Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics
- PDF / 555,264 Bytes
- 10 Pages / 562 x 810 pts Page_size
- 46 Downloads / 183 Views
~z
~n2
+ 2 ~ - -~~
= 0 ,
(1)
with =
R ~
R ~
- ~ ~-~
'
U
=
4
~
"
Here ~ is the stream function and R is the Reynolds number. The variables (~,q) are coordinates in some orthogonal curvilinear system, usually chosen so that the domain of the solution of (i) is a rectangle. In obtaining numerical solutions to problems governed by the two-dimensional Navier-Stokes equations, it is quite customary to solve equation (i) numerically in conjunction with the equation ~2~ + ~2~ + ~ / H 2 = 0 ,
~2
(2)
~n2
where H(~,N) is a function which depends upon the particular coordinate system used. In most of the earlier work on the numerical solution of the Navier-Stokes equation~ using finite-difference approximations, for example the work of Thom (1933), Kawaguti (1953) and Apelt (1961) on flow past a circular cylinder, it has been usual to approximate all derivatives of ~ in (I) by central-difference formulae. More recently Spalding (1967) and Greenspan (1968) have suggested a method in which, although the second derivatives of ~ in (I) are approximated as usual by central differences, the terms ~ / ~ and ~ / ~ q are approximated by forward or backward differences depending upon the signs of % and ~, i.e., depending upon the local direction of the flow at any given grid point. This method will briefly be reviewed in the next section, but basically the object is as follows. If % and ~ become large compared with the grid size in some regions of the flow field, as may happen if R is large, the matrix associated with the difference equations obtained by approximating all derivatives of ~ in (I) by central differences may fail to be diagonally dominant. Diagonal dominance is a sufficient condition for the convergence of the point Gauss-Seidel iterative procedure, and for the point SOR procedure for a well-defined range of the SOR factor (Varga, 1962), whereas these procedures may fail to converge for matrices which are not diagonally dominant. The matrix associated with the equations obtained by approximating ~ / ~ and $~/$q by forward or backward differences depending on the direction of flow is diagonally dominant under all circumstances. This method of approximation therefore leads to a successful computational procedure. It has been discussed in some detail by Runchal, Spalding and Wolfshtein (1968) and a number of applications of this and methods of sim$1ar type have been given, for example, by Kawaguti (1969), Thoman and Szewczyk (1969). The disadvantage of methods of this type is that the truncation error in approximating (i) is greater than that when all derivatives of ~ are replaced by central differences. *)
On leave from the University of Western Ontario, London, Ontario, Canada.
121
The object of the present paper is to discuss a method of approximating (i) in which the matrix associated with the difference equations is diagonally dominant and, further, the truncation error is the same as that of the fully central-difference approximation. The origin of the method is in a paper by Allen and Southwell (
Data Loading...
