Week 5: Prandtl Meyer Shock problem

Aim: To perform the Prandtl Meyer shock problem.

Introduction:

Prandtl Meyer expansion fan is a two-dimensional simple wave. It is a centered expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of a finite number of Mach waves diverging from sharp corners. When a flow turnaround a smooth and circular corner, these waves can be extended backward to meet a point.

Each wave in the expansion fan turns the flow gradually. Near the expansion fan, the flow accelerates and the Mach number increases, while the static pressure, temperature, and density decrease. As the process is an isentropic process, the total temperature and total pressure remain constant across the fan.

Shock wave:

In physics, a shock wave is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a medium but is characterized by an abrupt, nearly discontinuous, change in pressure, temperature, and density of the medium.

For the purpose of comparison, in supersonic flows, additional increased expansion may be achieved through an expansion fan, also known as a Prandtl–Meyer expansion fan. The accompanying expansion wave may approach and eventually collide and recombine with the shock wave, creating a process of destructive interference. The sonic boom associated with the passage of a supersonic aircraft is a type of sound wave produced by constructive interference.

Unlike solitons (another kind of nonlinear wave), the energy and speed of a shock wave alone dissipate relatively quickly with distance. When a shock wave passes through matter, energy is preserved but entropy increases. This change in the matter's properties manifests itself as a decrease in the energy which can be extracted as work, and as a drag force on supersonic objects; shock waves are strongly irreversible processes.

Boundary conditions:

Dirichlet boundary conditions: The Dirichlet boundary condition define the value of a function itself on the surface i.e. Y = f(t). The value of the dependent variable is specified on the boundary.

Neumann boundary conditions: The Neumann boundary condition define the value of a normal derivative of a function on the surface. dy/dn = f(t).The normal derivative of the dependent variable is specified on the boundary.

Cauchy boundary conditions: Both the value and the normal derivative of the dependent variable are specified on the boundary.

Robin boundary conditions: The value of a linear combination of the dependent variable and the normal derivative of the dependent variable is specified on the boundary.

Geometry:

Diagnosis result: (before case setup)

Simulation Case setup:

• Application type: Time based

• Materials:

        Predefined mixtures = Air

• Simulation Parameters:

        In Run Parameters two main things to be considered:- 

        (a) select the following solver settings

        (b) Steady-state monitor

• Simulation Time Parameters:

• Solver Parameters [Steady-State]:

• Boundary Conditions:

• Regions and Initialization:

• Physical Models:

        Click on Physical Models and select Turbulence Modeling in this case.

• Turbulence modeling:

• Grid Control:

• Now click on Grid Control > select Adaptive Mesh Refinement.

• Output Files:

• Case setup 

• Export data:

Results:

Case 1: Velocity = 680 m/s and SGS Temperature sub-grid criteria = 0.1

Mesh:

Cell Count:

Pressure Contour and plot:

Temperature Contour and plot:

Velocity  Contour and plot:

Case 2: Velocity = 680 m/s and SGS Temperature sub-grid criteria = 0.01

Mesh:

Cell Count:

Pressure Contour and plot:

Temperature Contour and plot:

Velocity  Contour and plot:

Case 3: Velocity = 680 m/s and SGS Temperature sub-grid criteria = 0.001

Mesh:

Cell Count:

Pressure Contour and plot:

Temperature Contour and plot:

Velocity  Contour and plot:

Conclusion:

1. Temperature Sub-grid-scale of 0.01 is able to capture the Mach waves better than 0.1 and 0.001 which are under-capture and over-captures respectively.

2. The cell-count increases when we decrease the temperature sub-grid scale criteria.

3. The pressure, temperature, and density decrease across the Mach waves, however the velocity increases, for the supersonic inlet.