Week 5: Prandtl Meyer Shock problem

Objective: The objective of this challenge is to perform a Prandtl Meyer shock wave problem at Mach number 0.3 and 2 (which means at the velocities of 100 m/s and 680 m/s) by observing the effect of SGS parameter a concept in the Adaptive Mesh Refinement. The case setup is made up of the Converge Studio and post-processing in the ParaView software.

Introduction:

In physics, a shock wave (also spelled shockwave), 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.

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 backward 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.

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 (e.g. the total pressure and total temperature) remain constant across the fan.

As there are an infinite number of Mach waves in the wave region, calculating the shock wave phenomena and properties using a coarse or fine mesh is a difficult process and if we do the results must be garbage. 

To overcome this problem, the adaptive mesh refinement option is implemented in the case set up of the problem to refine the mesh in order to capture as many Mach waves as possible in the transition region.

Boundary Conditions: 

Boundary conditions, which exist in the form of mathematical equations, exert a set of additional constraints to the problem on specified boundaries. The concept of boundary conditions applies to both ordinary and partial differential equations.

Dirichlet Boundary Condition:

 Dirichlet boundary conditions are where the values of the properties are directly given and it takes along the boundary of the domain. This is also called a type-1 boundary condition.

`phi=f(x)`

Neumann Boundary Condition:

Neumann Boundary conditions are called flux or gradient of the normal is specified. This is also called a type-2 boundary condition.

`(delphi)/(deln)=f(x)`

Robin Boundary COndition:

Robin boundary conditions are a weighted combination of Dirichlet Boundary Condition and Neumann Boundary Conditions. This contrasts with Mixed Boundary Condition, which are boundary conditions of different types specified on different subsets of the boundary.

In the field of heat transfer, we call this boundary condition as Convective Boundary Condition.

`phi+(delphi)/(deln)=f(x)`

Problem Description:

Here, we are studying the shock wave problem with two different velocities. One is 100 m/s which Mach number will be 0.3 and the other velocity is 680 m/s which Mach number will be 2.

We also study the effect of the SGS temperature value ranging from 0.1 to 0.025 on the Total cell count and the resolution of the shock in the simulation.

Geometry:

• The above figure is the geometry considered for the Shock Wave simulation. The visible brown colored boundary is the inlet of the duct, and opposite to inlet is the outlet. The blue-colored boundary is a wall.
• The dimension of the Duct is 65 * 45.15 * 8.15 meters. As we are performing the supersonic condition, the duct dimensions should be large.

Case Set-up:

Application Type: 

Time-Based

Materials:

Predefined Mixture: Air

Simulation Parameters:

Run Parameters: 

• Solver: Steady-state solver
• simulation mode: Fully hydrodynamic
• Miscellaneous settings: Toggle Steady-state.

1. Steady-State monitor:

• Choose the target and target type from the Regions and initialization.

Simulation time parameters:

• Start-time: 0 cyc
• End-time: 25000 cyc
• Initial time-step: 1e-7
• Minimum time-step: 1e-7
• Maximum time-step: 1.0

Solver Parameters (Steady state)

1. Navier-Stokes Solver

• Navier-stokes solver scheme - PISO
• Navier-stokes solver type - Density-Based

1. Steady-state solver control:

• Maximum convection CFL limit: 0.5

Initial Conditions and Events:

Regions and Initialization:

• Add a region or a volume to give in the boundary type.
• Pressure: 50000.0 pa

Boundary Conditions:

Boundary:

Physical Models:

• Turbulence Modelling: RANS K-epsilon

Grid Control:

Base-grid size - 0.8 m.

Adaptive Mesh Refinement:

• Min. cells = 1
• Max. cells = 200000
• Embedded type = Sub-Grid-Scale (SGS)
• Max. embedding level = 2
• sub-grid criterion = 0.05 (we have varied this value fro 0.1 to 0.025 to study the effect of SGS on cell count)
• Timing-control type = Sequential

• How the Refinement of grid size is done using Embedded sizing is:
• The data structure the Converge Software uses is the Octri-data structure.

Refinement Formula = Grid Size = `(N_(base size)/2^(Embed-Level))`

                                               = `(0.8)/(2^(2))`

                                               = `0.8/4`

                                               = 0.2 m.

Output/Post-processing:

• Output files: 

1. Time Interval for writing 3D files: 1000 cycles
2. Max. Number of restart files saved: 3

Results

Case 1: Velocity = 100 m/s (Mach = 0.3), SGS = 0.05

Mesh:

• As the flow is subsonic, there will be no much change in the temperature. So, the Adaptive mesh refinement has not appeared on the surface.
• The base-grid size is 0.8m.

Cell Count:

• A total of 4300 cells has been generated, and due to the subsonic flow phenomenon, there is no AMR.
• So, there is no disturbance in the curve of the cell-count.

Temperature:

• The inlet temperature is 286.1 K given from the inlet boundary condition.
• The temperature at the outlet is gradually decreasing and settles to 278.3 K due to the decrease of the pressure inside the duct.

Temperature Contour:

• The above contour shows how the temperature is behaving inside the duct when flow happens inside.
• The min. temperature recorded is 278 k and max. temperature is nearly 300 k.

Velocity Contour:

• The Inlet velocity given is 100m/s. Due to the viscous force and resistances the max. velocity recorded at the end of the simulation is 108.4 m/s.
• The min. velocity recorded inside is 91 m/s. 
• Because this is a subsonic problem, the shock wave is not produced. So that the AMR does not come into the picture for this simulation.

Pressure contour:

Results of different parameters: (Supersonic Flow)

Mach Number:

• The Mach number at the Inlet is 2.
• From the above graphs, it can be observed that the Mach number of the supersonic flow increases as the flow passes through the expansion wave.

Density:

Pressure:

Temperature:

Velocity:

Case 2: Velocity = 680 m/s (Mach = 2), SGS = 0.1

 Mesh: https://youtu.be/uGyEqO0N0ew

Temperature: https://youtu.be/BvIPwolcVRQ

• The MIn. temperature recorded is approximately 260 K and Max.temperature recorded inside the duct is 286.4 k.

Velocity:

• Due to the changes in the physical properties the change in Mach number has occurred. The Mach number varies from 2 to 2.15 because of the increase in the velocity.
• The max. velocity recorded is 740 m/s and it is recorded at the wedge region.

Pressure:

• As we know that the pressure decreases, the velocity increases.
• By observing all the above figures, we can say that Adaptive Mesh Refinement (AMR) comes into the picture and shows its effect.
• But due to the low Sub-Grid-Scale (SGS), it is not able to capture the far-field physics more accurately. 

• As the AMR comes into the picture, and as we switched to the supersonic flow the total -cell count is increased to 5000.

Case 3: Velocity = 680 m/s (Mach = 2), SGS = 0.075

Mesh: https://youtu.be/KvN8iSBju20

 Temperature:https://youtu.be/ppIG-Vgx8Wc

Velocity:

Pressure:

• As we know that the pressure decreases, the velocity increases.
• By observing all the above figures, we can say that Adaptive Mesh Refinement (AMR) comes into the picture and shows its effect.
• But due to the low Sub-Grid-Scale (SGS), it is able to capture the far-field physics but not completely.  

Total Cell-Count:

• As the AMR comes into the picture, and as we switched to the supersonic flow by reducing the size of SGS, the total -cell count is increased to 5812.

Case 4: Velocity = 680 m/s (Mach = 2), SGS = 0.05

Mesh: https://youtu.be/skB7eth2f8w

 Temperature:https://youtu.be/x43swbnv3HE

Velocity:

Pressure:

• As we know that the pressure decreases, the velocity increases.
• By observing all the above figures, we can say that Adaptive Mesh Refinement (AMR) comes into the picture and shows its effect.
• But due to the low Sub-Grid-Scale (SGS), it is able to capture the far-field physics. 
• By decreasing the SGS rate, the AMR reaches the far-field partially.

• As the AMR comes into the picture, and as we switched to the supersonic flow by reducing the size of SGS, the total -cell count is increased to 7220.

Case 4: Velocity = 680 m/s (Mach = 2), SGS = 0.025

Mesh:https://youtu.be/SsyDRWXtBVA

 Temperature:https://youtu.be/oDVCrJGuQnY

Velocity:

Pressure:

• As we know that the pressure decreases, the velocity increases.
• By observing all the above figures, we can say that Adaptive Mesh Refinement (AMR) comes into the picture and shows its effect.
• But due to the low Sub-Grid-Scale (SGS), it is able to capture the far-field physics. 
• By decreasing the SGS rate, the AMR reaches the far-field completely.

Total Cell Count:

• As the AMR comes into the picture, and as we switched to the supersonic flow by reducing the size of SGS, the total -cell count is increased to 12477.

Conclusion:

• This SGS temperature criterion is able to capture the shock wave from where it started and where it is going out of the flow domain.
• We can decrease the SGS temperature criteria to 0.001 so that the resolution of the shock wave will have a high resolution but at the same time there is an increase in the cell count too which is unnecessary.
• For the above reason, we can say that this SGS criterion is more reliable and has a limited number of cells to capture the shock wave with good resolution.
• A general phenomenon is studied, which is the pressure is gradually decreasing, in the same region, the velocity is increasing.
• The maximum number of cells generated for the respective SGS criteria is: