Prandtl mayer shock problem

Objective:

The objective of this project is to setup a Prandtl Meyer Shock problem and solve using Converge Flow Solver with different inlet boundary and initial condition:

• Subsonic at 100 m/s
• Supersonic at 680 m/s

The overview of this project is to understand the following:

• What is Shock Wave?
• About shock flow boundary conditions
• Effect of SGS parameter on shock location
• Effect of SGS temperature value on cell count and shock location

Boundary conditions in CFD:

Neumann and Dirichlet boundary conditions:

• When using a Dirichlet boundary condition, one prescribes the
  value of a variable at the boundary, e.g. u(x) = constant.
• When using a Neumann boundary condition, one prescribes the
  gradient normal to the boundary of a variable at the boundary,
  e.g.`del_n u(x) = constant`
• When using a mixed boundary condition a function of the form
  `au(x)+bdel_n u(x) = constant`
   Note that at a given boundary, different types of boundary
  conditions can be used for different variables.

Shock wave:

In physics, a shock wave , or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a medium but is characterized by an abrupt, nearly discontinuous, change in pressure, temperature, and density of the medium.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.

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 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 theory was described by Theodor Meyer on his thesis dissertation in 1908, along with his advisor Ludwig Prandtl, who had already discussed the problem a year before.

 In order to understand Prandtl Meyer shock wave, we first need to understand what oblique waves are. The normal shock waves are straight in which the flow before and after the wave is normal to the shock. It is considered as a special case in the general family of oblique shock waves that occur in supersonic flow. In general, oblique shock waves are straight but inclined at an angle to the upstream flow and produce a change in flow direction. An oblique shock generally occurs, when a supersonic flow is ‘turned into itself”.

Another class of two dimensional waves occurring in supersonic flow shows the opposite effects of oblique shock. Such types of waves are known as expansion waves . When the supersonic flow is “turned away from itself”, an expansion wave is formed. Here, the flow is allowed to pass over a surface which is inclined at an angle θ to the horizontal and all the flow streamlines are deflected downwards. The change in flow direction takes place across an expansion fan centered at point ‘A\'. The flow streamlines are smoothly curved till the downstream flow becomes parallel to the wall surface behind the point ‘A\'. Here, the flow properties change smoothly through the expansion fan except at point ‘A\'. An infinitely strong oblique expansion wave may be called as a Mach wave . An expansion wave emanating from a sharp convex corner is known as a centered expansion which is commonly known as Prandtl-Meyer expansion wave.

Mach Number

Mach number is a dimensionless quantity defined as the ratio of velocity of flow to the local speed of sound. In aerodynamic and fluid dynamic applications, Mach number and Reynolds number are the important parameters related to compressibility and viscosity .

Classification of flow based on Mach number –

1. Subsonic: M < 0.8
2. Transonic: M belongs to 0.8 – 1.3
3. Supersonic: M belongs to 1.3 – 5.0
4. Hypersonic: M belongs to 5.0 – 10.0
5. High-Hypersonic: M belongs 10.0 – 25.0
6. Re-entry Speeds: M > 25.0

In Prandtl-Meyer Shock Flow, supersonic waves are made to pass through the inlet of a duct that has a sharp corner. An infinite number of Mach Waves are created which spread out over a wide area or fan out creating a pattern that will be shown by simulating the flow in Converge Studio.

WORKFLOW FOR A CONVERGE CFD SIMULATION:

The workflow in CONVERGE consists of three steps:

1. Pre-processing (preparing the surface geometry and configuring the input and data files)
2. Running the simulation
3. Post-processing (analyzing the *.out ASCII files in the Case Directory and using a visualization program to view the information in the post*.out binary files saved in the output subdirectory).

1. Pre-processing:

Pre-processing involves three sub-steps

a. Preparing the surface geometry

The geometry dock provides us the option to create our desired geometry. In this case, it is the throttle valve. Converge studio though allows us to create a geometry it also allows user to import a geometry made in a cad program.

Steps for importing geometry: File>import>import STL

the window after importing geometry

Setting up the case:

Converge provides us with a handy wizard like feauture for setting up the case as sbown below:

1.  It is a general flow case.
2. Oxygen (O2) and Nitrogen (N2) are the two main species.
3. Solver: Density based steady state solver.
4. Simulation time parameters: 
   • Number of cycles: 25000
   • Initial and minimum time step: 1e-9
   • Maximum time step: 1
5. Boundary conditions:

   a.) Inlet: Inflow type; At a temperature of 286.1 Kelvin,velocity of air is 340 m/s, velocity along x-axis was kept 680 m/s making the inlet Mach number as 2 (680/340).

   b.) Outlet: Outflow type. Neumann Boundary Condition was used which is dependent on the initial conditions of the flow.

   c.) Top & Bottom: Wall boundary type with \'Slip\' condition. The slip condition causes the fluid to flow along a wall instead of stopping at the wall, which typically occurs along a wall. Fluid is prevented from flowing through the wall, however. 

   d.) Front & Back: Since the case was setup for 2-D flow, \'TWO-D\' conditions were assigned.

6. Regions and initialization: Species O₂ =23%; N₂=77%; Velocity =680 m/s Temperature= 300 K; Pressure= 50000 Pa.

7.Turbulence Modelling:RNG kε – Default values considered.

8. Base Grid: dx = dy = dz = 0.8

9.Adaptive Mesh Refinement:

Adaptive Mesh Refinement technique was used because the region where shocks would occur require the mesh to be refined locally. The mesh in the other regions have the same size as the base grid size whereas the region where shock occurs finer cells are used. Temperature based mesh refinement was used that would monitor curvature in temperature. Curvature means the `(del^2t)/(delx^2)`. When curvature is greater than a particular value defined by the user the mesh is refined. In the first run, SGS was taken as 0.05 K. So, when sub-grid temperature would be greater than 0.05 K, the mesh would refine. 

If the refinement level/max embedding level is 2(n), then Grid-Size would be, Grid-Size= base grid (0.8 m)/2^n(2), which would be equal to a grid-size of 0.2 m. That would be the smallest cell size used by converge in mesh refinement.

Maximum Cells = 200000

Active Regions = Region 0

AMR type = either based on Temperature or Velocity

Sub-grid criterion

case 1 : 0.05

case 2 : 0.03

case 3 : 0.02

Embed scale = 2

10.Select the required output variables like pressure, density, temperature etc. whose results need to be analyzed.

After setting up the case, the input files are exported in a directory.

The final geometry after applying the boundary conditions is shown below:

2. Running the simulation:

        After exporting all the required input files the simulation is ran by using cygwin.

        Cygwin is a POSIX-compatible environment that runs natively on Microsoft Windows. Its goal is to allow programs of Unix-like systems to be recompiled and run natively on Windows with minimal source code modifications by providing them with the same underlying POSIX API they would expect in those systems.

        The executable file provided by the convergent science is executed using cygwin with the help of Microsoft Message Passing Interface (MSMPI).

        Microsoft Message Passing Interface is an implementation of the MPI-2 specification by Microsoft for use in Windows to interconnect and communicate between High performance computing nodes.

        The output files are generated in the same folder where the input files are executed using the executable.

Time required for the simulation and the summary of usage is as folllws:

Case 1 : SGS = 0.05

Program used 800.506564 seconds.

Summary of total time for:
load balance = 14.05 seconds ( 1.76%)
solving transport equations = 568.61 seconds (71.03%)
move surface and update grid = 48.12 seconds ( 6.01%)
update boundary conditions = 39.90 seconds ( 4.98%)
combustion = 0.01 seconds ( 0.00%)
spray = 0.02 seconds ( 0.00%)
writing output files = 83.09 seconds (10.38%)

Case 2 : SGS = 0.04

Program used 903.606314 seconds.

Summary of total time for:
load balance = 17.16 seconds ( 1.90%)
solving transport equations = 647.89 seconds (71.70%)
move surface and update grid = 48.58 seconds ( 5.38%)
update boundary conditions = 44.90 seconds ( 4.97%)
combustion = 0.00 seconds ( 0.00%)
spray = 0.02 seconds ( 0.00%)
writing output files = 93.52 seconds (10.35%)

Case 3 : SGS = 0.03

Program used 1568.171756 seconds.

Summary of total time for:
load balance = 24.35 seconds ( 1.55%)
solving transport equations = 1148.09 seconds (73.21%)
move surface and update grid = 59.22 seconds ( 3.78%)
update boundary conditions = 73.96 seconds ( 4.72%)
combustion = 0.01 seconds ( 0.00%)
spray = 0.02 seconds ( 0.00%)
writing output files = 176.74 seconds (11.27%)

3) Post-processing:

• Meshing: Mesh is generated for the given STL file as shown below.

Case 1 :SGS = 0.05K

Case 2 :SGS = 0.04K

Case 3 :SGS = 0.03K

It can be seen clearly from the above images that as the sub grid criterian temperature valuve is reduced the AMR generates very fine cells in the regions of flow.

Cell counts:

Three cases have been setup with different SGS temperature namely 0.05,0.04 and 0.03 respectively. The total cell count for each case is shown below.

Changing the SGS values changes the number of cells that are formed during meshing. As mentioned above when the sub-grid temperature becomes greater than SGS value, the mesh is refined. Thus for a low value of SGS(0.03), more cells are refined than at a higher value of SGS (0.05).

Pressure, Temperature and velocity contours:

Case 1: SGS = 0.05

a. Pressure contour:

b. Velocity contour:

c. Temperature contour:

Case 2: SGS = 0.04 K

a. Pressure contour:

b. Velocity contour:

c. Temperature contour:

Case 3: SGS = 0.03

a. Pressure contour:

b. Velocity contour:

c. Temperature contour:

effect of subsonic inlet velocity:

When the inlet velocity of the flow is made to be subsonic there are no shock waves seen to be emanating from the convex point, instead it can be clearly seen that a small pressure drop can be located at the point of origin of the shock waves which occured in the supersonic initial condition. Thus the pressure contour of the subsonic flow condition is shown below from which the point of location of the origin of shock wave is seen.

Conclusions:

1. Prandtl meyer expansion fan was simulated using he given geometry successfully.
2. pressure, velocity, density and mach number change is studied.
3. various boundary conditions and their significance is studied.
4. Effect of Sub grid scale temperature on the mesh is understood.
5. Understood the importance of adaptive mesh refinement and used it to visualize the prandtl meyer expansion fan.

Link for youtube video:

https://youtu.be/EDqQ_KI4k0Q