Week 5: Prandtl Meyer Shock problem

Aim: To simulate the Prandtl Meyer shock problem and check the effect of SGS temperature value on cell count and shock location using Converge. Also, perform a literature review on flow boundary conditions used for shock flow problems.

Theory:

We need boundary conditions in order to solve ordinary differential equations and partial differential equations. Different types of boundary conditions can be imposed on the boundaries of the domain/geometry. The selection of boundary conditions is necessary for solving computational problems. Inappropriate selection of boundary conditions may result in an erroneous solution which is of no importance.

There are 3 types of boundary conditions which are commonly used:

• Dirichlet Boundary condition: This boundary condition specifies the value of the function or variable at the boundary. It is also known as a first-type boundary condition.

• Neumann Boundary condition: This boundary condition specifies the values in which the derivative of a solution is applied within the boundary of the domain. It is also known as a second-type boundary condition.

• Mixed Boundary condition: This boundary condition for a PDE defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated. In other words, if both Dirichlet and Neumann conditions are imposed on various parts of the boundary, whose boundary belongs to the same domain is known as Mixed Boundary condition.

What is a shock wave?

It is a thin region in a flow that is created when the velocity of flow exceeds or equals the speed of sound which we refer to as Mach Number. This phenomenon can be observed in a supersonic and hypersonic flow. In this region of shock wave the properties of a gas such as pressure, temperature, flow speed, etc. all change abruptly.

Prandtl Meyer Shock Wave: Mach wave deflects the flow through an infinitesimally small angle and is seen only in the supersonic flow. These waves can be either compressive or expansive in nature and the whole process is isentropic. It is also known as a supersonic expansion fan, which is a centered expansion process that occurs when a supersonic flow turns around a convex corner which can be seen in the figure below.

The fan consists of an infinite number of weak Mach waves diverging from a sharp corner. In the above figure, we can see the supersonic waves passes through the Mach waves at an angle `mu` which is the 'Mach Angle' at corner point A and the streamlines are deflected downwards. This expansion fan is known as the 'Prandtl Meyer Expansion wave'. Once the wave is deflected the flow velocity increases and hence the Mach number also increases.

Here 3 different cases are simulated based on the flow of inlet velocity and SGS Parameter.

1) Supersonic case of velocity 650 m/s with different SGS parameter.

a) SGS: 0.05.

b) SGS: 0.01.

2) Subsonic case of velocity 100 m/s.

Methodology:

1) Pre-Processing:

We will be importing the geometry in converge studio whose dimension will be scaled into meters and the dimensions are as follows:

• Length along x-direction (l_x) = 65 m.
• Length along y-direction (l_y) = 45.1525 m.
• Length along z-direction (l_z) = 8.151 m.

We will be flagging the domain into 5 different boundaries namely Inlet, Outlet, Front 2D, Back 2D, and Walls.

The next step is we will be moving on to the case setup where we will be setting up all the parameters.

• In this project we are interested in analyzing the shock flow. Hence we perform a steady-state analysis. Steady-state simulation computes the fully developed solution that does not change in time.
• This is a general flow problem and hence we will be selecting a Time-based application type.
• Moving on to the Materials we will be using predefined air whose composition consists of nitrogen and oxygen. The transport properties are set to the default values. The species composition consists of N2 and O2.
• Simulation parameters are set in which the required steady-state solver and a steady-state monitor is set up for the mass flow rate at the inlet and outlet boundary. Also, simulation time parameters are set accordingly.

- Start time : 0 cycles.

- End time : 25000 cycles.

- Initial time step : 1e-9 s.

- Minimum time step : 1e-9 s.

- Maximum time step : 1 s.

• Now we will be creating a region in order to group various boundaries to define a volumetric region under initial conditions and initialization tab.
• Moving on, we flag different boundaries and assign respective boundary conditions to create a meaningful Volumetric region.

  - Inlet Boundary :

  1) Boundary Type: Inflow.

  2) Pressure: Zero Normal Gradient (Neumann condition).

  3) Velocity (Dirichlet): 680 m/s in x-direction.

  4) Temperature (Dirichlet condition): 286.1 K.

  5) Species BC: air.

  - Outlet Boundary :

  1) Boundary Type: Outflow.

  2) Pressure: Zero Normal Gradient (Neumann condition).

  3) Velocity: Zero Normal Gradient (Neumann condition).

  3) Temperature (Dirichlet condition): 300 K.

  4) Species BC: air.

  - Front 2D:

  1) Boundary Type: Two_D.

  - Back 2D:

  1) Boundary Type: Two_D.

  - Walls:

  1) Boundary Type: Wall.

  2) Wall motion type: Stationary.

  3) Surface Movement: Fixed.

  4) Slip.

• Turbulence Model: RNG k-epsilon.
• A grid size of 0.8 m was used along with Adaptive Mesh Refinement. Adaptive mesh refinement is used to capture the information accurately were the shock occurs, in order to capture this information we need refinement of mesh in that particular area while keeping the mesh size of other area similar to the base grid. This indeed reduces the overall computational time compared to refining the overall domain. Converge adaptive mesh refinement has various techniques based on different parameters and in our case we will use temperature-based sub-grid-criterion with Max.embedding level set to 3. Converge will monitor the curvature of Temperature `(del^2T)/(delx^2)` provided in AMR and if the curvature is greater than a particular value defined by the user the mesh gets refined.

Refine Grid Size = `(Basegrid)/2^(embed-Level)`

• Finally, we will be setting up the parameters for printing the 3D and other output files in the post-processing tab. - Time interval for writing 3D output data files :500 cycles. - Time interval for writing text output:1 cycle. - Time interval for writing restart output:1000 cycle.

2) Running the solver:

• Once the case is setup we export the input files from the converge studio and save in a designated location.
• Then using Cygwin we will run the simulation.

3) Post-Processing:

• The output files are converted to a format in which ParaView can understand using Cygwin or converge studio.
• These converted files can be used for visualizing contours, animations, etc. using Paraview which is the post-processing software.

Post-Processing Results:

Case 1: SGS-0.05

Geometry Mesh:

A base grid of 0.8 m was provided with Adaptive Mesh Refinement. The mesh element of the geometry can be seen below.

SGS: 0.05

The total cell count of the geometry is as follows:

The graph pattern of the independent ranks is similar to the total cell of the geometry line graph. Since we used 4 processors to solve the problem we can observe 4 different ranks. The total number of cells of the geometry is distributed among these 4 processors.

Mass Flow Rate Plot:

Velocity Plot:

Pressure Plot:

Temperature Plot:

Mach Plot:

The Mach number at the inlet is 2 and the outlet Mach number at the outlet reaches a value of 2.15.

Density Plot:

The Mach number of the supersonic flow increases as the flow passes through the expansion while thermodynamic properties such as density, temperature and pressure tends to decrease.

Velocity Contour:

The effect of the SGS parameter near the shock location can be easily observed by looking into the below contours.

Velocity Animation:

Pressure Contour:

Pressure Animation:

Temperature Contour:

Temperature Animation:

Case 2: SGS-0.1

Geometry Mesh:

A base grid of 0.8 m was provided with Adaptive Mesh Refinement. The mesh element of the geometry can be seen below.

SGS: 0.1

The total cell count of the geometry is as follows:

The graph pattern of the independent ranks is similar to the total cell of the geometry line graph. Since we used 4 processors to solve the problem we can observe 4 different ranks. The total number of cells of the geometry is distributed among these 4 processors.

Mass Flow Rate Plot:

Velocity Plot:

Pressure Plot:

Temperature Plot:

Mach Plot:

The Mach number at the inlet is 2 and the outlet Mach number at the outlet reaches a value of 2.15.

Density Plot:

The Mach number of the supersonic flow increases as the flow passes through the expansion while thermodynamic properties such as density, temperature and pressure tends to decrease.

Velocity Contour:

The effect of the SGS parameter near the shock location can be easily observed by looking into the below contours.

Velocity Animation:

Pressure Contour:

Pressure Animation:

Temperature Contour:

Temperature Animation:

Case 3: SGS-0.1

Geometry Mesh:

A base grid of 0.8 m was provided with Adaptive Mesh Refinement. The mesh element of the geometry can be seen below.

SGS: 0.1

The total cell count of the geometry is as follows:

The graph pattern of the independent ranks is similar to the total cell of the geometry line graph. Since we used 4 processors to solve the problem we can observe 4 different ranks. The total number of cells of the geometry is distributed among these 4 processors.

Mass Flow Rate Plot:

Velocity Plot:

Pressure Plot:

Temperature Plot:

Mach Plot:

The Mach number at the inlet is 0.294 and the outlet Mach number decreases reaching a value of 0.283.

Density Plot:

The Mach number of the subsonic flow decreases as the flow passes through the expansion while thermodynamic properties such as density and pressure almost remain constant.

Velocity Contour:

Velocity Animation:

Pressure Contour:

Pressure Animation:

Temperature Contour:

Temperature Animation:

Conclusions:

• Differing SGS values have an impact on the number of cells formed during the meshing. From the above simulations, we can see that lower SGS value (0.05) tends to have an increased number of cells compared to higher SGS value (0.01).
• The mesh size is refined only when the sub-grid temperature is greater than the SGS value.
• From the plots of supersonic flow, we can see that the mass flow rate and the Mach number increases at the outlet and the other thermodynamic parameters such as pressure, temperature and density decreases near the outlet.
• From the plots of subsonic flow, we can see that the match number decreases as the flow passes through the expansion while thermodynamic properties such as density and pressure almost remain constant.