Prandtl Meyer Shock problem

Objectives :-

• What BC\'s are typically used for shock flow problems

• What is a shock wave?

• Describe the physics behind shock waves.

• Show the effect of SGS temperature value on cell count and the shock location

• Plots for velocity, pressure,mass flow rate and total cell count.

What BC\'s are typically used for shock flow problem

Generally three types of BC\'s are used to solve CFD problems :-

Dirichlet/Fixed BC\'s

• The Dirichlet boundary condition is a type of boundary condiion, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordiary or a partial differential equation, it specifies the values that a solution needs to take on along the boundary of the domain.

Neumann BC\'s

• The Neumann boundary condition is a type of boundary condiion, named after Carl Neumann. When imposed on an ordiary or a partial differential equation, the condition specifies the values in which the deravitive of a solution is applied within the boundary of the domain.

Mixed BC\'s

• A mixed boundary condition for a PDE defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary condition on disjoint parts of the boundary of the domain where the condition is stated. Precisely, in a mixed boundary value problem, the solution is required to satisfy a Dirichlet or a Neumann bounday condition in a mutually exclusive way on disjoint parts of the boundary.

What is a shock wave?

• Sound waves are caused by infinitesimally small pressure disturbances, and travel through a medium at the speed of sound.
• Under certain flow conditions, abrupt changes in fluid properties occur across a very thin section: shock wave.
• Shock waves are characteristic of supersonic flows, that is when the fluid velocity is greater than the speed of sound.
• Flow across a shock is highly irreversible and cannot be approximated as isentropic.

Shock waves can be:

Normal

• At 90° (perpendicular) to the shock medium\'s flow direction.

Oblique

• At an angle to the direction of flow.

Bow

• Occurs upstream of the front (bow) of a blunt object when the upstream flow velocity exceeds Mach 1.

Physics behind the shock waves

In physics, a shock wave, 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, temperatutre, and density of the medium.

Prandtl Meyer shock problem :-

A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, a two-dimensional simple waves, 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.

Case Setup :-

• Import the stl file format of elbow and placed it into converge studio.
• Give the proper boundary condition.
• Simulation done for 25,000 cycles.
• Using density based steady solver.
• Base grid size 0.8 m.
• In this case we use Adaptive mesh refinement

Inputs :-

• Air as fluid flowing through the duct.

Boundary condition :-

For this case we use Neumann boundary condition

For Inlet :-

• Pressure - Zero normal gradient(NE)
• velocity - 680 m/s in x direction
• Temperature - 286.1 K

For Outlet :-

• Pressure - Zero normal gradient(NE)
• velocity - Zero normal gradient(NE)
• Temperature backflow - 300 K

Front and back Face :-

• It behave like a 2D face.

Top and bottom wall :-

• In this case it behave as a slip wall.

Initial condition for the problem :-

case I :- Input velocity 100 m/s (subsonic flow)

Case II :- Input velocity 680 m/s (supersonic flow)

3D model for the simulation :-

Case I :- Input velocity 100 m/s (subsonic flow)

Fig 1:- Base grid (Normal)

Fig 2:- Adaptive mesh refinement

Animation 1:- Velocity contour for 0 to 25000 cycle

Animation 2:- Pressure contour for 0 to 25000 cycle

Animation 3:- Temperature changes in between 0 to 25000 cycle

Fig 3:- Mach Number

Note :- bound_id_1 represent Inflow boundary and bound_id_3 represents Outflow boundary.

Fig 4:- Total pressure

Fig 5:- Density

Fig 6:- Temperature

Fig 7:- Total Cells count

Case II :- Input velocity 680 m/s (supersonic flow)

a) Sub grid scale Temperature 0.05 K

Animation 4:- Velocity contour for 0 to 25000 cycle

Animation 5:- Pressure contour for 0 to 25000 cycle

Animation 6:- Temperature changes in between 0 to 25000 cycle

Fig 8:- Mach number

Fig 9:- Total pressure

Fig 10:- Density

Fig 11:- Temperature

Fig 12:- Total Cells count

b) Sub grid scale Temperature 0.03 K

Animation 7:- Velocity contour for 0 to 25000 cycle

Animation 8:- Pressure contour for 0 to 25000 cycle

Animation 9:- Temperature changes in between 0 to 25000 cycle

Fig 13:- Mach number

Fig 14:- Total pressure

Fig 15:- Density

Fig 16:- Temperature

Fig 17:- Total Cells count

c) Sub grid scale Temperature 0.1 K

Animation 10:- Velocity contour for 0 to 25000 cycle

Animation 11:- Pressure contour for 0 to 25000 cycle

Animation 12:- Temperature changes in between 0 to 25000 cycle

Fig 18:- Mach number

Fig 19:- Total pressure

Fig 20:- Density

Fig 21:- Temperature

Fig 22:- Total Cells count

Result and Conclusion :-

• As we shown above for the sub sonic flow, there will no any shock wave generated because shock wave only genereted if the fluid velocity is greater than the speed of sound.
• In case ii for super-sonic flow, shock wave started as the area suddenly increases because in super sonic flow velocity is greater than the speed of sound.
• In Animaion 1,4,7,10 shows how velocity changes if shock wave occured.
• In Prandtl meyer shock wave velociy and Mach number increases, and it will shown in fig 8,13,18.
• While in Fig 9,10,11,14,15,16,19,20,21 shows for the pressure,density and temperature and all are decreases.
• So, by the above two points it is proved that in the Prandtl meyer shock wave problem Mach number increases, while the static pressure, temperature and density decrease.
• As shown in Fig 12,17,22 after 5,000 cycle total cell started increasing because we seteup the adeptive mesh refinement after 5,000 cycle.
• By observing the above for the sub grid scale temp 0.05K the total cell is about 7,000, for the sub grid scale temp 0.03K is about 10,000 and for the sub grid scale temp 0.1K the total cell is about 5,200.
• So, if the subgrid scale temperature is less refinement takes place in more area. so the total cell increases.