convergence in unstedy problems

hedge

Hello

I'm trying to solve an unsteady problem with moving shocks in fluent. I'm using density based solver, second order upwind discretization and Roe flux difference splittiong.

I would like to check the convergence of the solution in the following way. I get a several grids (with cell sizes h of 1e-5 m, 5e-6 m 2.5e-6 m and so on). I get the strict location in the space, say x*=3e-3 m and specific time moment, say t*=2.8e-6 sec, after the shock wave passed this point. Another point was x**=-1.2e-3 m, where no shocks and only rarefactions are present.

For all the grids I made the runs and plotted the pressure as function of computational cell size at this time moment at this point.

I guess, that the dependence of the pressure on cell size should be as P(t*,x*) = A + B*h^2 (h - cell size, where the last term is relatively small), i.e. quadratic fuction parabola as I use the second order method.

But I don't. I actually has the linear dependence P(t*,x*) = a + b*h with a very good accuracy, beginning from cell size of 5e-6 m and less. So it means, that I probably have reached the convergence, but with the first order instead of second ones. For the case with the rarefection wave I got the same result.

Could anybody advise me in the follwoing quastions 1) Is this procedure strict for unsteady solvers? 2) does I actually reached the convergence? 2) Does it work in Fluent, when the shock waves are simulated? 3) Does this approach to convergence test applicable to commercial solvers? ;-)

Thanks in advance