Week 5: Prandtl Meyer Shock problem
Shock flow boundary conditions:
A flow problem is typically characterized by a mathematical equation. The boundary conditions exert a set of additional constraints to the problem on specified boundaries. A boundary condition is required to solve both ordinary and partial differential equations.
Following are the types of boundary conditions:
- Dirichlet boundary condition
- Neumann boundary condition
- Cauchy boundary condition
- Robin boundary condition
- Mixed boundary condition
The Dirichlet and Neumann boundary conditions are more predominant boundary conditions while solving a flow problem.
The Dirichlet boundary condition specifies the value of a function on surface.
ODE
For an ordinary differential equation, for instance,
the Dirichlet boundary conditions on the interval [a,b] take the form
where α and β are given numbers.
PDE
For a partial differential equation, for example,
where ∇2 denotes the Laplace, the Dirichlet boundary conditions on a domain Ω ⊂ ℝn take the form
where f is a known function defined on the boundary ∂Ω.
The Neumann boundary condition on the other hand specifies the normal derivative of a function on the surface.
ODE
For an ordinary differential equation, for instance,
the Neumann boundary conditions on the interval [a,b] take the form
where α and β are given numbers.
PDE
For a partial differential equation, for instance,
where ∇2 denotes the Laplace operator, the Neumann boundary conditions on a domain Ω ⊂ ℝn take the form
where n denotes the (typically exterior) normal to the boundary ∂Ω, and f is a given scalar function.
The normal derivative, which shows up on the left side, is defined as
where ∇y(x) represents the gradient vector of y(x), n̂ is the unit normal, and ⋅ represents the innner product operator.
The mixed boundary condition is a combination of both Dirichlet and Neumann boundary condition.
The Cauchy and Robin boundary conditions are different combinations of Dirichlet and Neumann boundary conditions.
For Shock flow problems usually the Neumann boundary condition is used.
(Source: Wikipedia)
Shock Wave:
When an object moves through a gas, gas molecules get deflected by the object. If the speed of the object is much less than the speed of sound of the gas, the density of the gas remains constant and the flow of gas can be described by conserving momentum and energy. As the speed of the object increases towards the speed of sound, we can see compresibility effects on the gas. The density of the gas varies as the gas is compressed by the object locally. When an object moves faster than the speed of sound with an abrupt decrease in the flow area,shock waves are generated.
Shock waves are very small regions in the gas where the gas properties change by a large amount. Across a shock wave, the static pressure, temperature, and gas density increases almost instantaneously. Because no work is performed by a shock wave and there is no heat addition, the total enthalpy and the total temperature are constant. But because the flow is non-isentropic, the total pressure downstream of the shock is always less than the total pressure upstream of the shock. There is a loss of total pressure associated with a shock wave.The Mach number and speed of the flow also decrease across a shock wave.
(Source:https://www.grc.nasa.gov/www/k-12/airplane/normal.html )
Geometry and Case Setup:
Dimensions:
Length=65 m, Width= 45.15 m, Height= 8.15 m
Case Setup:
General flow (Time based)
Fluid : Air
Steady state solver, Full hydrodynamic, Compressible
Start time: 0 cyc
End time: 25000 cyc
Initial time-step: 1e-07 s
Minimum time-step : 1e-07 s
Maximum time-step: 1 s
Sover type: Density based
Boundary conditions:
Inlet Pressure : Zero Normal Gradient
Inlet Velocity : Supersonic (680 m/s); Subsonic (100 m/s)
Inlet temperature : 286.1 K
Outlet Pressure : Zero Normal Gradient
Outlet Velocity : Zero Normal Gradient
Temperature Backflow: 300 K
Top and Bottom Walls: Slip
Front and Back : 2D wall
Turbulence model : Reynolds Averaged Navier Stokes (RANS)
Base grid : 0.8 m
Adaptive Mesh Refinement-
Embed type: Sub-grid scale (SGS)
Max embedding level: 2
Sub-grid criterion: 0.1 K,0.05 K,0.001 K
Timing Control Type: Sequential
Results:
Supersonic flow (Flow velocity: 680 m/s, SGS : 0.001K)
Mesh grid:
Velocity contour:
Temperature contour:
Temperature plot:
Mach number plot:
Cell count plot:
Supersonic flow (Flow velocity: 680 m/s, SGS : 0.05K)
Mesh grid:
Velocity contour:
Temperature contour:
Temperature plot:
Mach number plot:
Cell count plot:
Supersonic flow (Flow velocity: 680 m/s, SGS : 0.1K)
Mesh grid:
Velocity contour:
Temperature contour:
Temperature plot:
Mach number plot:
Cell count plot:
Subsonic flow (Flow velocity: 100 m/s, SGS : 0.001K)
Mesh grid:
Velocity contour:
Temperature contour:
Temperature plot:
Mach number plot:
Cell count plot:
Subsonic flow (Flow velocity: 100 m/s, SGS : 0.05K)
Mesh grid:
Velocity contour:
Temperature contour:
Temperature plot:
Mach number plot:
Cell count plot:
Subsonic flow (Flow velocity: 100 m/s, SGS : 0.1K)
Mesh grid:
Velocity contour:
Temperature contour:
Temperature plot:
Mach number plot:
Cell count plot:
Conclusion:
As the SGS value changes the cell count value changes.
For supersonic flow the change can be clearly seen in the cell count plot.
For subsonic flow , we can see that only for SGS 0.001 K there is a change in cell count.