Prantyl Meyer Shock wave problem

SITUATION 

            To simulate a supersonic flow through a Duct channel having the sharp corners to observe the formation of shock waves in converge studio and post-processes the results in Para View and CYGWIN was used to run the commands for the simulation and converting the output files for the post-processer. The airflow in the channel is turbulent flow.

TASK

        The simulation is to run for two different cases by varying the inlet velocity and different parameters are to be studied.

a) Observe the effect of the SGS parameter on the shock wave location and also the effect of SGS temperature value on the cell count.

b) Literature review on the shock wave and its boundary conditions.

Case 1: Supersonic (V = 680m/s) & SGS value - i) 0.05

                                                                      ii)0.1

Case 2: Subsonic (V = 100 m/s) & SGS value is 0.05.

Boundary Conditions

There are three types of boundary conditions used in CFD.

• When using a Dirichlet boundary condition, one prescribes the value of a variable at the boundary, e.g. u(x) = constant.
• When using a Neumann boundary condition, one prescribes the gradient normal to the boundary of a variable at the boundary,
e.g. ∂u(x) = constant.
• When using a mixed boundary condition a function of the form a*u(x)+b*∂u(x) = constant is applied.
• Note that at a given boundary, different types of boundary conditions can be used for different variables.

At the inlet of the flow, the Dirichlet boundary condition is used by specifying the inlet velocity for the flow, the Dirichlet B.C\'s condition helps us to calculate the value in the domain of influence. The value of velocity is specified instead of the pressure this is mainly due to the supersonic flow condition where velocity is important.

In the outlet, the value of the velocity & pressure is unknown so the value needs to be calculated from the inner nodes where the Neumann boundary condition is used which helps to interpolate the data from the inner nodes to calculate the value at the outlet. The pressure B.C\'s are used to specify the static pressure at the outlet only for the subsonic flow and since the supersonic conditions are used the pressure B.C\'s cannot be used.

The properties of the fluid vary as they flow across the flow due to the shock wave produced in the flow so the output properties cannot be found before the simulation, so Neumann B.C is used which specifies that value of the output properties depends only on the input parameters we set. 

Hence, the Neumann boundary condition is used at the outlet of the supersonic flow.

Shock wave

If the source travel at a speed greater than the speed of sound a wave will not be created at the front of the source but they pile together and form a compressed wave at the back. The pressure variation regions are formed around it. The properties of the fluid vary as the move from inside & outside to it. The wave energy dissipates as the distance increases and it then converted back into a normal wave.

ACTION

Workflow for a CONVERGE CFD Simulation

    Pre-processing(preparing the surface geometry and configuring the input and data files).

    Running the simulation.

    Post-processing(analyzing the *.out ASCII files in the Case Directory and using a visualization program to view the information in the post*.out)

Pre-processing

File --> import --> Import STL file, this option is used to import the 2D supersonic file.

Boundary dialog box 

Geometry

Select on the Normal toggle option and direction on the normal must be in the direction of the flow of air. The CONVERGE is mainly developed for running the IC engine simulation & other types of flow simulation are considered to be General flow. So General flow is selected.

1. Materials – Select Air as a predefined mixture & validate all the remaining files under it.  
2. Simulation Parameter – i) Run Parameters – Density-based steady solver is chosen.                                         ii) Simulation time parameter – start time – 0.

                                                                                        end-time  - 25000.

                                                                                        Initial time step – 1e-7.

                                                                                        Min time step – 1e-7.

                                                                                        Max\'s time step – 1.

1. Regions – Add a region, provide the value of the velocity as 680m/s & select air under the species area.                                                                                                                                                                       
2. Physical models – Select RNG k-e model is chosen under the turbulence model.                           
3. Grid control – i) Base Grid – the value of the grid depending is set to be 0.8.                                                  ii) Adaptive mesh refinement - Max cells - 200000.                                                                                                                 Active region - region 0.                                                                                                         Max embedding level - 2.                                                                                                         Subgrid criteria - 0.05.                                                                                                           Timing control type - Sequential.                                                                                             Start time - 5000.                                                                                                                   End time - 999999.

                                                                                                                                                                                                                          The adaptive mesh refinement technique is used where the shock wave occurs. The mesh grid at the shock regions gets refined. For our study, a temperature-based mesh refinement technique is used. The temperature curvature is d²t/ dx² , when the curvature is greater than the SGS value we set mesh gets refined.                                                                                                                                                       Grid size = base grid / 2^ embed level.                                                                                                                                                                                                                                                              = 0.8 / 2^2 = 0.2. This is the size of the least cell in our model.                                                          

    6. Boundary – i) Inlet – Type – inflow.

                                       Pressure B.C – Zero normal gradient(NE).

                                       Velocity B.C - Specified value & value depending on the case.                                               Species B.C’s – Air.

                          ii) Outlet – Type – outflow.

                                          Pressure & velocity B.C – Zero normal gradient(NE).

                                          Species B.C’s – Air.

                          iii) Slip – Type – wall.

                                        Motion type - Stationary.                                                                                                  Surface movement - Slip.

                         iv) Front_2D – Type – wall.                                     

                          v) Back_2D – Type – Wall.

    7. Output/post-processing – Validate all the files under it. 

Running the Simulation:

File --> export – To export the case set up for the running the simulation using CYGWIN.

Mpiexec.exe -n 4 converge-2.3.26-msmpi-win-64.exe </dev/null>logfile & - The command is used to run the simulation using 4 processors & the following command is used to store the file in the name of the log file and & symbol is used to store the files in the background.

Output data

i) Sub grid-scale - 0.05 & velocity - 680 m/s.

ii) Sub grid-scale - 0.1 & velocity - 680 m/s.

iii) Sub grid-scale - 0.05 & velocity - 100 m/s.

Results of various plot

1. Total cell count

i) SGS value -0.05 & Supersonic case.

ii) SGS value - 0.1 & Supersonic case.

iii) SGS value - 0.05 & subsonic case.

2) Pressure

i) Subsonic 

ii) Supersonic

3) Velocity

a) Subsonic

b) Supersonic

4) Mass flow rate 

a) Subsonic

b) Supersonic

5) Temperature

a) Subsonic

b) Supersonic

Post-processing:

Provide a suitable case name & select the Paraview VTK inline binary format as the file type and enter the directory which contains the output files which are to be post-processed & then click all in the files & cell variables and click convert.

Open the Paraview application and open the file using File -- > Open -- > test..vtm.

Click apply & select the slice option & choose the z normal to divide the geometry along the z-axis.

Element mesh

a) Supersonic

i) SGS - 0.1.

ii) SGS - 0.05.

b) Subsonic

Glyph plot

Profile contour

a) Pressure 

i) Subsonic

ii) Supersonic

b) Temperature

i) Subsonic

ii) Supersonic

c) Velocity 

i) Subsonic

ii) Supersonic

Youtube links

a) Velocity 

b) Pressure

c) Temperature

INFERENCE

The decrease in SGS temperature value increases the cell count by creating a finer mesh. The Subgrid criteria help us to create cells when the difference between the adjacent cells is greater than the value we set, so by decreasing the value the difference between is increased so more cells are formed. 

The number of cells is more in the subsonic flow this is mainly due to the variation along the wall as the area, but in the supersonic flow, the value of the cell count increases only in the shock region.

From the plot for supersonic flow, it is clear that only the mass flow rate at the outlet increases and other parameters like pressure, temperature, density as the flow progress.

For the subsonic flow, the value of pressure, temperature & velocity varies only at the wall which shows that the subsonic flow follows Bernoulli\'s principle (i.e) an increase in the speed of a fluid occurs simultaneously with a decrease in pressure.

The shock wave creates a pile of compressed wave and the pressure increases at the back of the flow and the velocity increases with the flow direction. The properties of the fluid vary at the shock region this forms a separation region that is depicted by the plot.