To visualize Prandtl Meyer expansion fan over an expansion corner using converge and understand the effect of SGS parameter on the shock location.
When a supersonic wave flows around a convex corner, a centered expansion occurs at the point of turn. This expansion consists of an infinite number of weak Mach waves diverging from the point of turn in the form of a fan. This is called as Prandtl-Meyer expansion fan.
The expansion is isentropic since the expansion waves takes place across a continuous succession of Mach waves and there is no change in entropy for each Mach wave. As the flow passes through the expansion wave, the Mach number increases and the pressure, density and temperature decreases.
The pre-modelled duct was imported into Converge and the geometry was inspected for any surface errors or open edges. After fixing all the errors, the boundaries were flagged, and the case was setup.
Velocity =680 m/s in x direction
Temperature= 286.1 K
Pressure- Zero normal gradient
Velocity- Zero normal gradient
Front and Back – 2D
Top and Bottom Wall – Slip Wall
Simulation Time Parameters:
Start Time: 0 Cyc
End Time: 25000 Cyc
Solver: Density based steady state solver
Fluid under consideration: Air (O2 + N2)
Mesh size of 0.8m was provided in X, Y, Z directions along with adaptive mesh refinement.
Temperature AMR was provided with maximum embedding level of 2 and sub-grid scale criteria as 0.05.
Note: Sub-grid scale is a parameter which will be used by converge for refining the mesh. Converge monitors the curvature of the fluid property provided in the AMR and refines the mesh when the property curvature variation between the consecutive grid is more than the defined SGS value.
Mesh with AMR (SGS = 0.07)
For Inlet velocity=680 m/s and SGS = 0.05
Temperature Contour with AMR
Velocity Contour with AMR
Flow property variations at Inlet and Outlet
From the above graphs, it can be observed that Mach number of the supersonic flow increases as the flow passes through the expansion wave whereas properties like temperature, pressure and density tends to decrease.
The flow simulation was carried out for different SGS parameters i.e. 0.05,0.07,0.10,0.30 and the results were determined.
Effect on expansion fan location
Below are the temperature contours for different SGS values.
SGS = 0.10
SGS = 0.30
From the post processing results, it can be observed that the expansion fan tends to become less narrow and distinct as the SGS parameter increase. This is due to the fact that the adaptive mesh refinement is dependent on the value of sub grid scale (SGS). Higher values of SGS of limits the mesh refinement required to capture the expansion waves effectively leading to a blur and not well-defined waves.
Effect on cell count
From the above graph, the simulation with the least SGS value had the largest number of cell count. This is understandable because lower SGS ensures higher refinement and higher refinement results in higher number of cells. For SGS = 0.30, the refinement criteria were so high that no refinement was achieved, it was as good as no adaptive mesh refinement.
Thus, we were able to design a geometry with an expansion corner, simulate flow over it and study the effects of SGS parameter. This can be beneficial when we design aircrafts that travel in supersonic speeds. We can simulate flow over various designs and modify the expansion corners in the design such that the expansion wave does not affect the airflow around the aircraft.
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