finite difference method for navier-stokes problem
 

hi all,

I have a problem in solving nonlinear PDE with finite difference method(unstaggered scheme).It is an 2D flow through a contraction in a pipe. Finally I ended up with 2 governing equation(x and y momentum equations) and 1 continuity euqation:du/dx+dv/dy=0, where u and v are velocity and x and y directions.

The two momentum equations contain velocity terms(linear and non-linear) and pressure terms. To my understanding, they needs iterative methods with an initial guess of velocity(u,v) and pressure(p).

here comes the problem:

with an initial guess of (u,v) field and pressure(p),I got a new set of (u,v). then how should I continue? what is the function of the continuity equation? I guess I need it to update the pressure(p),and continue using new sets of (u,v,p) to proceed with first two momentum equations , and so on.

but I cant see how to get pressure updated with continuity equation.

I read the book by H.K Versteeg and W Malalasekera, which used finite volumn methods,but it wont help me with finite difference methods, right?

anyone had any suggestion on the problem or any helpful resource?

thanks in advance.

Re: finite difference method for navier-stokes pro
 

For 2-d flow through a pipe (variation along the axis and radially from center line to wall edge), I would expect a cylindrical coordinate system to be used. Then the continuity equation would be

du/dr + u/r + dv/dz = 0

with r being the radius and z being the axial coordinate. To do an axisymmetric pipe flow in Cartesian (x,y) coordinates, you'll need to do a 3-d calculation with curved boundaries!

The extra term, u/r, arises from the cylindrical coordinate system. Similar changes occur in the momentum equations.

Patankar's book describes one way to extract pressure from the coupling of momentum and continuity. Earlier work at Los Alamos (MAC, SMAC, the SOLA schemes) show the original details.

There are detail differences between finite difference and finite volume implementation of these schemes (and formal accuracy estimates may be different), but either should be workable.

All of the schemes mentioned above were developed for a staggered mesh. I can't advise on the details of moving them to an unstaggered mesh, although apparently the work of Rhie & Chou (spelling?) addresses this problem.

Hope I haven't misunderstood your problem.

Re: finite difference method for navier-stokes pro
 

Here are two references that you may want to consider:

1. Computational Fluid Mechanics and Heat Transfer by Tannehill, Anderson, and Pletcher 2. Finite Analytic Method in Flows and Heat Transfer by Chen, Bernatz, Carlson and Lin

The second reference gives pretty specific details for implementing SIMPLE methods on both staggered and non-staggered grids. You might also want to considering "reformulating" the continuity equation into a Poisson pressure equation. This will give you a "definitive" equation for the pressure (just like the u momentum equation is a "definitive" equation for the u velocity, etc.) This formulation has nice convergence and stability properties however, continuity is not always exactly satisfied and large dilation (dilitation) values may appear (so it's a tradeoff!)

Hope this helps!

Ryoga

Re: finite difference method for navier-stokes pro
 

You may also search for the Uzawa iterations. A little bit slower than the approximate projection method as Ryoga suggested, but the divergence can be made free to machine accuracy.

Re: finite difference method for navier-stokes pro
 

Hi There!

Another alternative I can suggest you. It is possible to modify Continuity & Momentum equations from (u,v,p) to stream function and vorticity forms. In this case you only have two unknowns which are stream function and vorticity, no headache about pressure, P. For 2D case this process works nicely even in DNS methods.

Manosh

Re: finite difference method for navier-stokes pro
 

thanks a bunch guys! I am now learning to use vorticity-stream function approach.