Week 5: Prandtl Meyer Shock problem

OBJECTIVE: To study the prandlt Shock flow problem with the given geometry using COnverge CFD and Cygwin to run the simulations.

SOFTWARE USED: Converge CFD, Cygwin (Simulation Running)

LITERATURE:

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.

The expansion fan consists of an infinite number of expansion waves or mach lines. The first Mach line is at an angle with respect to the flow direction, and the last Mach line is at an angle with respect to final flow direction. Since the flow turns in small angles and the changes across each expansion wave are small, the whole process is isentropic. This simplifies the calculations of the flow properties significantly. Since the flow is isentropic, the stagnation properties like stagnation pressure , stagnation Temperature and stagnation density remain constant. The final static properties are a function of the final flow Mach number and can be related to the initial flow conditions as follows,

The Mach number after the turn is related to the initial Mach number

where, is the prandlt-meyer Function. This function determines the angle through which a sonic flow (M = 1) must turn to reach a particular Mach number (M). Mathematically,

By convention, {\displaystyle \nu (1)=0.\,}

Thus, given the initial Mach number (), one can calculate and using the turn angle find . From the value of one can obtain the final Mach number () and the other flow properties.

Boundary Conditions

1. Dirichlet Boundary Conditions

The value of the function is specified to the boundary of the case setup.

y = f

2. Neumann Boundary Condition

For neumann boundary conditions we specify the normal derivative of the case setup to the boundary

`del y/n` = f

3. Robin Boundary Conditions

This is the combination of the neumann and dirichlet boundary Conditions.

Supersonic Outlets

Neumann Condition is used for the boundary condition for supersonic outlets as the values (density, volume etc.) are not availible before solving the problem. So nothing is specified in supersonic outlet and all variables are extrapolated from domain interior.

SIMULATIONS

For this simulation we have selected a time based simulation.

We firstly Import the file to the converge workspace.

We then have to flag the boundary after creating different boundary conoditions.

In the above picture we can see the boundaries made for this geometry.

Inlet and outlet would be the inflow and outflow boundary conditions for this Model.

For this we have also specified a supersonic speed of 638 m/s.

This is a 2D simulations so we have given the back and the front walls as 2D outputs for this simulations.\

The top and bottoom wall is the slip boundary for this simulation.

Adaptive Mesh Refinement

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 constitute 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 which 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.

Run parameters for the Simulation.

Simulations time Parameters.

Temperature Contour for the shock problem

In the above image we can see that the temperature at the shock area is remaining constant thoughout the area.

Velocity Contour

We can observe in the video that the shock spreads evenly around the geometry. Due to the adaptive mesh refinement, we can see that the flow spreads evenly around the centre area.

Pressure Contour.

Mass Flow Rate Comparison

Total Pressure Plot at velocity 630 m/s

We can see iin the comparison fromthe 2 plots that there is not much change in the pressure in the inlet to th eeoutlet. This proves that the solutions have converged for this simulations for the pressure for this shock flow problem.

CONCLUTIONS

In Supersonic Flow the prandlt meyer expansion fan and the expansion line is also visible in the shock region.

on the other hand, in case of the subsonic flow, the region for the simulation is not visible. due to the absence of fluctuations in pressure this is observed in a subsonic flow.

The Link for the file is :

https://drive.google.com/drive/folders/1B-VPwiEIIGdAQRMk3WgfpllvztNiuJcM?usp=sharing