Week 5: Prandtl Meyer Shock problem

AIM -

1. To understand what is a shock wave.

2. To understand the what are the different boundary conditions used for a shock flow problems.

3. To understand the effect of SGS parameter on shock location.

4. To simulate Prandalt Meyer Shcok Wave.

OBJECTIVE - 

Que 1. What is Shock Wave?

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 , temperature and density of the medium.

Que 2. Types of Boundary Contitions used for a shock wave

generally the following type of boundary conditions are used.

1. Dirichlet Boundary Condition:

Here the values of function is known at the boundary, sometimes it is also known a fixed boundary condition.  

`F(x) = g`

for an interval {a,b} where a and b are the boundaries ,

then the dirichlet boundary conditions for interval {a,b} take the form as

`F(a) =alpha` & `F(b) = beta`

where `alpha` & `beta` are given numbers.

2. Neuman Boundary Condition:

For the Neumann Boundary conditions the values of the normal derivatives is known.

`F(x) = g` for an interval {a,b} where a and b are the boundaries ,

then the dirichlet boundary conditions for interval {a,b} take the form as

`F'(a) =alpha`& `F'(b) = beta`

where `alpha` and `beta` are given number

3. Robin Boundary Condition:

Robin boundary condition is the linear combination of dirichlet and newmann boundary condition together,

`a. F(x) + b F'(x) = g`

here a and b are the non zero constants.

4. Mixed Boundary Condition:

Here different boundary conditions are applied to disjoint part of the boundary where the conditions are stated.

Que 3. Effect of the SGS parameter on the Shock Location

SGS stands for Sub-Grid Scaling, the region of the shock wave has abrupt discontinuity in parameters like pressure, temperature and density. to capture this changes we will need to reduce the mesh size in these regions and this can be done using the SGS parameter. This helps us to capture the region affected the shock using a finer mesh refinement without refineing the mesh for the whole geometry. It saves a lot of computational power and time.

PRANDLT MEYER SHOCK FLOW SIMULATION:

Prandlt meyer shock is a supersonic expansion fan, where 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.

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 remain constant across the fan.

SETTING UP THE SIMULATION IN CONVERGE:

Geometry-

Boundary conditions-

Inlet

At the inlet has a flow velocity of 680m/s at 286 K. the inflow species is air

Outlet

The Outflow conditions are set to Newmann boundary conditions

On the top and bottom walls the conditions are set to Slip wall and the front and back is set as 2D.

The Base Grid is at 0.8m 

Here adaptive mesh refinement is used. Maximum cell count is limited to 200000 and we have used temperature variable to set up the Sub-Grid Scaling using the critrion value as 0.01 K , the adaptive mesh refinement will start after 5000 cycles.

Now the simualtion is setup and it it run using parallel processing with 6 processors. 

We have to see the effection SGS parameter so we have used 3 valuse of SGS criterion, these values are 0.01 , 0.001, 0.0001 

we will observe the results and analyse them.

Results-

For SGS = 0.01 K

Mesh Generated -

Pressure-

Temperature-

 Velocity-

Cell Count-

Mach Number-

For SGS = 0.001 K

Mesh Generated -

Pressure-

Temperature-

Velocity-

Cell Count-

Mach Number-

For SGS = 0.0001 K

Mesh-

Pressure-

Temperature-

Velocity-

Cell Count-

Mach Number-

Following is the table to analyse the effect of the SGS criterion-

Observation-

1. From the table we can observe that as the cirterion value if reduced for 0.01 to 0.0001 the mesh count increases,
2. the time required for the simualtion increases as the criterion value is reduced.
3. There is not much difference in the value of the Mach Number at the outflow.

Links to the flow animation :-

For SGS = 0.01 

For SGS = 0.001

For SGS = 0.0001