Incompressible Navier-Stokes equations for a backward facing step
I'm trying to write a program which calculates the velocities and the pressure for a backward facing step using a MAC grid and artificial compressibility for the continuity equation.
I'm using the non-dimensionalized equations, as a result the speed at the inflow boundary is 1.0. The problem is: which values should the velocities inside the grid have (and the pressures)? In a first, naive approach I have initialized them with zero but after just two or three time steps the values get out of control because of the big difference between the values at the inflow boundary and the values situated just to the right of the inflow boundary.
You might try setting up the narrow part as a channel flow (analytic solution is in Schlichting, Boundary Layer Theory. After the step (maybe one or two step heights downstream), initialize the velocities as a new channel flow starting and flowing to the outlet. Of course the recirculation region just downstream from the step is the toughest part. Don't have a good suggestion to initialize that.
I don't have experience using artificial compressibility. But a rule of thumb for starting calculations is that "if the IC values are smooth and grossly similar to the expected solution, that's good enough." Just spend a lot of time with small relaxation factors and small time steps to reduce the residuals, then gradually increase them as the flow settles out. NOTE that this is not the only "rule of thumb for starting calculations".
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