Prandtl Meyer Shock problem

Objective:

1. To uderstan the theory of formation of expansion fan due to infinite mach waves, originated at the sharp convex corner
2. To understan the reason for the usage of Neuman BC at the outlet for supersonic flow
3. To understan the effect of SGS parameters on shock location
4. To undestand the effect of using subsonic initial condition for supersonic flow

The physics behind the shock waves-

Shocks are generated in supersonuc flows where the information about the future disturbances can reach the domain only through packets of subsonic regions in the domain i,e near boundry layer etc. By using slip BC we further constrained the posibility of information flow about the future disturbance. So the flowing fluid in the domain cant able to predict the future disturbance, and so it cant make considerable change in its properties to match with the properties of the fluid at the region of disturbance. Therefore it encounters a sudden change in its properties while hitting the region of disturbance , which we observe as shocks

In our case at the convex corner infinite no of point disturbance creats infinite no of mach waves which fan out together and we reprent it with the name expansion fan.

Boundry conditions used for shock flow problems

Physical Reason:

As the future information cannot be carried inside the domain (information travels as pressure waves and the maximum speed at which they can travel is the speed of sound) the properties of the fluid in the domain cant be altered outlet to match the flow with the required outlet BC. So usage of Neumann BC indirectly says that the flow is completely dependent on the initial condition of the domain.

Mathematical Reason:

As all the characteristics lines travel out of the domain from the outlet boundry , the only way of defining the conditiuon at the outlet boundry is through Neumann.

The simulation was run under two different conditions:

1. Sub_Grid criterion(SGS) =0.1 K
2. Sub_Grid criterion(SGS) = 0.01 K
3. Subsonic flow = 100 m/s

Cell counts:

For SGS = 0.1 K Total cells = 5172

For SGS = 0.01 K Total cells = 15573

By changing the SGS values changes the number of cells that are formed during meshing as mentioned above when the sub-grid temp becomes greater than SGS value the mesh is refined. Thusfor a low value of SGS(0.01) more cells are refined than at a higher value of SGS(0.1)

The inlet mach number where the flow enters the duct is 2 and the outlet mach nuber at the outlet is 2.15 which corresponds to an velocity of 731 m/s as copared to the inlet velocity of 680m/s

Mesh formed with SGS = 0.1K

1. Temperature and pressure at SGS=0.1K

Animations for SGS = 0.1k

2. Temperature and pressure at SGS=0.01K

Animations for SGS = 0.01k

Effect of using subsonic initial condition at velocity of 100m/s (Mach no:0.29) for supersonic flow

The solution blows up at 100m/s because the difference between the inlet velocity at boundry condition being 680 m/s at intial condition being 100m/s leads the solution to diverge

The effect can very well be seen in the following graph. The origin is broken depicting a discontinuity in solution.