Week 5: Prandtl Meyer Shock problem

Aim:

• To study the flow over the Convex corner or expansion waves
• To study the influence of SUB-GRID-CRITERION on the results.
• Literature review on Boundary Conditions and Shock flow
• To simulate the Prandtl_Meyer expansion Wave

Prandtl-Meyer Expansion Fan:

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.

Prandtl-Meyer expansion fans:

To this point, the only flow phenomena that have been discussed are shock waves, which slow the flow and increase its entropy. It is possible to accelerate supersonic flow in what has been termed a Prandtl-Meyer expansion fan, after Ludwig Prandtl and Theodore Meyer. The mechanism for the expansion is shown in the figure below.

As opposed to the flow encountering an inclined obstruction and forming an oblique shock, the flow expands around a convex corner and forms an expansion fan through a series of isentropic Mach waves. The expansion "fan" is composed of Mach waves that span from the initial Mach angle to the final Mach angle. Flow can expand around either a sharp or rounded corner equally, as the increase in Mach number is proportional to only the convex angle of the passage (δ). The expansion corner that produces the Prandtl-Meyer fan can be sharp (as illustrated in the figure) or rounded. If the total turning angle is the same, then the P-M flow solution is also the same.

The Prandtl-Meyer expansion can be seen as the physical explanation of the operation of the Laval nozzle. The contour of the nozzle creates a smooth and continuous series of Prandtl-Meyer expansion waves.

Prandtl-Meyer Compression fans:

A Prandtl-Meyer compression is an opposite phenomenon to a Prandtl-Meyer expansion. If the flow is gradually turned through an angle of δ, a compression fan can be formed. This fan is a series of Mach waves that eventually coalesce into an oblique shock. Because the flow is defined by an isentropic region (flow that travels through the fan) and an anisotropic region (flow that travels through the oblique shock), a slip line results between the two flow regions.

Boundary Conditions:

In CFD, Boundary Conditions (BC) are equally important as the governing equations, even if most textbooks deal predominantly with the governing equations only. From a mathematical point of view, BCs are inherently linked with PDE’s. A PDE must have BCs to obtain a unique solution. In CFD, the various solutions to the same governing equations are distinguished by the BCS. In other words, the flow over a car differs from the flow over a Space Shuttle exactly because of the different BCs applied.

Specifying wrong boundary conditions in the CFD imposing the solution to blown up or given a wrong Convergence solution.

Dirichlet boundary Conditions:

In mathematics, a Dirichlet problem is the problem of finding a function that solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.

The value of the dependent variable is specified on the boundary.

y=f

For example, if one end of an iron rod is held at absolute zero, then the value of the problem would be known at that point in space

Neumann boundary conditions:

A boundary condition that specifies the value of the normal derivative of the function is a Neumann Boundary Condition or second-type boundary condition.

The normal derivative of the dependent variable is specified on the boundary.

dy/dx = f

For example, if there is a heater at one end of an iron rod, then energy would be added at a constant rate but the actual temperature would not be known.

Robin boundary conditions:

Robin boundary conditions are a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions. This contrasts with mixed boundary conditions, which are boundary conditions of different types specified on different subsets of the boundary. Robin boundary conditions are also called impedance boundary conditions, from their application in electromagnetic problems, or convective boundary conditions, from their application in heat transfer problems

It is imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain.

Cauchy Boundary Condition:

The Cauchy Boundary Conditions augment an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and normal derivative on the boundary of the domain. This corresponds to imposing both a Dirichlet Boundary Condition and a Neumann boundary condition.

Cauchy boundary conditions are simple and common in second-order ordinary differential equations.

What is meant by Shock Wave?

When the speed of the moving object or source in a medium exceeds the speed of the sound, then the wavefronts lag behind the object forming a cone-shaped region with the source at the Vertex behind the object. The edge of the cone forms a supersonic wavefront with a large amplitude which is called a Shock wave.

The Shock Wave carries the energy. Across a shock wave, the Mach number decreases, the static pressure increases, and there is a loss of total pressure because the process is irreversible. Through an expansion fan, the Mach number increases, the static pressure decreases, and the total pressure remain constant. Expansion fans are isentropic.

Shock waves are not conventional sound waves; a shock wave takes the form of a very sharp change in gas properties. Shock waves in the air are heard as a loud "crack" or "snap" noise

Importing Geometry and Assign the boundary:

Import the geometry into the Converge studio and it is already cleaned up so we do need not to clean up the geometry.

Enable the vectors on the geometry and make sure that all the normals should be facing inside the fluid domain The below figures shows the vectors outside the domain to convert them into inside where the fluid is going.

Now check for the errors in the geometry by opening the diagnosis dock and then find if no errors are there means we can proceed to the case setup.

Now Assign the boundaries name to the geometry

Case setup for Simulation:

• Application Type – Time-Based
• The flow is the general flow that contains the air as a Material.
• Air contains a Species mixture of O2 and N2 which will flow through the required geometry.
• Global Transport Parameters the Prandtl number is 0.9 and the Schmidt number is 0.78.

Simulation Parameters:

• In the Run Parameter: Solver we used is Steady state and the simulation mode = Full hydrodynamic in the mic state we enable the Solver Momentum, Solve Species.
• In Simulation time Parameters we set the values for start time as 0 cycles and end time as 25000 cycles and the initial and Minimum time setup for 1e-9 and remaining values are put to default.
• In the Solver Parameters, the Navier stock equation should be Density based solver, and the remaining should be the default.

Initialization and Boundary Conditions:

• In the initialization, region changes the name of the Region as a volumetric, and the values assigned for the Velocity as 680 m/sec temperature and pressure is 300k and 50000 Pascal and add the species O2 and N2 with the mole fraction of 0.23 and 0.77.

Inlet Boundary Conditions:

• The boundary type is INFLOW
• Pressure Boundary Conditions- Zero Normal Gradient (Neumann Boundary)
• Velocity Boundary Conditions-680 m /sec and the velocity of sound is 340 m/sec

Mach number =Velocity of the object / velocity of the sound =680 / 340 = 2

• The temperature should be 286.1k and
• Add species 02=0.23 N2=0.77

Outlet Boundary Conditions:

• The boundary type is OUTFLOW
• Pressure Boundary Conditions- Zero Normal Gradient (Neumann Boundary)
• Velocity Boundary Conditions- Zero Normal Gradient (Neumann Boundary)
• The temperature should be 300 k and
• Add species 02=0.23 N2=0.77

Front, Back, Top, and Bottom Boundary Conditions:

• For the Front wall, the boundary should be the 2D wall type.
• For the Back wall, the Boundary type should be the 2D wall type.
• For the Top and Bottom walls, the boundary type is WALL type with the slip condition as a boundary condition and the temperature should be 300k.

Physical Models:

• In the physical models enable the Turbulence Modelling and the solver should be selected as RNG K-e and put the remaining values as a default.

Grid Control:

• Element size dx=dy=dz=0.8 m

Adaptive Mesh Refinement:

In numerical analysis, adaptive mesh refinement (AMR) is a method of adapting the accuracy of a solution within certain sensitive or turbulent regions of simulation, dynamically and during the time the solution is being calculated. When solutions are calculated numerically, they are often limited to pre-determined quantified grids as in the Cartesian plane which constitutes the computational grid, or mesh. Many problems in numerical analysis, however, do not require a uniform precision in the numerical grids used for graph plotting or computational simulation and would be better suited if specific areas of graphs that needed precision could be refined in quantification only in the regions requiring the added precision. Adaptive mesh refinement provides such a dynamic programming environment for adapting the precision of the numerical computation based on the requirements of a computation problem in specific areas of multi-dimensional graphs which need precision while leaving the other regions of the multi-dimensional graphs at lower levels of precision and resolution.

In our case, we need more refinement of mesh at the shock were occur in that region we need more refinement of mesh in that area and the remaining should be the normal grid size mesh.

In converge we have the different methods which are used for the Adaptive Mesh Refinement Right now we are using the Temperature-based Sub-Grid-Criterion method with the 2 embedded layers. Thus the convergence helps the curvature of the temperature property provided in AMR and starts the refining of the mesh for the temperature curvature by checking the values provided in the SGS where the valve is greater than the provided valve in SGS. This property helps us with the cost of the computational domain and saves time by not doing unnecessary meshing for the entire domain.

• In the Grid control enable the Adaptive Mesh Refinement
• In Adapative Mesh Refinement the volumetric region is changed from the Available region to the Active Region.
• Then Go to Temperature
• Enter the number of Embedded level=2
• Sub_Grid_Criterion=0.001 k (Use different valves of SGS to study the changes in the problem)(the values are 0.01 and 0.1)

Output and Post Processing:

• Post-variable Selection and Output files are the default values.
• In the output files, we have given a time interval to write the 3D output data=250 cycles, thus the outputs of the simulation files are stored after 250 cycles.

Post-Processing:

I have done the post-processing of the results for all three cases using converge studio and ParaView. Post-processing is actually used in order to get the desired Output by running the simulation

Results:

Case –I for SGS Valve of 0.001:

Mesh Profile for SGS Valve 0.001:

Mesh size of 0.8 m along in all directions with the Adaptive mesh refinement of 0.001. we can observe the AMR with the blue-coloured mesh.

Mesh animation:

Temperature Contour for SGS Valve 0.001

The temperature contour shows the formation of the expansion fan at the expansion point.

Temperature Animation:

Velocity Contour for SGS Valve 0.001:

The Velocity contour shows the formation of the expansion fan at the expansion point.

Pressure Contour for SGS Valve 0.001:

The Pressure contour shows the formation of the expansion fan at the expansion point.

Plot for Average Mach number of SGS Valve 0.001:

Plot for Average Temperature of SGS Valve 0.001:

Plot for Static Pressure of SGS Valve 0.001:

Plots for the Total cell counts of SGS Valve 0.001:

• The total count of cells in this simulation is around 20,000 cells and the solution is reached convergence after the 20,000 cycles and it is a Steady-state Simulation.
• Apart from the total count cells, The Mach number of the flow is increasing after passing through the expansion fan, and the remaining properties like Static Pressure, temperature are declined we can clearly observe in the above plots

Case –II for SGS Valve of 0.01:

Mesh Profile for SGS Valve 0.01:

Mesh size of 0.8 m along in all directions with the Adaptive mesh refinement of 0.01

Mesh animation:

Temperature Contour for SGS Valve 0.01

The temperature contour shows the formation of the expansion fan at the expansion point

Temperature Animation:

Velocity Contour for SGS Valve 0.01:

The Velocity contour shows the formation of the expansion fan at the expansion point.

Pressure Contour for SGS Valve 0.01:

The Pressure contour shows the formation of the expansion fan at the expansion point.

Plot for Average Mach number of SGS Valve 0.01:

Plot for Average Temperature of SGS Valve 0.01:

Plot for Static Pressure of SGS Valve 0.01:

Plots for the Total cell counts of SGS Valve 0.01:

• The total count of cells in this simulation is around 16,000 cells and the solution is reached convergence after the 14,000 cycles and it is a Steady-state Simulation.
• Apart from the total count cells, The Mach number of the flow is increasing after passing through the expansion fan, and the remaining properties like Static Pressure, temperature are declined we can clearly observe in the above plots

Case –III for SGS Valve of 0.1:

Mesh Profile for SGS Valve 0.1:

Mesh size of 0.8 m along in all directions with the Adaptive mesh refinement of 0.1

Mesh animation:

Temperature Contour for SGS Valve 0.1

The temperature contour shows the formation of the expansion fan at the expansion point.

Temperature Animation:

Velocity Contour for SGS Valve 0.1:

The Velocity contour shows the formation of the expansion fan at the expansion point.

Pressure Contour for SGS Valve 0.1:

The Pressure contour shows the formation of the expansion fan at the expansion point.

Plot for Average Mach number of SGS Valve 0.1:

Plot for Average Temperature of SGS Valve 0.1:

Plot for Static Pressure of SGS Valve 0.1:

Plots for the Total cell counts of SGS Valve 0.1:

• The total count of cells in this simulation are in between 5200 -5400 cells and the solution is reached convergence after the 20,000 cycles it is a Steady-state Simulation.
• Apart from the total count cells, The Mach number of the flow is increasing after passing through the expansion fan, and the remaining properties like Static Pressure, temperature are declined we can clearly observe in the above plots

Effect of SGS on Total Cell Count:

• From the above all three plots we can clearly observe that as the SGS valve decreases then more refinement of cells takes place thus it indicates the increases in the cell count.
• For the low SGS valve, the number of cells is more due to higher refinement means a higher number of cells.

Conclusion:

• Different values of SGS are performed and observe the effects of the SGS valve on different parameters are.
• With the Fine mesh i.e low Valve of SGS, we can clearly observe the expansion fan with a little width.
• Simulation of Prandtl Meyer number is done
• As we observed in the above plots Across the expansion fan, the flow accelerates (velocity increases) and the Mach number increases, while the static pressure, and temperature decrease.