Prandtl Meyer Shock problem

SHOCK WAVE:

• A shock is a compression wave or thin region across which large gradients of the pressure, temperature, velocity occurs and also the transport phenomena of momentum and energy is considered important.
• Flow consists of large number of fluid molecules in unit volume and the transport of mass, momentum and energy occur due to the movement of these molecules. This is called as kinetic theory.

PHYSICS BEHIND SHOCKS:

Let V∞ represent molecules and a∞ represent the traces in the figures depicted above with varying velocities.

• When the flow is subsonic (a) the signals pass the information to the molecules upstream about the presence of the cylinder before hand so that they can align themselves to flow around the cylinder.
• When the flow is supersonic (b) signals cannot pass information sooner to the molecules as they travel at high speeds. The reflected signals all combine together at short distance to create a shock as shown in fig(b). Shocks tend to be discontintuities in flow that never go across the cylinder.
• The streamlines behind the shock compensate for the obstruction by a detached shock before the cylinder body where flow becomes subsonic. The formation of the shock takes place after the fluid molecules impinge on a surface and rebound.
• Taking an example from an acoustic point of view only after the source passes the point the sound waves can be heard by the point. i.e. an aircraft breaking the sound barrier.

BOUNDARY CONDITIONS:

A boundary value problem is a differential equation put together with additional constraints called the boundary conditions. The various types of boundary conditions are:

DIRICHLET BOUNDARY CONDITION:

When these conditions are imposed on an ODE or PDE, it specifies the values the solution needs to take on along the boundary of the domain.

NEUMANN BOUNDARY CONDITION:

When these conditions are imposed on an ODE or PDE, it specifies the values the derivative of the solution needs to take on the boundary of the domain.

CAUCHY AND ROBIN BOUNDARY CONDITIONS:

These boundary conditions are a weighted combination of both Dirichlet and Neumann Boundary Conditions

Why Neumann Boundary Conditions are used in shock flow problems?

Shock Waves are supersonic phenomena associated with breaking the sound barrier in general. In a normal atmosphere the signals always can travel as fast as the speed of sound. In a supersonic phase the fluid within the domain cannot travel as fast to match the flow properties in the outlet. Hence information at the boundary cannot be taken to the inlet, which therefore requires an additional set of conditions called the initial conditions.

PRANDTL MEYER SHOCK PROBLEM DESCRIPTION:

1. Supersonic inflow is expanded around a sharp expansion corner.
2. An expansion wave is made up of infinite number of infinitely weak Mach Waves fanning out in the corners.
3. The leading edge in which the angle makes a shock wave proportional to the inflow Mach number and the trailing edge of the shock is proportional to the outflow Mach number.

CASE SETUP:

• Start Time: 0 Cycles
• End Time: 25000 Cycles
• Initial Time Step: 1e^(-7)
• Minimum Time Step: 1e^(-7)
• Maximum Time Step: 1
• Species: O2 and N2
• Turbulence Model: RNG k-ε
• Base Grid:0.8m

The boundaries for the case are flagged respectively as shown in figure below:

MESH GENERATION:

• In certain areas the mesh is fixed whereas at places where the shock occurs we are using fewer cells.
• Temperature based Adaptive Mesh Refinement is where we monitor the curvature in Temperature i.e ∂^(2)*T/∂^(2)*x.
• Whenever the curvature of temperature is greater than a certain value the mesh is refined. This is known to be Sub Grid Scaling.

CASE 1: SUBSONIC FLOW (100 m/s ,SGS 0.07K):

• Here no adaptive mesh refinement takes place at low speeds.

• Here SGS temperature less than 0.07K therefore no Adaptive Mesh Refinement takes place.

• Velocity seems to strike the sharp corners at a high speed but it doesnt contribute to the overall changes in the average speed.

CASE 2: SUPERSONIC FLOW (680 m/s, SGS 0.03K):

• Here SGS temperature is more than 0.03K therefore Adaptive Mesh Refinement takes place.

• Now we can see the waves fanning out in the leading and trailing edges while running the simulation.

• The velocity is high at the trailing edge of the shock when compared with the leading edge.

CASE 3: SUPERSONIC FLOW (680 m/s, SGS 0.05K):

• Here SGS temperature is more than 0.05K therefore Adaptive Mesh Refinement takes place.

• The expansion waves can be seen fanning out now.

• The velocity is high at the trailing edge of the shock when compared with the leading edge similiar in case 2.

CASE 3: SUPERSONIC FLOW (680 m/s, SGS 0.07K):

• Here SGS temperature is more than 0.07K therefore Adaptive Mesh Refinement takes place.

• The expansion waves fan out in the trailing edge.

• The velocity is high at the trailing edge of the shock when compared with the leading edge similiar to previous supersonic cases.

SUBSONIC PLOTS:

For a subsonic plot we can see that the values plotted are similiar to a normal flow visualisation problem.

• The finer cells refinement is not present in the plot which indicates about no Adaptive Mesh Refinement.

• The average Mach Number lies entirely within the subsonic range for the inlet and outlet respectively.

SUPERSONIC PLOTS:

• The adaptive mesh refinement happens as the temperature moves beyond the SGS values indicated in the graph above.

• The Mach Number at outlet is higher than the initial value we gave for the inlet conditions.