Prandtl Meyer Shock problem
Shock Flow Boundary Conditions: For solving the steady state flow appropriate boundary conditions are needed. It is one of the required components of the mathematical model. On the other hand, for solving transient flow, appropriate initial condition is also required. Boundary conditions are generally three types. They are Dirichlet boundary condition, Neumann boundary condition and mixed boundary condition. When using a Dirichlet boundary condition, the value of variable is prescribed at the boundary. Neumann boundary Conditions, When Neumann boundary Conditions imposed on an ordinary or a PDE, the condition specifies the values in which the derivative of a solution is applied within the boundary of the domain.
Pressure outlet boundary conditions require the specification of a static (gauge) pressure at the outlet boundary. The value of the specified static pressure is used only while the flow is subsonic. If the flow is supersonic, the specified pressure will no longer be used,pressure will be extrapolated from the flow in the interior. This means that Neumann Boundary Conditions which say that the outlet condition is dependent on the initial conditions of the flow need to be used in this case, that will derive the value of pressure from the initial conditions.
SHOCK WAVE: When the speed of the moving object in a medium becomes equal to the velocity of sound(340 m/s) in the medium, the wavefronts cannot escape from the source and pile up. This results in a large amplitude sound barrier. When the speed of the moving object or source exceeds the speed of sound in medium then the wavefronts lag behind the source forming a cone shaped region with a source at vertex. The edge of the cone forms a supersonic wave front with an unusually large amplitude called a shock wave. A sonic boom is heard when the shock waves reach an observer. The occurence of shock waves can be characterized by instantaneous change in pressure, velocity and temperature in a fluid flow. The region between the vehicle and the shock wave known as the shock layer will be a region of high pressure, density and temperature than the free-stream flow conditions. When a fluid streamline crosses the standing shock wave, abrupt increase in the pressure, temperature and density of the fluid flow occurs with a decrease in velocity of the flow.
Prandtl Meyer Shock Waves: In order to understand Prandtl Meyer shock wave, we first need to understand what oblique waves are. The normal shock waves are straight in which the flow before and after the wave is normal to the shock. It is considered as a special case in the general family of oblique shock waves that occur in supersonic flow. In general, oblique shock waves are straight but inclined at an angle to the upstream flow and produce a change in flow direction. An oblique shock generally occurs, when a supersonic flow is ‘turned into itself”.
Another class of two dimensional waves occurring in supersonic flow shows the opposite effects of oblique shock. Such types of waves are known as expansion waves . When the supersonic flow is “turned away from itself”, an expansion wave is formed. Here, the flow is allowed to pass over a surface which is inclined at an angle θ to the horizontal and all the flow streamlines are deflected downwards. The change in flow direction takes place across an expansion fan centered at point ‘A'. The flow streamlines are smoothly curved till the downstream flow becomes parallel to the wall surface behind the point ‘A'. Here, the flow properties change smoothly through the expansion fan except at point ‘A'. An infinitely strong oblique expansion wave may be called as a Mach wave . An expansion wave emanating from a sharp convex corner is known as a centered expansion which is commonly known as Prandtl-Meyer expansion wave.
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 or fan out creating a pattern that will be shown by simulating the flow in Converge Studio.
The simulation was run under three different conditions:
1) Sub-Grid Criterion(SGS)= 0.05 K
2) Sub-Grid Criterion (SGS)= 0.1 K
3)Sub-Grid Criterion (SGS)= 0.01 K
4) Subsonic Flow= 100 m/s
After flagging the boundaries, the following conditions were assigned to them:
1) Inlet: Inflow type; At a temperature of 286.1 Kelvin,velocity is 340 m/s, velocity along x-axis was kept 680 m/s making the inlet Mach number as 2 (680/340).
2) Outlet: Outflow type. Neumann Boundary Condition was used which is dependent on the initial conditions of the flow.
3) Top & Bottom: Wall boundary type with 'Slip' condition. The slip condition causes the fluid to flow along a wall instead of stopping at the wall, which typically occurs along a wall. Fluid is prevented from flowing through the wall, however.
4) Front & Back: Since the case was setup for 2-D flow, 'TWO-D' conditions were assigned.
Adaptive Mesh Refinement technique was used because the region where shocks would occur require the mesh to be refined locally. The mesh in the other regions have the same size as the base grid size whereas the region where shock occurs finer cells are used. When curvature is greater than a particular value defined by the user the mesh is refined. In the first run, SGS was taken as 0.05 K. So, when sub-grid temperature would be greater than 0.05 K, the mesh would refine.
If the refinement level/max embedding level is 2(n), then Grid-Size would be, Grid-Size= base grid (0.8 m)/2^n(2), which would be equal to a grid-size of 0.2 m. That would be the smallest cell size used by converge in mesh refinement.
SGS= 0.05 K , Total Cells= 7094
SGS=0.1 K, Total Cells= 5172
SGS=0.01K, Total Cells:15573
Changing the SGS values changes the number of cells that are formed during meshing. As mentioned above when the sub-grid temperature becomes greater than SGS value, the mesh is refined. Thus for a low value of SGS(0.01), more cells are refined than at a higher value of SGS (0.1).
The inlet mach number where the flow enters the duct is 2 and the outlet Mach number at the outlet is 2.15 which corresponds to an outlet velocity of 731 m/s as compared to the inlet velocity of 680 m/s.
Mesh formed with SGS=0.05K
Mesh formed with SGS=0.1K
Mesh formed with SGS=0.01 K
1) Temperature and Pressure at SGS=0.05K
Youtube links for the animation at SGS=0.05k
2) Temperature and Pressure at SGS=0.1 K- The shock waves are not refined well where there is no mesh refinement along the path of shock.
Youtube links for the animation at SGS=0.1k
https://youtu.be/aVq_NNqtz34
3) Temperature and Pressure at SGS=0.01 K- Due to the mesh being refinement being larger in this case, the shock waves are also finer in this case
Youtube links for the animation at SGS=0.01k
https://youtu.be/-fGY3fHWFAE
Effect of Using Subsonic Inital Condition at a velocity of 100 m/s (Mach No:0.29) for Supersonic Flow
The solution blows up at 100 m/s because the difference between the inlet velocity at boundary condition being 680 m/s and at initial condition being 100 m/s leads the solution to diverge.
This effect can very well be seen in the following graph. The origin is broken depicting a discontinuity in solution.
