Prandtl Meyer Shock problem

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1. Literature review on Boundary conditions for shock flow

• Front and Back boundaries are assigned as 2 D boundaries.
• Top and bottom- slip boundaries
• Inlet boundary- supersonic inlet velocity is given with Mach number= 2, species= Air.
• Outlet bounday- Neumann boundary condition is applied, in which 1^st order derivative of dependent variable are used and equated to zero.

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Converge will behave unpredictably if part or all of fluid exiting the region assigned a supersonic outlet is subsonic. The solver might fail to converge or diverge in this situation. To correct this, you should move the outlet to a location where the flow is supersonic over the entire boundary. All values of dependent variable are extrapolated from upstream.

2. Shock Waves

When the speed of a source equals the speed of sound (v = c) the wave fronts cannot escape the source. The resulting pile of waves forms large amplitude that makes sustained flight at this speed difficult and risky.

These waves are generated when velocity of source is greater than velocity of wave (or sound velocity). At this point, very sharp changes in gas properties take place. When the speed of a source exceeds the speed of sound (v > c) the wave fronts lag behind the source in a cone-shaped region with the source at the vertex. The edge of the cone forms a supersonic wave front with unusually large amplitude called a "shock wave". These waves progresses with increase in amplitude of wavefront along common tangent (as shown in Fig 1). Angle Ɵ made by the cone is known as mach angle and Mach number is defined as ratio of speed of source to speed of wave (or sound) in the medium.

Fig. 1 Shock wave and mach angle

Fig 2 Subsonic Flow

Fig 3 Supersonic flow with shock

3. Effect of SGS tempeature value on cell count and shock location

Subgrid scale temperature refers to second order temperature gradient value between the neighboring grids for which mesh refinement occurs in adaptive mesh. Thus it is going to change the number of cell count in those locations and shock can be analysed more properly.

Maximum embedding level is set to 2, which means that mesh will be refined upto two sets and minimum grid size is thus given by:

Grid Scale= base grid/ 2^{embedding level}

Fig 4 Output at SGT 0.05 K

Fig 5 Output at SGT 0.1 K

As seen from Fig 4 and Fig 5, as Scale grid temperature is less, more mesh is refined and output is analysed more effectively.