Numerical Solutions 



Bt 



Bx 



3y 



3^ 



3x2 



32^ 



and the relation between vorticity and the stream function 



- ^ . 



B2^ 



3^0 



By 2 



(1) 



(2) 



These equations are dimensionless with reference to the mean velocity u^ and 

 the spacing Dq of the upstream portion of the nonuniform conduit, and the density 

 of the fluid. The vorticity is denoted by ^ , and the stream function by ^ . (i, j) 

 denote spatial coordinates in discrete form, n is a superscript which counts the 

 time intervals in the following finite-difference equations which were established 

 as counterparts of Eq. (1) and (2): 



^i.j ^h2 2St/ l2St ^'^ h2 V^^^i'^ ^-1-^ ^^.^-1 



- (^i,..x-^i,i-i)(^i.i,j- ^?-i,, I • 



Boundary conditions (for example the nonslip condition) were also expressed 

 by means of finite-difference schemes in the form of inward expansions. The 

 functions ^ and ^ were expanded by means of Taylor series from the walls 

 inwards. 



Because the computational technique is based on calculating the distribution 

 of the stream function during one of the steps, and subsequently that of the vor- 

 ticity function, expressions in difference form are necessary to calculate I, in 

 terms of •/- at the boundaries (the expressions at inner points have the standard 

 form for the Laplacian in two dimensions). One of the expressions used at the 

 boundaries, which can be considered as a typical one, is 



^T 



(■/-I 



) - 



B+1 



(^vv + ^v.) 



yy^B 



Here, B is a point at the wall and B+ i is a point one mesh inside. 



To begin the calculation we assumed that the flow would be started impul- 

 sively and that at the time O"*" the flow would be irrotational. Thus, the initial 

 values of ^ were given by irrotational flow without separation. 



The results given in Fig. 1 show how an eddy forms initially at the entrant 

 corner, and how it grows. Two eddies form at a certain time, but one is 



1610 



