Prandtl Meyer Shock Problem
Objective:
To visualize Prandtl Meyer expansion over an expansion corner using converge and understand the effect of SGS parameter on the shock location.
Introduction:
Shock Wave:
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a medium but is characterized by an abrupt, nearly discontinuous, change in pressure, temperature, and density of the medium.For the purpose of comparison, in supersonic flows, additional increased expansion may be achieved through an expansion fan, also known as a Prandtl–Meyer expansion fan. The accompanying expansion wave may approach and eventually collide and recombine with the shock wave, creating a process of destructive interference. The sonic boom associated with the passage of a supersonic aircraft is a type of sound wave produced by constructive interference.Unlike solitons (another kind of nonlinear wave), the energy and speed of a shock wave alone dissipates relatively quickly with distance. When a shock wave passes through matter, energy is preserved but entropy increases. This change in the matter\'s properties manifests itself as a decrease in the energy which can be extracted as work, and as a drag force on supersonic objects; shock waves are strongly irreversible processes.
Prandtl Meyer Expansion Fan:
A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, a two-dimensional simple wave, is a centered 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 because this would violate the second law of thermodynamics.Across the expansion fan, the flow accelerates (velocity increases) and the Mach number increases, while the static pressure, temperature and density decrease. Since the process is isentropic, the stagnation properties (e.g. the total pressure and total temperature) remain constant across the fan.
Types of Boundary Conditions:
- Dirichlet boundary condition: This condition specifies the value that the unknown function needs to take on along the boundary of the domain.
- Neumann boundary condition: When imposed on an ordinary or a partial differential equation, it specifies the values that the derivative of a solution is going to take on the boundary of the domain.
- Robin boundary condition: It consists of a linear combination of the values of the field and its derivatives on the boundary.
Shock Flow Boundary Conditions:
The boundary conditions which is typically used for shock flow problems is Neumann Boundary Condition. When imposed on an ordinary or a partial differential equation, the condition specifies the values in which the derivative of a solution is applied within the boundary of the domain. Here the inlet flow is supersonic and the outlet flow can be decided by the amount of Mach waves accompanying the shocks. So the outlet BC is based on the inlet BC. So Neumann Boundary Condition is used for shock flow problems.
Geometry:
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.
Boundary Conditions:
- Case 1
Inlet:
Velocity = 680 m/s in x direction
Temperature= 286.1 K
Exit:
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:
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.
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.
Results:
For Inlet velocity=680 m/s and SGS = 0.05
- Temperature Contour
- Velocity Contour
Flow Property Variations at Inlet and Outlet:
- Mach Number
- Density
- Temperature
- Pressure
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.
Boundary Conditions:
- Case 2
Inlet:
Velocity = 100 m/s in x direction
Temperature= 286.1 K
Exit:
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:
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.
Results:
For Inlet velocity=100 m/s and SGS = 0.05
- Tempurature Contour
- Velocity Contour
From the above figure we can infer that, when the subsonic initial condition (100 m/s) is set for the supersonic flow (Mach = 2) the solution will not converge. The difference between the Initial Condition and the Boundary Condition leads to an ambiguous condition and the solution will diverge.
Flow Property Variations at Inlet and Outlet:
- Mach Number
- Density
- Tempurature
- Pressure
From the above graphs, it can be observed that Mach number of the subsonic flow increases as the flow passes through the expansion wave whereas properties like temperature, pressure and density tends to decrease.
Sub Grade Scale(SGS) Parameter:
The flow simulation was carried out for different SGS parameters i.e. 0.05,0.07,0.10,0.20 and the results were determined.
Effect on Expansion Location
Below are the temperature contours for different SGS values.
- SGS = 0.07
- SGS = 0.10
- SGS = 0.20
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 = 0.20, the refinement criteria was so high that no adaptive mesh refinement was achieved.
Conclusion:
The Prandtl-Meyer Shock flow problem is one of the most important simulation test case in the CFD domain. In this project the shocks are observed with various SGS criterion and the temperature contour with lowest SGS value is finer than the others. Another major case was observed that, the subsonic initial condition for the supersonic flow will not converge.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 effect the airflow around the aircraft.