Numerical Solutions 



taking the divergence of the Navier-Stokes equations. The relevant equations in 

 Cartesian coordinates are 



^ 1 3p 



+ VV" u. , (1) 



where v and p are assumed constant for this discussion. Since neither the 

 pressure nor its normal gradient are known at a rigid wall, the application of 

 the boundary conditions becomes an integral part of the iteration process. The 

 exact procedure is as follows: One forecasts new values of u^ at time level 

 n+i, say, using their values (and that of p) at levels n and n-l, depending on 

 the scheme employed. Then one finds p to order n+1 by iterating (2), using the 

 values of the normal derivative 3p/3n on the boundaries and the source function 

 F evaluated at time level n+ 1 . The boundary values of 3p/^n at level n+ 1 are 

 obtained from (1) upon substitution of the u^+i into all the terms. 



2. V or ticity -Stream Function Approach: In this method one introduces a 

 vector potential -/- and a vorticity vector i . Defining u = v x and ?; = v x u, 

 one obtains the following set of equations: 



^l, 



+ (" • V) ^. - a ■^)^i = ^v2 ^. , (3) 



V' -/-; = - ^i , (4) 



where the condition V • </- = o is put on the vector potential. The numerical pro- 

 cedure is similar to the previous one, though the boundary conditions are again 

 troublesome. The components of the vorticity vector parallel to a rigid surface 

 are not known, whereas the corresponding stream functions and their normal 

 derivatives are known. It is clear that we cannot use both sets of conditions in 

 solving (4), because then the problem becomes over-determined. Rather, one 

 uses the boundary values of "/-. to solve (4), and the boundary values of d4'.^/dn 

 to find the vorticities ^. at the wall from a Taylor-series expansion. The 

 exact iteration procedure is then as follows: One forecasts new values of ^^ 

 at time level n + i on all interior mesh points and then finds -/-j by iterating Eq. 

 (4). Finally, the boundary values of ^. are found from a Taylor-series expan- 

 sion of the stream function values on mesh points adjacent to the wall, about 

 values on the wall, and utilizing the fact that, at a rigid wall, any parallel com- 

 ponent of vorticity is given by ^. - 3^0/3^2 evaluated at the wall. An alterna- 

 tive procedure would be to forecast 4 /on the wall itself, using one-sided spatial 

 differences and ensuring that the total forecast procedure remains conservative; 

 however, this procedure has not met with much success. 



In 1966, a paper appeared by Arakawa that showed how to difference the ad- 

 vective terms in two-dimensional, incompressible flow that conserves vorticity, 

 mean-square vorticity, and kinetic energy. An analog of this procedure for Eq. 

 (3) has not yet been found. 



The author is not aware of any published works relying on the velocity- 

 pressure approach, though recently successful use of it has been reported by 



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