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 *
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:
herein, \jJ is the streamfunction; Yi, the vorticity; ho, the mesh size; r, the radial distance;
ot,
the Hey- nolds 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! l' K+1 K-f-l l' (
l \\jJi+l,j
-1-
\jJi-l,j-1-
\jJi.i-l-1-
\jJi,Hl4' 2 l ' + l . K+1 l'
1.-
ho [l'i,jYii,j-1-
(\jJi,j--l'- \jJi,HI )/(2ruho]1
\
r r ') ')
[l -- 6t(\jJi~],r--\jJi~1,j)/2ho
-1-
41';'/h(~]-12 K 1\:+1 K+l K 2
l'i,j (Yii+],j
-1-
Yii-],j-1-
YiU-l-1-
Tli,j-l-l)/hol'-j-] l'
ru(Yii,j-] -- Tli.H]) /(2 ho)
K K K 1\.+]
. - Cn.J'i.j [(\jJ;,Hl -- \jJi,j-1) (Yi;-I-1,j -- Yii-1,)
K K K+l K 2
-1-
(\jJ':+1.j - \jJi-1) C'f]i,j-l - Yii,Hl)] /(4 ho)x
1':-fol
Yii,.i=
-,.,._---_._._-_..~---'--'---- - --
• Iowa Institutc of Hydraulie Hcsearch Uni\,cl'sity 01' Iowa, Iowa City.
During a computational investigation of viscous- fluid 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 uni- form flow. Itwas 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 eco- nomy 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 uni- form region to anothcr. The results obtained to date for expansions of ratio 2: 1confinll the sup- position.
'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 solu- tions for two-dimensional and axisymmetric Poi- seuille 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 hypo- thetically in error at the start of a computation.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
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 resi- dual 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 sta- bility 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 defi- nitely more advantageous than the one-way itera- tion. \Vithin the range of disturbances investigated a tenfold increment of the assumed initial devia- tions did not afIect the results to a noticeable degree. The most important result is that the ana- lysis of the purposely disturbed uniform llow gives a reliable indication of the mesh size that ensures com putational stability for a given Reynolds num- ber 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.
32 36 28
24 28 32 36
,ç>, Four-way iteration
'0' Itération -4voies One-way iterafion
o Itération voie unique
20 24
D/h D/h
20
16 16
8 12
Flow with seporation Écoulement avec déco/lemenl
o Divergence -Divergence
• Convergence -Convergence
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.
Thom's GrIterion Cn/ère de Thom
Disturbed uniform flow Écoulement uni/orme perturbé
30'---='---'---'---~----'---'---'---
4 8 12
40 50 80
40 70 80 90 90 100 -
1·2/
70
cR60
60
ik
50
74
