Prandtl Meyer Shock Wave - 2D simulation

OBJECTIVE

Simulation of Supersonic flow through a duct channel with sharp corners to observe the formation of shock waves.

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.

1. 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.
2. 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.
3. 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.
4. Hence, the Neumann boundary condition is used at the outlet of the supersonic flow.

SHOCK WAVE

1. 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.
2. The pressure variation regions are formed around it. The properties of the fluid vary as the move from inside & outside to it.
3. The wave energy dissipates as the distance increases and it then converted back into a normal wave.

GEOMETRY

SIMULA/TION SETUP

• Fluid for the current simulation is air as predefined mixture.
• Solver Type: Density Based Solver
• Simulation Time Parameter

1. 1. start time = 0
   2. end time = 25000
   3. Initial time step = 1e-7
   4. Minimum Time Step = 1e-7
   5. Max Time Step = 1

• Turbulence Model: RNG k-epsilon
• Base Grid: The value of grid dependence is set to be 0.8
• Adaptive Mesh Refinement: The mesh grid at the shock regions get refined. For our study, a temperature-based mesh refinement technique is used. The temperature curvature is d²t/dx². The mesh is set te get refined when the curvature is greater than the SGS value.

1. 1. Maximum Cells = 200000
   2. Active region = region 0.
   3. Maximum Embedding Level = 2
   4. Subgrid Criteria = 0.05 & 0.1
   5. Control Type = Sequential
   6. Time = Start time: 5000 ; End time: 999999
   7. `Grid Size = (Base Grid)/2^(embed l e vel) = (0.8)/2^2=0.2`. This is the size of the least cell in the model.

• Inlet Conditions:

1. 1. Pressure: zero normal gradient (NE)
   2. Velocity: 100 m/s for subsonic case & 680 m/s for supersonic case.

• Outlet Conditions:

1. 1. Pressure: zero normal gradient (NE)
   2. Velocity: zero normal Gradient (NE)

• Slip

1. 1. Wall Boundary Condition.
   2. Law of wall function is selected as turbulence model is utilized in the current study

• Front_2D & Back_2D

1. 1. Wall Boundary Condition.
   2. Temperature: 300 K
   3. Law of wall function is selected as turbulence model is utilized in the current study.

SIMULATION OUTPUTS

CASE 1: SUBSONIC FLOW (WITH SGS VALUE = 0.05)

1.1. OUTPUT DATA

1.2. TOTAL CELL COUNT

1.3. ELEMENT MESH

1.4. PRESSURE

1.5. TEMPERATURE

1.6. VELOCITY

1.7. MASS FLOW RATE

CASE 2: SUPERSONIC FLOW (with SGS =0.05 & SGS = 0.1)

2.1. OUTPUT DATA

2.1.1. SGS = 0.05

2.1.2. SGS = 0.1

2.2. TOTAL CELL COUNT

2.2.1. SGS = 0.05

2.2.2. SGS = 0.1

2.3. ELEMENT MESH

2.3.1. SGS = 0.05

2.3.2. SGS = 0.1

2.4. PRESSURE

2.5. TEMPERATURE

2.6. VELOCITY

2.7. MASS FLOW RATE

TEMPERATURE CONTOUR ANIMATIONS

NOTE: 4 DIFFERENT SGS VALUES WERE TESTED TO GET A BETTER INSIGHT OF THE EFFECT OF THE SGS PARAMETER

INFERENCE

1. 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.
2. 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.
3. From the plot for supersonic flow, it is clear that only the mass flow rate at the outlet increases unlike other parameters like pressure, temperature, density as the flow progress.
4. 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.
5. 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.