Prandtl Meyer Shock wave Case study

PRANDTL MEYER SHOCK WAVE SIMULATION

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.

In this project we are going to simulate this shock wave problem in a model specifically designed with a convex corner. We will simulate for four specific cases, by changing the Sub-grid criterion(SGC) in the adaptive mesh refinement tab under base grid and a subsonic flow input value. The different values to be tested are:

1. SGC = 0.05 Kelvin
2. SGC = 0.01 Kelvin
3. SGC = 0.1 Kelvin
4. Subsonic flow velocity = 280 m/s

We know that at normal temperature, the speed of sound is approximately 340 m/s. Hence for our supersonic inlet velocity, we take 680 m/s as the input velocity. This results in a Mach number of 2.0. Before moving on to case setup, let\'s us do a literature review about what type of boundary conditions are assumed as default for shock wave problems.

LITERATURE REVIEW

What is a shockwave?

When a source moves faster than the waves it produces, no waves will form in front of the source but will pile up behind and become compressed. The waves protruding are confined to a cone that narrows as the speed of the source increases and the waves bunch up, creating high-pressure regions outside the compressed waves. This border from inside to outside is the shock wave. The strength of a shock wave dissipates greatly with distance, much more so than a regular wave, as heat and other energy are more quickly transferred into the surrounding environment. Once enough energy has dissipated, the shock wave will become a regular wave such as a sound wave.

Boundary conditions:

There are three broad classes of boundary conditions:
a) Dirichlet boundary conditions: The value of the dependent variable is specified on the boundary.
b) Neumann boundary conditions: The normal derivative of the dependent variable is specified on the boundary.
c) Cauchy boundary conditions: Both the value and the normal derivative of the dependent variable are specified on the boundary.

Other boundary conditions can also be used a mix of one or two known boundary conditons.

As we give input values of velocity of flow into the problem, that is a Dirichlet boundary condition, where the value of the dependent variable is specified directly. We could also give pressure outflow as input values too, but that is applicable only for subsonic flows. When it comes to supersonic flows, the values of pressure outflows are negelected and the values are derived from the flow interior itself. That\'s why we have to give Neumann boundary conditions for shock wave problems.

Shock waves basically occur when the waves propagate faster than the speed of sound and forms a cone like structure, restricted or compressed because of the faster wave propagation. Basically after the waves reach a lower velocity, they become subsonic and dissipate as normal waves rather than shock waves.

CASE SETUP

The model designed is imported in CONVERGE through a STL file format:

As we had discussed earlier, the convex corner produces a sudden shift in which the supersonic flow is \"flipped\" against itself. Unlike normal shock waves where propagating waves and shocks are normal to each other, a special class known as oblique shock waves are produced in this kind of setup. The Prandtl Meyer expansion fan is the normally refered term here.

Given changes in parameters are:

• Fluid medium: Air
• Inlet velocity: 680 m/s in X direction
• Outlet: Neumann Boundary Conditions given. (All values are assigned as Zero gradient)
• Grid size: dx = dy = dz = 0.8m for the given scale.
  • Adaptive mesh refinement:
    • Embedding level: 2
    • SGC: All three cases - 0.05, 0.01, 0.1 K
    • Select available regions and take it to active regions to activate
    • We can select AMR with respect to Temperature or Velocity. Here we have selected Temperature for our project.
    • Maximum cells: 200000

We have to initialize velocity values in the regions tab too.

• Solver used: Density-based

Now, there is a reason why the solver used must be density-based. A solver is required to solve the transport equations in order to obtain the results.

Converge has 3 solvers, 2 for the steady state solver and one for transient state solver.

For the steady state solvers, there are 2 types,

• • Pressure based steady state solver &
  • Density based steady state solver.

In general, the pressure based solver is used for incompressible flows whereas the density based solver is used for high speed flows, generally ones with Mach number greater than 0.3 in order to account for the effects of compressibility.

Both solvers use the momentum equation to obtain the velocity field. While the pressure based solver obtains the pressure field by solving an equation obtained by the manipulation of the momentum and continuity equations, the density based solver obtains the pressure field from the equation of state, with the continuity equation providing the density.

Though the density based solver was designed for high speed compressible flows, it also holds good for low speed flows. Using the pressure based solver for high speed flows may give a warning as shown below since it treats the flow as incompressible and density is taken to be constant, which it is not. The solver still runs and solves the equations, though incorrectly since the wrong assumptions are made.

Hence, for this problem, we select density-based solver.

• Time parameters:
  • Minimum time step: 1e-7 secs
  • Initial time step: 1e-7 secs
  • Maximum time step: 1 secs
  • Cycles: 0-25000
• Processors used: 4
• All other parameters are default.

To set up the subsonic flow case, just change the inlet velocity values at boundary tab and regions tab too.

After the case has been completely set up with appropriate boundaries, the input files are exported to the specific directory-> shock_flow -> sgs_0.05

Then, converge executable is run through Cygwin terminal. After the output files have been written, they are post processed using CONVERGE itself.

After that, group files in .vtm format are loaded in paraview to view the 3D output files and simulation configured.

RESULTS

The simulation results are rendered using Paraview after post processing using CONVERGE.

Mesh

CASE 1: SGC = 0.05 K

With \'surface with grids\' display, we can see how the adaptive mesh refinement changes dynamically with respect to how the simulation acts on temperature and velocity differences during each time step. A more generic look would be the surface display alone, were the AMR is not visible.

As we can see, the green color code is the shockwave propagating obliquely, with respect to the convex angle of depression of the geometry. Hence, the angle of depression is also a criterion for this type of profile. The theory matches with the simulation as the velocity or Mach entering at inlet, M1, is dispersed across by the shockwave, compressed into mach waves unconuntable, and a higher velocity, M2 goes across the depressed surface.

Pressure

During simulation, we have to rescale it again and again because the pressure is constantly being derived from the interior flow due to Neumann BCs. The pressure differences are clear with this contour plot. There is a significant pressure drop as the flow moves along, after the oblique shock wave.

CASE 2: SGC = 0.01 K

Considering the reduction in SGC number, we can clearly see how the AMR has adapted for even very minor changes in temperature. This results in a larger region covered with very fine mesh, which gives more accurate data, but might take more time to compute due to fine mesh. The surface display option hasn\'t changed much but the image is far more smoother due to a finer mesh and sensitivity of SGC reflecting on the solution.

CASE 3: SGC = 0.1 K

As we can expect, a larger AMR number is less sensitive to changes in temperature and eventually velocity, and hence even though we get a fairly accurate profile, it is not smooth and not as accurate as a finer mesh.

Effect of SGC on AMR:

Therefore, the value of AMR with respect to temperature denotes how sensitive will the adaptive mesh become, with temperature changes along the simulation for each time step. This determines how accurate a simulation can be, since a smaller AMR might include finer mesh as we process and smoother results. The only drawback might be the increase in time taken to complete the simulation due to dynamic moving mesh. For this case however, the time taken is faster than previous simulations due to lesser moving mesh. The shock location doesn\'t change much in every case considered.

CASE 4: Subsonic = 280 m/s

For this case, the default SGC value 0.05K is used. The subsonic velocity is simulated here to show the effects of shockwave generation failure when the mach number is less than 1. Here as the mach number is almost equal to 1, we could see a faint attempt of shockwave generation but is the faded or dispersed along with the air. A lower value of subsonic velocity will eventually show an almost unchanged velocity profile with slight changes in the convex area of cross section. Even AMR isn\'t present in the expansion fan zone as there is no significant activity happening in the vicinity.

PLOTS

Cell counts:

As we can see from the cell count plot, the highest cell count is for the finer mesh generated values or low AMR\'s, i.e, 0.01 Kelvin. The lowest cell count needed is for the subsonic flow simulation as it needs less computational work and shockwave generation doesn\'t happen. This explains that the more computational load comes from finer mesh were every detail is counted in the simulation and the dynamic moving mesh area is larger.

Mach number:

The two velocities input are 680 and 280 m/s which have distinct mach number variations as it should have. As we can see, 280 m/s is almost 0.9 mach, lesser than 1 but almost equal and that\'s why it was selected to see how the shock wave generation propagates only after mach number crosses 1.

ANIMATION:

This consists the animations of each SGC case studied in the project in the \'surface with grid\' display to see the AMR in action. The animations are merged in the respective order of SGC: 0.1, 0.05, 0.01 Kelvin.