Dissipation in upwind scheme- remedies!!
 

Hi,
I am currently working on numerically solving partial differential equation (basically Navier-Stokes equations ) for a particular problem. I wrote a Matlab code.

For higher Reynolds number, I am using first order upwind scheme. But I am getting quite a dissipative solution. I searched on the internet and came across the use of deferred correction for convective terms. But now, the iterations are diverging.

Can anyone suggest me what other schemes can i use? or what changes can i consider in the upwind scheme ?

Thanks in advance. :)

have you tried to simply switch to central second order discretization? If you have high Re number, your flow maybe is turbulent and you must use some modelling formulation. On the other side, if you reach the DNS requirement, the second order discretization is stable

Thanks for your response.

I would like to add that it is assured that the flow is laminar.

I tried using central difference scheme but the solution is diverging for high reynolds number. I am getting solution for high reynolds number for 1st order upwind scheme but as i mentioned they are dissipative. I was thinking about high resolution schemes ->
http://www.cfd-online.com/Wiki/Appro...emes_.28HRS.29
But i dont know which one to choose.

Thanks.:)

if your flow is laminar, simply refine the grid in such a way that your cell Re number is <2 and the solution with central second order discretization will be stable. There is no need to use higher order scheme for laminar solution

hello again,

As suggested, i tried to decrease the grid size such that the cell Pe <2 .
I ran into two problems.
1. The solution converged only for low reynolds number, not for medium to high value of Re.
2. I ended up with large number of linear equations are to be solved. As such large system necessarily require the iteration solvers, I am using Gauss Siedel with relaxation. Is there any other faster iteration solvers? what about Multi-grid method?

Thanks.

1. if the cell Re number is less than 1, you are solving all the scales of the flow, provided that the dt is within the stability constraint your numerical solution must be stable.
2. the SOR method is quite useful also for large system. Often, more sophisticated methods such as Multigrid, BCGSTAB, GMRES, etc, are efficient in terms of reducted number of iterations but the computational cost of the single work-unit can be much more expensive than tenth of SOR iteration.

Thank you for your response.

However, the problem is steady. I am basically solving stream vorticity equations(so an outer iterative procedure is also required). Is there any other condition for steady problem ?

On the other hand, I also wrote a code for 2nd order upwind scheme. There the solution is converging for the required Reynolds number. But the solution is highly dampened (or dissipative). Any insights?

Thank you for your response.

However, the problem is steady. I am basically solving stream vorticity equations(so an outer iterative procedure is also required). Is there any other condition for steady problem ?

On the other hand, I also wrote a code for 2nd order upwind scheme. There the solution is converging for the required Reynolds number. But the solution is highly dampened (or dissipative). Any insights?

Have you tried to reduce the relaxation factor?

be careful, laminar solution can be unsteady and maybe your steady state ad high Re number can be simply due to strong numerical dissipation

yes, i have tried to reduce the relaxation factor.
This problem on which i am working on is two dimensional. I have read about effectiveness of upwind scheme in one-dimensional from Patankar. While using an iterative procedure to solve a set of PDEs, I am getting negative coefficients of the variable (as in ApTp=AeTe+AwTw; Ae <0) for some of the grid points.

I don't know if this happens because i am using upwind in 2D problem. How do i tackle this "negative coefficient" problem?

Thanks in advance.