Prandtl Meyer Expansion Waves
Prandtl Meyer Expansion Waves
Objective: To perform a CFD simulation of the Prandtl Meyer Mach waves manifested in the Prandtl Meyer expansion fan. To explore the effect of Sub-grid Criterion Scale in adaptive mesh refinement.
Shocks: When an object travels supersonically in a medium, it projects influence on particles within the zone of influence in the form of disturbances after the object has passed said point. For objects travelling supersonically, when the cross-section area abruptly changes in the direction of the motion, shocks are manifested as propagating disturbances traveling faster than the local speed of sound in the medium. These shocks entail abrupt changes in static conditions of pressure, temperature, and density as effects of shock manifestation.
For a supersonic object, when the flow area abruptly expands in a setting that causes the flow to turn away from itself, Mach waves in the form of expansions waves begin to fan out bound by front and rearward Mach waves. These are manifestations of isentropic expansion waves. Each wave is a Mach-wave and the expansion waves are divergent because as the flow expands the Mach number increases, causing the Mach angle to decrease and consequently, diverging the flow.
Boundary Conditions: Boundary conditions are fundamental backbones of correctly converging CFD simulations. In order to attain tangible results, the correct boundary conditions must be employed based on the type of geometry, simulation, and flow-type. Boundary conditions are a combination of Dirichlet and Neumann boundary conditions. In Dirichlet boundary conditions, the value of the parameter at a known state is defined and utilized to compute all other parameters in the simulation. In Neumann boundary conditions, the states through the simulations are devised as differentials of the initially defined flow characteristics and are thus extrapolated. In supersonic flow, characterized by abrupt changes in state parameters, a steady value of parameters post flow are a rarity and thus extrapolated Neumann boundary conditions are employed.
Adaptive Mesh Refinement and Subgrid-Criterion: In COVERGE, like other CFD packages, adaptive mesh refinement can be deployed. The adaptive mesh refinement setting can be set as function of the sub-grid criterion i.e. the mesh will refine wherever the differential change in a parameter (e.g. Temperature) is of magnitude of defined value. Larger or smaller SGC values govern the differential change for which mesh refinement will be deployed. This setting of the SGC criterion effects the grade of mesh refinement in regions of interest in order to capture to most differential of changes in parameter and greatly affect the cell count and precision in location of event.
Three progressively refining SGC settings were utilize to capture differential change in temperature of 0.5, 0.05, 0.005 K. It was observed that as smaller and smaller differential changes are targeted, the cell count vastly increases. Furthermore, the spikes in the cell count as represetatives of manifestations of abrupt changes in states due to the manifesations of Mach waves. As the mesh becomes more refined, the location of the shocks moves from vicinity of the expansion vertex to the exact incidence of the expansion vertex.
0.5
Figure 1: Mesh for SGC 0.5 K
Figure 2: Cell Count for SGC 0.5 K
Figure 3: Temperature Contour for SGC 0.5 K
0.05
Figure 4: Mesh for SGC 0.05 K
Figure 5: Cell Count for SGS 0.05 K
Figure 6: Temperature Contour for SGC 0.05 K
0.005
Figure 7: Mesh for SGC 0.005 K
FIgure 8: Cell Count for SGC 0.005 K
Figure 9: Temperature Contour for SGC 0.005 K
Data and Analysis:
Isentropic expansion waves manifest when supersonic flow turns away from itself. It is characterizd by the expected drop in static pressure, temperature, and density across the shock. The Mach number and velocity increase accross shock and this is characteristic of expansion Mach waves. The total pressure remains identical at the inlet and outlet, characteristic of isentropic flow.
1. Pressure
Figure 10: Total Pressure Plot
Figure 11: Pressure Contour
2. Temperature
Animation 1: Tempearture Contour for Prandlt Meyer Expansion
3. Density
Figure 12: Density Contour
4. Mach
Figure 13: Mach Plot
5. Velocity
Figure 14: Velocity Contour
Intial Condition 100 m/s :
For the simulation with initial interanl flow condition of 100 m/s, we observe these conditions prevalent while flow is not yet established. Once flow is established and completes first cycle, the intial conditions get washed away and flow conditions as per the Boundary Conditions become prevalent. This can be seen in the velocity contours for cycle step 0 and 1, respectively.
Figure 15: Velocity Contour at Step 0
Figure 16: Velocity Contou at Step 1