Week 5: Prandtl Meyer Shock problem
Aim:- The aim of this project is to simulate a Prandtl Meyer wave. The location and nature of shock formed is examined. Also impacts of adaptive Mesh refinement is studied.
Theory:-
Shock Waves:
Shock waves are essentially non linear waves that propogate at supersonic speeds.They can be visualized as a very sharp and thin wave front across which pressure, temperature, density, entropy and velocity of flow change abruptly. This means that the region between the body and the shock wave usually referred to as shock layer will be a region of high pressure, temperature and density compared to the free stram flow conditions. This sudden change in the medium of propogation is one of the unique feature that characterizes the shock wave. Due to the sudden changes in flow properties, shock waves are often regarded as discontinuities with highly localised irreversibiliities.The typical thickness of the shock in standard atmospheric air is about 2.5 Angstrom, very small compared to other characteristic lenghts in fluid flow.
Types of Shock Waves
1. Moving Shock- Consists of a shock propagating into a stationary medium
2. Detonation Wave- A detonation wave is essentially a shock supported by a trailing exothermic reaction. It involves a wave travelling through a highly combustible or chemically unstable medium, such as an oxygen-methane mixture or a high explosive. The chemical reaction of the medium occurs following the shock wave, and the chemical energy of the reaction drives the wave forward.
3. Bow Shock- These shocks are curved and form a small distance in front of the body. Directly in front of the body, they stand at 90 degrees to the oncoming flow and then curve around the body.
4. Attached Shock-These shocks appear as attached to the tip of sharp bodies moving at supersonic speed.
Shock Flow Boundary Conditions:-
Neumann Boundary Condition
The typical boundary conditons which are used to solve shock flow problems are Neumann boundary conditions. In this boundary condition, one prescribes the gradient normal to the variable at the boundary. Since the pressure values are not defined at boundaries, we use Neumann boundary conditions. The values are extrapolated to boundaries from the internal regions.
The other boundary conditions that are available are
1, Dirichlet Boundary Condition:-
When using a Dirichlet boundary condition, one prescribes the value of a variable at the boundary.
In a typical CFD code, plane two-dimensional flows can be simulated using a single layer of cells bounded on its sides by symmetry planes, since there is no velocity component in the spanwise direction.
• Symmetry boundary condition can also be applied to axially symmetric flows, for instance by generating a single segment of cells and applying the symmetry conditions on the segment surfaces.
3. Periodic Boundary Condition:- This type of boundary condition consists of pairs of geometrically identical boundaries at which all flow conditions are matched.
4.Wall Boundary Condition:-
• Wall boundaries can be defined as no-slip or slip walls
• In viscous flows, no-slip conditions are applied at walls.
• The wall thermal boundary conditions are usually defined as constant temperature or constant heat flux
An expansion fan, sometimes called a Prandtl Meyer expansion waves, is considered as a continuous sequence of infinitesimal Mach expansion waves. It is a 2D simple wave that occurs when a supersonic flow turns around a convex corner. Some qualitative aspects of flow through an expansion waves are given below
1. An expansion corner is a means to increase the flow Mach Number.
2. The pressure, temperature and density decrease through an expansion wave.
3. Streamlines through an expansion waves are smooth curved lines
4. Since the expansion takes place through a continuous succession of Mach waves and ds=0, for each Mach wave the expansion is isentropic.
Case Setup:-
Case Setup:-
- Geometry Creation:- The geometry in STL format was imported into CONVERGE for analysis.
- Creation of Boundaries: In the geometry which is created above, boundaries such as inflow, outflow , top and bottom walls, front and back faces are labelled.
- Checking Orientation of Normals:Orientation of Normals were checked to ensure all the normals are pointing inwards. This was done using the Normal Toggle
- Finally a Diagnosis check was run to ensure no errors in the geometry.
B Case Setup
- Application Type:- This was set to Time based
2.Materials:- This was set to Air
- i) Gas Simulation: Default properties were retained
- ii) Global Transposrt Parameters:- Default properties retained
- Simulation Parameters
- Run Parameters:- Steady state solver was used. Simulation mode was set to Full Hydrodynamic. Gas Flow solver set to Compressible
- Steady State Monitor:- Minimum no of cycles for to be executed for Steady state solver -500
Monitor Start Delay, cyc :- 200
Sample size cyc :- 100
- Simulation Time Parameters:- The parameters were set as displayed below. Simulation was run for 25000 cycles.Max CFL limit set to 1.0
- Solver Parameters:- A Pressure based PISO solver was used, Maximum Convection CFL limit set to 0.8.
- Boundary
- Inlet:- Boundary flow - Inflow
Inlet Velocity:- 680 m/s (Supersonic)
- Outlet- Boundary - Outflow with zero normal gradient for both Pressure and Velocity Boundary condition. We use a Neumann boundary condition since at the supersonic boundary condition it is not possible to predict the flow properties.
- Top and Bottom Walls:- Stationary wall with Fixed surface movement with slip boundary condition is used
- Front and Back faces set to Two D
- Initial condition and events
The created geometry was selected as the Volumetric Region. The velocity was set to 680m/s
- Physical Models:- RNG k epsilon used
- Grid Control:- Base Grid. Here the size of the mesh is entered.
Adaptive Mesh Refinement was also enabled. A temperature based adaptive mesh refinement technique was used in which the curvature of temperature `(del^2T)/(delx^2)` is monitored. Whenever the curvature of temperature is greater than a particular value the mesh would be refined. It can be thought of as sub grid scale temperature.
In the below snapshot, we see that the sub grid scale criterion is set to 0.05K. This implies that whenever the temperature difference is greater that 0.05 K, AMR will refine. Maximum embedding level is set to 2, this means that there will be two layers of refinement. Converge uses an octri data structure. In this data structure, the grid size can be calculated by the given formula
In this case the base grid size is 0.08. The smallest grid size will be 0.08/(2^2) which means the smallest grid size will be 0.02.
- Output/Post Processing:- The variables that we intend to see during post processing are selected. In this case Density, Pressure , Temperature and Velocity were selected.
Also the frequency of generation of output files were set below
After setting the above, the input files were exported to a directory
Results
Base Grid- 0.8 m SGS 0.05 K
Case 1
Velocity-680m/s
SGS=0.05K
Meshing:- The refined areas are those where temp difference is greater than 0.05K
Velocity Plots:- From the below plot, it can be observed that the expansion wave casuses an increase in velocity from the inlet to the outlet
Temperature Contour:- From the below plot, it can be observed that the expansion wave casuses a decrease in temperature from the inlet to the outlet
Pressure Contour:- From the below plot, it can be observed that the expansion wave casuses an inrease in pressure from the inlet to the outlet
Cell Count:- The total cell count for simulating Sub grid scale temperature of 0.05K is around 8000.
Average Velocity:- It is observed that across the expansion wave there is an increase in velocity . Consequently the Mach number also increases.
Average Temperature:- There is a decrease in temperature across the expansion wave. The decrease is to the tune of 16 K.
Average Pressure- The pressure decreases across the expansion wave.
Mach number increases across the expanison wave. The magnitude of increase is around 0.15
SGS 0.04K
The mesh under a sub grid scale factor of 0.04K is depicted below. There is a notable increase in cell count as compared to the previous scale factor of 0.04K.
Velocity Contour:- With the increase in region of Adaptive Mesh refinement, it is observed that the variation in velocity across the expansion wave can be observed more clearly. The same trend of increase in velocity from inlet to outlet is observed.
Pressure:- With the increase in region of Adaptive Mesh refinement, it is observed that the variation in pressure across the expansion wave can be observed more clearly. The same trend of decrease in pressure from inlet to outlet is observed.
Temperature:- With the increase in region of Adaptive Mesh refinement, it is observed that the variation in temperature across the expansion wave can be observed more clearly. The same trend of decrease in pressure from inlet to outlet is observed.
Cell Count:- It is noted that the number of cells in the simulation is around 12000 cells
SGS 0.02
The mesh under a sub grid scale factor of 0.02K is depicted below. There is a notable increase in cell count as compared to the previous scale factor of 0.04K.
Velocity Contour:- With the increase in region of Adaptive Mesh refinement, it is observed that the variation in velocity across the expansion wave can be observed more clearly. The same trend of increase in velocity from inlet to outlet is observed.
Pressure:- With the increase in region of Adaptive Mesh refinement, it is observed that the variation in pressure across the expansion wave can be observed more clearly. The same trend of decrease in pressure from inlet to outlet is observed.
Temperature:- With the increase in region of Adaptive Mesh refinement, it is observed that the variation in temperature across the expansion wave can be observed more clearly. The same trend of decrease in temperature from inlet to outlet is observed.
Cell Count:- It is noted that the number of cells in the simulation is around 14000 cells
SGS 0.01
The mesh under a sub grid scale factor of 0.02K is depicted below. There is a notable increase in cell count as compared to the previous scale factor of 0.04K.
Velocity Contour:- With the increase in region of Adaptive Mesh refinement, it is observed that the variation in velocity across the expansion wave can be observed more clearly. The same trend of increase in velocity from inlet to outlet is observed.
Pressure:- With the increase in region of Adaptive Mesh refinement, it is observed that the variation in pressure across the expansion wave can be observed more clearly. The same trend of decrease in pressure from inlet to outlet is observed.
Temperature:- With the increase in region of Adaptive Mesh refinement, it is observed that the variation in temperature across the expansion wave can be observed more clearly. The same trend of decrease in temperature from inlet to outlet is observed.
Cell Count:- It is noted that the number of cells in the simulation is around 17000 cells
Animations
1) SGS 0.05 K
- Velocity
- Pressure
2) SGS 0.04 K
-Velocity
- Pressure
3) SGS 0.02 K
-Velocity
- Pressure
4) SGS 0.01
- Velocity
-Pressure
Observations:-
1. Across the expansion wave, velocity and Mach number increase, wheras pressure, density and temperature decrease from inlet to outlet.
2.With the decrease in SGS factor from 0.05 K to 0.01 K, we note that the variation of properties such as velocity, pressure . Mach Number and density can be observed more clearly.
3. There is an increase in the number of cells when the sub grid scale factor is 0.05K to 0.01K. Also the area of Adaptive mesh refinement region increases with reduction in SGS value.
4. The location of the expansion waves shifts slightly rightwards with the reduction of SGS value