Before running a simulation for turbulence (e.g Rayleigh-Benard Convection), how do we check for well-posedness of Navier-Stokes with other equations for a given boundary condition??
Can someone point to any reference to read on this topic??
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Sign up to join this communityBefore running a simulation for turbulence (e.g Rayleigh-Benard Convection), how do we check for well-posedness of Navier-Stokes with other equations for a given boundary condition??
Can someone point to any reference to read on this topic??
The following paper gives a good overview of well-posedness for the compressible Navier Stokes equations.
https://link.springer.com/content/pdf/10.1007%2Fs10915-016-0303-9.pdf
The basic building block is the energy method, where you linearize the equations, then left multiply with the primitive variables, and then integrate. This gives you a norm which you check for stability. The number and form of boundary conditions can be determined from the eigenvalues of the boundary operator.
This gives a necessary condition for well-posedness, and other equations can be similarly treated. However, I am not sure if this is sufficient. There are often other norms that can be used such as entropy, but the above approach seems to be a robust recipe.