Prandtl Meyer Shock flow
Prandtl Meyer Shock flow
Objective:
- To do a literature survey on shock flows and shock flow boundary conditions.
- To explain the physics behind the shock waves
- To study the effect of SGS parameter (SGS temperature value) on cell count and the shock location
Introduction:
Shock waves are the sound waves produced by a medium due to sudden dissipation of mechanical energy in a medium enclosed in a small space.
Any fluid that travels at supersonic speeds, give rise to a shock wave. Shock waves are produced during earthquakes and lightning. Shock wave cannot be heard, but we can hear a booming sound, when velocity of a body increases from subsonic to supersonic. It is called ‘sonic boom’. When a shock wave is formed, there is a distinct surface created by the medium itself called shock front. Typical thickness of a shock front is few micrometres. Within the shock front, the pressure, temperature and density of the gas undergoes sudden changes.
The conservation of mass, momentum and energy are the three fundamental principles of classical physics.
- Law of conservation of mass: “The total mass of any isolated system remains unchanged and is independent of any chemical and physical changes that could occur within the system”. It is also sometimes referred to as conservation of matter.
- Law of conservation of momentum: “In a closed system, the total momentum remains a constant”. It can also be stated as, “when two objects collide in an isolated system, the total momentum of the objects before collision = the total momentum of the two objects after the collision”.
- Law of conservation of energy: “The total energy of a closed system remains constant and is independent of any changes occurring within the system”.
A shock wave propagation can be characterized by change in pressure, temperature and density of medium. Simulation of such a shockwave was carried out to analyse its behaviour.
A supersonic expansion fan, also known as Prandtl Meyer expansion fan is a centred expansion fan that occurs when a supersonic flow turns around a convex corner. This fan consists of an infinite number of Mach waves diverging from a sharp point.
Figure 2 Shock wave
The boundary conditions used for the shock flow problems are:
- Dirichlet BC
- Neumann BC
- Robin BC
Dirichlet boundary condition is given as a specified value as:
đťś™=f
Neumann boundary condition is given as a specified value of the first-derivative as:
Robin boundary condition is a mix of Dirichlet and Neumann boundary conditions given as:
where Ď• is a solved quantity (pressure, energy, velocity or species)
f is the specified derivative on the boundary
As in a shock flow, the downstream flow variables are highly dependent on the upstream flow variables, thus they are calculated by the Neumann boundary conditions with a zero gradient
Geometry
Figure 2 Profile geometry for shock flow
A predefined air mixture containing Nitrogen and Oxygen with 0.77 and 0.23 concentrations is used.
Three different cases were discussed here as mentioned below:
- Supersonic flow with sub-grid criterion 0.03K
- Supersonic flow with sub-grid criterion 0.01K
- Subsonic flow with sub-grid criterion 0.01K
A density based steady state solver is used to obtain the pressure field from the equation of state, with the continuity equation providing the density. Such solver holds good for high speed flows with Mach number greater than 0.3 in order to account for the effects of compressibility.
As the solution proceeds to be solved, the regions requiring more resolution are identified and adaptive mesh refinement plays its role here. Finer and finer sub-grids are added until a given maximum level of refinement is reached. Therefore, in adaptive mesh refinement, computation grid size is fixed for base grid only and is determined locally for sub-grids according to the requirement of the problem.
For the supersonic criterion, a velocity component in x direction as 680 m/s was used as initial boundary condition.
For the subsonic criterion, a velocity component in x direction as 100 m/s was used as initial boundary condition.
A base grid size of 8e-3 metres was applied to the geometry.
dx = dy = dz = 0.008 metres
Figure 3 Temperature contour with sgc 0.03 K (supersonic)
Figure 4 Temperature contour with sgc 0.01 K (supersonic)
Figure 5 Temperature contour with sgc 0.01 K (subsonic)
As seen in Figure 5, no mach waves were observed because the flow is subsonic in nature.
Figure 6 Velocity contour with sgc 0.03 K (supersonic)
Figure 7 Velocity contour with sgc 0.01 K (supersonic)
Figure 8 Velocity contour with sgc 0.01 K (subsonic)
Figure 9 Average Mach with sgc 0.03 K (supersonic)
Figure 10 Average Mach with sgc 0.01 K (supersonic)
Conclusions
Adaptive Mesh Refinement is an important tool to capture the complex phenomena such as shockwaves effectively with the use of sub-grid criterion based on temperature or velocity. The value of sub-grid criterion determines the level of detail to be captured with change of variables such as temperature or velocity based on the requirement of the problem.
As the value of sub-grid criterion was reduced from 0.03 K until 0.01 K based on temperature, more level of details were captured into the contours as observed from the results. The SGS temperature value of 0.01 was found to be ideal for capturing the flow generated over the wedge.
Animation links
https://youtu.be/ciTTkD1EDJY