# a few questions on understanding geometric conservation law

I was wondering if anyone could help with understanding the geometric conservation law for moving domains. I came across Link1, and have tried to understood the paper by Farhat et al Link2.

So far my understanding is that the two things to consider are 1) the time step to which the cell areas using for integration of flux terms correspond to; 2) the time step at which the mesh velocity ( $\dot{x}$ ) is being evaluated. Finally, the ALE equations must also be satisfied for a uniform flow case.

The paper states a GCL law as:

$$A_{i}x^{n+1}-A_{i}x^{n}=\int_{t^{n}}^{t^{n+1}}\int_{\partial C_{i}(x)}\dot{x}\overrightarrow{n}d\sigma dt$$

where $A$ refers to the area of cells, $C$ refers to the cell areas swept by the fluxes, t is time, x is the spatial coordinate at the current configuration, and I think $\sigma$ refers to the surface area of integration corresponding $C$.

I have a few questions that I would really appreciate some help with:

1) what is $C$ in 1D. Is C referring to the length of the cells or the area, which is set to 1 in 1D.

2) in the paper (just before equation 16), for a uniform flow, it sets the variables being solved for at different time steps equal to each other. Is that not the definition of steady flow?

3) Finally, is there a way to determine whether a set of numerical results in 1D, obey GCL or not, without actually going through the equations. For instance, my understanding is that the results should stay independent of the moving domains, so if I compared the results obtained using $\dot{x}$ =0 and $\dot{x} \neq 0$ and showed that they were not the same?

If there are any simpler papers or examples on GCL, please let me know.

1) In 3D, C is a swept volume, in 2D a swept area and in 1D a swept length, i.e. the change in length of a cell between $t^n$ and $t^{n+1}$.

2) Uniform flow is steady, the fact that the grid deforms implies that the simulation is unsteady, but if the GCL is satisfied then the flow itself shouldn't change.

3) Yes, that would be enough.

As far as I know, the first (and easiest) paper to introduce the idea of swept volumes is Demirdzic & Peric, 1988 for incompressible flows and finite volumes. It seems to have been reinvented later in the context of compressible flows and finite elements.

In Etienne e.a. 2009, you will find a nice overview. They introduce 3 levels of GCL compliance: exact solution of no-flow, exact solution of uniform flow and identical grid/timestep convergence rates on fixed and deforming grids. Manufactured solutions are used to verify that a code complies.

• @ Chris, thanks. I've now read the papers and some other papers. 1) I noticed that calculation of uniform flow is tested using the flow in a square domain test case, I was wondering how this could be done in 1D? 2) The fact that the results with fixed mesh and moving mesh are different, does not imply which produces the correct prediction, right? I have a set of results which match analytical results on a moving mesh but not on a fixed grid. – Hooman Jan 12 '16 at 0:30
• 1) Incompressible uniform flow in 1D means $u$ and $p$ constant. But beware: methods which do work in 1D might fail in 2D. – chris Jan 12 '16 at 7:49
• 2) Strange. If it works for a moving mesh, then it should also work for mesh velocity zero (a fixed mesh). – chris Jan 12 '16 at 7:51
• on question 2) I know the problem originates from a boundary condition at the fluid solid interface, where I've set the mesh velocity, fluid velocity and solid velocity to be equal. so say I have a (v-v_mesh) at the BC, then for a moving mesh I have v=v_mesh hence v-v_mesh = 0, and for a fixed mesh v-v_mesh = v, and so they give different results but no idea how to fix it. – Hooman Jan 12 '16 at 8:52
• I tried a constant solution problem and have similar problems with the BC. are the BC's not supposed to be effected by the mesh velocity from the terms including the relative velocity (v-v_mesh) ? Thanks. – Hooman Jan 12 '16 at 23:28