.-o 4_• uJ

t,,_ uJ A_

w_Ju •

• _ --') -.)

Z • •

uJ • _4 H

_- o >

tic CI • -J N •

4- =2E O • >- •

_ . _. -_,IIi J v

+ • • L_ _ Z ..J_ __ _.

_E LLI lU p If] CJ • %0 % _ _ _ (j I-

..... H z ........ H

_ v LiLl H Z H _4 II II II

P_ ILtL k- O _ O (_ OOC_O > _>

I

I

I

L3

0_

O

Q

r._

ul

,._o _

_-. [,.... Lii UI

*--I (,-- _r,.- r-_

-4-

v H _l _ rj _ Z _ _

f_ UU _r _L 0 0 0 ,"_ O0 Z

O0 0_0___000

L_ _oOOC)(-J

O O O r_ O O O c3 f_ O O r3 _3 C) _3 L30 _-) C} O IT) (_ c_ O O O (3 13 O O

O r30 C_ O O O O E) (_ r) O O n O O O r_ C) {7) _30 O FD (-J C) C} r_ O O

45

O0 00000 000 O0

H H H_

00000000000 _ 0

000000_ oo°°_°°°°_oooooo_oooo _

46

?U

Yl'u

Q. I•- L;_

• ,_ • ÷

N

_lJO

_Ua

o_V_

1-11o.{i. I I I H I-- I-- rn_ _- I-- n9

_,,___.,,.......,_,,__,,....._,,_,_ __o_......... ;

_ T .... _,It _ "1 II II II ^

I

I

I

N I

I

U

I I I

_ H b4 _H _

vvv_v

_-,gg"

_ .....

,, . ,, ,,. ,,,,. ,,. z_.

_oooooooooooooo_oooo, ooo_oo _ooooooo_ _ _ _.......ooooNoooooo_oo ooooooo _ ooooooo ogooo3ooo0000000 000000 000000000000 00000000000 000000000

47

I I I

2,2,''

, ^ , ^ . _,_._..,__ , ^

' ' ' .... ' ' ......... _2'_.-_ .'_......... _' '' _....

• _ I,-- I_ _ • %1_- F-- mH -.b_ M i,-,l_l _-+ U.ltLIUJ IU I--I Ml'-'ll'-¢ I-'4M I--I_. 0,. I'$. I I I H l--4H • % %1,- I.-.- iJ)_

............. ' ........ o_o +"-: -'::+_"''--'_ .................... " ........ " "" " "-_...:...... ,,,, ,,'--+o++I

I

I

P-+l I

!

t.J

_ _ H H H H H _ H H H _I _ H H _ H F_

,r__ r.z, o 0 o o _ ro o o o r-j ,o ,__ o +-I t_,-_

0 r-J 0 C'b r_ 00000 C 000 0 +D +_ _ C_

I-4 I+-I I.--I +,+ I-4 i-i i.-+ I--,I I+-+ i-i i.-,i I-4 H _-,,II-I

(_, {_- t_ _ o, _, 0- _- .el _ ._1 "el _'_ .el

U _') I'+'+I _Ir .(:i ,_i_ 0 ..+_) +"_I .,0 .fat +,.J ,0('__ ,.._i. (j,. ,t-1 m U ii i 0 w _i_ _irj t.10 (-] i,.%1 l_l-j I__ L/_

+n _ iJJ ill LU IU ill Ul _ IL IJL U- U- U- 0 0 0 P P p

+m_+" ++"-'"+"-'___0 C-s 0 r-) C-,) 0 r'J _ C_ 0 L-) 000

0 0 c-} 00 C_ 00 C'1 0000000

48

! I !

_<<_

11,,_"

_UO __ _ U _UO

_vv _ _v _vv

_U_ • _ _ _ l I I _ vvv _ _ _ v HH_

_vvfg_ll N_HHU H _I M v _ I _ 1 _ _ _4XX_C_ N _ • • • vc_ $_II Hvvv_(_ll H

II II _ _4 Z v t_ _ U U | I _ v v _ I! II I_ II II II II I! II II _ _ Z II II II II II II _ v I| I| II II II _

_f_om q U _NHUH I m_ _ H n _ _ g _I_HHH_ _ g _ _ _ _ N _ N _ m_ O f_(_ U _ U _ _CJ_m _ U_

:!

49

! ! I

IIIIII

44"''4

• "2 " "2H _4 _ H H _4

o IllIII

° _;77 .......

VHHU HHHV_I$1 IIYNNXX_

I

LL U. tJl. U.. U. U.. U. U- LL IL 0000 OC_ _ _ ',..t %_ ",,I" _t ,.,.'It ,,..r ,,._ br'_ o_ U"_ _ ",0 _D ,{) I_. I_- P,,.- r',.- O0 O0 rJO _0 i_,. (_P, _ ,_[ eli _ _d[ ,q'

oooo ooogooooooOOOoOO oo ooooooo oooo_ goooo ooooooo ooOOOO OOO OOOOOOO_'JO O (3 oorD C._OO OOOOOO O C3 OOOOOOOOOEDOOO OO

5O

H_

HH _

_-4 _¢ lU ul 1L; Ul

_.._. ,. ,, .... v .... ,m_

I I I

,,,,,, : :m m

NNN I_

I_ U FIuO

m

N )" X _4 :=,- X m I'_1 m _-I

.... _ _'_ +N N N_J _1N _ N

.... o...... .....+ _ vvv _ _ P U _ J

_ _ I I I _HHH_ _O_H_HH_ _HHHZ

,............. ,,_ =,_..,,,,';;,,-_,_. ...... _,.-, ,.-_,.-o0,. i_ (_ n_ o. o,_ oc o- {_ ,,/) V'_ C]_ CI' CI ,,,_ _..) ro L._ )- C_ f] C_ n_l "_ U :_" *_ H H _-¢ tL el: tU

0 0

0

_goog_oooooOOooo _O0 0000

........ _______C_Q_O000 0_0000000 OC) O00000 O00C3000000000000000000000000000000

*1

U

_0

• . 0

_ ** :3

O_

U

H :3

UJ

W_

0

o_

Z

0

! 0

0

_ °.

°° ij

_-- L_

o_

51

ORIGINAL PAGE" IS

OF POOR QUALITY

oo o .... _oo _ _ o o o oooooooo o _:, o oo o o o o o _ _oo_oo RO0 _ _ 0000 0 0 0 0 0 C_ 0 O0 OC)

52

O0 '_ 0 _ _-_ (9 c_ 0 C_ C3 0 c_ (-_ 0 C_ Oc_ (DOOOO COOOOOOO

OOOOOOC90 O OO C9 CI_ O¢_ C_ O O O C_ O OOO CDOOOO OOOOOOO

53

_ 0000

NN N_mml_lu _

÷÷÷÷÷÷

............................................ "i"000 000000000() O0 0000 OLq 000 _000 000 O0 000000 000r30 _I _ Z _ _ _ _ _ _

vvvvv_v

x Xx x x X_

I1 II II #1 I111 I1

oo000o0ooooo0 oOOOOoOOOr>OO oOor_ ooo00o00o0o0ooo00ooo00000o0oo oO0 00000000000000000000000000000r ) 00000000000000000000 O0

54

leiii

00000

N_NNN

N_NN

XXX_X

_UJtU

XX

II II II II

_ LU Z _

XN

XX_Nr4_XXXX_NNN

Z_ ........ _q ......

ii: ........

NNN_N_N_N_f_gJgI_

x C) O0000 C3£)0( r_ 00 C_ C) C_

II

2 _

O

O0 00000000000 0000000

0000000000000000000 [)or3 O000

____o__

nO

0

u_

uu

J

¢0

.._

or-)

** 0

0

_o_

yu_U

f_ _ w

f_J _ uJ

rj

_)

_- (_

u_ul

Uir_

w I--

o_r_

wo.

Z

°_

g

0 IL

t|l C)

."-,_ Utu

I,.-L_

55

• 0

I-*¢_

_ 21 _..4 L2_

•.I $o_ Io

0 sAJ _ _ • _ Z

o'J • _ i,-i _.

v IUNmO rJ_.l

-3 t& c_ .J

_1_ -Jr

0 0 {"_

O0 fD

o°o_

58

U

Hv

v I

:Ir ::E i.u t/1

t.o t/3 r3 _-4

•_ --) _ v

)-- v _ "-_ I-4

H [J CJ L_ _ a

v _

|U UJ If) ",_ 'lj_

• _

UIJ It It II • • • L) _ _ ill --1 I.II

............. _v v v t/1 t/3 _ - Z

J_ _ (J U (J .-*-_ _-- _ I,-.- lU I*-- J

':> 0 O 0 u- _L U. --_ t3 _) O I_-I z3 ,u tZt

I

I

iJ

O0

O0 0

O0 0

0 0 cD

_NNN_N_N_

U.I

_J

,e(

It

t_

tU

.J

t_

,c[

I-"

r_t

t_J

,.V1

"oO

O- J

C)

O

4) r*d H

_..,:3

r_ (3. LII

(')'./1

L) tt

I--_V3 tll

t--I

UJ --_

2: (_

_,') IJI

U*(_

ItJ I--

Z >-t-

C) r'a 0

at) _ t_

_)_t/)v) t-

Z

UHL_

r_JZ

Or"

c) r4 ,,:1

t_ U

O U

t_ 0,

t_ Z

UJ O

(3, tt. I--I

e. tL _et

tt :3

w P-tu ._

57

O • • O

• • O .'IU

::bOOl-- _. • _ H¢:I

e-- L5

........ __ _^

• • _-- ,._OP_:! PQ. • • •

_uo _ _ *,o _. _ _ao _._ ).-_r_r,_ _._

o

O

_o?oooo• ,r- q-- _

-J

Z q. • C:)_2 "2 op _--q.-

)-- NNNNN

_l _ nO _x) ii0 IX) H u_

N N Z > t,,-

, "-. oo,°'

H _J

U

N, _ __

+++++_

r_l u, + I • • • •

• _ %,-,_ _,- _-_-.........

..... _. .....

..... z ......_- _XNZ II . _-,'_t.ILI • I-- P" I--- _--q N N fM O r_) C_ O O O E:) C30

I*,,-h P- I-- ,_i tL Ivl V) _-_ _[_ i"%1 ..8 N :L I%1 _ _ N IM l_.l J N r-d i'd II II II ii II II

.......... ; II ..... , OZ v<l IJJ 31: Z _'I I-- I'0

! !

I I 'I I N r_ I

I I I

U U U U

000 E') rj_

C'J 0(3 0 _

(3 ('30 0 0

O_11 nD _1_ _n n'l C0 m _1 (1_ _t_u U U _J UQ O Lu [ll [u

O O E) (-'J O O C) O O O O Q E) f__ 0000000(7)

58

ORIGINAL PAGE" iS

OF POO r+ .... '+' "_

<,0 +,. o _ ,+.,,.-,++,,,+,..=++o_,,+,+,.+,,,+

7 7 = _ _ 7 ......v

},},;_+ _ = = ooooooHHHHHHX X _ N N

_+_++++_++++_++_+_+_+_+_+++ _ _ _ _ _ ................0000 00000000000000 OttO0 0 OC_O 0000000(_0000000

000000000000000000000(_000000_00 0 0 0 c_ 0 000000000000000

59

÷

÷ ÷ -l- +"_

+, _ + ,-1

T T T _""71 .` .` 41

• • • 1-+

- _ _ v

•41 + _

iii÷ 4- .41 + .J+ + + + tu÷ i+ + + 0_ _ _-_

+ _ + + _ _, + . + + + _- 41. I +..< _- +"_ 2_7 .• • • • -I- I • • • •41 1 • • • •4- 1 4- I I+ I • •

:_ i( .`_HI+4 l--+.`Hl,-4l-lk--I .` I 1+--1 _I_',_ _.i_ _,,"

I-4 _I • ,i + 41 II 41 II il "41 it _k 41 41 41 1 _i il il il 41 v • m till Ill i_ll

t%l _ _I+ i_I _i O r+_ f70 Cl (-7 (l n L_ i • (L_ n E) __7 t I 4- .41 .41 •

>- I+- I-- _- ul .41 • I_ + • *t_ .41 I_ .41 .41 ÷ iii ÷ i- ÷ ÷ +

I-- ,=1 <It (.+ • ÷ ÷ ÷ () ¢- .41 .41 ÷ _l • + .41 ÷ ÷ _ _ _ _ _ •

_ _'_ _ -_ .... " .... '-," ..... ' -2.......• -) "-) • • I _- I • • • I I- I • • • I + I • • H H I-_ _ _I _-1 CL

I uv _t li- II U u u IJ_)) _)))_4_ 11 1 1 ]1: ¢_ (3 0 C1

+_

UU_JU_UU_U

+ + ÷ .41 ÷ ÷.41 ÷ ÷

Hj_HHH_HH

v ._+,')v ,Iii _i 41 Ii 41 (i

_- I t-- LI (3 _ (1 #_-1 It

II II II II • •

I-- it_ I-- Ill _-+ H

r_

+

t3 kl ¢ i rJ (i

r) fl tJ r_ (1

li_ iu ill ul Ill

c+ • ) 0 _ 0 _ C1 0 0

tu ul ill t_J li_ Iu lu Itl tU

t ÷ .41 ÷4- ÷.41 + ÷

u I i 1_ r} fl (;_ tJ () f_ (7 (_ (i II II

_ I + 7 7 7 ,>_.> ,_i, -;,V ....... V_,I

H _I II II II II II II II II II U U t..i U U U U U U I-4

+_i +--I vv_l *_L +-+ H<[ HH ":t I--4 --) .J J I_ +> ::_ J[ j J[V

v

÷

X_

_P..*

• .41

UJ

2.1

gu

IJ_

+-+

tu •

U U

+__v;#

+++->o 7,+Z J Z

0'1:1 0 ()

! l

I

I I

U U U

6O

ORIGINALPAGE I$

OF POOR QUALITY

^ _ _ N NI_.I N NN NNNI_N N NN NI_N N I%I N N t_J I"_I I'_ N f_I N N

• • • • • • • • • • • • • • ° O • • • • • • • • • • • •

+ U'l +, _ + er_ OOOOOO OOOOOO OOO OO OOOO(_O OOOO_,r: r,d r*J -,r:_r'.J ,I_ _ _ _ .st _ ,l ,k IE ,Ii ,I_ • If ,If ,I_ _ ,II "It ,Ii _ .Ii ,IX ,If .(I ,Ii ,If .Ii

.... ' " '' ' _.Z'-_''" '' " '' '÷÷+++ ....... _ ...... --, _T_ T_--_.... _._ ,-, ... ., ..., , ., .... , ,

,..., _,_ ,_, _, .... >_?Z,?^? 7 ,_

"=_ I 1-- I _- ! ÷ _ • • I • 4- • • _ I • • ÷ • _ _

...... , , , , ÷ . .-+ ... , , , , + . _-+-_ ' ,vvvvvv vvvv

_._._._._._,_ ?????, ??,_??? , , , , , , _,>>_ _7_'_._'? _'%+'_._.... . . .T "_'_.. _.T_';';-_- "_._T_._,. _.T+_;; _.,..

H÷ ÷ ÷ ÷ ÷ ÷ • • • _-4

• _ _l _ _• o._ c_ H H _ + v + v 4. _ _ _ v _ v _ v _ H v v

,r-_ l_-'_-l_ l_,- I x: I_¥ I v • + _ _ v _ _ ,,_ ÷ _ ,,. _ I- _ v v •

_ _ II II II II II II l--l <_L H II lW II II II II II II II II II II If II II II II II II II II I# #I II U II II II II II II II II II II II II II II II

oooooooooo o o oooooo_,_ooooooooooooooooo_oOOoOOOOOoooooooooo000000°0('3 - _'3 0 ()00r)O0 0 ('-_ 0 (:3 E3 C3. 0 El 00 L_ 000 (-_ 0000 0000000('3

61

0000 00000_00 O0000f_O0 O0 O0 0000000 O0 O000000000000000000000000000000 000000000000000000000000

62

I

v

+

v

_ _ _ V W _ N _ UI _ _ _ _ v Im v III LU IU Ill _ _l N N _I N tll

....... ....

|1 II l! II I| II I1 II II II II II II II I1 II _

.......

v v

v v

N+ +

Z

•=: .... ;.-uJ 3r Z _,_ G3

lu "_ u_ uJ I_ uJtlJ ILl

7cO I Ul_ Z ¢,'_*- m

IIJ s*l LU lU _L_ LEt t._ tlJ UJ

+_+ _ + ÷ + + _

• ,j .... _ -

L._

II II II U II II I! II II II O II It l! II tl II Oo

t_

+ • * . • • • • * * • . * • * • • * .N

_ Lu

oU'_._,. __, _, _=,.

H Ig It II II II II II II II II II II II el II II II II

O0O0

O0000000000000000000000OO00000000000000000000000000000000

63

CJ cJ _ A _

H

! I '_ •

v v _ I',"I C) E) 0"ILL U. O.I( 41 U

s

• " (_ -1 11. o. (_L L_ ¢-)0

IJJ tU UJ U.I UI 3£ UI Ill _J [11 LIJ _LI _ UI ILl I1_ eli U,I UJ _ _ I--- 41 4" _ _" | I'-- I'--

_Jf U U UU % L)(-) U U UU % L.P U U U L) U _ II II II Q _1 I_ II _ _1 _/) II I.,1 V*_ Q II --1 II II II

.........._._._',',.........._':'.......:'-......._ _o a}" _o_§ _o_ ....?_s __-_ , _z ...... _ ...... o, o o_, o o ,o oo_ _o ,I I I ! I

, ,, : :I I I I I

L) L) U L_ U

ooooo,_OOoO_Oo,.},-_o__,o ...... o ........ _, o_ooooo oo,__(;,oooo oo ooOOOoOOOoOO _O00O 000 (7) ,_'J (_ r_ 0 r'_ 0 c-) 00 r'_ _) 0 _ (_ f'} C) (-I 0 r) O0OOO(3 _ O0 00000 E) C) 0

64

ORIGINAL PAGE IS

OF POOR QUALITY

C_ 0 _-_

_-_ f- _ +

O H

_ ,_) "-) II • II II II _ L_I t/t II • II II II J • U _ _ LL -") _,_ II • • II II II II II If (I •

.......... _,_,_ _,...... __....... _ .................... _ ......H _ U_I _ • • O H _ U_ _ • II • • • _-4 _'_ LL O H _ O O (-} • II • II L) • • • • • • • • II H C'," • • • Ire

000 _ ..3 n,. ---) 13- 0 t 0 L'_ 0 D :;_- (]l- :30,. _ tJ_ _[ I:E O0 _ IL I_- I./1LIJ 1--- G") _) E} 0 _ IOI. LL 0

I °, ' ' '.., _o o, , o: ol I -.,:I ! I u'_l_ ) u'_

O0 0£_ O0 (_Or_ 00000 000 0000 0000000{30000 O_

000000 O000EJO O00fJorJooorJO 000 (_ 00000 0000000000000 00000

65

A

Ilk

"5

I,.-H

0 •

J

gr,_

Z J

i

', o

000000 OClOr_O0 _ ooooooo_oooooooooo_oooo_ooooooooo_OOOO0000 OOOOOOO000 0000 000000000

66

N_lu_ _ _N _OOu N t3_ _ _ _ _ N _0_ _ N _ +_ U+_ _O_UN_ m_P + _ _

oooog_oo_gg ooooooo_ooo_ooooooooooog_goooooo ooooooooooO00OO O0 00000000000 00000000000 O00OO000OOOO00000

67

Z

0

I.-4

I---

_- II II II _ v

i1_ uJ

U I I II

g, ooo_ zuJ

_ _ r3 C_l n r3

!

!I

!

I

U

T,TT• • • • O I/%

_''" _ d o

........ 7 .... d! I I I I ! I I I 1-4 _ _ _I _d ,II

...... 44 "'4 .... 4

_27

"'+ ' ._?._? :.>,+>_> 111_ ......

........ 2q _Tqq ......÷ • + ÷ • I • • ÷ I 4-

.... ........ , , , , , , , , ......... _ _ _

--; -') "-_ -') -) "3 "-_ -_ "-) -3 _ H H _--_ t- HH _ "-)

Z Z Z Z 2 "_ "-) "_ -_% "n 2 ..................... Z ........

, ..... , ........ L_; ....... _ +z_v,+--_ I ._ 7 " 222v_u_*.'_ • 4vvvv • ,,. _vvv • • U II II II H_-_HHH

• • • . .,. • u_ _ u_ u_ _ • i._ i_+_ _-, ir_ • • ,r_ L_ L..', If' 'P. I"+'+ _ U". l+r+_ IZ++ I/'+ L/_ U", ill Jl: z ,tl t-- (n • L_ (.+) -LL) 5J .3 /3:3

II II iii II vv , • • ,v_ • • , ,vv , , • ,C_O000EgO00_IIUJ_UIIJU_II 4*_ C) CD(_ l I I v_vvv_ _n iJ 5 jjutj*jiJ_ _>OUUU_ _u_Juk) H II II II II II II II (Ju C}_JIJ{J_$- _-4 "_ II II II It II

g 7__ 7 g l, Iii ........... ! .............................. • ('i ..... , I, II ......... OZ Z I ..... , ,lJ _ ._Z mlU tlJ i-- i i _ t/) ill _1 I.-- M __ _1 i/J 7_ I.-- f1[} Z I_I ULI ]_ X X X )- )'- _'+ +-J l-J N llJ l_ Z L./I I'-_ l_t i"IL O O O _-.4 --_ _i_ ._[ _:[

I

I

t.J

U-_L U_ _L U+U tL U_U_ II U u_u lt. ll IL U IL U If+ II IL tL _LC_£)OOOEIrJOOOOOC_OOOOOOf') 00(--300000000

OO OO g'3 r3 C-" _'_O O .... O ,_r_r_ _ r'_O C, r30 oO .... gr'JO OOC'J r O C__ gf) ll) 1_ L_ O C-J r_ r3 Lq) _r'_ r'J r9 r363 O O r'J O O C9 (:30 O O O r/) r_ OO O O OOO £_.]O E] OO

68

ORIGINAL PAGE ISOF POOR QUALITY

.o,._+oo.o_N,.,.,_,.o,.._o. o_...,,.,oooo_ o..- ++ ,,.o§

v

0 +

÷ v

H + _ --'J _ ") ÷ --'_ +

" T'-' _'-' _T =

_ + =,=, ..+,_,+,o .... + +, --, _

_ H i--_ =,--e e--i H _# •

:_ a=,.- e.+.IN • U.I 31: Z,v_,f-- n"_ . • • • • .,_r NO_-_He-'I=--+ • I ,,::I • tLIV'I

.._v _v _ _ _ _ (+O_(_=_c_ +It t'], +. ÷ i'D o. cl_ c'.,. o- _ l'x. • • • "_ __ ;_'- _ " IO11" • " " • IO1IO1" " _Z)I'- XHvILl."¢_'um_,-- _I I-"I-- _-I-- _- I-" • , . , . ._,_ v •,+/,ll.rl i..,i _,_ - ClOOO ,_xIJL H ] _1_ • II II II II II P

x X N ¢._I H l,Jl uJ III Iii LII ILl _ _ IL II II II H _ -"J O Ul II I.LJ

("Jl O (-_1 Cjl CJl I"lJ' I,- • • • • • • C_ • • • • • • d ,'_ _ ,l:..Jr • 4_._ • H P_ _ _ II • U • • • • % • • • H '_[ It'+

....................... g_o _" _ _ _ _ ='_g¢,I-- I,-- I--I,.-- ,,rl • <_.+I_ r_ _ _ o i_ i_ ¢++I-+HI*-+II,._++I_ t+--I '...m ,._J . IL_.'t ,'.I. •at "_ ":I _ U -J

I _ p P_J Pe_ _._ u+_ I I

O I O O

I U I I_- I e,_

u ,j i_

69

70

; o

e.- r_l oc) co

_zO

m _ mo_

H s"%J _ IJJ

x m _ ,,..ej[

ZO _0

._Iv _0

tu _

_,o

÷

H

tt.

• 4-

÷

^ "2v ! v

÷

Z • -') I I •

......... _ __-_ _ _ _. _. _ _, _ _ ....

Z_

O O O O • M I O O I O II II II II p II II II O 41"

000

000

000

000

000

_U P _ 0 u_ I1|_

0 O0o ooggg gooooogggooooo ooooooog°°°°°°°

3ul

tu

lu

CL I--

o

c_ H -I_ I I + 1-4_ 411--I_ l I _ II

1-4 _" E H O II II II II (:30 II II II M O II II O II II II • O

I

0 I _a 0 0 0

O

_ _N ooo°ooooooo°o°°°°o000000000000000000

?1

I,_. ¢00_ 0 i_,-. r,_,l _.I ..,f u_ ,,0

UI

u,I

.J

OE_

¢1

I--

".dr

O" 0

• . C)

01_ :]1

C)

O- z

O- 0

•0_ b-4

o°

_iu_

_tn

u H I.'_

0_12"

0 i._ ¢II ()

cl_ Z

iiJ 0

IL t_

£_ _-- r_

_.- t.')

72

ORIGINAL PAi=£ IS

OF POOR QUALITY

ooo o o__ _: ......... o_ _0(_0 0 000000 O0 _ _ __

ooo o oo_oo_ _oooo_ oo ooo_oo _000 _ O0 0(_ 0000 O0 000000000 000000 000000 O0 000000 O0

73

_+ m-4 • ,q _ J

_ _ v v ,uj -,1

_-- >- II II tl _ -1

I

I

!I

!

u

• . _, ,, , _, ,- .....

................... 4-_4 .... 4 " ":("

................ ; ...... 7_ ......LIIV V_ I ;_- luvvv 4. _ llJ_v v ill _ _ _ % _

............. 7 ..... 2 _,....... _,_

•;. \ _. ........................

..... 2" i_ _? (17 II I 0 I-_ Z I)T _l;'_ ..... ()_lOZ_l or) , ......

g ..... ,_ ,,, :__-+ g ..... _ ...... _ ................0 Ill I"11_ _ -1{ (.) 0 r,.. rt_ 0 L._ (_) 0 0 0,- O, (_l r3 {-_ _ 0 u Z

I I

I I I _'J

tJ u tJ

+J (:) ,::) r-) (:'_ ,+_ L) +-7' _ _'-) r) ,._ r+_ C) i_ r3 c) C-) 0 C) +':l :::) +-_ (-J 0 t.TJ c_ ¢+) _"I C) c) tD 0

?4

75

0

0

>orJI--

ZOO,.,

.elOnO')l--I

_.1 .J__.=ll.-.I •tL ,..4 • ,_¢l_

Z _ 0 rJ •

_ uj_vv

III H • • _ •

o .,,_ ....iii 0 2_ ,x TM _ • I_

_,_ .¢ (.j • v (_l (J LJ

'_ tD ":> >, •

-J Z _ • • •

_ U. >- 31 i> c_ v

0 0 Q

I UT_ _ •

L') • CJ r__ L) •

Z

v_O m .11 >-

_- (J L.J _ _, • r,.. rL (.J eL.J

-,_ c_, * _ n )< tL _t e_- i.- >- II *-.1 IL ,_ • _) --1

C,} _ v tL .7. II II J 11 I! II -_ f_- 3 CI_ {Z _'l _ Z Z I| --) (IL 3 Z Z _'- II _') i_ 3 Z Z Z _ Z II J C_ L_

v j _ J I-- _- I.-. _ _ I-- J ,_ II II v) N rM II J I_- j 11 II J _-- j i II J _- J I.-- i_. )- _,. _- ,]L

, , : , , :I I _ oj I I

0

8, oorj

76

•.J -4' -..Tr .4" -*..e Mr u'_ u'_ _r_ u_ I.r_ Lr_ It_ u_ u_ .O._O ,,04[.)

3

_J cJ

4 •

I'M r_J

# #

a a

C-_ rJ o_

:,, .,_:_'_'i*: '_,-

U r_3 r_3 r_3 r_ C'_ O >.-

f%l

U7

tU

I11

I--I

.11

.11

_MO:3

= .__.),-C)

0 Z _

.11

41UL3 L_ eL ¢3

_.°, _-_.H • -"_ _. uul

t9 Z -_ ") .ZZ _t I Z Z Z

4_c 1.11 tl II .J e-" :_. I .J J

I I

' :I

I !

L) U

ooooooooooo_,oooo+._ _oooo ooO000+_or'_oC3C_OC3 . ,_(_)r_t:) (-3 r-'JO 0

0 0 0 04_._ 0 0 0 0 0 0 IC} r_ 0 0 Ir_) rJ 0 0 O0 0 O0

77

0

er_ •

0 0 C.)O 00C_

I uJ ! ! I I I I N

! _ ! I I ! I ! (31

_ "_, "" " " "131> X >- _,J X >- _-4 _

÷ _-I t I I I I I (31

I1 I Lu _, • • ! I I C_

• 4 I I I •

._' _ ,; ,-,. ;. ,.,_-_ I I I Z L)

u_

.... T_ _T_ '-

°,_............. o ?_.g227222_-g°..°_~_._. _,

,_g ...... ;,+LZ b _ l-.I M H v v v _ b_; l'-'l0 f) II ,_ II II II II II II ¢J z _ H

0 0 0 II OOvvv_ _ lJJO,'d[ II J _ 0 0 U lllm _J _ _ N _ CI: j _ 0 N 0 0 vv v

I

! i

0 0

_ _ oo oo ooo o _oooooo_oooooooooooooo_ _oo_ooooo_ oooooo oooooooooooooo_O0 0 O0 00000 0000000000000000000000

78

ORIGt_AL PAGE • iS

OF POOR OUALITY

O0 000 O -.,It ,_r ,_ i'_l ,,.1r 0 t+j co OU_

o+'i I,I+'1U llJ lj,. +-- oO rO (,0 ill (0 0 N p,- 0'_i 0 iI_ eli ill I,i'I i+"i

OOCO01'_ r-boO0 0 C'I 0 0 C_ 0 0 I"_ 0 0 0_._0 O0000CD 00000 O0000C'd 0

o,

IlJ

2

J

00 0

too

r--i ii Ul

ne

_ 0

0 _

lll IP_

+-I III

[IJ _

0 r%l ,,_

I 0

i_ ..J

0 U.

U.. vl

• o ffl _..I

o_-_

79

-1

3b4

.jen

LL

IU

_ iLJ IL

Ill I[ _j_4

0

I-J

mO

v • •

_" O0 ;--

ZOOH

vv_

_,-,uz_n_

L _C1_

OC-) fJ

vt.J_

_ XH_

U Z • •

_ r_ _,'_ .It

• 4 _4 _4

OOr_

C) C) 0

(:l r_ 0

0 0_0

t) O0

Lu

u_

Ul

J

eL)

F-

f_C:l

n,! _ iu

_g

m ul

LUI_tu

0 n,/ _,

m_m

o-Ji

() u

(1: ,°

tu o

u_ _

°° _

ul iu J

0 H

8O

APPROVAL

A COMPUTER CODE FOR THREE-DIMENSIONAL INCOMPRESSIBLE FLOWSUSING NONORTHOGONAL BODY-FITTED COORDINATE SYSTEMS

By Y. S. Chen

The information in this report has been reviewed for technical content. Review

of any information concerning Department of Defense or nuclear energy activities orprograms has been made by the MSFC Security Classification Officer. This report,in its entirety, has been determined to be unclassified.

G. F. McDONOUGH

Director, Systems Dynamics Laboratory

"_'U.S. GOVERNMENT PRINTING OFFICE 1986--631-058/20122

81

Documents

NASA CONTRACTOR REPORT NASA CR-178818 A COMPUTER … · 2013-08-30 · NASA CONTRACTOR REPORT NASA CR-178818 A COMPUTER CODE FOR THREE-DIMENSIONAL INCOMPRESSIBLE FLOWS USING NONORTHOGONAL