Prandtl Meyer Shock Simulation

Shock wave:

Shock waves occur when the flow of a compressible fluid is supersonic in nature (i.e., velocity of flow > velocity of sound in the fluid medium). They are characterised by a sudden change in temperature, pressure and density of the medium.

When a object travels in air with a velocity greater than the velocity of sound, the wavefronts formed due to the object lag behind it and shock waves are formed in the shape of a cone as shown in the images.

The properties of fluid such as pressure, temperature and density vary drastically at the edges of the cone.

Shock flow boundary conditions:

Shock flow requires the pressure bounadry conditions to be Neumann boundary condition at both inlet and outlet and the velocity bounadry condition to be dirichlet at inlet to account for the supersonic flow and neumann at the outlet.

This means that the pressure values at the boundaries are calculated from the pressure conditions inside the region of flow and it is the same with the velocity at the outlet.

The wall is set to a slip boundary condition so that there is no shear stress acting along the wall of the object and hence no turbulence near the wall.

Prandtl Meyer shock problem:

The shock problem is set up in 2D with the following boundary conditions:

Inlet - Pressure ~ Zero normal gradient (Neumann)

          Velocity ~ 680 (m/s) in X-direction (Dirichlet)

          Temperature ~ 286.1 (K)

          `Mach=(v)/(c)`

          where v - velocity of flow & c - velocity of sound in the medium(340 m/s at 286.1 K)

          So, the inlet Mach no. is approximately 2 making it a supersonic flow

Outlet - Pressure ~ Zero normal gradient (Neumann)

            Velocity ~ Zero normal gradient (Neumann)

Wall - Slip boundary condition to make the flow take turn along the wall at the convex corner

We are going to simulate the flow using the following Sub grid criterion values of temperature for adaptive mesh refinement with an embedding level of 2:

1. SGS = 0.1 K

2. SGS = 0.05 K

3. SGS = 0.01 K

Mesh formation

The mesh gets finer at the shock location as the SGS value decreases

1. SGS = 0.1 K

2. SGS = 0.05 K

3. SGS = 0.01 K

Total cell count

The total cell count has increased for lower SGS temperature values which accounts for improved accuracy at the shock and larger simulation times

Shock location

The location of shock did not change for different SGS temperature values however for SGS > 0.1, there is no mesh refinement and the shock has moved towards the inlet which means that the solution is inaccurate.

Temperature contour

Velocity contour

Pressure contour

Density contour

Animation: