In 2D compressible flows, we have two momentum equations. Using no slip at stationary wall, we get two relations between the pressure-gradients and shear stresses, that is, one relation from each of the momentum equation. My proble is: whcich relation is to be used to update pressure and why?

Thanks in advance.

The pressure gradient alone will not give you the correct pressure. Compressible flows depend not only on the gradient but also on the absolute pressure, unlike incompressible flows. That's why in compressible flow computations we usually apply an equation of state (for example for ideal gas) in order to obtain the pressure as a dependent variable. Using the equation of state, the pressure is written as a function of density, momentum and energy.

The 'pressure equation' (in the Los Alamos techniques and the more modern codes from Patankar and the Imperial College researchers) is derived from the mass conservation equation. The boundary conditions are based on the mass flux conditions - that is, on the pressure gradients normal to the local boundary.

Mani is absolutely correct that, in compressible flow, the EOS must be taken into account, either in terms of boundary temperature, density, or some other condition.

Bottom line on this: you really need to think physically in developing your boundary conditions. The math tells you how many conditions, but how to express them is a physical consideration.

Mathematically you do not need a pressure boundary condition. But when you solve numerically, and if you have a grid point on the wall, then you need to update the pressure at that point. While I have not come across any good mathematical justification, taking the normal pressure gradient equation gives good results, that is, you take the momentum equation and project it onto the normal at a wall point. Then you will get an equation like

∂p/∂n = S

where S will involve velocity gradients.

Here are a couple of references to a widely used approach to apply compressible boundary conditions:

Kevin Thompson, ``Time-Dependent Boundary Conditions for Hyperbolic Systems, II'', J. Comp. Phys., 89, 439-461 (1990)

T. J. Poinsot and S. K. Lele, ``Boundary Conditions for Direct Simulations of Compressible Viscous Flows'', J. Comp. Phys., 101, 104-129 (1992)

There are one or two other approaches but this should be a good place to start.