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 fa, 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 singlefshocle 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 pressurebernperature and density decrease. Since process H is entropic, 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 for M to accelerate supersonic flow in what nos 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 Mn through a series of isentropic Mach waves. The expansion Ion' 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 comer equally, as the increase in Mach number is proportional to only the convex angle of the passage (b). rho expansion corner that produces the Prandtl-Meyer fan can be sharp (as illustrated in the figure).
Prandtl-Meyer compression fans:
A Prandtl-Meyer compression is an opposite phenomenon to a Pron. -Meyer expansion. If the flow is gradually
turned through an angle of 8, 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 blow that travels through the fan) and an isentropic region (floe, that travels through the oblique shock), o slip line results between the two flow regions.
Boundary conditions:
In CFD Boundary Conditions (cc) are equally important as the governing quaff one even S most textbooks deal predominantly with the governing equations only. From the mathematical point of view, 130.5 are inherently linked with SCE, A PDS must have BC, to obtain a unique solution.ln CEO, 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 ac's applied.
Dirichlet boundary conditions:
In mathematics, a Dirichlid problem is a problem of finding a function that solves a specified partial differential equation (POE) 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
Neumann boundary conditions:
A boundary condition that specifies the value of the normal derivative of the function is Neumann Boundary Condition or second-type boundary condition. The normal derivative of the dependent variable is specified on the boundary.
dy/dx = f
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.
Cauchy Boundary Condition:
The Cauchy Boundary Conditions augments an ordinary differential equation or partial differential equation with conditions that the solution must satisfy on the boundary ideally so as 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.
Shock wave:
When the speed of the moving object or source in a medium exceeds the speed of the 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 St the cone forms a supersonic wavefront with a large amplitude which is called a Shock wave.
The Shock Wove 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 fon, the Mach number increases, the static pressure decreases, and the total pressure remain constant Expansion fans are isentropic.
Importing Geometrgn the boundary:
Import the geometry into the Converge studio and it is already cleaned up so we need not clean up the geometry.
cases setup for simulation:
• Application Type - Time-Based
• The flow is general now which contains the air as a Material.
• Air contains a Species mixture of 02 and N2 which will now through the required geometry.
• Global Transport Parameters the Prandtlnurnber is 0.9 and the Schmidt number is 0.73.
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 le-9 and remaining values are put to default
• In the Solver Parameters, the Navier stock equation should be a Density-based solver, and the remaining should be the default.
Initialization and Boundary Conditions:
• In the initialization region change the name of the Region as a volumetric and the values assigned for the
Velocity as 680 m/s temperature and pressure is 300k and s0000 Pascm and add the species 02 and NO
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/s
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 boundary should be the 20 wall type
• For the Bock wall Boundary type should be the 20 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
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 a better suit., 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 as 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 I multi-dimensional graphs at lower levels of precision and resolution.
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 converge helps the curvature of the temperature property provided in OMR and starts the refining of the mesh for the temperature curvature by checking the values provided in the SOS where the valve is greater than the prodded valve in Set Ibis property helps us in the cost of the computational domain and saves the time by not doing the unnecessary meshing for the entire domain.
Output and post-processing:
• Post variable Selection and Output files are the default values.
• In the output files we have given time intervals to write the 3D output data=250 cycles, thus the outputs of the
simulation files are started after 250 cycles.
Post-Processing:
I have done the post-processing of the results for all three cases using the converge studio and ParaView. Post-processing is actually used in order to get the desired Output by running the simulation.
Results:
Case-1 for sgs Value of 0.001:
Mesh Profile for sgs Valve 0.001:
Temperature contour for sgs valve 0.001:
Velocity contour for sgs valve 0.001:
Pressure contour for sgs valve 0.001:
Plots for Average Mach number pf sgs valve 0.001:
Plots average temperature of sgs valve 0.001:
Plots for static pressure of sgs valve 0.001:
Plots for the total cell count of sgs valve 0.001:
Case 2 for sgs valve of 0.01:
Temperature contour for sgs valve 0.01:
Velocity contour for sgs valve of0.01:
Pressure contour for sgs valve of 0.01:
Plots for Average Mach number of sgs valve 0.01:
Plots for Average temperature of sgs valve 0.01:
Plots for static pressure of sgs valve 0.01:
Plots for the total count of sgs valve 0.01:
Case 3 for sgs valve 0.1:
Mesh profile:
Temperature contour of sgs valve 0.1:
Velocity contour for sgs valve 0.1:
Pressure contour for sgs valve 0.1:
Plot for Average Mach number for sgs valve 0.1:
Plots for Average temperature of sgs valve 0.1:
Plot for static pressure of sgs valve 0.1:
Plots for a total count of sgs valve of 0.1:
