Week 5: Prandtl Meyer Shock problem

OBJECTIVE:-

To simulate Prandtl Meyer shock problem using CONVERGE, and learn the effect of SGS temperature value on cell count and the shock location.

1. Shock Flow Boundary Condition:-

• The boundary condition need to be defined for the surface boundary the entire domain.
• In a shock wave flow, the flow field gets separated into two region due to the generation of an expansion fan - the flow field before the expansion fan called the upstream region and the flow field after the expansion fan called the downstream region.

A. Inlet Boundary Condition:-

• Initally, in the upstream region, the flow need to be supersonic in order to generate an expansion fan.
• This is done by specifying Dirichlet boundary condition at inlet.

B. Outlet Boundary Condition:-

• Once the flow field passes the expanison fan, the fluid properties changes and hence are unknown across the expansion.

2. Shock Wave:-

• In physics, a shock wave or shock is a type of propagating disturbance that moves faster than the local speed of sound in the medium.
• A shock wave is propogating disturbance that where the supersonic flow decelerate to subsonic flow, the process is adiabatic but non-isentropic.
• The shock wave are characteristized by an abrupt change in the characteristic of the medium.
• For eg, across the shock, there is always a rapid rise in pressure, temperature and density of the flow.
• For the purpose of comparison, in supersonic flow, additional increased expansion may be achieved fan, also known as Prandtl Meyer expansion fan.
• The accompanying expansion wave may approach and eventually collide and recombine with the shock wave, creating a process of destructive interface.
• The sonic boom associated with passage of supersonic aircraft is type of sound wave produced by constructive interference.

Types of Boundary Condition:-

A. Dirichlet Boundary Condition:-

This condition specifies the value that the unknown function need to take on along the boundary of the domain.

• Slip Boundary Condition:-

The velocity normal to the boundary is set to zero, while the velocity parallel to the boundary is let free.

• No Slip Boundary Condition:-

Both the velocity normal to the boundary and the velocity parallel to the boundary are set equal to zero.

Atleast one homogenous boundary condition on the pressure has to be imposed as reference for open domain for the instance in the highest boundary of the air domain.

B. Neumann Boundary Condition:-

When imposed on an ordinary or a partial differential equation the Neumann Boundary condition specifies the value that the derivative of a solution is going to take on the boundary of the domain

3. Simulation of Prandtl Meyer Shock Problem Using CONVERGE:-

GEOMETRY:-

CASE SETUP:-

Application type:-

1. Time Based

Material:-

1. Predefined Mixture - air
2. Gas simulation
3. Reaction Mechanism

Global Transport Parameter:-

1. Turbulent Prandtl Number - 0.9
2. Turbulent Schemidt Number - 0.78

Simulation Parameter:-

1. Solver - Steady State Solver

Simulation time Parameter:-

1. Start time - 0 cycle
2. End time - 25000 cycle
3. Initial time step - 1e-9
4. Minimum time step - 1e-9
5. Maximum time step - 1

Solver parameter:-

1. Navier Stokes Solver Scheme - PISCO
2. Navier Stokes Solver type - Density Based

Boundary:-

1. Inlet Velocity - 680 m/s
2. Inlet Temperature - 286.1 K
3. Outlet Pressure and Velocity - Zero Normal Gradient
4. Front and Back - 2 D
5. Top and Bottom wall - Slip wall

MESH:-

Grid size dx = dy = dz = 0.8 m

Adaptive Mesh Refinement :-

1. Temperature AMR Group - Active Region
2. Minimum cell - 1
3. Maximum cell - 200000
4. Maximum embedded level - 2
5. Sub-grid criterion - 0.05
6. Timing control type - SEQUENTIAL
7. Start time - 5000 cycle
8. End time - 999999999 cycle

VARIATION IN FLOW PROPERTIES AT INLET AND OUTLET:-

1 Mach Number:-

2. Pressure:-

3. Temperature:-

4. Density:-

From the above plot we can observed, the Mach Number of the subsonic flow increase as flow passes through the expansion wave where other properties like Pressure, Temperature and Density decrease.

CELL COUNT:-

The simulation run for around 23500 cycle. From the cell count plot we can observed that the Adaptive Mesh Refinement takes place after 5000 cycle.

RESULT:-

Temperature Contour:-

Temperature Contour Animation:-

Density Contour:-

Density Contour Animation:-

Velocity Contour:-

Velocity Contour Animation:-

Pressure Contour:-

PRESSURE CONTOUR ANIMATION:-

Effect of SGS Parameter:-

In temperature SGS Parameter, the simulation is run with different SGS parameter like 0.01, 0.05, 0.1 and 0.3

SGS = 0.3

SGS = 0.1

SGS = 0.05

SGS = 0.01

From the post processing result, we can observe that the lesser the value of SGS which result in higher refinement of the mesh which lead result in less blur of the expansion wave and it well defined which can be seen when SGS = 0.01. As the SGS value increase the expansion wave becomes more blur.

CELL COUNT:-

From the above plot we can observed, the lesser the SGS value the mesh becomes more refined which result in increse the number of cell, hence the solution becomes more accurate. For SGS = 0.3, the adaptive mesh refinement does not take place and hence no refinement takes place.

CONCLUSION:-

Hence, we can conclude that we are able to design geometry over expansion corner, simulate flow over it and study the effect of SGS parameter.