Prandtl Meyer Shock problem simulation using Converge and Paraview

When an object moves through a medium faster than the speed of the sound in that medium, the pressure waves generated by the object moving through gets piled up and compressed to form large amplitude pressure waves this is called as a shockwave and takes the form of a cone which narrows down as the speed of the object increases further. The reason why shockwaves are generated past the speed of the sound is because sounds are typically weak pressure waves. There two types of shockwaves namely: oblique shock waves and expansion waves. Infinite number of expanding waves emenating from a sharp convex corner flowing at mach speed make up a supersonic expansion fan or a Prandtl Meyer fan.

In this project this Prandtl Meyer fan is simulated using Converge CFD. For the objective of the simulation a geometry sharp corner is utilised. The image and the dimensions of the geometry are given below:

To simulate shock flow appropriate supersonic boundary conditions need to be given, before discussing the boundary conditions used in this project let’s look at the types of the boundary conditions available:

Dirichlet condition: In this condition the values of the dependent variables are specified at the boundaries.

E.g., T = 200k at x=0

T = f(t) at x=L

Neumann condition: In this condition the derivative of the dependent variable is given as a constant or as a function of the independent variable on one boundary.

E.g., `(deltaT)/(deltax)`=0 at x=L

Cauchy condition: It is a boundary condition that combines both Dirichlet and Neumann condition.

E.g., T=300k at x=0

`(deltaT)/(deltax)`=0 at x=L

Robbins condition: The derivative of the dependent variable is given as a function of another dependent variable on the boundaries

E.g., k`(deltaT)/(deltax)`=h(T-`T_infty`) at x=0

Boundary conditions used for the simulation are:

Inlet: A dirichlet based velocity boundary condition is given at the inlet with inlet velocity as 680m/s which has a Mach number of 2 considering the dirichlet based temperature boundary condition as 286.1k at which the speed of the sound is nearly 340m/s. For pressure a Neumann boundary condition is employed with zero normal gradient.

Outlet: For outlet for both pressure and velocity Neumann boundary condition is employed with zero normal gradient.

Top and bottom walls: Both the top and bottom walls are set to slip condition.

Front and back walls: For both front and back walls 2D boundary is set.

A steady state simulation is carried out for 25,000 cycles using density based solver as the flow becomes compressible at supersonic velocities. A temperature based adaptive mesh refining technique is employed with the base grid size for the geometry is taken as 0.8m. The sub grid criterion for the temperature based adaptive mesh refining is varied and the results are captured.

With temperature based adaptive mesh refinement having sub grid criterion of 0.05k:

Mesh:

Cell count:

Temperature contour animation:

Velocity contour:

Temperature contour:

With temperature based adaptive mesh refinement having sub grid criterion of 0.02k:

Mesh:

Cell count:

Temperature contour animation:

Velocity contour:

Temperature contour:

With temperature based adaptive mesh refinement having sub grid criterion of 0.01k:

Mesh:

Cell count:

Temperature contour animation:

Velocity contour:

Temperature contour:

Converge calculates error between actual temperature and resolved temperature at each cell. If this error is larger than sub grid criterion, converge refines the cell automatically until maximum cell count is reached. If the error is smaller than 20% of sub grid criterion, embedding on this cell will be released. By giving different sub grid criterion, the results can change as the error introduced to the simulations system varies.

We can observe that due to reduction of sub grid criterion the shock narrows down because as we reduce the sub grid criterion more accurate results are obtained and so the edges of the shock waves are clearly defined resulting in the narrowing down of the shock waves. We can also observe that as we reduce sub grid criterion the total cells also increases for 0.05k sub grid criterion the total cells were between the range of 7000 to 7500, for 0.02k sub grid criterion the total cells were between the range of 13000 and 13500 and lastly for 0.01k sub grid criterion the total cells were between the range of 16000 and 16500.