2-D simulation of Prandtl Meyer shock wave using CONVERGE CFD

INTRODUCTION:

When a gas is flowing at subsonic speeds (slower than the speed of sound), the density of the gas can be assumed as constant throughout the doman. However, when a gas is flowing at supersonic speeds (faster than the speed of sound), the density of the gas changes locally as it gets compressed or expanded. If this supersonic flow passes through a sudden expansion or contraction, it leads to the development of shock waves at the location of the area change. When an external supersonic flow passes across a wedge, since it cannot flow directly over the convex wedge due to change in the boundary orientation, it creates infinite number of Mach waves where the flow changes gradually with each Mach wave resulting in an expansion fan propagating from the wedge corner. This behavior is similar to a flow passing through a convergin-diverging nozzle.

Since there are infinite number of Mach waves generated between the flow transition, capturing the shock wave through a coarse mesh is very difficult as the flow beheavior changes by a huge amount gradually. For this purpose, adaptive mesh refinement (AMR) techniques are used to refine the mesh in areas of interest in order to capture as many Mach waves as possible.

For CFD analysis, there are three major ways in which any boundary condition for a variable `phi` is applied:

1. Dirichlet Boundary Conditon

This method directly specifies the value of the variable at the boundary: `phi = f(x)`

2. Neumann Boundary Condition

This method specifies the gradient of the variable across the boundary normal: `{partial phi}/{partial n} = f(x)`

3. Robin Boundary Condition

This method is a mix of the Dirichlet and Neumann boundary conditions: `phi + {partial phi}/{partial n} = f(x)`