Prandtl Meyer Expansion Fan

Objectives:

• To simulate the Prandtl Meyer expansion fan using converge and analyse the results
• To understand the physics behind shock waves through literature review and study the boundary conditions required to simulate such a flow
• To study the effect of SGS parameter on the location of shock wave and its effect on cell count

Background:

Shock Wave and expansion fan:As a fluid/gas flows through a sharp corner, if the speed of fluid/gas is much less than the speed of sound (subsonic flow), the flow of gas can be described by conserving momentum and energy. As the speed of the gas approaches the speed of sound, we must consider the compressibility effects on the gas. When a fluid/gas flows faster than the speed of sound (Supersonic flow), and there is a sudden decrease in the flow area, shock waves are generated. However, if the flow takes place across a convex corner, the flow area in this case increases suddenly and a different phenomenon is observed which is termed as a centered expansion fan.There are some marked differences between shock waves and expansion fans. Across a shock wave, the Mach number decreases, the static pressure increases, and there is a loss of total pressure because the process is irreversible. Through an expansion fan, the Mach number increases, the static pressure decreases, and the total pressure remains constant. Expansion fans are isentropic.

Boundary Conditions: To solve the steady state flow equations, appropriate boundary conditions need to specified for the flow problem. There are essentially three types of boundary conditions:

• Dirichlet Boundary Conditions: In this case, prescribed value of the variable is specified at the boundary of the problem domain.
• Neumann Boundary Condition: In this case, the gradient of the variable is specified at the boundary of the problem domain.
• Mixed or Cauchy Boundary Conditions: It is a combination of the above two boundary conditions.

In the case of shock flows, since the flow is supersonic, the boundary condition at the outlet should be kept such that the flow variables are interpolated from the interior of the flow domain. A specified value of static pressure at outlet is only used for subsonic flows. Hence for this reason, Neumann boundaary conditions are specified at the outlet in case of shock flows.

Case Setup:

Initial and Boundary Conditions:

• Inlet Velocity: 680 m/s
• Temperature: 286.1 K
• Outlet: Neumann Boundary Conditions for Pressure
• Sub Grid Criterion value for Adaptive Mesh Refinement (Temperature based): 0.05

Mesh size:

dx	dy	dz
0.8	0.8	0.8

Base Mesh:

Mesh Development:

Remark: Based on the SGS value, the mesh refinement takes place based on the second derivative of temperature (curvature) every time there is a change in curvature of 0.05 K

Temperature, Pressure and Velocity Contour:

Temperature, Pressure and Mach Number plots:

Remark: As described above, when the fluid passes through an expansion fan, a decrease in pressure and temperature is observed and an increase in the Mach number at outlet is seen.

Flow Animation:

Effect of SGS parameter on shock location and Cell count:

The temperature contour for different values of SGS parameter have been analyzed below to visualize the change in the expansion fan.

Case 1: SGS = 0.05

Case 2: SGS = 0.1

Case 3: SGS = 0.3

Variation in Cell count with SGS Parameter:

Conculsion:

• As the value of SGS parameter is increased, the expansion fan tends to become wider and less distinct. This can be explained from the cell count plot, since increasing the value of SGS parameter results in lesser refinement i.e lesser number of cells and thus is unable to capture the expansion profile accurately