Simulating Prandtl Meyer Shock Flow using Converge CFD
OBJECTIVE:
1. To simulate a 2D shock flow on a plate, post process the results using converge CFD and analyse the fluid property variations using paraview
2. To study the effect of SGS parameter on shock location
INTRODUCTION:
A shock is basically a discontinuity in a fluid flow. When a fluid flows at a speed of a sound and strikes a sharp object disturbances are produced in the flow medium. These disturbances propagates at speed of sound and might spread to the rest of the flow field causing abrupt changes in fluid properties.
In this project we are going to set up a fluid flow which travells at a speed of 680 m/sec and strikes a sharp convex corner in a flat plate. We will observe that when the supersonic flow strikes the convex corner, it forms a fan like structure which consists of infinite number of mach waves diverging from the corner. This fan like structure is more commonly known as Prandtl-meyer Expansion fan. Each wave in the expansion fan turns the flow in small steps and hence the changes across each expansion wave are small, so we can assume it\'s an isentrophic process. We will observe that across the expansion fan there is sharp increase in velocity and mach number where as static pressure, temperature and density decreases.
FLOW PROPERTIES:
Suppose the initial mach number of fluid is `M_1` and the mach number after sharp turn is `M_2`. The first mach line is at an angle:
`mu_1=arcsin(1/M_1)` with respect to flow direction and the last mach line is at an angle
`mu_2=arcsin(1/M_2)` with respect to flow direction.
- `T_2/T_1=((1+(gamma-1)/2M_1^2)/(1+(gamma-1)/2M_2^2))`
- `P_2/P_1=((1+(gamma-1)/2M_1^2)/(1+(gamma-1)/2M_2^2))^(gamma/gamma-1)`
- `rho_1/rho_2=((1+(gamma-1)/2M_1^2)/(1+(gamma-1)/2M_2^2))^(1/gamma-1)`
BOUNDARY CONDITIONS FOR SHOCK FLOW PROBLEMS:
Boundary conditions are necessary to set up a simulation, so that based on these boundary constraints we can find solution of the boundary value problem. Both Ordinary as well as partial differential equations need boundary conditions to be solved. A bad choice of b.c could lead the solution to be unstable or diverge or even converge to wrong results. Choosing appropriate boundary conditions for a specific problem is a must need for proper resolution of a computational problem.
1. Drichlet Boundary Condition: It specifies a value which the unknown function will take along the boundary of the domain. In fluid problems drichlet boundary conditions consists of the value of velocity or pressure taken by a set of nodes or is associated with a surface. There are two types of drichlet b.c
- Slip Condition: In slip conditions, zero normal gradient i.e velocity normal to the boundary is set to zero while the velocity parallel to to the boundary is set free.
- No Slip Condition: In this case velocity normal to the boundary as well as velocity parallel to the boundary are set to zero.
2. Neumann boundary conditions: This type of boundary condition are generally the value of the derivate forms of velocity or pressure fields that is going to be carried on the boundary of the domain. They are appplied in addition to a drichlet boundary condition.
CASE SETUP:
1. Geometry
2. Boundary Conditions
INLET CONDITIONS:
Inlet Velocity - 680 m/sec
Temperature - 286 k
OUTLET CONDITIONS:
Pressure - Zero Normal Gradient
Velocity - Zero Normal Gradient
Front and Back - 2D
Top and Bottom wall - Slip Wall
3. Simulation Time Parameters
Start Time - 0 cycles
End Times - 25000 cycles
Solver Used - Density Based Steady Solver
Working Fluid - Air
4. Geometry Mesh
Base Mesh Size - 0.8m in X, Y and Z direction
Adaptive Mesh Refinement (Temperature) - Embedding level (2) and sub-grid scale (0.05)
RESULTS for SGS=0.05 :
1. Mach number vs No of Cycles
2. Average Temperature vs No of Cycles
3. Average Pressure vs No of Cycles
From the above graphs it is observed that the average mach number of supersonic flow increases as the flow crosses the expansion fan whereas fluid properties like temperature, pressure and density decreases.
5. Velocity Contour (SGS - 0.05)
6. Pressure Contour (SGS - 0.05)
7. Temperature Contour (SGS - 0.05)
F
EFFECT OF SGS PARAMETER ON MESH :
1. SGS - 0.05
2. SGS - 0.01
3. SGS - 0.1
4. SGS - 0.4
EFFECT OF SGS PARAMETER ON TEMPERATURE CONTOUR:
1. SGS - 0.05
2. SGS - 0.01
3. SGS - 0.1
4. SGS - 0.4
From the above plots we can clearly see as the SGS parameter increases, the temperature plots gets distorted and mesh is no more finer as compared to the ones with less SGS value.
EFFECT OF SGS PARAMETER ON CELL COUNT:
1. SGS - 0.01
2. SGS - 0.05
3. SGS - 0.1
4. SGS - 0.4
From the above graphs we can clearly see that as we increase the SGS parameter the cell count decreases and becomes constant when SGS-0.4.
CONCLUSION:
- The simulation over the geometry was succesfully done and the effect of SGS parameter can be well studied from the above results obtained.
- Research is being done in minimizing the effect of shock waves over flow field so that fluid properties are not altered and can\'t demage the geometry.
- Care should be taken in implementing proper boundary conditions for example in this case running the same simulation with pressure based steady solver won\'t work. Since, it will assume density is constant which is not true.
- Proper mesh refining is important as we have seen in the case of AMR (adaptive mesh refinement) with diffirent values,or else contours obtained would be distorted.