The code provides support for the variable density incompressible Euler equations. Does that mean you can input the density of the fluid density (air, water, etc…)?
Not really if what you mean is a constant density throughout the domain. In the case of the incompressible constant-density Navier-Stokes equations, the density is irrelevant. It is only a scaling factor for the pressure. What this really means is that Gerris can deal with flows where the density varies across the domain (e.g. a mixture of two miscible fluids, or density variations due to salinity variations in the sea for example).
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