NASA Technical Memorand,um 103141 _

tF

' An Unsteady Time Asymptotic Flow

in the Square Driven Cavity

John W. Goodrich .......

Lewis Research Center ................

Cleveland, Ohio

(NASA-TM-103t4[)

ASyMpTGTIC FL_W

(NASA) 12 p

AN UNSTEADY TIME

I_ THE S_UARE DRIVEN CAVITY

CSCL 20D

Nq0-22016

Unclas

G3/34 0280229

Prepared for the

IMACS 1st International Conference on CompUtational Physics

cosponsored by the International Association of Mathematics and Computers

in Simulation and the University of Colorado at Boulder

Boulder, Colorado, June 11-15, 1990

https://ntrs.nasa.gov/search.jsp?R=19900012700 2017-09-11T14:09:45+00:00Z

An Unsteady Time Asymptotic Flow in the Square Driven Cavity

John W. Goodrich

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135

Summary

This paper presents summary details of an aperiodic time asymptotic numerical solution for the square drive cavity at Re -- 10000. The data presented is for

6100 < t <: 7100, and is representative of the data that characterizes the aperiodic asymptotic state. Complete details are not given in this paper for either the

transient evolution or the asymptotic state.
,°

Introduction

A persisting final oscillation in the aspect ratio two driven cavity was shown

in [5] at Re --= 10000 and t _-, 300. A periodic time asymptotic flow in the aspect

ratio two driven cavity was presented in [3] at Re = 5000 and t _ 3800. This

paper presents summary details of an aperiodic time asymptotic numerical solution

for the square drive cavity at Re = 10000 and t _ 6000. Further details are in

[2]. Unsteady time asymptotic calculations for the square regularized driven cavity

have been presented in [7].

Numerical Method

The velocity field for two dimensional time dependent incompressible fluid flow

in a spatial domain n may be written in terms of the streamfunction ¢(x, t) as

u(x,t) = _-v and v(x,t) = _o¢o,. The flow equations may be written as

aA¢ 1 at.at a¢ a¢

-- ]_e/k:¢+_xx/--Xayy _yyAOxx, forxinfl, andt>O. (1)

The data for the impulsively started driven cavity consists of the initial values

¢(x,0) = 0 for x in fl and t = 0, and the boundary conditions ¢(x,t) = 0 for x in

an and t > 0, and _0¢ (x,t) = 1 for x on the cavity lid and t > 0, and _-_n¢(x, t) = 0

o

for x on the cavity walls and t > 0, where _ is differentiation in the exterior normal

direction at the boundary, and where the lid is moving from left to right for t > 0.

Let a discrete approximation for the streamfunction on a square uniform grid

be _. = {g'_ : m = 0, 1,... }. If La is the conventional five point centered difference

approximation to the Laplacian, if Bi is the conventional thirteen point centered

difference approximation to the Biharmonic operator, and if 6. and 6y are the

conventional centered difference operators, then a discretization of equation (1) is

At Bi(_,+ 1)

La(_.+ 1)_ 2R----e

= La(g") + 2_eBi(_. ") 32at [6. (6,(£")La(g"))- 6, (6. (i")La(_."))]

T

(2)

This algorithm is second order accurate in both time and space. Standard finite

difference approximations are used to incorporate the boundary condition. The

velocity components are directly recovered from the discrete streamfunction solution

using central difference approximations for the _ and _-. o¢ The velocity solution
is exactly discretely divergencefree with respect to a central difference formulation

of the mass conservation equation. Further details are given in [2] and [4].

The solution at time tn+ 1 of the implicit equation (2) is obtained by a multigrid

method, as in [2]. The Biharmonic operator is factored as two Laplacians, point

Gauss-Seidel relaxation is used for the smoothing operator, and linear restriction

and prolongation operators are used. A V-cycle iteration scheme is used, with 3

iterations per grid level while coarsening, and none while refining. At each time

step, 10 to 15 iteration cycles are used to reduce residuals to less than 5.0 × 10-1_

Numerical Results

Computations are reported for the impulsively started square driven cavity at

Re 10000 on a uniform 128 × 128 grid with At - i These computations

show that the time asymptotic state for this combination of algorithm, data, and

grid is an aperiodic laminar flow. Among the indicators that have been used [3]

for tracking convergence to an asymptotic state are the relative L1 norm of the

streamfunction change per time step _,,_. iCn,,_.+l _ ¢_,. I/_,,s I¢_. + l I, the total kinetic energy _axay__,_.# u "+l i,j 2 2 '

and the minimum streamfunction value, each

of which is obtained from the data on the interior grid.

The relative LI norm for the streamfunction change per time step is O(10 -8)

for t >_ 100, so that virtually the entire time evolution of the system and its time

asymptotic state are in a range of very small change per time step with respect to

the streamfunction solution surface as a whole. The initial transients completely

dominate the time evolution of this system until at least t _ 500. The relative L1

norm for the streamfunction change per time step seems to approach its asymptotic

pattern by t _ 1600, with persisting variation in the range [0.00006, 0.00009]. The

total kinetic energy of the flow on the interior grid in the cavity is greater than

0.036 by t _ 600, but there is clearly evident transient change until at least t _-, 1900

when the asymptotic evolutionry pattern is approached. Persisting variation for the

total kinetic energy is in the range [0.03624, 0.03630]. The minimum streamfunction

value approaches its asymptotic time evolution pattern and range by t _ 1900, with

persisting variation in the range [-0.10823,-0.10804]. The global stream_function

minimum is also point data, since the global minimum for the asymptotic state is

at a fixed point in the very center of the primary circulation. The combination

these three indicators suggests that the asymptotic attractor for this flow has been

approached by t _ 2000.

There were small transient changes in phase portraits for this data, so the

computation was run until t = 7100 in order to ensure that the asymptotic state

had indeed been reached. There is a period of T _ 200 for the asymptotic flow.

The resolution of the asymptotic attractor with phase portraits and spectral analysis
required at least 1000 nondimensional time units, or data from 160000 time steps.

Figure 1 shows the time series data for the relative L1 norm of the streamfunction

change per time step, and Figure 2 shows the streamfunction minimum, both for

6100 < t < 7100. This data shows approximately five complete periods of the

asymptotic state. Notice the complicated beating of multiple frequencies. Figure 3

shows the power spectrum for the total kinetic energy, for normalized frequencies

less than 5. This spectrum is representative of the complicated spectra for the data

from this flow, with multiple discrete spectral lines, with a primary frequency of

c_ _ 0.237, with various harmonics, and with a minimal significant frequency of

cm _ 0.005. For the relative LI norm of the streamfunction change per time step

the ratio of spectral weights for c_ and c,_ is 61.3, for the streamfunction minimum

it is 132.0, and for the total kinetic energy it is 5.66. It seems likely that there

are at least two incomensurate frequencies with c,_ approximately equal to their

difference. The power spectra for all three of the scalar time series also reflect

a complicated substructure in their spectral lines, with the peaks in the power

spectrum away from c a tending to spread into a cluster of subpeaks, with frequency

seperations that are approxunately 0.005. Figure 4 shows a phase portrait of the

relative L_ norm of the strearnfunction change per time step versus the total kinetic

energy, for 6100 < t < 7100. This time interval was required to define the entire

phase portrait. This phase portrait clearly shows a complex but regular dynamical

process. Figure 5 shows three representative streamfunction contour plots for the

asymptotic flow. The unsteady dynamics are laminar, and are visibly concentrated

in the relatively weak secondary flow structures. In each of the secondary flow

structures weak tertiary circulations form underneath the separated primary flow

near the point of separation, and then are convected into the secondary flow as a

whole, to interact with and join the focus of the secondary flow.

Discussion

There are well known steady solutions for the square driven cavity at Re =

10000 in [1] and [6]. These steady results do not contradict the current work since

they were obtained with steady state codes. Theoretical expectations [8], the unsteady asymptotic results in [7] for the square regularized driven cavity using a

Chebyshev-Tau approximation, and the results in [3] for the rectangular driven cavity using the same method as in this work all support these unsteady asymptotic

results in the square driven cavity at Re = 10000. Continuing work in preparation

[2] suggests that the flow in a square driven cavity has a Hopf bifurcation to a

periodic asymptotic state for Re < 9000.

4
Bibliography

[1] U. Ghia, K. N. Ghia and C. T. Shin, J. Comput. Phys. 48, 387 (1982).

[2] J. W. Goodrich, in preparation.

[3] J. W. Goodrich, K. Gustafson and K. Halasi, "Hopf Bifurcation in the Driven

Cavity, _ ICOMP Report 89-21, NASA Lewis Research Center, Cleveland, 1989,

to appear in J. Comput. Phys., (1990).

[4] J. W. Goodrich and W. Y.Soh, J. Comput. Phys. 84, 207 (1989).

[5] K. Gustafson and K. Halasi, J. Comput. Phys. 70, 271 (1987).

[6] R. Schreiber and H. B. Keller, J. Comput. Phys. 49, 310 (1983).

[7] J. Shen, "Hopf Bifurcation of the Unsteady Regularized Driven Cavity Flows, _

Report No. 8906, The Institute for Applied Mathematics and Scientific Computing, Indiana University, Bloomington, 1989.

[8] R. Temam, Infinlte-Dimensional Dynamical Systems in Mechanics and Physics,

(Springer-Verlag, New York, 1988).
L

i=

m

m_

,,=

i =,

0

0

o

0

O

r_

0

0

0

_D

0

0

r_

0

0

O

O

0

o

0

o

m 0

O000 O_ O0 OO0

_000000_O0000000

0

0

vl

V

%..

I.-

e,L

!

N

1.1_
p

-=-,

P i , _ i i i l , i , _ i , , _ , , , ,

_o _o____ _

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

I I | I I I I ! l I i I I I I I i I I I

0

0

p-.

0

0

0

0

CO

¢D

0

0

0

o

o

u')

o

o

o

o

Q_

o

0

p..

vl

V

m--

W

_w

W

Q.

Z

c.J

_J

p..

I

_J

U.
-r

01

I

U"I 0

.,,.I

"r i!

r"1'

T0

,..._

,__..,7J

,C. 'r T I 1 I I I ! I

T

r,..

vlV8

',.D

r,.-

l.,lJ

l-- Z

rJ ",.,"

tD I-,- k O

ktI'--I

e_

8
TKI

0.03630

0.03629

0,03628

0.03627

0.03626

0.03625

0.03624

0.000060

I I --I ................. 7"

0.000065 0.000070 0.000075 0,000080 0.000085

SFCNG

FIGURE q. - STREAMFUNCTION CHANGE VERSUS TOTAL KINETIC ENERGY, GIO0 < t_7100.

I

0.000090
OR|GINAL PAGE IS

OF. POOR QUALITY

O. _0

PSI -- -0,110 ---O,Oto 0.070 --*0_050

-0.030 -- -O_OIO 0,001 ....... 0.000

....... 0.001 ....... 0.002 ....... 0 00)

oso

x

(a) t = 7101.0.

050

(b) I = 7101.S.

lOO o:00 o:_S

FIGURE 5. - STREA/fUNCT[ON CONTOURS.

0.5O

(c) t = 7102.0.

10
Report Documentation Page National Aeronautics and

Space Administration

1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.

NASA TM-103141

4. Title and Subtitle 5. Report Date

An Unsteady Time Asymptotic Flow in the Square Driven Cavity

7. Author(s)

John W. Goodrich

9. Performing Organization Name and Address

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C. 20546-0001

6. Performing Organization Code

8. Performing Organization Report No.

E-5493

10. Work Unit No.

505-62-21

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15. Supplementary Notes

Prepared for the IMACS 1st International Conference on Computational Physics cosponsored by the International

Association of Mathematics and Computers in Simulation and the University of Colorado at Boulder, Boulder,

Colorado, June 11-15, 1990.

16. Abstract

This paper presents summary details of an aperiodic time asymptotic numerical solution for the square drive

cavity at Re = 10000s, The data presented is for 6100 < t < 7100, and is representative of the data that

characterizes the aperiodic asymptotic state. Complete details are not given in this paper for either the transient

evolution or the asymptotic state.

17. Key Words (Suggested by Author(s))

Incompressible viscous flow; Navier Stokes equations;

Time dependent; Unsteady asymptotic state; Driven cavity

19. Security Classif. (of this report)

Unclassified

18. Distribution Statement

Unclassified-Unlimited

Subject Category 34

20. Security Classif. (of this page)

Unclassified

21. No. of pages

12

22. Price*

A02

NASA FoRM 1626 OCT 86

*For sale by the National Technical Information Service, Springfield, Virginia 22161