Prandtl Meyer Shock Problem

Boundary Conditions:

Boundary Conditions are used to solve a boundary value problem in space. In this type we know a set of conditions which is opposite to initial value problem. When we solve a boundary value , we need as many boundary conditions as much the highest order in a space variable X. Both ODE and PDE needs boundary conditions to get solved. Boundary Conditions are very important for a computational problem as selection of wrong boundary condition will lead to wrong result.

Types of Boundary Conditions:

1. Dirichlet Boundary Condition
2. Neumann Boundary Condition
3. Robin Boundary Condition
4. Mixed Boundary Condition and
5. Cauche Boundary Condition.

Neumann Boundary Condition:

Essential boundary conditions are conditions that are imposed explicitly on the solution and natural boundary conditions are those that automatically will be satisfied after solution of the problem. Here Neumann BC is a natural BC.

Wave:

Wave is a phenomena created when the particles transmit the energy to other particles by its oscillatory motion in the same place after the system gets disturbed.

Shock wave:

As we move forward at a speed less than sound the source which creates the wave will move such that the wave transmits in all directions. But when the source moves at a speed more than the sound speed the wave fronts cannot escape from the source which creates a sound barrier. This was mistook by the people of second world war era that we can,t break the sound barrier. But in later days it was found only at the transonic region i.e., 0.8 -1.2 mach the drag created will raise to a maximum and then fall back once we crossed this transonic region which made it possible for the present day supersonic and hyperonic flights.

Shocks are of 2 types:

1. Normal Shock - Shock created at 90 degrees.
2. Oblique Shock - Shock created at an angle as per the body\'s deflection angle.

Prandtl Meyer Shock Problem:

Across shock wave static property increases except Mach number and total property decreases except entropy and total temperature remains constant.

But across expansion wave static property decreases and dynamic property increases whereas stagnation property remains constant.

Observations:

With 680 m/s as initial condition:

Sub grid criterion (0.05)K

Sub grid criterion (0.1)K

Sub grid criterion (1) K:

With 100 m/s initial condition:

From all the above abservations on mesh generation sub grid criterion is based on temperature difference. As we provide very less temperature difference the mesh gets finer in the shock region and as the temperature difference goes up the mesh refinement gets less.