Prandtl Mayer Shock Simulation

Shock Wave: The speed of a moving body if exceeds the sound barrier i.e the velocity of the sound, the wavefronts formed will be lagging behind the body and create a cone like structure. The edge of cone will form wavefront with large amplitutdes. It is called Shock Wave. There will be sudden change in pressure, temperature and velocity of the fluid.

When a supersonic flow is turned around a convex corner the mach cone having wave fronts will diverge from the the sharp corner forms an expansion fan containing expansion waves. This is known as Prandtl Mayer Expansion Fan. Also observed from the results that the total pressure, total temperature remain same in the expansion zone but the static temperature and pressure change.

Boundary Conditions for Shock Flow: For solving shock flow problems, Boundary condition can be Dirichlet which is the specified value or Neumann which is specified value of first derivative and lastly mixed boundary conditions. If Normal Neumann boundary condition is given, it constrains the direction of it to the normal of the boundary.

Here a density based steady-state solver is used in this case as the velocities involved have Mach numbers equal and above 2, higher than 0.3 which is normally suitable for pressure based solver. As the Neumann condition is used, a specified value of pressure value is not mentioned as inlet condition but a velocity in x-direction is given. But in application density based solvers are less efficient than pressure based.

Geometry: Maximum values in the x,y and z directions shown using the bounding box are I_x: 65, I_y: 45.15 and I_z: 8.15

Boundary Conditions:

Inlet-286.1 K, Velocity in x-axis is 680 m/s (Mach number - 2)

Outlet- Neumann Boundary Condition

Top and Bottom- Wall boundary type with \'Slip\' condition.

Front and Back- TWO-D

Adaptive Mesh Refinement- Temperature based mesh refinement used which changes with the change in temperature for three different cases of Sub-Grid Criterion (SGS) say 0.1, 0.05 and 0.01 with Max.Embedding Level of 2 are attempted in this project. So the grid size that is least observed would be 0.2m.

Initial Pressure of 0.5 bar, Base Grid Size- 0.8m and Total of 25000 cycles.

Case 1- SGS 0.05 and Initial Velocity of 680 m/s

Fig 1: A adaptive mesh for SGS 0.05 shown with surface edges of the duct

A more amount of refinement is observed at an angle i.e obliquely at an angle to the convex corner of the geometry.

Fig 2: A Total Cells of 7288 for case 1

Fig 3: Velocity observed for case 1 showing maximum at the sharp corner

For the fully converged solution, the peak velocity at the corner of the duct is observed as more tuned and is not more than value of 740 m/s i.e around Mach number of 2.15,. The exepansion fan zone has slighter high velocity than the remaining zone as observed and the waves caused due to the corner separates these two zones.

Fig 4: Temperature profile observed for case 1

Fig 5: Pressure profile observed for case 1

Case 2- SGS 0.01 and Initial Velocity of 680 m/s

Fig 6: A fine mesh observed if the SGS is set to 0.01 along the shock waves path from corner

Fig 7: The observed cell count is around 12030 representing a good value for a size of this geometry

The SGS 0.01 being less than the value in case 1 resulted in a finer adaptive meshing at the critical zone i.e where the changes in temperature and velocity are most observed for the case of prandtl mayer shock wave study and thus resulted in more number of cells.

Fig 8: Velocity profile for case 2 showing fine changes in the values of velocities

Fig 9 : Average Mach Number Plot

The simulations having input of velocity 680 m/s i.e. Mach number 2 initially, changes due to the shape of geometry resulting in slight increase to a value of 2.15 on an average after the solution converges at the outlet boundary region.

Fig 10: Temperature Profile for case 2 showing maximum of 290K

Case 3: SGS 0.1 and Initial Velocity of 100 m/s

Fig 11:

Fig 12:

Fig 13: A sudden change in Mach Number observed in the beginning due to subsonic velocity (100 m/s) as initial condition

Fig 14: Velocity profile of case 3

Fig 15: Pressure Profile of case 3

From aforementioned observations, with decrease in the value of SGS, the solution converges to yield better results with more number of cells. Also, the temperature observed at the locations is far more precise with more cells all though out the region of shock wave separating regions of expansion fan and remaining. The SGS 0.01 gives more accurate values of temperature, velocity and pressure at the outlet region also due to distribution of cells undergone during mesh refinement compared to SGS 0.1.