Simulation of Prandtl Meyer Expansion Fan
Objective:To simulate a Prandtl Meyer shock problem and study the effect of SGS temperature value on cell count and Shock Location
Introduction:
When an object moves through gases, the molecules are deflected around the object. If the speed is less than the speed of sound,density remains constant. However when the flow is supersonic, the gas density will vary locally as the gas gets compressed. In such a flow when the flow area abruptly changes, the phenomenon of shocks occurs.
When the flow area decreases suddenly, shock waves are generated and when the flow area increases suddenly, an expansion fan is generated.
Shock can be classified as Normal Shocks and Oblique Shocks.
Theory:
Oblique Shock Waves:
When the fluid flowing at supersonic speeds flows over a wedge with a half wedge angle `delta` , as it cannot flow through the wedge, the fluid turns around and results in a shock wave which is called the oblique shock wave as it is alligned at an angle to the initial flow direction.
The flow turns through an angle`theta` and results in an oblique shock wacve at an angle `beta`.
For Oblique shocks the downstream flow can be subsonic, sonic or supersonic depending on the inlet Mach Number and Turning angle. (The boundary layer effects are neglected along the wedge)
Prandtl Meyers Expansion Fan:
The flow that produces an oblique Shock wave is called a compressing flow.When the flow changes direction to conserve mass but does not form a shock wave ie when the area of flow suddenly increases. An expansion fan appears in suck cases. The expansion fan consists of an infinite number of Mach waves. Instead of turning suddenly the flow gradually changes with each Mach wave by moving an infinitesimal amount.
The flow in the entire fan is isentropic. The behaviour of the flow is similar to that in a supersonic convergent divergent nozzle.
The turning angle theta `theta=nu(M_1)-nu(M_2)`
where`nu(M)` is called the Prandtl Meyer Function defined as follows:
`nu(M)=sqrt((k+1)/(k-1))tan^-1(sqrt((k-1)/(k+1))(M^2-1))-tan^-1(sqrt(M^2-1))`
The physical meaning is the angle through which the supersonic flow has to be expanded in order to obtain a given Mach Number at the downstream end.
Boundary Conditions for Supersonic Flow:
To obtain a unique solution to any problem, flow variables are defined at the domain boundaries.Boundary conditions exist in the form of mathematical equations.There are 5 types of boundary conditions, Dirchlet, Neumann, Robin, Mixed and Cauchy conditions.The Dirchlet and Neumann conditions are most used.
Dirchlet Boundary Condition:
The Dirchlet boundary condition is defined for the transport property,by specifying the physical quantity over the boundary or the profile of the physical quantity can be defined.
`phi=f(x)`
Neumann Boundary Condition:
The Neumann Boundary condition, prescribes the derivative of the physical quantity at the boundary.
`(delphi)/(deln)=0`
For Supersonic flow,
the Dirchlet boundary condition is applied to the inlet.
The outflow in supersonic flow is not influenced by downstream conditions. The static pressure cannot be defined in the supersonic cases at the outlet and hence, all the properties are extrapolated from the upstream values.
Case Setup:
The simulation is setup using CONVERGE Studio. The geometry is as follows:
The boundaries are flagged as Inlet, Outlet, Wall, Front and Back.
Boundary Conditions:
Inlet:
Inflow Boundary is chosen with application of Dirchlet Boundary condition.
Two cases of Inlet Flow are simulated. Subsonic Inlet flow with a velocity of 100m/s and Supersonic Inlet flow with velovity 680m/s.
Inlet Temperature is 286.1K. Speed of sound at this temperature is 340m/s.
For the Supersonic Case the inlet Mach Number is 2 and for the subsonic case the inlet Mack Number is 0.3
Zero Normal Gradient for Pressure
Outlet:
Zero Normal Gradient for both Pressure and Temperature
Wall:
Slip wall is used.
Front and Back Faces:
Turbulence Model: RNG k-epsilon model
Initial Conditions:
Air is used for simulation with the velocity of 100m/s for subsonic and for supersonic case the velocity is 680m/s
Simulation Run Time:
A steady state simulation is carried out using the density based solver for 25000 cycles.
Grid Size:
Base grid size dx=dy=dz=0.8
AMR refinement is applied.
AMR Refinement:
The mesh changes in places where the selected property changes its value beyond the specified value.
In this case Temperature based AMR is used. The curvature in temperature is monitored and the mesh is changed accordingly.
Subgrid Scale Temperature:
When the curvature of temperature can be said to be the subgrid scale temperature. A value is specified and when the subgrid scale temperature is greater than the specified value the mesh is refined.
Different values of SGS Temperatures are implemented and the influence on the cell count is studied. The number of levels of refinement is set to 2.
Results:
Subsonic Inlet Flow (100m/s):
The pressure is lowest near the expansion point and the velocity is maximum.The velocity further reduces at the lower end of the duct.This is because the flow area has increased.
Evolution of Temperature Contours:
The temperature gradually reduces as the flow proceeds through the tube.
The cell count can be observed to be constant.
Supersonic Flow Inlet Velocity 680m/s:
Case 1: No AMR
The pressure and velocity contours both show the formation of the expansion fan at the expansion point.No AMR is applied in this case and hence the cell count is constant throughout the simulation.
The following cases show the formation of the expansion. However AMR refinement is employed in these cases with different vales of SGS temperature.The results are all similar for the pressure and velocity profiles. The cell count however can be observed to be variable
Case 2:SGS=0.01
Case 3:SGS=0.025
Case 4:SGS=0.05
Case 5:SGS=0.075
Case 6:SGS=0.1
The evolution of the temperature gradient for all the cases can be observed for all the cases. The variation of the mesh with temperature can be observed. The final temperature for all the cases can be observed in the figure below.
From the above temperature gradients it can be noted that the width of the expansion fan slightly changes with the refinement applied. The higher the SGS parameter the more spread out the fan seems. This is becaus ethe grid is not refined. At lower values of SGS parameter the grid is more refined at the points of formation of the expansion wave and hence the accuracy is greater.
The below table shows the cell count for different SGS temperature. As the velue of the SGS temperature reduces, the cell count can be observed to increases. This is because the mesh gets refined for even small changes in curvature of the temperature for lower values of SGS temperature and thus increases the cell count.