Prandtl Meyer Shock Problem
SHOCK WAVE:
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.
PRANDTL SHOCK WAVE:
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 backwards to meet at a point.
MACH NUMBER:
It is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound.
where:
- M is the local Mach number,
- u is the local flow velocity with respect to the boundaries (either internal, such as an object immersed in the flow, or external, like a channel), and
- c is the speed of sound in the medium, which in air varies with the square root of the thermodynamic temperature.
- Subsonic Flow: When the fluid velocity is lower than the acoustic speed (M<1) then the fluid flow is called subsonic. However Mach number of the flow changes while passing over an object or through a duct. Hence for simplicity, flow is considered as subsonic if the Mach number is in the range of 0-0.8.
- Transonic flow: When the flow Mach number is in the range 0.8-1.2 it is called transonic flow. Highly unstable and mixed subsonic and supersonic flows are the main features of this regime.
- Sonic flow: When flow Mach number is 1 it is called sonic flow.
- Supersonic Flow: When the flow Mach number is more than everywhere in the domain then it is called as supersonic flow.
- Hypersonic Flow: When the flow Mach number is more than 5 then it is called as hypersonic flows.
TYPES OF BOUNDARY CONDITIONS:
The concept of boundary conditions applies to both ordinary and partial differential equations.
There are five types of boundary conditions:
Dirichlet
Neumann
Robin
Mixed
Cauchy
Within this, Neumann and Dirichlet boundary conditions are predominant.
SHOCK FLOW BOUNDARY CONDITION:
For solving the steady state flow appropriate boundary conditions are needed. It is one of the required components of the mathematical model.
If the flow is supersonic, the specified pressure will no longer be used,pressure will be extrapolated from the flow in the interior.
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. ¶nu(x) = constant.
When using a mixed boundary condition a function of the form au(x)+b¶nu(x) = constant is applied.
For this shock flow, Neumann boundary condition is used because the outlet conditions are dependent on the initial conditions so it helps in movement of the shock wave from inflow to its outlet without any changes or effect from the external conditions outside the flow domain.
Effect of SGS parameter on shock location:
In Prandtl-Meyer Shock Flow, supersonic waves are made to pass through the inlet of a duct that has a sharp corner. An infinite number of Mach Waves are created which spread out over a wide area creating a pattern that will be shown by simulating the flow in Converge Studio.
1.The simulation is ran under three SGS values in supersonic flow condition:
CASE 1 - SGS VALUE - 0.005
CASE 2 - SGS VALUE - 0.003
CASE 3 - SGS VALUE - 0.001
2.The simulation is also ran in subsonic flow condition at 100 m/s
CASE SETUP:
Air is used as predefined mixture.
Density based steady state solver is used.
The simuation runs for 25000 cycles.
In volumetric region the air flow of initial temperature of 300K and a pressure of 500000 Pa and the initial velocity is changed to 100 m/s and 680 m/s depending on the nature of flow.
Boundary conditions:
| BOUNDARY | TYPE | PRESSURE | TEMPERATURE | VELOCITY |
| INLET | INFLOW | ZERO NORMAL GRADIENT | 286 K |
680 m/s (supersonic) 100 m/s (subsonic) |
| OUTLET | OUTLET | ZERO NORMAL GRADIENT | 300 K | ZERO NORMAL GRADIENT |
| FRONT | 2D | - | - | - |
| BACK | 2D | - | - | - |
| TOP & BOTTOM | WALLS | - | ZERO NORMAL GRADIENT | SLIP |
ZERO NORMAL GRADIENT is used to specify Neumann boundary condition.
Rng `k-epsilon` turbulent model is used here.
The base grid is 0.8m in all three directions
Adaptive mesh refinement is used here and the properties are
Maximum cells - 200000
Criteria - temperature
Embedding level - 3
SGS parameter - 0.05,0.03,0.01 depending on the cases.
Start cycle - 5000
End cycle - 9999999
1. SUPERSONIC FLOW:
CASE 1: SGS - 0.005
MESH:
MACH NUMBER:
TEMPERATURE CONTOUR:
TEMPERATURE ANIMATION:
TEMPERATURE PLOT:
PRESSURE CONTOUR:
CASE 2 : SGS - 0.03
MESH:
MACH NUMBER PLOT:
TEMPERATURE CONTOUR:
TEMPERATURE ANIMATION:
TEMPERATURE PLOT:
PRESSURE CONTOUR:
CASE 3: SGS - 0.01
MESH:
MACH NUMBER PLOT:
TEMPERATURE CONTOUR:
TEMPERATURE ANIMATION:
TEMPERATURE PLOT:
PRESSURE CONTOUR:
TOTAL CELL COUNT FOR ALL THREE CASES:
OBSERVATION:
By varying the SGS value we obtained the plots and mesh in a supersonic flow. We have used the same mesh size but with different AMR value we obtained different number of cell count. When the flow entering the domain the temperature, pressure and velocity plots are predominant when the AMR value is less( i.e. CASE 3). Thus the mesh generated in the first case with SGS criteria 0.005 has less number of cells generated near the shock fan. The SGS temperature criteria captured the flow from its origin till the point when they go out. Thus we can obtain the smoother resolution with low AMR value as in case 3.
SUBSONIC FLOW:
MESH:
TEMPERATURE CONTOUR:
MACH NUMBER PLOT:
PRESSURE CONTOUR:
TOTAL CELL COUNT:
OBSERVATION:
Thus in a subsonic flow, the plots look like it is not converged well and there is no effect due to change in SGS temperature criteria.