Prandtl-Meyer Expansion Shock problem using Converge

In this project the Prandtl-Meyer expansion fan was simulated to study compressible flow. Air flowing with a velocity of 680m/s (M=2) was simulated over a wedge to study the expansion.

Shocks are created when an object moves at a speed greater than that of sound in that medium. Shock are sudden change in the physical properties (pressure, temperature and density) as well as velocity of the fluid. Shocks are formed as waves which could interfere and get amplified.

A 2D steady-state case was setup for the Prandtl-Meyer expansion fan. Adaptive Mesh Refinement (AMR) feature in CONVERGE was used to refine the mesh only when the temperature gradient reached a pre-set value (Sub-grid Criterion - SGS). A velocity of 680m/s (Dirichlet), temperature of 286.1K (Dirichlet) and Neumann pressure boundary were set for the Inlet, and Neumann boundary conditions were set for both velocity and Pressure at the outlet. The simulation was run for 25,000 timesteps and converged results were obtained.

Dirichlet boundary condition: This type gives a specific value for the quantitiy at the boundary. E.g., velocity = 680m/s.

Neumann boundary condiition: This type makes sure that the derivative of a quantity is constant at the boundary. E.g.,`(delv)/(delx)`.

Neumann boundary is selected in shock flow problems because of the uncertainity about the shock location. The shock wave might reach the boundaries as well (which can be seen in this case). If Dirichlet condition is specified, the solver keeps the boundary at a constant value and the shock waves obtained might be inaccurate. 

In this project, the effect of SGS in AMR on the shock location was studied in specific. Four different SGS values were chosen and simulations were run. The following table shows the SGS, cell count at the end of simulation and time taken for solving the PDE.

The mesh at the end of simulation for all four cases can be seen below. It can be seen that the mesh resolution is improved near the shocks with decreasing SGS value. For 0.01, the mesh can be said to be optimal, because the 0.001 SGS refines the mesh where it is of no use to this study.

The Pressure, Density, Temperature and Velocity color maps can be seen below. It is clear again that the SGS of 0.01 is optimal for this study. There is a huge difference between the shocks as the SGS value is decreased. But there is no significant difference between cases having SGS of 0.01 and 0.001. From the table given above, it takes more time to solve as the SGS decreases. The value of SGS shiould be chosen wisely for accurate and faster results.