Prandtl Meyer Shock problem

OBJECTIVES:

• Setup Prandtl Meyer Shock model in CONVERGE.
• Study the variation of flow parameters.
• Post process the results and documentation.

SCOPE:

• Clean up CAD and extract fluid domain.
• Prepare and verify FV model of Expansion fan.
• Show pressue and velocity contours.
• Show the plots of velocity, pressure, mass flow rate and total cell count.

SOFTWARE USED:

• Converge, Case setup - To prepare FV model and setup cases of Shock Model.
• Converge, Post-Processing 3D - To convert .out files of converge to Paraview readable format.
• Converge, Line Plotting - To plot mass flow rate, pressure, velocity, temperature and density at Inlet and Outlet boundary conditions.
• ParaView 5.7.0 - To perform Post Processing.

INTRODUCTION:

When the speed of the moving object in a medium becomes equal to the velocity of sound(340 m/s) in the medium, the wavefronts cannot escape from the source and pile up. This results in a large amplitude sound barrier. When the speed of the moving object or source exceeds the speed of sound in medium then the wavefronts lag behind the source forming a cone shaped region with a source at vertex. The edge of the cone forms a supersonic wave front with an unusually large amplitude called a shock wave. A sonic boom is heard when the shock waves reach an observer. The occurence of shock waves can be characterized by instantaneous change in pressure, velocity and temperature in a fluid flow. The region between the vehicle and the shock wave known as the shock layer will be a region of high pressure, density and temperature than the free-stream flow conditions. When a fluid streamline crosses the standing shock wave, abrupt increase in the pressure, temperature and density of the fluid flow occurs with a decrease in velocity of the flow.

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 backwards 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.

Fig 1. Pictorial View at corner edge where shocks are developed.

In this project the following trials for the following parameters are made:

• Trail 1: For inlet velocity of 100 m/s
• Trail 2: For inlet velocity of 680 m/s.

and SGS parameter is studied with the following trails:

• Trail 1: For inlet velocity of 680 m/s, SGS parameter of 0.02
• Trail 2: For inlet velocity of 680 m/s, SGS parameter of 0.05
• Trail 3: For inlet velocity of 680 m/s, SGS parameter of 0.08
• Trail 4: For inlet velocity of 680 m/s, SGS parameter of 0.1

CAD MODEL DESCRIPTION:

Fig 2. CAD Geometry of Expansion Fan

Fig 3. Input Conditions for Expansion Fan

MESH:

Fig 4. Mesh generation in Expansion Fan

RESULTS AND DISCUSSION:

1. GEOMETRY

Fig 5. Expansion Fan Geometry

The figure shows a Expansion Fan, with a sudden sharp corner suddenly after the inlet region.

2. Velocity Contour:

Vid 1. Velocity Contour developed for Inlet velocity of 100 m/s

Vid 2. Velocity Contour developed for Inlet velocity of 680 m/s

Since the velocity at the inlet in case 1 (100 m/s), hence no shock is observed but for case 2 (680 m/s) the velocity at inlet is very high and due to the sudden sharp corner shocks are observed.

3. Temperature Contour:

Vid 3. Temperature Contour developed for Inlet velocity of 100 m/s

Vid 4. Temperature Contour developed for Inlet velocity of 680 m/s

Since the velocity at the inlet in case 1 (100 m/s), hence no shock is observed but for case 2 (680 m/s) the velocity at inlet is very high and due to the sudden sharp corner shocks are observed and sudden temperature change is oberved.

4. Mass Flow Rate at Inlet:

Fig 6. Mass Flow Rate at Inlet for velocity at 100 m/s

Fig 7. Mass Flow Rate at Inlet for velocity at 680 m/s

From the above graphs we can clearly observe that the Mass Flow Rate at the inlet has a high variation when compared with different velocity inlets. The mass flow rate at inlet increases rapidly for supersonic inlets.

5. Mass Flow Rate at Outlet:

Fig 8. Mass Flow Rate at Outlet for velocity at 100 m/s

Fig 9. Mass Flow Rate at Outletlet for velocity at 680 m/s

The mass flow rate at the outlet increases rapidly for a supersonic model hence the mass flow rate at the oulet is less when the inlet velocity is 100 m/s.

6. Static Pressure at Inlet:

Fig 10. Static Pressure at Inlet for velocity at 100 m/s

Fig 11. Static Pressure at Inlet for velocity at 680 m/s

The static pressure for inlet velocity decreases since most of the molecules are provided with energy to move but for a velocity of 680 m/s the pressure exerted by the molecues are high because if sudden change in the geometry and high velocity at the inlet.

7. Static pressure at Outlet:

Fig 12. Static Pressure at Outlet for velocity at 100 m/s

Fig 13. Static Pressure at Outlet for velocity at 680 m/s

The static pressure in the model for a inlet velocity is high since more number of molecules are at stationary at the sharp corner and the pressure exerted by them is more than for inlet at 100 m/s because of high velocity and mass flow rate at inlet.

8. Total Pressure at Inlet:

Fig 14. Total Pressure at Inlet for velocity at 100 m/s

Fig 15. Total Pressure at Inlet for velocity at 680 m/s

Total pressure is the sum of static pressure and dynamic pressure. For case 2 (680 m/s) the static pressure increases, hence the total pressure also increases.

9. Total Pressure at Outlet:

Fig 16. Total Pressure at Outlet for velocity at 100 m/s

Fig 17. Total Pressure at Outlet for velocity at 680 m/s

Total pressure is the sum of static pressure and dynamic pressure. For case 2 (680 m/s) the static pressure increases, hence the total pressure also increases.

10. Velocity at Inlet:

Fig 18. Velocity at Inlet for velocity at 100 m/s

Fig 19. Velocity at Inlet for velocity at 680 m/s

Form the graphs its evident that the velocity at inlet for case 1 is 100 m/s and for case 2 is 680 m/s. Negative sign idicates that the flow is entering in the domain.

11. Velocity at Outlet:

Fig 20. Velocity at Outlet for velocity at 100 m/s

Fig 21. Velocity at Inlet for velocity at 680 m/s

The velocity at the outlet for case 1(100m /s) increases and then becomes constant but for case 2(680 m/s) the velocity and a small fluctuations are observed.

12. Mach at Inlet:

Fig 22. Mach at Inlet for velocity at 100 m/s

Fig 23. Mach at Inlet for velocity at 680 m/s

The mach number at inlet is 0.3 for case 1 (100 m/s) and is 2 for case 2 (680 m/s).

13. Mach at Outlet:

Fig 24. Mach at Outlet for velocity at 100 m/s

Fig 25. Mach at Outlet for velocity at 680 m/s

The mach number for case 2 (680 m/s) increases rapidly due the development of sudden shock at the sharp corner in the geometry.

14. Shock Flow Boundary Conditions:

For solving the steady state flow appropriate boundary conditions are needed. It is one of the required components of the mathematical model. On the other hand, for solving transient flow, appropriate initial condition is also required. Boundary conditions are generally three types. They are Dirichlet boundary condition, Neumann boundary condition and Robin boundary condition.

i. Dirichlet Boundary Condition:

The Dirichlet boundary condition, credited to the German mathematician Dirichlet, is also known as the boundary condition of the first kind. In this case, the value of the dependent variable, φ, is prescribed on the boundary. In the finite difference method, since nodes are located on the boundary, the Dirichlet boundary condition is straightforward to apply. For scientific or engineering applications, it is often desirable to determine the first derivative of the dependent variable at the boundaries. This is because the de­rivative is related to the so-called flux at the boundary. By flux, we mean the amount transferred or transported per unit area per unit time.

ii. Neumann Boundary Condition:

The Neumann boundary condition, credited to the German mathematician Neumann, is also known as the boundary condition of the second kind. In this type of boundary condition, the value of the gradient of the dependent variable normal to the boundary, `(del phi)/(del n)`, is prescribed on the boundary. In the context of the finite difference method, the boundary condition serves the purpose of providing an equation for the boundary node so that closure can be at­tained for the system of equations. Thus, one approach to treatment of the Neumann boundary condition is to derive a discrete equivalent.

iii. Robin Boundary Condition:

In this type of boundary condition, a linear combination of the value of the dependent variable and its normal gradient is specified at the boundary. This type of boundary condition is credited to the French mathematician Gustave Robin. It is also known as the boundary condition of the third kind and sometimes referred to as the Robbins boundary condition.

When using a Dirichlet boundary condition, the value of variable is prescribed at the boundary. Neumann boundary Conditions, When Neumann boundary Conditions imposed on an ordinary or a PDE, the condition specifies the values in which the derivative of a solution is applied within the boundary of the domain.

Pressure outlet boundary conditions require the specification of a static (gauge) pressure at the outlet boundary. The value of the specified static pressure is used only while the flow is subsonic. If the flow is supersonic, the specified pressure will no longer be used,pressure will be extrapolated from the flow in the interior. This means that Neumann Boundary Conditions which say that the outlet condition is dependent on the initial conditions of the flow need to be used in this case, that will derive the value of pressure from the initial conditions.

15. Effect of SGS parameter on Shock Location:

Grid scaling refers to changing the base grid size at specified times during a simulation. Grid scaling can greatly reduce runtimes by coarsening the grid during non-critical simulation times and can help capture critical flow phenomena by refining the grid at other times. We use Adaptive Mesh Refinement (AMR) to automatically refine the grid based on fluctuating and moving conditions such as temperature or velocity. This option is useful for using a highly refined grid to accurately simulate complex phenomena such as flame propagation or high-velocity flow without unnecessarily slowing the simulation with a globally refined grid. For a scalar, the sub-grid field is defined as the difference between the actual field and the resolved field or

`phi ' = phi - bar phi`

where `phi` is the actual scalar field, `phi '` is the sub-grid scalar field and `bar phi` is the resolved scalar field. The sub-grid scale can be expressed as an infinite aeries given by:

`phi ' = alpha_([k]) (del^2 bar phi)/(del x_k del x_k) + 1/(2!) alpha_([k])alpha_([l]) (del^4 phi)/(del x_kdel x_kdel x_kdel x_k) - 1/(3!) alpha_([k])alpha_([l])alpha_([m]) (del^6 bar phi)/(del x_kdel x_kdel x_kdel x_k del x_k del x_k)`

where `alpha_([k])` is `dx_k^2/24` for a rectangular cell and the brackets [] indicate no summation. Since it is not possible to evaluate the entire series, only the first term is=n the series is used to approximate the scale of the sub-grid.

i. Temperature Variation

Vid 5. Temperature Contour developed for Inlet velocity of 680 m/s and SGS of 0.02

Vid 6. Temperature Contour developed for Inlet velocity of 680 m/s and SGS of 0.05

Vid 7. Temperature Contour developed for Inlet velocity of 680 m/s and SGS of 0.08

Vid 8. Temperature Contour developed for Inlet velocity of 680 m/s and SGS of 0.1

ii. Velocity Variation

Vid 9. Velocity Contour developed for Inlet velocity of 680 m/s and SGS of 0.02

Vid 10. Velocity Contour developed for Inlet velocity of 680 m/s and SGS of 0.05

Vid 11. Velocity Contour developed for Inlet velocity of 680 m/s and SGS of 0.08

Vid 12. Velocity Contour developed for Inlet velocity of 680 m/s and SGS of 0.1

From the above set of videos, we can clearly observe that as the SGS parameter increases the mesh gets much refined and captures the flow parameters more clearly by using smaller grids.