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Objective To simulate the Prandtl Meyer shock flow and analyse the effects of various sub grid criteria sizes on the results of subsonic and supersonic inlet velocities. Shock wave A shock wave is a type of propogating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave,…
Prashanth Barathan
updated on 15 Feb 2022
Objective
To simulate the Prandtl Meyer shock flow and analyse the effects of various sub grid criteria sizes on the results of subsonic and supersonic inlet velocities.
Shock wave
A shock wave is a type of propogating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy that can propogate through a medium, but is characterized by an abrupt, nearly discontinuous change in pressure, temperature and density of the medium.
Shock flow boundary conditions
2.1 Dirichlet boundary condition
In the Dirichlet boundary condition, the value of the function is given at the boundary.
y(x)=f(x)
2.2 Neumann boundary condition
In the Neumann BC, the value of a normal derivative of the function is specified.
∂y∂n=f(x)
2.3 Robin boundary condition
It is a weighted combination of the Dirichlet BC and the Neumann BC.
c1y+c2∂y∂n=f
where c1 and c2 are constants.
2.4 Mixed boundary conditions
In mixed BC, the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated.
Reason for Neumann BC at outlet
For a supersonic outlet, the characteristic curves propogate outside the domain, along with the streamline. Hence the values of the variable need to be floated, ie, we calculat eteh values using linear extrapolation. Hence, we use the zero normal gradient as the boundary condition for pressure and velocity.
Solution
Prandtl Meyer Shock
When an object moves faster than the speed of sound, and there is an abrupt decrease in the flow area, shock waves are generated. If the flow area increases however, a different flow phenomenon is observed. If the increase is abrupt, we encounter a centred 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. As the flow across a shock is highly irreversible, it cannot be approximated as isentropic. Through an expansion fan, the Mach number increases, the static pressure decreases, and the total pressure remains constant. Expansion fans are isentropic.
A Prandtl-Meyer expansion wave centred at the point of expansion corner is illustrated in the figure above. An expansion wave which is made up of several Mach waves fans out from the corner. The leading edge of the expansion fan makes an angle of μ1 with the upstream flow direction and the trailing edge of the wave makes an angle μ2 with respect to the downstream flow direction. These angles are called Mach angles and are defined as
μ1=sin−1(1M1);μ2=sin−1(1M2)
where M1 and M2 are upstream and downstream Mach numbers respectively. As the flow passes through the expansion wave, the Mach number increases, and pressure, temperature and density decrease. The flow upstream and downstream are always parallel to the wall.
Geometry setup
The stl file of the geometry is imported into converge studio and scaled to convert units from metres to millimetres.
After using the diagnosis tool to ensure correctness of geometry, the case can be setup for simulation.
A time based simulation is selected with the material as Air.
The values for the global transport parameters and reaction mechanism are kept default.
Solver setup
A steady state, full hydrodynamic compressible solver is selected as shown below. The steady state monitor is turned on, and the use of shared memory is unchecked.
The steady state monitor can be defined as shown below
Simulation time parameters
An end time of 25000 cycles is selected with minimum time step of 1e-7 s and max time step of 1.0 s. The convection CFL limit it set as 1.0. The values of the other parameters are maintained default.
A Navier Stokes density based solver is selected, and under steady state solver control, the maximum CFL limit is given as 1.0 to reflect the value given under the simulation time parameters.
Boundary conditions and initialization
The sides of the geometry are flagged with the appropriate boundary, and assigned relevant boundary conditions.
The initial pressure is given as 101000 Pa, velocity as 680 m/s and the temperature as 286 K.
Inlet
The pressure at the inlet is defined with Neumann BC - Zero gradient, with INFLOW boundary type and air as medium. The velocity is 680 m/s, which is a mach number of 2.0
Outlet
The pressure and velocity at the outlet are defined with Neumann BC - Zero normal gradient, with Outflow boundary type.
Wall
The walls are defined with law of wall - slip condition and zero normal gradient for the TKE.
The top and bottom faces are treated as 2D faces.
Physical models
The RANS K-Epsilon model is selected as the turbulence model, with the values for the constants as shown below.
Grid size
The base grid is defined with the required size along all three axes. The size here is 0.8.
Adaptive mesh refinement with the required sub-grid criteria can be defined, for refining the mesh, when the mach wave travels over the face.
The output and post processing parameters are kept default, and the setup is completed for simulation.
Four different simulations are carried out using Cygwin, and the results are illustrated below.
Results
Mass flow rate
Pressure
Temperature
Velocity
Mach number
Density
Number of cells
Paraview
Mesh
Pressure contour
Velocity contour
Temperature contour
Density contour
Animation pressure
Animation velocity
Mass flow rate
Pressure
Temperature
Velocity
Mach number
Density
Number of cells
Paraview
Mesh
Pressure contour
Velocity contour
Temperature contour
Density contour
Animation pressure
Animation velocity
Mass flow rate
Pressure
Temperature
Velocity
Mach number
Density
Number of cells
Paraview
Mesh
Pressure contour
Velocity contour
Temperature contour
Density contour
Animation pressure
Animation velocity
Observations
The pressure, temperature and density decrease across the Mach waves. However, the velocity increases, reaching a maximum Mach number of 2.15, which is approximately 730 m/s.
The temperature sub grid of 0.1 is not able to capture the physics exactly, since the values of pressure and density are different from expected results.
Mass flow rate
Pressure
Temperature
Velocity
Mach number
Density
Number of cells
Paraview
Mesh
Pressure contour
Velocity contour
Temperature contour
Density contour
Animation pressure
Animation velocity
Observation
In the case of subsonic inlet, there is no formation of Mach waves around the expansion corner as expected. Since the cross sectional area increases past the corner, the flow velocity close to the corner increases rapidly locally, thus conserving the mass - since the mass flow rates will be equal at the inlet and outlet.
However the velocity decreases again downstream, following Bernoulli's equation. There is a local reduction in pressure at the corner due to the local acceleration. At the outlet, we get an increase in pressure and velocity. The temperature decreases along the domain.
Comparison
Number of cells - Supersonic flow
As the SGS size decreases, the number of cells increases (ie) the mesh becomes finer. The physics of the flow are best captured however, with an SGS of 0.01, as can be seen in the mesh.
Shock location - Supersonic flow
SGS - 0.1
SGS - 0.05
SGS - 0.01
By looking at the temperature distribution of all three sub grid sizes, it is clear that the expansion fan tends to become less narrow and distinct at the SGS parameter increases. This is because the adaptive mesh refinement value is dependent on the value of sub grid scale.
Higher values of SGS limits the mesh refinement required to capture the expansion waves effectively, leading to a blur and not well-defined waves.
Conclusion
Thus the Prandtl Meyer shock flow for supersonic and subsonic velocities have been simulated, and the effect of the temperature SGS on the parameters have been illustrated.
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