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 shock expansion:

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 supersonicflow 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 physicallyimpossible for the flow to turn through a single "shock" wave because this would violate thesecond law of Thermodynamics.

Across the expansion fan, the flow accelerates (velocity increases) and the Mach numberincreases, while the static pressure,temperature and density decrease. Since the processis isentropic, the stagnation properties (e.g. the total pressure and total temperature)remain constant across the fan.

To this point, the only flow phenomena that have been discussed are shock waves, whichslow the flow and increase its entropy. It is possible to accelerate supersonic flow in whathas 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 ofisentropic Mach waves. The expansion "fan" is composed of Mach waves that span fromthe initial Mach angle to the final Mach angle. Flow can expand around either a sharp orrounded corner equally, as the increase in Mach number is proportional to only the convexangle of the passage (δ). The expansion corner that produces the Prandtl-Meyer fan canbe sharp (as illustrated in the figure) or rounded. If the total turning angle is the same, thenthe P-M flow solution is also the same.

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

Prandtl-Meyer Compression fans:

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

Boundary conditions:

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

Specifying of wrong boundary conditions in the CFD imposing the solution to blown up orgiven a wrong Convergence solution.

Dirichlet boundary Conditions:

In mathematics, a Dirichlet problem is the problem of finding a function which solves aspecified partial differential equation (PDE) in the interior of a given region that takesprescribed 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 problemwould be known at that point in space.

Neumann boundary conditions:

A boundary condition which specifies the value of the normal derivative of the function is aNeumann 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 aconstant rate but the actual temperature would not be known.

Robin boundary conditions:

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

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

Cauchy Boundary Condition:

The Cauchy Boundary Conditions augments an ordinary differential equation or a partialdifferential equation with conditions that the solution must satisfy on the boundary; ideallyso to ensure that a unique solution exists. A Cauchy boundary condition specifies both thefunction value and normal derivative on the boundary of the domain. This corresponds toimposing 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 is exceeds the speed ofthe sound, then the wave fronts lag behind the object forming a cone shaped region with the source at the Vertex behind the object. The edge of the cone forms supersonic wavefront with a large amplitude which is called Shock wave.

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

Shock waves are not conventional sound waves; a shock wave takes the form of avery sharp change in the gas properties. Shock waves in 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 that we neednot to clean up the geometry.

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

After correcting normal:

Now check for the errors in the geometry by opening the diagnosis dock and then find if noerrors 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 general flow which 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 = Fullhydrodynamic 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 time25000 cycles and the initial and Minimum time setup for 1e-9 and remaining values are putto default.
• In the Solver Parameters, the Navier stock equation should be Density based solver andremaining should be default.

Initialization and Boundary Conditions:

In the initialization region change the name of the Region as a volumetric and the valuesassigned for the Velocity as 680 m/sec temperature and pressure is 300k and 50000Pascal 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
• 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
• species 02=0.23 N2=0.77

Front, Back, Top and Bottom Boundary Conditions:

• For the Front wall boundary should be the 2D wall type.
• For the Back wall Boundary type should be the 2D wall type.
• For the Top and Bottom wall the boundary type is WALL type with the slip condition as a boundary condition and 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 theaccuracy of a solution within certain sensitive or turbulent regions of simulation,dynamically and during the time the solution is being calculated. When solutions arecalculated numerically, they are often limited to pre-determined quantified grids as in theCartesian plane which constitute the computational grid, or mesh. Many problems innumerical analysis, however, do not require a uniform precision in the numerical grids usedfor graph plotting or computational simulation, and would be better suited if specific areasof graphs which needed precision could be refined in quantification only in the regionsrequiring the added precision. Adaptive mesh refinement provides such a dynamicprogramming environment for adapting the precision of the numerical computation basedon the requirements of a computation problem in specific areas of multi-dimensionalgraphs which need precision while leaving the other regions of the multi-dimensionalgraphs at lower levels of precision and resolution.

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

In converge we have the different methods which are used for the Adaptive MeshRefinement Right now we are using the Temperature based Sub-Grid-Criterion methodwith the 2 embedded layers. Thus the converge helps the curvature of temperatureproperty provided in AMR and starts the refining of the mesh for the temperature curvatureby checking the values provided in the SGS where the valve is greater than the providedvalve in SGS.This property helps us in the cost of computational domain and saves the timeby not doing the unnnecessary 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 regionto 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 time interval to write the 3D output data=250 cycles, thusthe outputs of the simulation files be stored after 250 cycles.

Post-Processing:

I have done the post processing of the results for all three cases using the converge studioand paraview. Post processing is actually used in order to get the desired Output byrunning 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 canobserve the AMR with the blue coloured mesh.

Temperature Contour for SGS Valve 0.001

Temperature animation:

Velocity contour for SGS Valve 0.001:

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

Pressure Contour for SGS Valve 0.001:

The Pressure contour showing 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 cells in this simulations are around 20,000 cells and the solution is reachedconvergence 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 passingthrough the expansion fan and the remaining properties like Static Pressure,temperatureare declined we can clearly observed 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

Temperature Contour for SGS Valve 0.01:

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

Temperature animation:

Velocity Contour for SGS Valve 0.01:

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

Pressure Contour for SGS Valve 0.01:

The Pressure contour showing 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 cells in this simulations are around 16,000 cells and the solution is reachedconvergence 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 passingthrough the expansion fan and the remaining properties like Static Pressure,temperatureare declined we can clearly observed 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.

Temperature Contour for SGS Valve 0.1:

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

Temperature Animation:

Velocity Contour for SGS Valve 0.1:

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

Pressure Contour for SGS Valve 0.1:

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

Plot for Average Mach number of SGS Valve 0.1:

The Pressure contour showing 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 cells in this simulations are in between the 5200 -5400 cells and thesolution 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 passingthrough the expansion fan and the remaining properties like Static Pressure,temperature are declined we can clearly observed in the above plots.

Effect of SGS on Total Cell Count:

• From the above all three plots we can clearly observed that the as the SGS valve decreasesthen the more refinement of cells takes places thus it indicates the increases in the cell count.
• For the low SGS valve the number of cells is more due to higher refinement higherrefinement means higher number of cells.

Conclusion:

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