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Wiki/Galilean Invariance

Solutions to the fluid equations should be Galilean invariant, that is the same in every inertial frame. Recently, it has been reported that solutions generated by grid codes (like Athena) are not Galilean invariant. In fact, however, the test problems themselves are flawed. They involve unresolved interfaces, so solutions are dominated by truncation error. It is the truncation error which violates Galilean invariance. We show below resolved solutions generated by Athena are invariant.

KH Instability Test

Initial Conditions

One of the test problems used is the Kelvin-Helmholtz instability between two fluids with different densities. The interface between the fluids is an unresolved discontinuity (a slip-surface). The total velocity difference is equal to the sound speed Cs. The test is based on the KH instability test in the Athena test suite; the figure to the right sketches the initial conditions.

Results with Unresolved Interfaces

The images below show results typical of a grid code when the solution is computed in the rest frame (left), in a frame moving at the sound speed to the right (center), and a frame moving at ten times the sound speed to the right (right). Note the instability seems to disappear in the moving frames. This is because numerical diffusion at the interface damps the unstable modes.

Of course, numerical diffusion is not only a problem in the moving frames. It also dominates the solution in the rest frame. The images below show the same test run at different resolutions in the rest frame. Note the solution is vastly different at each resolution, and seems to disappear at the lowest resolution (where the truncation error is largest).

So which solution is correct? That is, which solution should the code reproduce in any inertial frame?. Of course, this question cannot be answered unless the test is based on a resolved solution.

Results with a Resolved Interface

The images below show the streamwise (x-direction), and spanwise (y-direction) velocity fluctuations from a test in which the interface is spread out using a hyperbolic tangent function, with a characteristic scale a equal to twice the grid size. This replaces the discontinuity at the interface by a very sharp but smooth profile, and introduces a minimum laengthscale a in the problem that can be resolved. The images on the left are for a calculation in the rest frame, on the right from a calculation moving at 100 Cs. This is even more extreme than the test above''

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