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syoo April 12, 2011 02:22

deforming mesh and freestream preservation

I am currently trying to code an 2D Euler-eq solver using the Beam-Warming scheme for a given deforming mesh. So far, I can't preserve freestream using my solver (embarrassing ...) for deforming grids. The freestream is conserved for a non-deforming grid.

My current understanding is that the mesh deformation (mesh velocity) is accounted for in the solution through the Jacobian which transforms the grid from the cartesian coordinate to generalized body-fitted coordinate. I also satisfy the Geometric Conservation Law by calculating the grid metrics written the conservative form, suggested by Thomas and Lombard.

Should I be including the grid velocity in the flux terms? I have went to many literatures that use Beam-Warming scheme and none of them mention this.

Any help, suggestion, or comment will be very much appreciated.

thank you.


agd April 12, 2011 14:11

You need to include the grid velocity terms in the fluxes, since the flux across the face depends on the fluid velocity relative to the face. Additionally, Thomas and Lombard describe a procedure for ensuring geometric conservation when the grid is moving that involves the proper construction of the time derivative of the Jacobian.

syoo April 14, 2011 00:05

thank you. I'll try it.


syoo April 18, 2011 11:12

I would like some clarification.

I have seen many papers using ALE formulation which includes the grid velocities in the fluxes to accommodate for non-rigidly deforming grids. But I also see papers (ex. Visbal, "On the Use of Higher-ORder Finite-Difference Schemes on Curviilinear and Deforming Meshes") that does not mention including the grid velocities in the flux. In Visbal paper, he uses the same time-implicit method that I use which the Beam-Warming scheme.

so is including the grid velocity necessary? or does Visbal imply that grid velocities are already included in the velocity terms?

thank you.


agd April 18, 2011 12:42

Sorry - I should have been clearer in my original post. The grid velocities are included in the flux terms, but they appear in the form of the grid time metrics (the time derivatives of the grid transformation terms). The Jacobian relationship in Thomas-Lombard connects the time derivative of the Jacobian to the spatial variation of the time metric terms. All of this is necessary for GCL.

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