Prandtl Meyer Shock Simulation

Objective:-

• Simulate Prandtl Meyer Shock problem
• Literature review on shock flows and assosiated boundary conditions
• Effect of SGS parameter on shock location

What is boundary conditions?

Both ordinary and partial differential equations need boundary conditions to be solved. Different types of boundary conditions can be imposed on the boundary of the domain. The choice of the boundary condition is fundamental for the resolution of the computational problem a bad imposition of b.c. may lead to the divergence of the solution or to the convergence to a wrong solution.

Types of Boundary conditions are as follows:

• Dirichlet boundary condition:When this boundary condition is imposed on ODE or PDE its specifies the value of the function or variable at the boundary
• Neumann boundary condition:When this boundary condition is imposed on ODE or PDE the condition specifies the values in which the derivative of a solution is applied within the boundary of the domain.

What is shock wave?

When the speed of the moving object exceeds the speed of the sound in a medium, then the wavefront lag behnd the object forming a cone shaped region with the source at the vertex behind the object.The edge of the cone forms supersonic wavefront with a large amplitude which is called as shock wave.

Prandtl Meyer Shock wave:It is also called as supersonic expansion fan this occur a supersonic flow encounters a convex corner as shown in the below figure, it forms an expansion fan, which consists of an infinite number of expansion waves centred at the corner.The angle at which supersonic wave allow to pass is called as shock angle.

Across the expansion fan, the flow accelerates (velocity increases) and the Mach number increases, while the static pressure,temperature & density decrease.

Case Setup

Step1.Geometry

First the geometry has been imported in converge which is shown below

Boundary:Before proceeding for case setup we need to assing boundary selection which is shown below

Normal toggle

As we can see in above image normal toggle is pointing outward which is wrong its should point towards the fluid region and this can been done by using transform option.

Step2.Case Setup

• Application:General Flow
• Analysis:Steady State
• Type:Density type
• Material:Air
• Species:O2 & N2
• Simualtion mode:Full hydrodynamic

Simulation time parameters:

• Start time:0cycles
• End time:25000cycles
• Initial time step:1e-7sec
• Minimum time step:1e-7sec
• Maximum time step:1sec

Region & Initialization:

In this step we will create a region of the fluid and initializing the species value

• O2=0.77
• N2=0.23

Boundary Conditions:

• Inlet
  • Type:Dirichlet condition
  • Velocity:680m/s in x-direction
  • Temperature:286.1k
  • Species condition:O2=0.23,N2=0.77

• Outlet
  • Type:Neumann condition
  • Pressure:zero normal gradient
  • Temperature:300k
  • Species condition:O2=0.23,N2=0.77

• Front-2D & Back-2D
  • TWO_D

• Top-Bottom-Wall
  • Slip condition

Apart from assigning the boundary conditions we will also assign repective Region to each boundary

Grid Control:

• Element size:dx = 0.8m, dy = 0.8m,dz = 0.8m

Adaptive Mesh refinement

• Its essential to capture the data accurately were the shock occur,so for this we need more refine mesh or local mesh in that area while keeping the mesh size in the other area as same as base grid size so,in order to do this we are using Adaptive mesh refinement technique.
• In converge adaptive mesh refinement gives variety of based technique in our case we will use temperature based sub-grid-criterion with two layer.Converge will monitors the curvature of Temperture property provided in the AMR and refines the mesh when the property of temperature curvature between the consecutive grid is more than the defined SGS value.
• AMR use below formula to determine the refine grid size
• Refine grid size=`frac{Base grid}{2^(embedd-Level)}`

Output Files:

• Time interval for writing 3D output data files:250 cyc(So,that we get atleast 100 3D output files)
• Time interval for writing text output:1e-6 cyc(So,that we get every minute data)
• Time interval for writing restarting output:250 cyc(So,that we get every minute )
• Maximum number of restart file saved:3

Step3.Simulation

• After setting case setup in converge now we will run simulation in terminal called cygwin

Step4.Post-process

3D output files

• As we can see in the above image cygwin created 3D output files in order to view this files in paraview we need to convert this file so,that paraview can read those files this can either be done using cygwin or post converge.
• I have use post converge to convert the files as we can see in the below image

Step5.Result

Mesh

SGS-0.03

SGS-0.02

SGS-0.01

Mach number Plot

Temperature Plot

Density Plot

Pressure Plot

From the above plots, it can be observed that Mach number of the supersonic flow increases as the flow passes through the expansion while other thermodynamics properties like density,temperature & pressure tends to decrease.

Effect of SGS parameter on Total Cell

From the above plots, it can be observed that least value of SGS gives large number of cell,which is obivous as we decrease the SGS value AMR will increase refinement and that will increase the number of cell.

Effect of SGS parameter on shock location

The Effect of SGS parameter on shock location can easily been observed by looking temperature contour.Below images shows temperature contour for different SGS value

SGS-0.03

SGS-0.02

SGS-0.01

From the above plots, it can be observed that least value of SGS gives clear and sharp weak mach waves.

Temperature Animation

SGS-0.03

SGS-0.02

SGS-0.01

Conclusion

• Succesfully Simulated Prandtl Meyer Shock problem in Converge.
• Studied shock flows and assosiated boundary conditions
• Studied Effect of SGS parameter on shock location