Computational stability of explicit difference form ... - La Houille Blanche

2 downloads 0 Views 121KB Size Report
OF EXPLICIT DIFFERENCE. fOR'M OF EQUATIONS .... line separating stable from unstable points is thus ... band defined hy the dashed lines in'])()th figures.
LA HOUILLE BLANCHE/N° 1-1967

COMPUTATIONAL STABILITY OF EXPLICIT DIFFERENCE fOR'M OF EQUATIONS FOR' VISCOUS FLUID FLOW

SY ENZO O. MACAGNO * AND W. T. K. HUNG *

During a computational investigation of viscousfluid flow in conduit expansions, a preliminary study has been made of the stability and accuracy of the calculations. This analysis was bascd on applying the same numerical techniques to a flow with separation and to a purposely disturbed uniform flow. It was conjeetured that the behaviour of thediscretized form of the equations would not be too difl'erent for uniform and nonuniform conduits, and it could th en he established with a great economy of computer time by studying the simpler case. The conjecture seemed very plausible, because experimental kno,vledge indicated that steady flow with separation at not-too-Jow Heynolds numbers consisted of a weak eddy-or two--and a Inain flow with a smooth transition l'rom one uniform region to anothcr. The results obtained to date for expansions of ratio 2: 1confinll the supposition. '1'0 efl'ect the disturbance of the uniform flow, a length of conduit slightly greater than one conduit ,vidth was subdivided by means of a square-mesh net; the location of a mesh point was indicated by the pair of integers i, f. The values of the stream function and the vorticity ,vere then calculated for aIl the i, j points according to the lmovv exact solutions for two-dimensional and axisymmetric Poiseuille flows. The values of the vorticity ,vere numerically disturbed by a factor (1 -1- e) at inner points with an even value of i, and by (1- e) at odd values of i; the values of e used were 0.1, 0.2, 0.50, 0.75, and l.OO. The result of this operation was considered as a vorticity distribution hypothetically in error at the start of a computation. -,.,._---_._._-_..

~-------'--'----

-

--

• Iowa Institutc of Hydraulie Hcsearch Uni\,cl'sity 01' Iowa, Iowa City.

The fînite-difference formulas utilized were oÎ the same type for aH conduit geomefries considered; only the formulas for the circulaI' tube are given here: 1': -fol

Yii,.i =

r

r

')

')

[l -- 6t (\jJi~],r-- \jJi~1,j)/2 ho -1- 4 1';'/h(~]-1 2

K

1\:+1

l'i,j (Yii+],j

-1- Yii-],j -1-

l'-j-]

K+l

YiU-l

K

2

-1- Tli,j-l-l)/h o

l'

1

ru (Yii,j-] -- Tli.H]) /(2 ho)

x

K

. - Cn.J'i.j

-1!

K

(\jJ':+1.j l'

1.- ho

1\.+]

K

[(\jJ;,Hl -- \jJi,j-1) (Yi;-I-1,j -- Yii-1,)

K

l \ \jJi+l,j 4' 2

K

-1-

K+l

\jJi-1) C'f]i,j-l K+1

K-f-l

K

l'

-1- \jJi.i-l -1- \jJi,Hl l ' + l . K+1 l' [l'i,j Yii,j -1- (\jJi,j--l'- \jJi,HI )/(2 \jJi-l,j

\

2

Yii,Hl)] /(4 ho)

(

ru ho]

herein, \jJ is the streamfunction; Yi, the vorticity; ho, the mesh size; r, the radial distance; ot, the Heynolds number. All quantities are dimensionless. 13 indicates points on the wall. K is the iteration index. Thê first formula is a cliscretized form of the vorticity diffusion equation, while the second expresses the vorticity, and the thinl the non slip condition at fixed walls. Of the many difl'erent ways in which the field can be swept when iterating the previous formulas, two were triecl. In the first, the field was always swept, say, from the SW corner of the field to the NE corner; in the second, from 73

Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/1967006

Enzo O. MACAGNO and W. T. K. HUNG

110 100 -

90 80 Thom's GrIterion

Cn/ère de Thom

70

cR

60 50 40 30

D/h '---------='------'------'-----~----'-----'-----'---

4

8

12

16

24

20 ,ç>,

Disturbed uniform flow

'0'

Écoulement uni/orme perturbé 90 80 70

o

28

32

36

Four-way iteration

Itération - 4 voies One-way iterafion

Itération voie unique

Flow with seporation

Écoulement avec déco/lemenl o Divergence - Divergence • Convergence - Convergence

60

ik 50 40 8

1·2/

74

12

16

20 D/h

24

28

32

36

Two-dimensi6l)al flow and axisymmetrie flow. Points below fuÎl lines are numel'Ïeally stable. Ecolliement plan et écolliement axisymétriqlle. Les points sitllés all-desSOllS des li(Jnes continlles sont nllmériqllement stables.

the S\V corner 10 Ihe NE corner and bad:, then from the SE corner to the N\V corner and back, and so on until the results either settied within an assigned margin or diverged. As should be expeeted, the accuracy improved with increased number of meshes. Constant residual errors, ho"\vever, were found 10 persist aner the numerical solution was settled through a large number of iterations. Those residual errors were smul! enough 10 consider the result satisfaetory; al! other things being equal, hU'ger residual errors resulted for the axisymmeiric flow th an for the Iwo-dimensional flow. Findings concerning the stability of the numerical solution are represented in Figures land 2. Results of calculations for conduit expansions are also shown; they consume an order of magnitude more time, and theref'nre only few of Ihese computations have been performed. The line separating stable from unstable points is thus not so sharllly defined as in the case of disturbed uniform flo\v,' but it must certainly be within the band defined hy the dashed lines in'])()th figures. From this study it has been found that the influence of the method of sweeping the field is not negligible; the four-way iteration (with practically the same programming and computing time) is definitely more advantageous than the one-way iteration. \Vithin the range of disturbances investigated a tenfold increment of the assumed initial deviations did not afIect the results to a noticeable degree. The most important result is that the analysis of the purposely disturbed uniform llow gives a reliable indication of the mesh size that ensures com putational stability for a given Reynolds number in a 1l0w with a region of separation. II remains to be determined how much more a nonuniform conduit can depart from the uniform before the indication is invalidated.