Prandtl Meyer Shock problem

What is a shock wave?

• The shock wave is produced in an elastic medium when an object moves faster than the local speed of sound. It creates a sudden change in speed, pressure, the temperature of the medium so this can be generated only when the speed of the object reaches above supersonic.
• The generation of shock waves creates a disturbance in the medium with a certain energy and most of it is disputed in the form of heat diffusion.
• In the analysis of a physical model of an object, the generation of the shock wave has treated a loss of energy and it is an irreversible process.
• The model dealing with high velocities should be designed in such a way that there should be a minimum chance for generating a shock wave if it is generated care should be taken such that there should be no damage to the model.

Prandtl Meyer expansion Fan:

When the sudden expansion as shown in Fig is isentropic it forms a fan of centered Mach waves emanating from the corner. Note that the flow on any streamline passing through the fan changes smoothly to higher Mach number and velocity. In the limit as we near the corner the flow expands almost discontinuously at the surface. This type of case is called the Prandtl Meyer expansion fan. Note that none of this discussion makes sense if the upstream Mach number is subsonic since Mach wave and shock wave patterns cannot exist in subsonic flow.

Type of Boundary Conditions:

1. Dirichlet: This condition specifies the value that the unknown function needs to take on along the boundary of the domain. This boundary condition is generally used while solving linear problems.
2. Neumann boundary condition: It is imposed on an ordinary or a partial differential equation, it specifies the values that the derivative of a solution is going to take on the boundary of the domain. This boundary condition is generally used in non-linear problems.
3. Robin boundary condition: The Robin boundary condition is a weighted combination of the Dirichlet boundary and the Neumann boundary condition in all the parts of the boundary.
4. Mixed boundary conditions: The mixed boundary condition refers to the cases in which Dirichlet boundary conditions are prescribed in some parts of the boundary while Neumann boundary conditions exist in the others. It is important to notice that boundary conditions must be applied to the whole boundary.
5. Cauchy boundary conditions: The Cauchy boundary condition is a condition on both the unknown field(Dirichlet) and its derivatives(Neumann). It differs from robin because it is also used in second-order differential equations. Therefore, the Cauchy boundary conditions correspond to imposing a Dirichlet and Neumann boundary conditions simultaneously.

In this challenge, an example of Prandtl Meyer expansion Fan model is given to determine the shockwave region:

The model is shown in the above figure

The simulation process is done in 4 cases:

• case 1: The SGS criteria is set for 0.05
• case 2: The SGS criteria is set for 0.1
• case 3:The SGS criteria is set for 0.01
• case 4: The initial region is set for a subsonic condition (100m/s)

The general procedure for setting up the simulation:

1. The AIR is given as a predefined mixture.
2. The Density-Based Steady-State solver is used for the simulation.
3. The End time is set for 25000cycles with a minimum time step of 1e-7sec.
4. The Neumann boundary conditions are given at both inlet and outlet and the inlet velocity is given as 680m/s.
5. The Front and Back geometries are given as 2D boundary conditions. and the TOP& Bottom wall is given as slip wall boundary conditions.
6. For the first three cases, the velocity of the region is specified as 680m/s and for the case-4 it is given as 100m/s.
7. The RNG k-epsilon turbulence condition is given for the model.
8. The base grid is set to 0.8m in all directions.
9. Adaptive mesh refinement is given as per the cases.

case 1: When the SGS criteria is set for 0.05K

The above figure shows the Adaptive mesh refinement for case:1

https://youtu.be/o24RrrRO7AY

The above link is the temperature contour animation for case1

Case2: When the SGS criteria is set for 0.1K

The above figure shows the Adaptive mesh refinement for case:2

https://youtu.be/PO3Owk_-jQ0

The above link is the temperature contour animation for case2

Case3: When the SGS criteria is set for 0.01K

The above figure shows the Adaptive mesh refinement for case:3

https://youtu.be/AVzbzchHb7A

The above link is the temperature contour animation for case3

case4: The initial region is set for a subsonic condition (100m/s)

The above figure shows the Adaptive mesh refinement for case:4

https://youtu.be/Smx8TWxV1NM

The above link is the temperature contour animation for case4

When we compared the first three cases the number of cells increases with the decreases in the SGS parameter because it is obvious that on decreasing the sensitivity value of the curvature of the temperature profile the number of cells increases as per the condition.

If we see the case4 animation file in the first few seconds the AMR didn\'t form because the flow need some time to wash out the initial subsonic conditions that we are provided.