Week 5: Prandtl Meyer Shock problem

Aim :- To perform prandtl meyer shock problem.

Introduction :-

Prandtl Meyer expansion fan is a two dimensional simple wave. It is a centered expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of a finite number of Mach waves diverging from a sharp corners. When a flow turnsaround a smooth and circular corners, these waves can be extended backwards to meet a point.

Each wave in the expansion fan turns the flow gradually. Near the expansion fan,the flow accelerates and the Mach number increases, while the static pressure , temperature and density decreases. As the process is isentropic process, the total temperature, total pressure remains constant across the fan.

What is a Shock Wave?

 It is a very thin region in a flow which gets created when the velocity of flow exceeds or equals to the speed of sound which we refer as a Mach number, Hence it occurs in a supersonic and hypersonic flow

In this region of the shock wave the gas properties such as pressure, temperature.

Shock flow boundary conditions 

Even the density of air changes which is the function of time.

 Shock wave gets created when huge energy explodes from a tiny spot and shakes the material around it. This disturbs the atmosphere around it which leads to change the gas parameters and parameters li pressure, temperature, flow speed, etc.

 It is a quick phenomenon which doesn't last long and it flows through any kind of medium

Physics  behind the shock waves ?

When an obstruction is suddenly introduced into the flow, something interesting happens. The particles in front of it slow down instantaneously, creating an increase in local pressure.
Since the fluid is a continuum, the information of whatever happens in one place is transmitted
To oher region in the form of pressure waves this pressure waves then cause the incoming fluid to against its properties. Travels in the form of compression and rerefractions happens in acoustic waves.

Boundary Condition:

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

are:

Types of Boundary Conditions:

1. Dirichlet Boundary Condition.

2. Neumann Boundary Condition.

3. Robin Boundary Condition.

4. Mixed Boundary Condition.

5. Cauche Boundary Condition.

Dirichlet Boundary Condition.

The values of dependent variables are provided along the boundary of the domain. It is also known as a fixed boundary condition. T specified at the boundary.

Neumann Boundary Condition.

The value of the gradient of the dependent variable (flux) is provided along the boundary of the domain. Because the heat flux is related to temperature in Fourier's law.

Cauche Boundary Condition.

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

Mixed Boundary Condition.

In this condition, neither temperature or q is specified but one is expressed as a function of others at the boundary condition.

Why Boundary Conditions are used in shock flow problems?  And why Neumann? 

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.

What is Prandtl–Meyer shock?

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

Procedure :- 

• The  geometry is imported and the boundary flagging is done.  

• Once the boundaries are allotted then we proceed for the case setup.

Setup:

• Material - Air
• Species - N2 and O2
• Run parameters

•      Simulation parameters

• regions and initialisation

• Grid size of 0.8m is used and adaptive mesh refinement is also done to the geometry.

• output files

Results:

case-1: velocity = 680m/s and SGC = 0.05

mesh

Temperature contour:

Animation:-

Plots:

Mach number:

Total cells:

Case-2: velocity = 680m/s and SGC = 0.1

mesh

temperature contour

Animation :-

Plots

total cells

mach number

case-3: velocity = 100m/s and SGC = 0.05

mesh

temperature contour

Animation :-

Plots:

Mach number

 Total cells

Conclusion:

From the above results we can conclude that

• as the sub grid criterion value decreases,cell count increases.
• in case-3 which is subsonic,we do not see any shock waves getting generated.
• using SGC value of 0.05 gives us better and accurate results than 0.1 value.