While deriving integral form of NS equation, we use Gauss divergence theorem to convert volume integral to surface integral. Why we are doing this?
Divergence term has derivative calculation. In high-speed flow there will be discontinuity in flux, so calculating derivative is meaningless. If this is a reason resulting form of NS equation after divergence theorem also looks like equivalent finite difference form of derivative calculation.
We can evaluate flux across boundary easily if we use divergence theorem. Using that we can ensure conservativeness of flux. But conservativeness of flux does not ensure conservativeness of mass, momentum etc. for transient case (because fluxes are discontinuous in transient stage but mass, momentum etc. are not) . Because conservativeness is based on combined spatial and temporal parts. I believe only doing spatial will not ensure conservativeness except steady state case.
Dear,
See the book of Patankar. If you want, use the Hybrid scheme. Good luck
Regards,
The weak form of the differential equation is a mathematical formulation in which the original equation is projected along some shape functions that must have some requirements. The Gauss theorem is applied to lower the degree of derivation so that for first order equations (like Euler), you have no longer the differentiability requirement. For the second order equations (like NS) the differentiability remains required but, differently from inviscid flows, viscous flows do not actually produce a mathematical discontinuity. This is the reason why the weak form is particulary relevant for Euler equations. For the NS equation the reason to use the weak form is no longer for a mathematical constraint but for numerical properties.
Now, one of the weak forms that can be formulated corresponds to the integral formulation of the conservative equations. In such case, conservation of mass, momentum and total energy is fulfilled even for the unsteady case. It states that the total conserved variable in a volume can vary in time only if there is a net flux difference.
I agree, I would only highlight that the Integral formulation could be seen as a sort of FInite Element method, considering the proper choice of the shape function.
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