Prandtl Meyer Shock problem

1. Shock Flow BC\'s:

* As Shock occurs at supersonic flows (M>1), for which the Mach no is defined using the flow velocity and velocity of sound at the specified temperature in the given medium, the Inlet Temperature is one BC to be defined and the Inlet Velocity is another BC to be defined, which are Dirichlet BC\'s. Hence, the Inlet Velocity is calculated from Mach No & Temperature using

* As it is supersonic flow, the Pressure BC\'s at Inlet and Exit are set to Neumann BC, which are calculated during the simulation.

* The reacting species in the region is also defined along with Back flow conditions at the outlet, in casem the flow reverses direction in specific problem solutions.

The Neumann BC properties are calculated during the solver iterations using the Rankine Hugoniot Equations which are:

Upon solving the above, unknown terms are exprapolated and utilised for subsequent interations.

2. Physics behind Shock Wave:

Any body moving in a fluid medium tend to propagate waves around it. When the velocity of body is less than the phase velocity of sound in the medium at the given temperature, the body is said to be at subsonic speed and when its equal or greater, its Sonic or Supersonic respectively. In case of supersonic speeds, the waves lag behind the moving body as the body moves as higher velocity than wave. This tends to a rapid change in physical properties of the medium around the body. Hence, these waves are called Shock Waves. As we draw a tangent using the waves generated at equal interval of time as the body moves, it creates a conical shape called as mach cone, whose semi vertical angle varies with the Mach no.

For subsonic speeds, the Disturbance doesnt affect thermodynamic changes in temperature and pressure. However, in case of Supersonic Flows, huge difference in the thermodynamic properties are found at the region of Wave or Shock Wave. The Compression and rarefraction process is irreversible and the entropy of gas changes.

3. Effect of SGS Parameter on Cell Count and Shock Location:

With a larger SGS Temperature Value, the cell count decreased and as the cell count increased, the cell count also increased during the iterations.

S.No	Temperature SGS (K)	Highest Cell Count Observed
1.	.05	7263
2.	.01	17231

The shock location according to the given geometry, which is an expansion fan, occurs in infinite numbers emerging from the divergent corner at the bottom. However, to determine from the output, we utilise the change in thermodynamic properties of the medium to observe the shock location & Mach no. So, we obtain more accurate shock wave angle with higher SGS value due to more refined solutions at critial cells and vice versa.

For SGS = .05:

For SGS = .01:

From the above plot, the Entry and Exit Mach No. obtained by finding the shock wave angles for SGS = .05 are:

M1 - 28.7 deg - 2.08 mach no
M2 - 27.6 deg - 2.15 mach no

For SGS = .01, the Mach no obtained are directly obtained from the post processing result as follows: