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Outflow Boundary Conditions on Staggered Grid

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Old   October 15, 2016, 13:56
Smile Outflow Boundary Conditions on Staggered Grid
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I am trying to implement a 3D finite difference Navier-Stokes solver. I'm having trouble with implementing the boundary conditions, especially around corners and edges.

Right now I have a separate function called apply_boundary_conditions that seems to work for the Drichlet boundary (even on corners and edges); the next step is to add outflow boundary conditions (normal derivative of the velocity is 0).

Adding it on the faces of the boundary isn't too difficult, but I'm having difficulties in picturing how to deal with the different combinations of boundary conditions around corners and edges. There's a lot of different possible combinations for example consider a corner cell. In this case we have the following cases to consider:
- Dirichlet on all three sides
- outflow on all three sides
- outflow on z, Dirichlet on x,y
- outflow on z and x, Dirichlet on y
- ....

And in addition there's similar combinations for edges. As it stands my apply_boundary_condition function is already several hundred lines of code.

I know this is a somewhat vague question, but is there a well known way to structure the application of boundary conditions using staggered grid finite difference? Or is it just something that has to be done using brute force, considering every possible case?
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Old   October 15, 2016, 14:14
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Filippo Maria Denaro
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As a general rule, staggered grids works in such a way that on the boundaries you set the normal component of the (divergence-free) velocity field. Therefore, is zero on a not-permeable wall, known component for an inflow profile and Neumann condition for the normal component (or DIrichlet condition if known) on the outflow. Values of corners are not required since FD works on each coordinate direction and no mixed derivatives for velocity and pressure are required.
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finite difference, outflow boundary, staggered grid

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