Prandtl Meyer Shock problem

OBJECTIVE

1. Shock flow boundary conditions

2. What is a shock wave?

3. Effect of SGS parameter on shock location

 

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, the appropriate initial condition is also required.

Boundary conditions are generally three types

  • Dirichlet boundary condition.
  • Neumann boundary condition.
  • Mixed boundary condition. 

When using a Dirichlet boundary condition, the value of the 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, the 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 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 the medium then the wavefronts lag behind the source

the 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, an 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: To understand Prandtl Meyer shock wave, we first need to 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 the flow direction. An oblique shock generally occurs, when a supersonic flow is 'turned into itself '

A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, a two-dimensional simple wave, is a centred 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.

Each wave in the expansion fan turns the flow gradually (in small steps). It is physically impossible for the flow to turn through a single "shock" wave.

 PROFILE GEOMETRY:

           

 

INITIAL BOUNDARY CONDITIONS:

  • The flow is general flow.
  • Air contains a mixture of O2 and N2 which will flow through the required geometry.
  • The pressure at the inlet and outlet of the geometry is 100001 Pascal and 100000 Pascal.
  • The Prandtl number is 0.9 and the Schmidt number is 0.78.
  • The total number of cycles performed is 15000.

BASE GRID:

dx=dy=dz=0.8

 

AT SGS 0.05 TEMPERATURE VARIATION:

           

AT SGS 0.05 PRESSURE VARIATION:

           

PLOTS:

AT SGS 0.05 AVG TEMPERATURE VARIATION:

                   

  • Avg temperature along with the boundary id 2 decreases upto 270 k abd after that it gets constant

 

AT SGS 0.05  AVG MACH NUMBER VARIATION:

                   

 

                     

Temperature contour:

                     

Velocity contour:

                     

PLOT FOR TOTAL CELL COUNT AT TWO DIFFRENT SGS

Changing the SGS no changes the number of cells. The number of cells is refined at SGS 0.02  is more than SGS 0.05 because the sub-grid temperature becomes greater than the SGS value.

CONCLUSION

  • Adaptive mesh refinement is an important tool to capture the shockwave effectively using a subgrid criterion based on the curvature of the parameters provided.
  • It shows the abrupt effect of oblique shock.
  • The cell counts were observed to decrease an expected for higher SGS of 0.05 as can be seen from the contour that its accuracy decreases due to less no of cells

https://www.youtube.com/watch?v=A8xhKBNyw_Y

https://www.youtube.com/watch?v=PQvW5lHJcys


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