SIMPLER Algorithm

Bharath

Saturday, 22 April 2006

Hello everyone

I'm trying to solve 2D incompressible flow problem in a channel using SIMPLER algorithm on a staggered grid approach using Finite volume discretization using central differences. I have some issues with the proper implementation of boundary conditions. If you have some insight to the questions i have below, please reply to this message.

Problem setup: Flow is incompressible and is from left to right. The top and bottom are walls.

Boundary conditions: Left side: I specify a value of pressure and v-velocity at the inlet and allow u-velocity to float. Right side: I specify fully developed flow for u and v velocities and specify a value of pressure Top wall: No slip and normal pressure gradient is zero Bottom wall: No slip and normal pressure gradient is zero

1. If I'm right, one cannot simultaneously specify pressure and velocity at a boundary. In SIMPLER, since we need to solve the pressure equation using the estimated pseudo-velocities, we need to see the pressure equation as a boundary value problem. So, we need to specify pressures at the inlet and exit and apply the normal derivative of pressure at the top and bottom walls. This is fine. The question I have is- how can one consistently estimate the boundary velocities at the left side? Right now, I'm using a linearized form of NS equations like the Darcy's law to estimate the boundary velocities and then use the full blown NS equations for the interior volumes. The cell continuity is not satisfied and the pressure drop in the domain always stays linear. Is this right? Or, to be more accurate one can use Bernoulli equation to estimate the boundary velocities but again this prescription is not straight forward to implement and becomes too complex. Also, if I use the Darcy law kind of linear momentum equation connecting pressure gradient and velocity, the continuity is not satisfied.

In addition, I'm using Darcy law kind of linear momentum equation both in the u-pseudo-velocity routine to estimate the u-pseudo-velocities and in u-star-velocity routine to estimate the u-star-velocities. Is there any other consistent form of momentum equation to correctly estimate the boundary velocities? This special treatment of the boundary volumes is due to the fact that we need to specify a value of pressure (that has physical significance) in SIMPLER as against just specifying p' (mathematical artefact) in SIMPLE. Please let me know if you have any comments on this thought.

2. The over-specified and may be the inconsistent formulation would be to specify pressure and velocities at all sides. If this is the case, the continuity in each cell will not be satisfied.

In other words, what are the consistent? boundary conditions in SIMPLER algorithm?

Thanks