The problem of how a boundary layer responds to the motion of a convexed vortex on a porous wall was investigated. The wall velocity is approximately given by Darcy's law. The vorticity-stream function approach was adopted for solving Navier-Stokes equations of two dimensional incompressible viscous flows. The weighted-mean scheme was used for constructing finite difference approximations of spatial derivatives. Several test problems were solved and numerical results demonstrate clearly the accuracy, stability, and efficiency of the scheme. The weighted mean scheme then can be applied to the vortical flow problem

Huynh, Q. Q.

English

NASA Technical Reports Server

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3 1176 00161 6896 
NASA Contractor Report 165747 
WEIGHTED-MEAN SCHEME FOR SOLVING 
INCOMPRESSIBLE VISCOUS FLOW 
Quyen Q. Huynh 
SYSTEMS AND APPLIED SCIENCES CORPORATION 
17 Research Drive 
Hampton, Virginia 23666 
NASA-CR-16S747 
19810021847 
Contract NAS1-15604 
May 1981 JUL 2 1 19t1l 
NI\S/\ 
National Aeronautics and 
Space Administration 
Langley Research Center 
Hampton, Virginia 23665 
\ . . _ t 
https://ntrs.nasa.gov/search.jsp?R=19810021847 2020-03-21T12:51:43+00:00Z
TABLE OF CONTENTS 
PAGE 
SECTION 1 - INTRODUCTION . . . . . . 
· 
. 
· · · · 
1-1 
.0 
SECTION 2 - THE WEIGHTED-MEAN SCHEME 
· 
. 
· · 
2-1 
SECTION 3 - TEST PROBLEM 1 
· · · 
3-1 
SECTION 4 - TEST PROBLEM 2 . . 
· 
4-1 I • 
SECTION 5 - TEST PROBLEM 3 
· · · 
5-1 
SECTION 6 - COMMENTS ON THE VORTEX PROBLEM . . . 6-1 
SECTION 7 - CONCLUDING REMARKS 
· · · · 
7-1 
REFERENCES . . . . . . . . . . 
· · · 
. . . R-l 
i 
Page intentionally left blank 
• 
· ... _------ - -'_._ --
FIGURE CAPTIONS 
Figure 1. Isotherms for the Benard convection cell with 
P = 10. 
Fi gure 2. Isotherms for the Benard convection cell with 
P = 80. 
Fi gure 3. Driven cavity . 
Fi gure 4 . Stream function contours at R = 100 ( 41 x 41 
Figure 5. Stream function contours at R = 1000 ( 61 x 61 
Figure 6. Single vortex convected over a porous wall. 
. - .'-~ 
grid) . 
gr id) . 
iii 
Nfl - 303fflo~ 
SECTION 1 - INTRODUCTION 
The turbulent bursting phenomenon is reco nized as a 
dominant feature of time - de pendent turbule t boundary-layer 
flow . The nature of the boundary laye r formed on a flat plate 
located beneath a convecting rectilinear ortex embedded in a 
uniform flow was investigated by Doligalski and Walker (1). 
They obtained numerical solutions for the temporal development 
of the boundary laye r induced by the motion of a rectilinear 
vortex. The boundary layer was expected to erupt from the wall 
into the inviscid flow. 
In the present study the flat plate is replaced by a porous 
wall where the velocity at the wall obey s Darcy 's law. The 
point of inflexion of the boundary layer velocity profile is 
controlled by varying the strength and / or the location of the 
vortex. Our interest lS focused on how the inflected velocity 
profile responds to wall mass transfer induced by the motion 
of the vortex. 
The governing equations for a two-dimensional incompressible 
viscous flow are the vorticity transpor t equation and the stream 
function equation. These equations can be solved simultaneously 
at each time step by means of a finite difference scheme. We 
p ropo se to use the wei ghted-mean scheme of Fiadeiro and Veronis 
(2) for the vorticity transport equation. This scheme is first 
applied to a series of test problems to determine its accuracy, 
stability and efficiency . For the stream function equation , we 
use the usual central difference scheme. 
1-1 
--.- ---- --~ 
• 
1 
_---I 
• 
, 
SECTION 2 - THE WEIGHTED-MEAN SCHEME 
The advection-diffusion equation is writte n in conservative 
form: 
aw + a (uw) 
at ax + 
a (vw) 
= ay v (1 ) 
where v is the diffusivity constant 
u and v are the velocity components in the 
x and y directions, respe ctively and 
w is the vorticity 
For cases when the vorticity field is subject to strong advection, 
c e ntered-finite differe nce procedures are inefficient; consequently, 
the weighted-mean scheme is adop ted for the present investigation. 
An explicit (leap-frog) c e ntered-t ime difference in equation 
(1) is used except for the c en tral 
the spatial differencing evaluated 
n term ( w . . ) 1J 
n+l 
as ( w .. + 
that results from 
n-l 
w . . ) / 2 1J 1J 
where the superscript denotes t he t ime step. When the weighted-
me an scheme is applied to t he s patial deri v atives the equation 
f o r the calculation of n+l w . . 1J 
n+l 2 11 t ( ex .' 1 n U,) • . = w. 1 . 1J l+ a li t 1- 1- , J 0 
I-a li t 
n 0 
+ a j +1 Wi ,j+l) + l+a li t 0 
+ 
n given v alue s at w .. 1J 
n 
+ a i +1 w. . a. 1 1.+1,J J-
n-l 
w .. 1J 
where li t is the time increment 
h is the spatial increment 
2-1 
----------
and 
n 
w .. 1 1, J-
n-l (jJ . . 
1J 
(2 ) 
is 
with all a's evaluated at time step n, we have: 
u . ~ h u . ;" 1+ [coth ( 1+ 2 1] a . 1 :: 2h 2v ) -1+ 
U. ~ h u. ;" 1- [coth ( 1- 2) 1] a i - l = 211 + 2v 
( 3) 
v. k h v. ;" 
a j + l = ~ [coth ( 
J+2 
- 1 '] 2h 2v ) • 
v. ;" h v . ;" 
a. 1 = ~ [coth ( J - 2) + 1 ] J- 2h 2v 
The error of approximation 1S O( 6t 2 , h 2 ). The formulation 
uses only a five point operator for two-dimensional flow and 
is antisymmetric in relation to the velocity field. When the 
component change sign, the c oefficients upstream a nd downstream 
of the point are automatically reversed, a feature particularly 
useful in computer progr ammi ng because the sign of the velocity 
components is not generally known a p r iori . 
• I 
2-2 
• 
I • 
SECTION 3 - TEST PROBLEM I 
The one -dimensional advection-diffusion problem presented 
by Roache (3) to illustrate the weakness of the standard central 
difference relation is as follows: 
I aw u aw 
v at + v ax = o 
with boundary conditions 
w = 0 at x = 0 t > 0 
w I at x = 1 t > 0 
( 3) 
Regardless of the initial condition, the steady state 
solution for inc ompressible flow is 
I - e ux/v 
w = 
I - e u/v 
The time derivative is written as first - order forward-dif-
ference approximation and the weighted-mean scheme is applied 
t o the spatial derivative s. 
Steady state results appear in Table I for weighted-mean 
scheme, upwind differencing scheme and central difference 
scheme for various values of cell Reynolds number 
uh Rc = -
'J 
The steady state form of e q uation (3) lS 
= o (4 ) 
It can be sho~m that the solution of the finite difference 
analog of this equation agrees identically at the points x . 
l 
with the solution of the differential equation (4). We may 
3-1 
obse rve this rema rkable re sult in Table 1. Table 1 also reveals 
that the central di fferen ce scheme produce s wiggles for values 
of Rc greate r than 2.0. The upwind differencing scheme, although 
stable , is considerably in error at all value s of Rc. 
3 - 2 
• 
I 
• i I 
• 
• 
SECTION 4 - TEST PROBLEM 2 
The problem is to determine the temperature field in a 
two-dimensional square, Benard convection cell for a fluid 
heated uniformly from below and cooled from above. The length 
scale L and the maximum ampli tude U of velocity are used in 
the non-dimensionalization. 
The non-dimensional temperature equation has the form: 
where 
P [ a (uT) + 
ax 
a (VT)] = 
ay 
P = Pee let number = UL 
K 
K = thermal conductivity 
The dimensionless velocity field lS given by 
u = sin x cos y v = cos x sin y 
Boundary conditions on Tare 
T = 1 at y == 0 
T == 0 at y = n 
c T = 0 at x = O, n 
ax 
O<x<n 
O<y<n 
A relatively co~rse grid s ys tem of 25 x 25 mesh points was 
used for different values of P. Although the scheme assures 
convergence to all iterative methods, we would like to compare 
the rates of convergences in Table 2. Line SOR combined with 
multigrid techniques are approximately four times faster than 
ADI method. Figures 1 and 2 show isotherms of the fluid for 
Peclet numbers 10 and 80. The intertwining tongue for the 
4-1 
case P = 80 indicates the effect of stro ng convection. A final 
point that we would like to bring out for the strongly con-
vective case (P = 80) is that distortions of isotherms were not 
introduced with relatively coarse grid (25 x 25). 
4-2 
• 
• 
J 
. . , -' 
SECTION 5 - TEST PROBLEM 3 
We consider the flow in a driven square cavity. This problem 
is a standard test case for evaluating the accuracy, siability 
and efficiency of numerical schemes. For simplicity we choose 
a square with sides of length L = 1. The upper surface (fig. 
3) moves to the right with a constant transverse velocity U = 1. 
The flow of a viscous incompressible fluid in a square 
cavity is governed by the following coupled equations: 
dW 
at = 
+ 
a (uw) 
dX 
a (vw) + 1 
dY R 
= - w 
where ~ = stream function 
w = vorticity 
R Re yno lds number 
u = d~ 
ay v = - E1. , ax 
Boundary conditions: 
upper surface 
bottom, left, right 
surfaces 
UL 
-
v 
=~ y 
(5 ) 
( 6 ) 
v = kinematic viscosity 
o 
The steady state solution of the s ys tem also may be reached 
by replacing equation (5) by its steady state form (the unsteady 
term is omitted) 
_ a (uw) 
ax 
--~--
a (vw) 
dY 
1 
+ -R 
5-1 
---------
The vorticity transport equation (5) becomes equation (2) in 
finite difference approximations . The stream function equation 
(6) can be discretized using central difference formula. For 
c oncreteness, we will describe the computation cycle for the 
time-dependent approach. 
The calculation starts at t = 0 where ~ and w are known 
eve r ywhere . From equation (5) we obtain the vorticity w for L 
all interiors points at t = 6 t. The numerical solution of (6) 
gives ~ at t = 6 t and hence u and v. The last step of the 
computational c ycle is to update the boundary values of w 
using the most current values of ~ and w at the interior points . 
This computational c ycle is repeated until the steady state 
is reached to a specified convergence level. If we use the 
steady state form of equation (5), instead of "marching" the 
vorticity transport forward in time, new values of w are obtained 
by an iterative process . The way in which the boundary value s 
of the vorticity are approximated affects both the rate of 
convergence and the accuracy of the solutions. This fact has 
been noted b y man y authors. In the p rese nt case we use a second 
order form for e valuating wall vorticity given b y 
w/w 1 = 
2h2 
(8 ~/w+l - (7 ) 
Studies indicated that instead of uSlng formula (7) at time step 
I
n I n-l 
n , the average (w wall + Wj wall) / 2 yielded a more stable proces s 
• 
for the time-dependent and steady state approaches . Contour 
plots of stream function are shown 'in figure 4 and 5 for various 
5-2 
---- - ~ .. -----
grid sizes and Reynolds number. Numerical values also are listed 
in Table 3 for comparison. 
5-3 
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• 
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! 
SECTION 6 - COMMENTS ON THE VORTEX PROBLEM 
The pro?lem is shown schematically in figure 6. An isolated 
vor tex embedded inside a boundary layer convects to the right 
over a porous wall. The velocity at the wall is given by 
Darc y 's law: 
where c 1S a constant 
Pr is a reference pressure 
The wall mass transfer mechanism is driven by the wall 
pressure gradient induced by the motion of the vortex . Conseque!ltly , 
it is of interest to examine the effect of wall p ressure 
fluctuations on veloci t y profile with an inflexion point inside 
the boundary laye r. The point of inflexion initially may be 
created by varying the position and / or the strength of the 
vortex . 
It is worthwhile to first consider a non-porous wall. 
The effect of two counter-rotating vortices of opposite and 
equal strength is equivalent to having an impermeable wall 
between them. In an inviscid flow , a dimensionless stream 
function of the vortices may be defined by 
2 2 2 
r r + (x-x ) + (y - Yo) 
\)J v £'n 
c 0 
= 2 2 n 2 2 
r + (x-x ) + (y+y 0) 
c 0 . I 
where r is the strength of the vortex 
r is the core radius 
c 
(xo'Yo) is the c en ter of the vortex 
6-1 
Let the vortex be convected in an inviscid flow. The convection 
speed of the vortex is composed of 2 parts consisting of the local 
mean velocity and the sel f induced velocity due to the vortex 
i mage below the wall. Note that the inviscid solution predi c t s 
that the vortex will remain at constant heiaht y . 
~ 0 However , 
t he inv iscid solution is not valid for o ur realistic problem 
be cause it fails to satisfy the no-slip boundary condi t ion on 
t he wall. 
We have also tested the possibility of satisfying the no -
slip c ondition by means of a smoothing function, s. The modified 
stream functi on is de fine d as: 
1); = s 1); 
modif orig 
and s vanishe s strongly ne ar the wall, to ensure zero derivatives 
of 1); d' ' f (z e r o ve l ocity ). 
mo l 
Far from the wall, s is identically 
1. However, th i s new st ream function p roduce d a reverse flow 
near t he wall , introducing new compl ication s i n the f low pattern . 
It wa s t here f o r e a b a ndoned . 
6 - 2 
• 
SECTION 7 - CONCLUDING REMARKS 
The weighted-mean scheme has several significant advantages: 
it is locally a conservative scheme neither creating nor 
" 
destroying the advected quantity artifically. 
I 
it is stable for all grid spacings or cell Reynolds number s. 
I ~ The scheme thus has the practical advantage of yielding 
solutions for relatively large values of g~id s ize and i s, 
therefore, economical with computer time. One can use this 
crude solution to construct a good initial guess solution 
for a finer grid. 
it becomes a central difference scheme for strongly dif -
fusive cases and an upwind difference scheme for strongly 
advecti ve case s. Furthermore, when the components of the 
velocity change sign, the coefficients upstream and downstream 
of the point are automatically reversed, a feature particularly 
useful in computer programming because the sign of the 
velocity components ne ed not be known a priori. 
The only major disadvantage is the lo'ss in speed due to 
the computation of the hype r bolic tangent at each grid point 
ln multidimensional problems. 
For the vortex proble m, the initial conditions need to 
correspond to some real initial situation Slnce we are interested 
in the transient solution. Therefore one needs to satisfy 
initially the no - slip condition. One suggestion is that the 
vortex at time t equals zero, should be far away from the wal l 
in the inviscid region so that the velocity induced by t he 
vortex vanishes at the wall . 
7 - 1 
REFERENCES 
1. Doligalski , T. L. and Walker , J . D. A., "Shear Laye r 
Breakdown due to Vortex Mo t ion ," Proceed ings of the AFO SR 
Workshop on Coheren t Structure of Turbulent Boundary La yers, 
edited by C. Smi t h and D. Abbott , Lehigh Un i v ., Nov . 1978, 
pp . 288 - 3 39 . 
2. Fiadei ro, M. E. ·, and Veroni s, G. , Tellus 29 (1977), 512. 
3. Roa che, P. J ., "Computational Fluid Dynamics ," Hermosa 
-- i 
Publishers , New Mexi co, 1972 . 
4 . Burggraf, O. R., J . Fl u id . Mec h . 24 (1966) , 1 13 . 
5. Bozeman , J . D. and Dal t on , C., J . Computational Phys . 12 
(1973), 348 . 
6. Hall asamy , M. and Kr isna P r asad, K., J . Flu i d. Me ch. 79 
(1977) , 391. 
7. Gosman A. D. et a1., "Heat and Mass Transfer in Re circulating 
Flows ," Ac ademic Press , ew York , 1969 . 
8 . Lustman , L ., Huynh , Q., and Hussaini , - "Multigrid Method s 
with Weigh t ed - Mean Schemes , " To be submitted at Sympo sium on 
Multigrid Me thods , NASA - Ames Research Center , Oct . 21 - 22 , 
1981. 
-, 
r 
R- l 
Table 1 
Rc = 1.0 
Grid point 
1 2 3 4 5 6 
ES 0 .0 0 .0 1 17 0.0433 0.1295 0 . 3636 1.0 
WMS 0.0 0 .0 117 0 . 0433 0.1295 0.3636 1.0 
UDS 0.0 0.0323 0 . 0968 0 . 2258 0.4839 1 .0 
CDS 0 . 0 0.0083 0 . 0331 0 . 1074 0.3306 1.0 
Rc = 2 . 0 
Grid point 
1 2 3 4 5 6 
ES 0 . 0 0 .0 003 0 . 0024 0 . 0183 0 . 1353 1.0 
WMS 0 . 0 0 . 0003 0.0024 0 . 0183 0.1353 1 . 0 
lDS 0 . 0 0 . 0082 0 . 0331 0.1074 0 . 3306 1.0 
CDS 0.0 0.0000 0 . 0000 0 . 0000 0.0000 1 . 0 
Rc = 4 . 0 
Grid point 
1 2 3 4 5 6 
ES 0 .0 0 . 0000 0 . 0000 0 . 0003 0 . 0183 1.0 
WMS 0 . 0 0 . 0000 0 .0 000 0.0003 0.0183 1.0 
eDS 0 .0 0.0013 0.0077 0.0397 0 . 1997 1.0 
CDS 0 . 0 0.0164 -. 0328 0.1148 - 0 . 3279 1.0 
ES exact solution 
WMS = weighted - mean scheme 
.. UDS = upwind differencing s cheme 
CDS = central difference scheme 
Table 2 
Ben ard :::e11 ( 25 x 2 5 ) 
MULTI GRID (8 ) 
P SOR ADI i n work uni t (wu ) 
SOR Line SOR 
1 0 140 8 5 38 20 
20 188 1 10 35 27 
4 0 6. 140 4 4 3 8 
80 6. 2 05 83 51 
160 6. 101 96 
6. The r e sults f o r t ho s e speci f ic Pec1e t numbe r s we r e 
not c omputed . 
r . 
Table 3 
Reference Grid \jI 
vc 
Iterations 
100 WMS 21 x 21 - 0.094 55 (TD) 
WMS 4 1 41 - 0.103 120 (s s) 
Burggraf (4 ) 41 4 1 - 0.101 
B + D (5 ) 51 5 1 - 0.103 
N + KP (6 ) 51 51 - 0.102 
• 
1000 "(",MS 61 61 - 0 . 092 210 (ss) 
B + D ( 5 ) 51 51 - 0.081 
N + KP ( 6 ) 51 51 - 0.097 
Gosman et al . ( 7 ) 81 81 - 0.099 
WMS Weighted - mean scheme 
TD = time dependent app r oach 
ss steady state approach 
= stream functi on at vortex center 
vc 
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1. Report No. 2. Government Accession No. 
NASA CR-165747 
4. Title and Subtitle 
WEIGHTED-MEAN SCHEME FOR SOLVING INCOMPRESSIBLE 
VISCOUS FLOW 
7. Author(s) 
Quyen Q. Huynh 
3. Recipient's Catalog No . 
5. Report Date 
May 1981 
6. Performing Organization Code 
8. Performing Organization Report No. 
R-SAL-05jSl-Ol 
t-----------------------------~ 10. Work Unit No. 
9. Performing Organization Name and Address 
Systems and Applied Sciences Corporation 
17 Research Drive 
Hampton, VA 23665 
'2 . Sponsoring Agency Name and Address 
~ational Aeronautics and Space Administration 
Washington, DC 20546 
15. Supplementary Notes 
Langley Technical Monitor: Dr. Julius E. Harris 
Final Report 
16. Abstract 
11. Contract or Grant No . 
NASl-15604 
Contractor Report 
, 4. Sponsoring Agency Code 
The problem of how a boundary layer responds to the motion of a convected vortex 
on a porous wall is being investi gated. The wall velocity is approximately 
given by Darcy's law. The vorticity-stream function approach was adopted for 
solving Navier-Stokes equations of two-dimensional incompressible viscous flows. 
The weighted-mean scheme was used for constructing finite difference approximations 
of spatial derivatives. Several test problems were solved and numerical results 
demonstrate clearly the accuracy, stability and efficiency of the scheme. The 
weighted-mean scheme then can be applied to the vortical flow problem. 
7 I<f'Y W'wls (Suggested hy Author(s)) 
Fluid Mechanics 
Viscolls Flow 
Numerical Analysis 
1!) Securi ty Oassif. (of his report) 
Ilncl rtSS i fi ed 
18. Distribution Statement 
20. Security Classif . (of this page) 
Unclrtssifiecl 
IJnclrtssif i f'd - Unlimited 
Subject Category 34 
21. No . of Pages 
27 
22. Price 
A03 
'j 1" For sale by the National Technical Inf(1rmatlon Service . Sprlngfleln. Virginia 22\61 


Weighted-mean scheme for solving incompressible viscous flow

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