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Old   June 5, 2008, 06:26
Default Finite volume inflow/outflow BCs
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I am trying to figure out what boundary conditions to apply at inflow and outflow boundaries for my 3D finite volume solver. I came across this old paper:

Ph. Rostand, B. Stoufflet, "Finite Volume Galerkin Methods for Viscous Gas Dynamics", INRIA, 1988

Section II.3 mentions a mechanism whereby such BC's can be imposed by means of a split of the inviscid flux Jacobian A = dF/dU according to the sign of the eigenvalues:

A = A+ + A-


A+ = R(D+)L, A- = R(D-)L

and D+, D- contain only the non-negative and non-positive eigenvalues respectively. I realize that for a homogenous function of degree 1, as is the case for the Euler flux function, the fluxes can now be calculated as

F(U) = (A+)U + (A-)U

However, the paper says that the boundary fluxes at an inflow or outflow can be calculated as:

F(U) = (A-)U_external + (A+)U_internal

U_external is the state exterior to the computational domain, U_internal the state just inside the boundary and where the flux Jacobians are evaluated at U_internal. I don't understand. Intuitively, I would have it this way:

F(U) = (A-)U_internal + (A+)U_external

because A+ and A- correspond to characteristics entering and leaving the computational domain. For example, at an inflow I would like extererior information to propagate into the domain.

I tried to find the background paper by Steger and Warming, but failed. So if anyone can explain why I am wrong, that would be great.

Thanks, Martin
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Old   June 5, 2008, 09:20
Default Re: Finite volume inflow/outflow BCs
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Hi Niels,

instead of "F(U) = (A-)U_external + (A+)U_internal " consider "F(U) = (A-)U_right_cell + (A+)U_left_cell " for time being, and convensions are face normal is defined such that it propogates in to the right_cell.

Now see,

INCELL (A+)-> |(face)| <-(A-) OUTCELL at a given face A- will always points towards face from outcell and A+ will always points towards face from incell

Finally , at inflow, INCELL will be the exterior domain and hence you use A+, you are right here but you confused because in that formula (F(U) = (A-)U_exterior + (A+)U_interior) external cell means external to the face with the normal direction I mentioned and hence external cell what he is refering is OUTCELL which is a interior zone not the exterior zone with respect to the computational domain..........

You have just confused with the word external which does not mean exterior domain to the computatinal domain but cell which is right side to the face (with face normal that i defined) and all your "physics of upwinding" understanding is right.

If you dont understand what I said please draw it on a paper while reading, and even then you dont understand it means that I dint expressed thing properly and I am sorry for that.

Thanks, Vinayender.
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Old   June 6, 2008, 01:34
Default Re: Finite volume inflow/outflow BCs
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Sorru in above discription Incell corresponds to left cell and Outcell correspoond to right cell for a particular face.

Thanks, Vinayender.
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Old   June 6, 2008, 03:50
Default Re: Finite volume inflow/outflow BCs
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Hi Vinayender,

Thank you very much for your answer. It totally cleared things up!

It seems that my text books, although they never explicitly mention it, consistently adhere to the convention of traversing boundaries so that the interior of the volume always lies on the left hand side of the boundary. Just like you say.

Regards, Martin
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