Prandtl Meyer Shock problem   (Converge CFD, Skill Lync, Medium)
Aim:
To simulate Prandtl Meyer expansion fan problem using Converge CFD.
Introduction:
When a fluid flowing at supersonic velocity passes over a convex corner of surface, it produces a discontinuity in flow properties which is called the expansion fan. The fluid is flowing at so high speed that it tries to not follow the surface curve and break off. It is not possible for flow to seperate directly so it expands suddenly and this causes shockwaves of different mach numbers spreading out like a fan. Expansion fans are isentropic processes that generate continuous and smooth changes in the flow, causing the flow’s total properties to be conserved. The shock wave created is a type of propagating disturbance that moves faster than the local speed of sound in the medium. The shockwave due supersonic vehicles are created because the pressure variation created at local point is more strong and faster than its neighbouring points.
The use of this problem in CFD domain is for validation purposes. The results from inviscid compressible fluid theory are used for comparision of results from our CFD code.
For these kind of supersonic flows, the pressure values are not defined at boundaries. We rather provide zero gradient or Neumann boundary condition at the inlet boundary. The pressure values are extrapolated to boundaries from internal regions. The region is initialized with proper velocity and pressure so as to not cause solution instability when the simulation starts. In this report, the region is initialized with air at 50000 Pa and 680 m/s. So, when the simulation would start, the solver would have a better initial guess. Also, at the outlet, the pressure condition is defined as zero gradient boundary. For this report, following geometry was used for 2D analysis.
Dimensions are: (in x, y, z): 65 * 45.1525 * 8.151 metres
Following are the details of simulation setup:
Mesh Settings:
Global element size in all directions = 0.8 m
Adaptive mesh refinement settings:
Minimum cells: 1
Maximum cells: 200000
Refinement based on: Temperature
Max Embedding level: 2
Sub grid criteria: 0.05 and 0.01 (Two cases)
Start at: 5000 cycles
Refinement cycle = 100 cycles
Solver: Steady state, compressible, density based, PISO solver with step flux limiter.
Cycle settings:
Minimum cycles = 0
Maximum cycles = 25000
Starting time step = 1e-9
Minimum time step = 1e-9
Maximum time step = 1
Max Convective CFL = 1
Boundary conditions:
Inlet: Pressure condition as zero normal gradient, velocity of 680 m/s in x direction, temperature of 286.1 K
Outlet: Pressure condition as zero normal gradient
Top and bottom walls: Slip wall
Front and back walls: 2D flow condition.
Region initial settings: Pressure at 50000 Pa, Velocity of 680 m/s in x direction, temperature of 300 K.
Turbulence model: RNG k-e model with standard wall functions.
3D Output files generated at every 1000 cycles.
(The adaptive cell refinement creates cells based on sub grid criteria. It determines the location of refinement based curvature (y'') of temperature field. The Converge CFD manual states that the algorithm calculates the solution on base grid and then on refined grid and calculates the difference in values. If the values are less than sub grid criteria mentioned by user, the cells are removed. These calculations take place every after refinement cycle time as mentioned by user. In this report, two values of subgrid criteria are used inorder to understand its effect on grid. The region and inlet are initialized with same values of velocity inorder to provide smooth start. The walls are having slip conditions inorder to simulate inviscid flow.)
Results:
1) Mesh:
The two images above show mesh at final cycle. The left one is of Subgrid criteria of 0.05 and right one is of 0.01. We can clearly see the difference that refinement is more in right image. The embed level is 2, so we can see cells of size 0.8, 0.4 and 0.2 m. The embed level deciedes how much times should a cell be divided into. The following video shows development of mesh.
2) Total cells
The plot shows total number of cells generated for both cases. The AMR started at 5000 cycles as specified and we can see steps of 100 cycles between refinements. The cell number also rose to a high number in second case of SGS = 0.01.
3) Pressure:
If we look at a particular case, we can say that the pressure suddenly drops in the expansion fan region. But when we compare the pressure for both cases. The case with SGS = 0.01 has lower pressure than other case. The reason for this can be the cell size. The finer mesh takes more time to converge. So it might come to a same value after few more iterations, but I think it won't stop increasing. The Finer mesh gives accurate contours in expansion fan.
4) Velocity
The velocity at outlet is more than inlet. The velocity variation poorly captured in coarse grid compared to fine grid. Looking at the plot, the velocities in both case are equal.
The vectors show that the flow below the expansion fan follows the downward curve of wall and the flow above the expansion fan moves in straight line.
5) Temperature:
The temperature of air lowers down after expansion. The finer grid gives better variation of expansion fan. The Temperature difference is also the same for both grids. The following video shows temperature contour animation.
6) Mass Flow Rate:
Similar to pressure, the mass flow rate also evolves slowly in finer mesh and would require more iterations to come to same level as case 1.
Conclusion:
Simulation of Prandtl Meyer expansion fan problem is done using Converge CFD and the effect of SG criteria is also studied.