Prandtl Meyer Shock problem

In this project, the Prandtl Meyer Shock problem is studied by using Converge 3.0, and the effect of Sub-Grid Scale (SGS) Temperature on the cell count and shock location are studied.

Introduction:

Shock Wave:

A shock wave or shock 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.

A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, a two-dimensional simple wave, is a centered expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of an infinite number of Mach waves, diverging from a sharp corner. When a flow turns around a smooth and circular corner, these waves can be extended backward to meet at a point. The expansion is isentropic since the expansion waves take place across a continuous succession of Mach waves and also there is no change in entropy for each Mach wave. As the flow passes through the expansion wave, the Mach number increases, and the pressure, density, and temperature decreases.

Boundary Conditions for Shock Waves:

For solving the PDE's we need the sufficient and necessary boundary and initial conditions. If the model under study is steady-state we need only boundary conditions as no time derivative is involved. If it is transient model, we need the appropriate initial conditions of the domain as well.

Types of Boundary Conditions usually encountered are:

• Dirichlet ( specify the value of the function like T= constant. or T=f(r,t))
• Neumann ( specify the normal derivative of the function on a surface. `(∂T) /(∂n)=f(r,t)` or `(∂T)/(∂n) = constant`
• Mixed (combination of above two like `(∂T) /(∂n)=f(r,t,T)`)

The pressure at the outlet boundary conditions requires the specification of static pressure. The value of the specified static pressure is used only while the flow is subsonic. If the flow is supersonic, the specified pressure will no longer be used, the pressure will be extrapolated from the interior flow cells. This means that the Neumann boundary conditions which say that the outlet condition is dependent on the initial conditions of the flow need to be used in this case, that will infer the values of pressure from the initialization or initial conditions.

Geometry:

A duct with the expansion corner (convex corner) is considered as shown below to visualize the expansion fan.

Mesh Generation:

A base grid (0.8m) with Adaptive Mesh Refinement for temperature (AMR) with SGS (0.05) is used. Though the selected geometry is 3D, we are performing only a 2D simulation, where the front and back walls are selected as 2D in the boundary conditions section.

AMR technique is used to refine the mesh locally where the curvature (`(∂^2T)/(∂n^2)`) is beyond the specified SGS value.

Physical Model:

General	Steady State (25000 cycles) Density-Based
Model	RNG k-ε">ε
Material	Air (77% N2, 23% O2)
Monitors	Average Temperature at Outlet
Boundary Conditions	Inlet (680 m/s, 286.1 K, pressure-Zero Normal Gradient) and Outlet (Pressure and Velocity are Zero Normal Gradient) Top and Bottom (Slip walls) Front and Back (2D) Initialization (Pressure-5e4 Pa, Velocity-680 m/s and T= 300K)

Flow of Project in Converge:

1. Provided geometry is dropped in the Converge workspace.
2. Boundaries are named using appropriate names.
3. Normal directions are checked to be in line with the geometry and flow.
4. Diagnosis is performed to check any discrepancies in geometry.
5. Then appropriate species are selected for the fluid.
6. Simulation Parameters along with the steady-state monitor are set.
7. Boundary conditions are set to the appropriate regions and their corresponding values.
8. The base grid of the considered step size is inserted along with respective directions and associated temperature AMR with the desired SGS value.
9. Variables for post-processing and time intervals for writing the output and restart files are set and selected.
10. All the input files are stored at the desired location.
11. All the stored input files are executed using the command editor. These input files are executed by Converge Solver in the background.
12. Then the executed files generate the output folder in which data files after every selected cycle intervals are stored which are post-processed for the graphical results.

Results:

The results below describe the variation of parameters over the simulation with the SGS of 0.05.

Mesh:

Total Cells:

Pressure Contour:

Inlet and Outlet Pressure:

Velocity Contour:

Inlet and Outlet Velocity:

Inlet and Outlet Mass Flow Rate:

Temperature Contour:

Inlet and Outlet Temperature:

Mach Contour:

Density contour:

Inlet and Outlet, Mach and Density:

From the above graphs, it can be observed that the velocity and the Mach Number of the supersonic flow increase as the flow pass through the expansion wave whereas temperature, pressure, and density tends to decrease.

Comparative Study and effect of SGS:

In this study variation of the SGS parameter on mesh refinement and temperature are studied.

Four different cases are taken into consideration with SGS of 0.02, 0.05, 0.1, and 0.3.

With the decrease in SGS value cells are more refined locally and the number of cells through the simulation is increased which is as shown below.

Temperature Contour:

Variation of Mesh:

Variation of Temperature:

From the results, it can be observed that the expansion fan tends to become less narrow and distinct as the SGS parameter decreases. This is due to the fact that the adaptive mesh refinement is dependent on the value of SGS. Higher values of SGS, limits the mesh refinement required to capture the expansion waves effectively leading to a blur and not well-defined waves.

Conclusion:

Thus, we were able to simulate flow over the expansion corner and study the effects of the SGS parameter. This can be beneficial to study the new and modified designs of aerofoils.