Week 5: Prandtl Meyer Shock problem
Prandtl Meyer Shock problem
Aim:
This particular project aims at obtaining a steady-state simulation of flow of a supersonic wave through a sharp corner which is famously known as the Prandtl Meyer shock problem.
Governing Equations:
The simulation is run on the basis of the Navier Stokes equations. The NS equation consists of three momenta, one continuity, and one energy equation. The energy equation is solved only during the time where there is a heat transfer taking place. This is done so to reduce unnecessary computational time.
The outlet boundary conditions will be set to Neuman boundary conditions. It is a condition named after Carl Neumann. The condition specifies that the derivative of the solution is going to take place in the direction normal to the boundary of domain D. This is mathematically described as,
∂u∂n=grad(u).n
Objective:
The objective is to obtain a steady-state simulation of flow of a supersonic wave through a sharp corner. The project later demands to analyze the role played by the sub-grid-scale and compare results between them.
The workflow involves:
- Creating a geometry of the shock problem model using the create function in the converge studio.
- Flagging the boundaries starting from the flow into the model and the fluid exiting from it.
- Creating the required case setup and setting up parameters according to the requirements.
- Transporting the input files and running the simulation.
- Convert the obtained 3D files into a corresponding viewable form.
- Obtaining the contours and animation and drawing results from them using a 3D viewing platform such as ParaView.
The case setup:
The simulation was run with a base mesh size of 0.8m. The boundary was first flagged into 4 different regions. The four different regions being the inlet, outlet, top, and bottom, front and back 2D Figure1. The regions have been indicated based on the colors. Once the boundary flagging is finished, the case setup is started, starting with time-based as the application type. Air is selected as a gas mixture. Global transport parameters were selected to be the default with the Prandtl number being 0.9 and Schmidt number being 0.78.
The solver was selected to be a steady-state solver. The simulation was run for 25000 cycles with an initial time step 1e-7 seconds. The solution was obtained using the NS equation. The solver scheme was selected to be PISO. PISO starts for Pressure Implicit with Splitting of operator. All the boundaries were dropped under a single region.
Coming to the boundary setup, the inlet (refer Figure2) was selected to be a supersonic flow with a velocity of 580m/s and a temperature of 286.1K which gives us an approximate mach number 2. The outlet was selected to be the Neumann boundary conditions as seen in Figure3. The condition specifies that the derivative of the solution is going to take place in the direction normal to the boundary of domain D. Supersonic flows have a tendency to run out of the domain usually and are hyperbolic in nature. Hence using a Neumann boundary condition helps in obtaining calculations at that particular instant rather than other boundary conditions that follow the ones that are set before starting the simulation.
Turbulence modeling was run according to the RNG k-ε equation. Coming to setting the grid control, the base grid was set to be 0.8m. Adaptive Mesh Refining (AMR) was used in this simulation and was set as seen in Figure 4. The output tab was selected to be the same all default. Once the setup was completed, the files were exported to a particularly saved output folder.
Figure1. Screenshot involving boundary flagging window during a shock flow simulation
Figure2. Screenshot involving inlet boundary setup window during a shock flow simulation
Figure3. Screenshot involving outlet boundary setup window during a shock flow simulation.
Figure4. Screenshot involving Adaptive Mesh Refinement (AMR) setup window during a shock flow simulation.
Results and Figures:
Shock waves are disturbances propagating through a medium and which when compared to a normal wave also carry energy with them during their movement. But they are characterized by traveling at speed greater than the speed of sound and having an abrupt change in pressure, temperature, and density of medium.
As mentioned previously, the inlet was set to flow at supersonic speeds, speeds greater than Mach1. This fluid comes in contact with a sharp edge surface inside the body and as a result, the flow is scattered with the waves having a spectrum of Mach numbers as seen in Figure5. The figure indicates the surface with edges for the setup having a base grid of 0.8m and a sub-grid criterion 0.05. Figure5 shows the extent to which the cells near the spectrum have been sized. This spectrum is known as the expansion fan. The inside of the spectrum starts with the inlet Mach number which is approximately 2 and the outside or trailing of the expansion fan has the outlet Mach number.
Figure 5 shows how closely packed the cells are near the expansion fan. Figure6 indicates the total cells used during the simulation which is around 7000 cells. The simulation ran for around 450 seconds.
Figure5. Screenshot involving surface with edges for a sub-grid criterion of 0.05 during a shock flow simulation.
Figure6. Plot involving Total number of cells with a sub-grid criterion of 0.05 during a shock flow simulation.
Now the sub-grid criterion is changed from 0.05 to a high number of 1 and the simulation was run again. The simulation just ran for a mere 300 seconds. The number of cells reduces to around 4000 which is a 42% reduction compared to the sub-grid criterion of 0.05 as seen in Figure7. The capture of the 3D output (Figure8) shows that there is absolutely no refinement taking place with the increase in criterion number. This results in a highly inaccurate value compared to the first setup which is highly undesired.
Figure7. Plot involving Total number of cells with a sub-grid criterion of 1 during a shock flow simulation.
Figure8. Screenshot involving surface with edges for a sub-grid criterion of 1 during a shock flow simulation.
The sub-grid criterion is changed from 1 to 0.01 and simulated again. This resulted in a high simulation time of around 1000 seconds. The Figure9 below indicates how close the cells are packed. This results in a highly accurate value but at the cost of computational time. The total cell count is around 17000 cells (Figure10) an increase of 1.42 times the one compared with the 0.05 cell grid criterion. The location of the shock is clearly captured by this simulation as compared to the other two.
Figure8. Screenshot involving surface with edges for a sub-grid criterion of 0.01 during a shock flow simulation.
Figure7. Plot involving Total number of cells with a sub-grid criterion of 0.01 during a shock flow simulation.
Conclusions:
The sub-grid criterion is a key factor in capturing the flow precisely. Hence care should be taken during the setup of the AMR. A low sub-grid criterion number results in a highly accurate value with a precise capture of the changes that are bound to happen. The use of Neumann conditions made sure that the calculations were made at that particular time instant resulting in desired values. Finally, the shock flow problem was physically sound meaning that the behavior of the flow was as expected. The animation link would be https://youtu.be/GtbdleAxF0k