An incompressible Navier-Stokes flow solver in three ...

AIA A-84-0253
An Incompressible Navier-Stokes Flow Solver
in Three-Dimensional Curvilinear Coordinate
System Using Primitive Variables
D. Kwak, NASA Ames Research Center
Moffett Field, CA; and J.L.C. Chang and S.P.
1 Shanks, Rockwell International, Canoga Park,
CA; and S.R. Chakravarthy, Stanford Univ.,
Stanford, CA

AlAA 22nd Aerospace Sciences Meeting

January 9-12, 1984/Reno, Nevada

A N INCOMPRESSIBLE NAVIER-STOKES FLOW SOLVER IN THREE-DIMENSIONAL

CURVILINEAR COORDINATE SYSTEMS USING PRIMITIVE VARIABLES

Dochan Kwak'
NASA Ames Research Center, Moffett Field, California
W

James L.C. Changt and Samuel P. Shankst
Rocketdyne Division, Rockwell International, Canoga Park, California

Suhmar R. Chakravarthyt
Stanford University, Stanford, California

Abstract three-dimensional, incompressible, Navier-Stokes sol-
ver cast in generalized curvilinear coordinates using
An implicit, finite-difference computer code has primitive variables.
been developed to solve the incompressible Navier-
Stokes equations in a three-dimensional, curvilinear Incompressible flow phenomena are frequently
coordinate system. The pressure-field solution is based encountered in many engineering applications, espe-
on the pseudo compressibility approach in which cially, in hydrodynamics and in certain classes of
the time derivative pressure term is introduced into aerodynamic problems such as dynamic stall and low-
the mass conservation equation to form a set of hy- speed wind-tunnel test problems. For most two-
perbolic equations. The solution procedure employs dimensional flow simulations, computer time and
an implicit, approximate factorization scheme. The memory requirements are not major limiting factors,
Reynolds stresses, that are uncoupled from the implicit and various numerical techniques have been imple-
scheme, are lagged by one time-step to facilitate im- mented quite successfully. For example, a stream
plementing various !evels of the turbulence model. Test function-vorticity formulation is frequently used for
problems for external and internal flows are computed, solving lwo-dimensional, viscous, incompressible now
and the results are compared with existing experimen- problems (for example, see Refs. 2-5). The
tal data. The application of this technique for general three-dimensional extension of this method is not
three-dimensional problems is then demonstrated. straightforward. Various three-dimensional Navier-
Stokes codes ha.ve been developed, mainly, for com-
I. Introduction pressible flow. A few examples follow: Shang et
al! utilbed MacCormack's explicit scheme; Hung and
The development of new solution methodologies Kordulla7 developed a code based on MacCormack's
is one of the primary pacing items in computa- implicit scheme; Pulliam and Steger8 implemented the
tional fluid dynamics today.' With the current rate of Beam-Warming algorithm for a fully implicit code;
progress in this discipline, as well as with grid genera- and Briley and McDonald' independently developed
tion techniques, and with the enhancements in com- a similar AD1 scheme. Implemeuting these codes for
puter capability, it is now practical to simulate compli- simulating incompressible flows is not efficient and is
cated fluid dynamic phenomena associated with realis- generally not recommended. Therefore, in the present
tic geometries. To date, large computing times and work, an efflcient three-dimensional Wavier-Stokes sol-
memory requirements have been major difficulties in ver, using primitive variables, is developed for incom-
producing successful results from full Navier-Stokes pressible flow problems.
codes. Even though the economic aspect should still
be a primary concern in developing a fully three- One of the major problems to be addressed in
dimensional production code, the time has come to solving incompressible flows that use primitive vari-
utilize the available algorithms and high-speed com- ables is making the decision about which pressure solu-
puters to develop a useful tool for analysts and desig- tion method should be used to guarantee a divergence-
ners. The present paper presents the development of a free velocity fleld. The method of solving Poisson's
equation for pressure was developed by Harlow and
'4 *Research Scientist. Welch," and has been used frequently for obtaining
tSenior Member of Technic81 StaR. the pressure field, mostly using explicit methods. The
usual computational procedure is to choose the pres-
$Senior Research Associate. Member AIAA. Present address, sure fleld such that continuity is satisfied at the next
time-level, so that the new flow fleld will be divergence-
Roekwell Scienw Center, Thousand Oaks,Calilornia. free. This procedure normally requires a relaxation
scheme iterating on pressure until the divergence-free
This paper is declared a work of the US. Government and

therefore is in the public domain.

1

condition is reasonably satisfled. This approach can be To implemeaL an implicit, approximate fac- d
very time consuming, and thus the computing time re- torization schemeI3 to the above set of equations,
quired for simulating three-dimensional flows has been the continuity equaion is modifled according to the i/
prohibitively large. To accelerate the pressure-fleld procedures of Refs. 11 and 12 as follows:
solution and alleviate the drawback associated with
the Poisson's equation approach, Chorin" proposed to + + +-aatp B(,au -8a.n aw = ap* (14
use artiflcial compressibility in solving the continuity
equation. A similar method was adopted by Steger Here, t is time; x, y\ and z are Cartesian coordinates;
and KutlerI2 using an implicit approximate factoriza-
tion scheme by Beam and Warming.I3 To implemrnt u,v, and w are corresponding Cartesian velocity com-
the implicit time-differencing, they fabricated a hyper-
bolic time-dependent system of equations by adding a ponents; p is the pressure; and ~ ,ijs the viscous stress
time-derivative of the pressure term to the mass con- tensor. The parameter 1/,9 is the pseudo compres-
servation equation. These hyperbolic equations pos-
sess characteristics that are not present in the usual sibility, and p' is the value of p at the previous itera-
Poisson's equation for pressure (see Ref. 14 for a
comprehensive analysis). This approach has been a p tion. The p and p' terms, and the numerical algorithm
plied to simulate laminar, incompressible now within
are chosen to satisfy the continuity equation (la);they
liquid fllled shell^'^. Presently, this procedure has been will be discussed in more detail later.

extended to a three-dimensional flow solber cast in The equations are written in dimensionless form
generalized curvilinear coordinates. with

In Sec. II of this paper, the governing equa- u. =- u , -v = - v , -w=-, w
G.1
tions and turbulence model are presented. The finite- UV.1 U?tJ
difference algorithm is described in Sec. KI, and iu
- ' 2_ ,Z "# .z = - Z
See. IV,results are presented for internal and external &I=-,

flow test problems that verify the accuracy of the code. Z,.J 27.1 2r.J' (14
Additional examples of a more practical nature are in-
cluded to show the versatility of the current soher. - -tU,.f P-PIel - = 7PUiLiI '
p = -,r;j
,t = - pu:.,
G*I

; = R e - ' = - Y , B- = - B
z7clUrel

lI.Governing Equations and Turbulence Model The subscript ref denotes reference quantities and for
convenience the tildes ( - ) are dropped from the equa-
Governing Equations tions.

Unsteady, three-dimensional, incompressible The viscous stress tensor can he written in the
flow with constant density is governed by the follow- following form:
ing Navier-Stokes equations, written in Cartesian coor-
dinates:

Here, R,;is the Reynolds stress, and L/ is the coeflcient

of viscosity. Combining Eqs. (Ib)and (Id), the govern-
ing equations are written as

where

2

Coordinate Transformation For orthogonal coordinates, Eq. (7)can be further

To accommodate fully three-dimensional geometries, simplified to read

'httrhoedfuoclelodwwinhgic etnrearnaslfizoermd itnhdeeppehnydsiecnatl cvoaroiradbtlneasteasreininto- k. = (.;&v 2 T Cs2 r C s 2 J fIJmD-~r( & . . . ~ ~ r ~ ~ . s j

general curvilinear coordinates:

LJ

T=t

F = t(z3Y,2,t) (3)

11 = 11(% Y, 2, 0

f = f b >Y,2,t) &bulence Model

Applying this transformation to Eq (2a), the following Various levels of modeling are available (see Ref.
form of the governing equations is obtained
16 for a review), most of which require considerable
-+aD a(&-&..) a ( P - P d + a(&-&) experimental inputs. In the present code, however,
aT an (4)
+ the turbulence model is c o n h e d to an algebraic model
1 ap. to maximize the cost effectiveness of the flow solver.
= -(--,O,O,O)~ Even though the accuracy of the result could be limited
J ST by the algebraic model, economy is still an overriding
factor in many engineering applications.
where
In the present code, turbulence is simulated by
J = Jacobian of the transJormation,
eian al ebraic eddy viscosity model, uslng a constitutive
D =DIJ,
e uat on involving a "mixing length" that is a measure
o?the turbulence length scale. A generalization of this
approach is given by the following equation:

1
-Rkk6ij
-Rij 2utSij (10)
3

Here, is the turbulent eddy viscosity, and Rkk is the

noma1 component of the Reynolds stress. The strain
rate tensor is de6ned by

+ z)sij i
Here, the contravariant velocities, U,V, and W withdut = 421 -s8 uX auj
j
metric normalization, are deiined as
+ + +u = ( I
+y = rlt
+ + +w = ft
tdJ l u v Czw (6) +By including the normal stress, R k k , in the pres-

n.u+llvv+ 11.w sure, u can be replaced by (u ut). For the turbulent
f.U <uv c z w
Viscosity, the algebraic model of Baldwin and Lomax17
The viscous terms are is implemented in the present code. Following their
formulation, the turbulent viscosity for incompressible
flow can be written as

+ (R.,...terms) y1= (V1).nnrr, Y<Yc

>(ut).atrr, Y Y c (12)

where y is the normal distance from the wall, and ye is
the smallest value of y a t which values from the inner
and outer formula are equal.

In the inner region,

where (ut)inncr = PlwI (13)

= I 1,I, 0 1 0 0 I,,,-=8D -,etc where (w( is the magnitude of local vorticity. The tur-
0010 BE
BC (8) bulence length scale 1 and the nondimensional distance
from wall y+ are deflned as

v Looo1-I

3

where

In the outer region, 1La LlW 4
d
(vt)orier = KGpF,.r.Fdy) + +(15) Q = Lo LIU LZVfLSW

where Lo = ((i)t, L1 = (ti)., L2 = ( t i ) U , La = ( t i ) Z

ti= e,?, ot $ Jot A , B , or C, respecttvely

Fw.+ = minimum[(ym..Fm~.),( C , r ~ ~ ~ ~ U ~ d i l / F r n6~< =~ ]Jinite diJJerence form of -a8t ' etc.
(lab)
Here, F,,, is the first peak value in a profile given by The superscript n denotes nth time step, and the vis-

F ( d = ylwl[l- e w - y +/ A+11 - +cous terms are given from Eq. (9) as
r l cVc(V(ilmGc,) (R,,...terms)
and ymoz is the value of y at that point. The Klcbauofl J
intermittency factor is written as (Ru,...terms) (18c)
+-rl - 7uvtp(V(irms,,) (R,,...terms)
+F.br.b(y) = (1 5 . 5 ( C h r . b y / 3 1 n ~ ~ ) ~ l - ' (16)
- +rS -UVc+'(,rm6,,)
and J

+ + + +Udi, = (JUZ Approximate Factorization
-Y Z wz)rn.,z ( J U Z YZ w")m,n

(17) 'The full viscous terms &, kv and 6, in Eq. ( 7 )

In wakes, the exponential term in F(y) and the second produce non-tridiagonal elements in the left-hand side

term in Eq. (17) are set equal to zero. of Eq. (18a). Therefore, to implement an approximate

The constants appearing in the model were deter- factorization scheme, only orthogonal terms are kept
mined by requiring agreement with the Cebeci formula-
tion for constant-pressure boundary layers18. These on the left-hand side. For steady-state solutions, this

values are em be done since the left-hand side approaches zero as
a steady state is approached. For a time-accurate solu-
A = 26, Ccp= 1.6, Cxlpb= 0.3, Cwx= 0.25, tion, this approximation procedure needs to be further
lnvesti ated when a nonorthogonal grid is used. For
k = 0.4, K = 0.0168 the ri %t-hand side, the full viscous terms may be in-
cludefj. In the present version, a nearly orthogonal grid
m.Numerical Algorithm
is used, and the viscous terms in Eq. ( 7 )are simplified
Time Advancing to E* = -vj(VcV<,)imS,D = rJrn6cD

The numerical algorithm used to advance Eq. k v = U .Vq, ) I m 6, D = 7d,,,S,D (19)
(4) in time is an implicit, approximately factored,
flnitedifference scheme by Beam and Warming13. By -J(Vq
combining trapezoidal-rule time-differencing and the 6" -- J-U(V<.V<,)Im6,D = 7slm6,D
difference form of Eq. (4), the following governing
equation in delta-form is obtained: After adding smoothing terms to stabilize the com-
putation, the approximate-factored form of the govern-
ing equation becomes

+[If
h ' &( y -712-8~) c.V<A<].
-J"+
2

[ I + 2hJ"+16,(b" - 721-6,) fe,V,,A,].

I+ 6,(? - r,)] (D"+' - D") + +[r 2h~ " + 1 6 , ( Z- 7arms,) ~ . V , A J ( D " +-~ D")

E IRHS (1841 - cG[(VcAC)'+ (V,A,)*+ (V7,A,)21Dn

(20)

where e , and c e are implicit and explicit smoolhing
terms (which are explained in the next section), and

4

h = AT= time- step, where

- -V c D = Dj Dj-1, A q D =DK+I Dk, A u = u"+' - un

6cD (DI+I-D I - I ) / ( ~ A ~ ) E. = explicit smoothing coeJJicient

+6 , 7 8 4 = I(7*+1 7k)(Dk+l- D I )
-v
( 7 k f 'Yk--1)(Dk -D I - I ) ] / [ ~ ( A V ) ' I In a similar manner, the implicit second-order

smoothing term can be obtained by starting from

Analogous terms in the q and { directions are deflned C ~ = U -2(AuC,x+, -- u t - , ) -2(IAu4,2+, +- 2u, u,--1)
similarly.

Higher Order Smoothing Terms 1\-24-a-l ,
This can be put into a numerical scheme, in delta form,
It has been found that a higher-order smoothing
term is required t o make the present algorithm stable as
(see Refs. 12 and 13). In this section, smoothing
terms used in the code are described in relation to the +AIL -2c(AAAxtuj+i -Auj-I)
usual upwinding scheme as well as their behavior near

computational boundaries.

For simplicity, the following equation is con- --r.lclAt +- ~ U J - I 6uj - 4 U j + i +UJ+~)"
~ A (XU j - 2
sidered: -aa+ut c-=oaaux where 124b)

(21)

The usual upwind differencing of the convective term f, = implicit smoothing coejjicient
results in

+C ~ =~ c(U3uj - 4Uj-1 uj-z)/(2Az), c > 0 f, = explicit smoothing coe//icient

~ ( - 3 ~f 4j u j + i - ~ j + 2 ) / ( 2 A z ) , c < 0 ( 2 2 4 It is interesting to observe how a small perturbation, u',
propagates under various smoothing terms. Following

where 6, is a difference lorm of BulBz. This equation the fluid, the second-order smoothing acting on the
perturbed quantity is
can be writlen, for all c, as

+ +c8,u = -[C(3uj - 4uj-1 uj-2) _aut - av =0 (254
4Az
+ + +W at f- 8 %

(-3Uj 4Uj+l - Uj+z)l -4[I(AC31xUj - 4Uj--1 Uj-2) then +-uI - e-coat (ae'*' be-'"')

- (-3Uj f 4Uj+l- Uj+z)] For the fourth-order smoothing term,

Replacing the flrst [ ] term of the above by the usual

central difference formula results in -BU+' r - = o3a%a'X

c6,u = -2(aCux -,+I U j - 4 at

+ -(IUC1j-z + -- ~ U J - I 6 ~ 2 ~ U J + I + U J + Z ) then
4Ax
=ul e-<"4t(aeid" + be-'"")

(22b)
.;Therefore, a flnite-difference form of Eq. (21) with a
On the boundary, the flow variables are often ex-
fourth order explicit smoothing term can be written as trapolated to maintain the same order of differencing.
-,;+I -u?+-l u?+l
1+1 j-1 = However, a lower order of differencing could be used
At -kc 2 A z
f*kI +- 4Uj--1 by backward or forward differencing. And it should
-6uj 4 ~ j + 1 + uJ+z), @3a) be noted that the sign changes when a lower-order
--(uj-p smoothing is incorporated. In the present work, the
0 < f,<l second-order smoothing behaves well on inflow and
4Ax

and, in delta form, it can be written as outflow boundaries, and the fourth-order extrapolated
rmoothing term is better on solid boundaries.

+ - -AU -2c(AhAtxu~+I
Au,--1) = --( 2cAAtx UJ+I ui)" Pseudo Compressibility and the Pressure Field
Solution
- - + - +_feLlt(uAjt-,
4Ax 4uj--1 6uj 4uj+1 uj+z)" In solving the incompressible Navier-Stokes
equations, exact mass conservation is of crucial im-
(23b)

portance in order to obtain a stable solution. In the

5

present study, the continuity equation is modified to of the code. For this discussion, a $ = const surface
a hyperbolic- type [Eq. (Id)], thus, introducing pres-
sure waves of flnite speed; the wave speed is inEnity for is considered next. On this surface, Eq. (37) can be
truly incompressible flow. Wave propagation depends
written as
on the magnitude of the compressibility parameter, P,
V f.vp= 0 (283)
which, if chosen to be too large, would at the Fame
Expanding this,
time contaminate the accuracy of the approximate fac-
+ ++f.(E.PC
torization. Therefore, a correct specification of ?I, is of 7sPn IrPc) = 0 (28L)
prime importance to the success of this approach. For
Since a nearly orthogonal grid is assumed near the
inslance, a certain value of lhe pseudo comyressibdily surface, this can be further simplified to
suitable for a particular problem can be totally inade-
This condition is implemented in the c-directional
quate for other geometries and flow speeds, and can sweep. From Eq. (20), the matrix equation on a solid
surface (at L = l ) is written as follows:
even cause instability.
where AD = D"+' - D". To implement Eq. (28c)
In Ref. 14, wave-propagation characteristics are
analyzed using a one-dimensional form of the govern- explicitly, one simply sets
ing equations. By compwing the velocity cf prpssure-
wave propagation and fhr_ rate of vorticity spryading,
the following criterion for the artificial compressibility
was derived:

where 26 and ZL are the characteristic lengths that then the pressure is updated at the end of each step.
the vorticity and the pressure waves have to propagat,e The same procedure is used for fixed boundaries such
during a given time span. During a duct-flow simula-
tion, X L is equal to the total length of the duct, and z6 as a free stream boundary. Equation (ZSc) is applied
is half the distance between the two walls of the duct. J , q 0I,in an implicit manner by setting
The computational experiments showed this depen- j=[oi,h=l 0 1 0 0
dency of the pseudo compressibility on the Reyi~olds 0 000
number, as well as on the characteristic lengths of the
geometry. 0010 000

For the near-field in external flows where pres- 0001 0 000

sure waves propagate out to inEnity the choice of B is IV.Computed Results

less restrictive. The pressure buildup in the near-field

region is only temporary, even if p is not optimum.

However, for internal flows, local pressure buildup can
be serious when?!, is not properly chosen (see Rrf. 14
for details).

Boundary Condition The flow solver was verified by solving a few
simple test problems. To test the code on an ester-
Once the numerical algorithm has been developed nal flow simulation, flow over a circular cylinder was
computed. To test this program for internal Bows, a
the next most important aspect of solving a Ruid channel-flow problem was computed. Various compli-
dynamics problem is the proper implementation of the cated three-dimensional flows were then computed; a
boundary conditions. There are several different types couple of examples are described here to demonstrate
the capability of the present code.
of boundaries encountered in numerical simulations:
1) solid surface, 2) far-field, 3) in-fiow and Flow over a .Circular Cylinder
out-flow, and 4) symmetric or reflective boun-
daries. All of these are required in the present Flow over a circular cylinder has been a rich
source of various fluid dynamic phenomena (see Ref.
code. 19). The Bow over an impulsively started circular
cylinder at a Reynolds number of 40 based on the
On a solid surface, the ususal no-slip condit,ion is diameter of the cylinder is chosen as a verification
applied. By taking advantage of the viscous sublayer case. The three-dimensional coordinate system chosen
assumption where the pressure is constant normal to is shown in Figs. l a and lb. Figs. 2a and 2b show
the surface, one obtains the steady-state velocity vectors and stream function

P" = o (27)

The boundary condition can be implemented cither J
explicitly or implicitly. The latter enhances stability

v contours. The steady-state pressure eoefflcient on the one station in Fig. 7b. This clean-duct sirnulation
cylinder surface is then compared in Figs. 3a with that shows the nature and magnitude of the secondary Bow,
" of Mehta (private communication), who used a stream separated region, and substantial variation in mean
function and vorticity formulation in two dimensions. velocity vectors (Figs. 8a and 8b) from t o p or bottom
Various experimental and computztional studies on wall to the center of the duct.
this bench-mark case have been reported. The quan-
tities, that are presently used to verify the physical Figures 9a and 9b show the geometry of an
phenomena are the wake length, L,uvobet;he separation annular duct with a 18Cf' bend. This configuration
angle, BSep (see Fig. 2b); pressure drag, Cdp, and the represents the turn-around duct in the hot-gas
pressure coefflcients at the forward and rear stagnation manifold of the Shuttle engine. The laminar-flow
points, C p a~nd C,,, respectively. The Row field com- (Re=1,000) solution in Fig. 9c shows a considerable
puted by the present code INS3D compares quite well nonuniformity after the 180' bend. In this case, an
with those reported in the literature (see Table 1). For adverse pressure gradient is developed after the bend,
this computation, a nondimensional-time step of 0 1 and subsequently a large area of separation is formed
was used, and the steady state was reached in 300 itera- along the inner wall. Further numerical simulation of
tions, To simulate the impulsive start, the number of the SSME will be used to redesign and to optimize the
pressure iterations was increased to 6ve at each time-
step; the pressure-drag history is compared in Fig. 3b hot-gas manifold. For this computation, a 50 x 21 x
with the time-accurate computation of blehta Even 16 mesh was used with a non-dimensional time-step
though the present algorithm is geared for a steady- of 0.1. A converged solution was obtained after 300
state solution, this example demonstrates the potential
for an efflcient time accurate procedure. time-steps, starting from a uniform velocity Geld. The
computing time for each time-step was 1.443-04 sec
Channel Flow per mesh point on the Cray I S computer at Ames.
Because of the axisymmetric nature of the present
Just as the circular cylinder problem is the configuration, only 16 mesh points were used in the
simplest representation for external flows, channel flow circumferential direction, However, for nonsymmetric
is perhaps the simplest representation for internal flows, more mesh points and proper local elustcring
flows. However, the chanuel-fiow problem still provides of grid lines will be required. The three-dimensional
the essential features for testing the present algorithm. computation of the hoegas manifold of the SShIE has
Figures l a and 4b show the developing laminar chan- been performed using the present code. Details of this
nel 60w. To reduce the channel length for obtaining numerical simulation, as well as comparison with ex-
fully developed flow, a partially developed boundary perimental results, will be presented in a future paper.
layer pro6le is used as an inflow condition. 7Jsing a
nondimensional time-step of 0.1, fully developed flow V. Concluding Remarks
is obtained after about 100 steps for a Reynolds num-
ber of 1,000, based on channel width. The Baldwin- This paper presents the development of an
Lomax turbulence model is tested by incrensing the efflcient and, robust computer code for incompressible
Reynolds number t o 100,000. The velocity vectors for Navier-Stokes Rows (WS3D). In this work, the basic
this case are shown in Fig. 5a. The mean velority formulation and the algorithm are described. mS3D
defect is compared, in Fig. 5b, with Prandtl's universal has been applied to various geometrically complex
law, Comte-Bellot's experiment for channel flow,2oand flows. Other aspects, such as a multiple-zone computa-
Klebanoffs well-known boundary layer profile. The tions, implementation of smoothing terms near inter-
convergence histories for both laminar and turbulent sections of solid walls, symmetry boundary conditions,
cases are shown in Fig. 6. and analysis of differencing errors, will be presented
in a future paper. The algebraic turbulence model
Three-Dimensional Flow Examples implemented in the present version of the code needs
more development for solving the flow with massively
One of the prime objectives in developing the separated regions, as in the case of the SSME power
present code is t o analyze and verify the Bow field in head simulation. Higher-level turbulence models are
the Space Shuttle Main Engine (SSME) power head. being investigated in an efIort to achieve better es-
The examples shown in Figs. 7, 8 and 9 exhibit typical timates of the turbulence length scales.
flow characteristics locally encountered in the SSME
simulation. References

The rectangular duct shown in Fig. 7a is similar 'Kutler, P, " A Perspective of Theoretical and
t o part of the new 80 x 120 low-speed wind tunnel at
Ames Research Center (guide vanes are not shown). Applied Computational Fluid Dynamics," AWIZ Paper
The flow, in the form of velocity vectors, is shown at 83-0037, Reno, Nev., 1983.

T

2Mehta, U. B., " Dynamic Stall of an Oscillating "Chorin, A. J., " A Numerical Method for Solving
Airfoil", Paper No.23, AGARD Conference
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Computational Physics, Vol. 2, 1967, pp. 12-26.

3Lecointe, Y. and Piquet, J., On the Use of 12Steger, J . L. and Kutler, P., " Implicit Finite- ,J
Difference Procedures for the Computation of Vortes
Several Compact Methods for the Study of Unsteady .
Incompressible Viscous Flow for Outer Problems. E," Wakes," AIAA Jounal, Vol. 15, No. 4, Apr. 1077, pp.
Eighth International Conference on Numerical Method \J
581-590.

in Fluid Dynamics, Aachen, Germany, June 28-July 2, I3Beam, R. M. and Warming, R. F., "An Implicit
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4Bramley, J. S. and Dennis, S. C. R., " A Conservation-Law Form," Journal of Computational
Numerical Treatment of Two-Dimensional Flow in a _Phy_sics, Vol. 22, Sept. 1976, pp. 87-110.

Branching Channel," Eighth International Conference

on Numerical Method in Fluid Dynamics, Aachcn, '*Chang, J. L. C. and Kwak, D., " O n the Method

Germany, June 28-July 2, 1982, pp. 155-160. of Pseudo Compressibility for Numerically Solving

Incompressible Flows," AIAA Paper 84-252, Reno,

5Rosenfeld M. and Wolfstein, M., " Numerical Nev., 1984.

Solution of Viscous Flow around Arbitrary Airfoils in

a Straight Cascade," Eighth International Conference "Chakravarthy, S. R., Numerical Simulation

on Numerical Method in Fluid Dynamics, h c h c u , of Laminar Incompressible Flow within Liquid

Germany, June 28-July 2, 1982, pp. 433-439. Filled Shells," Report ARBRL-CR-00491, U S . Army

Ballistics Research Laboratory, Aberdeen Proving

%hang, J. S.,Buning, P. G., Hankey, \V. I,., Ground, Md., Nov. 1982.

and Wirth, M. C., " Performance of a Vt,cI<~rii,t~l

Three-Dimensional Navier-Stokes Code on the (:ray-1 "Reynolds, W. C., " Computation of Turbulent

Computer," AIAA Journal, Vol. 18, No. 9, Scpt. 19x0, Flows,"~Annual Review of Fluid Mechanics, Vol. 8,

pp. 1073-1079. 1976, pp. 183-208.

7Hung, C. M. and Kordulla, W., " A Time-Split, "Baldwin, B. S. and Lomax, H., '' Thin Layer
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Field Simulation," AIAA Paper 83-1957, Danver, Approximation and Algebraic Model for Separated
Mass., 1983.
Turbulent Flows," AIAA Paper 78-257, Huntsville,
Ala., 1978.

*Pulliam, T. H. and Steger, J. L., Implicit '*Cebeci, T., " Calculation of Compressible
Finite-Difference Simulations of Three-Dimensional
Compressible Flow," AIAA Journal, Vol. 18, No. 2, Turbulent Boundary Layers with Heat and Mass
1980, pp. 159-167. Transfer," AIAA Paper 70-741, Los Angel??, Calif,

1970

BBriley, W. R. and McDonald, H., I' Solution '"Morkovin, M. V., Wow around Circular Cylinder
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2189.

8

v 'dp CPf 'pr gsep Lwake
0.93-1.05 1.14-1.23 -0.47-(-0.551 50r53.9' 1.8-2.5
Summary of
literature data 1.03 1.15 -0.51 52" 1.9

Present result

I." I

PHYSICAL DOMAIN COMPUTATIONAL
DOMAIN

1. J 0 .6 1.2 1.8 2.4 3

,I -1.75 z

q.K KMAX J=K=L=l -a) velocity vectors,
(JMAX.KMAX.IMAX1 = (3.80.41)
STRETCHING RATIO= 1.15

a) coordinate system.

0.967- -0.721-0.721 ---0.721-

-/o,&8,-0.481- -0.481 0.481

1.05 0.241-0.241
o.24,---0.241-

.1.25 -.65 -.05 0.55 1.15 1.75 -1.75 -.05 *.65 1.35 2.05 2.75
-.75
2

b)grid. b)stream-function contours.

Fig. 1 Topology for flow over a circular cylinder. Fig. 2 Steady state solution for flow over a circular
cylinder at Re=40.
U

1.2 2h
Y
T
-.2
I
0 .2 .4 .6 .a 1 -1.6 I
15
DISTANCE FROM THE LEADING EDGE 0 3 6 9 12

a) coordinate system, x

a) velocity vector.

- 2D BY MEHTA USING STREAM-

FUNCTION VORTICITY Cdp

0 INS3D 1.25
8

1.00 -

-,c"

? .75

50 -

0 10 20 30 40 v//2hI

TIME, DIAMETERS TRAVELED b) fully developed velocity proflle.

b)grid Fig. 4 Developing laminar channel flow at Re = 1000.
(Re based on channel width and average velocity.)
Fin. 3 Flow over an impulsively started circular

s~~

cylinder at Re=40.

10

!i- ~ ~ = 1 (0TU5RBULENTI
e Re=103(LAMINAR1
-.5

v

26-

12-

- - - - - - -V
s4mL9t4s~:
2h 2

l-- 2 -

-1 6 7 36 9 12 15 -2 5 50 100 150 200 250 300
0 0 NUMBER OF TIME STEPS
x
Fig. 6 Convergence history comparison for laminar and
a) velocity vectors. turbulent channel flow.

R25

20

v PRANOTL'S UNIVERSAL LAW
0 PRESENT RESULT (BALDWIN-LOMAX

TURBULENCE MODEL)

x COMTE-BELLOT (Re=1.14X 1051, CHANNEL
a COMTE-BELLOT ( R ~ 2 .x4 1051, CHANNEL
0 KLEBANOFF (Re=1.5x l o 5 ) ,B.L.

FLOW

0 . 2 .4 .6 .8 1 a) Duct geometry.

vh

b) mean velocity defect comparison

Fig. 5 Developing t.urbulent,channel Bow at, R e = 10'.

x + -..
b) secondary now at z=O plane (view from down

stream).
W Fig. 7 Flow through a rectangular duct with a 45'

bend.

11

__.-

a) Velocity vector at x=0.5 plane.

----.................................SEPARATION
POINT
\
s ; ~ $ ~ : t t ~ - ~a-* ~ - s - -

b) velocity vector at x=0.025 plane (top view).
Fig. 8 Flow through a rectangular duct with a 450
bend.

12

a) Three-dimensional grid.

2.0

1.3

W .6
V

-0.1

-0.8

-1.5 4.7 5.4 6.1 6.8 7.5 3.0 4.2 5.4 6.6 7.8 9.0
4.0
X x

b)two-dimensional crass section. c) typical flow pattern with separation.

Fig. 9 Flow through a turn-around duct.

13