Aim: Perform a Prandtl Meyer Expansion Simulation

Objectives:

1. Shock flow boundary conditions

Do a literature search on what BC's are typically used for shock flow problems

2. What is a shock wave?

In your own words, describe the physics behind shock waves

3. Effect of SGS parameter on shock location

In the Prandtl Meyer shock problem, look at the effect of SGS temperature value on cell count and the shock location

It is expected to show the mesh generation using ParaView.

Also, create animation and upload it on Youtube and provide the Youtube link for an animation in your report.

Theory:

Shock flow boundary conditions:

Boundary conditions bring the influence of the outside world into the simulation domain. On a theoretical level, they are constraints applied to the governing equations of the system and are categorized accordingly:

• Dirichlet boundary condition specifies the value of a variable at a boundary. It may also be called a fixed condition. Therefore, it is used when a value needs to be imposed at a boundary. For example, a no-slip condition in fluid mechanics is a Dirichlet condition because it sets the value of the velocity to zero. In solid mechanics, prescribing a certain load or displacement, and in heat transfer—setting the temperature at a surface are other examples of this type of condition.
• Neumann boundary condition specifies the derivative of a variable at a boundary. It is used rather than the actual value when a certain rate of change in value needs to be imposed for a variable. A common example in fluid mechanics is the fully developed condition at an outlet where the gradient of flow variables is set to zero. Traction conditions in solid mechanics and insulated surfaces in heat transfer are other examples.
• Robin boundary condition specifies a combination of the value and the derivative of a variable at a boundary. Therefore, these conditions are suited for more complex behavior. For example, in heat transfer, Robin conditions are used to model Newton’s law of cooling, where heat flux (derivative) is proportional to temperature (value). However, Robin conditions are not the only class of boundary conditions that utilize value and derivative to specify a condition, but this is another discussion.
• There are other boundary conditions besides Dirichlet, Neumann, and Robin that exploit certain flow features. For example, in scenarios where the domain features spatial symmetry, the size of the domain, and therefore the computational effort, is reduced by using symmetry boundaries. Similarly, periodic conditions are used to simulate periodicity effects without having to run computations on the full domain. Thus boundary conditions can be used to effectively gain an advantage.

What is a shock wave?

Shock waves are large amplitude pressure waves produced by an object moving faster than the speed of sounds like a boat or plane, or other things such as an explosion, lightning or even a moving piston.

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

LIFT AND DRAG AT SUPERSONIC SPEEDS

Air acts much differently at supersonic speeds than it does at subsonic speeds.

When an aircraft approaches the speed of sound, the airflow over the wing reaches supersonic speed before the airplane itself does, and a shock wave forms on the wing. The airflow behind the shock wave breaks up into a turbulent wake, increasing drag.

When the airplane exceeds the speed of sound, a shock wave forms just ahead of the wing's leading edge. The shock wave that formed on the wing is now at the trailing edge.

When the wing is tilted upward, a shock wave forms below its leading edge, and an expansion wave forms above its leading edge. The higher pressure behind the shock wave and lower pressure behind the expansion wave result in a single force that pushes the wing up and back.

The upward part of this force is lift; the backward part of this force is drag.

The drawings show waves produced at regular time intervals by an object moving through a fluid. When the object speed is less than the wave speed (top), the waves run out ahead. When the object speed equals the wave speed (middle), the waves pile up right in front of the object. When the object speed is greater than the wave speed (bottom), the waves form a wake.

To understand these shock waves, take a look at the diagrams below, which show waves spreading out from an object moving through a fluid. Such waves are produced continually as the object moves. Each diagram shows a series snapshots taken at regular time intervals. These waves spread out in circles, like ripples from a stone dropped into a pond, and move at the speed of the waves in the fluid. Notice that if the object moves faster than the speed of the waves, the circles form a “V”-shaped wake, like the wake of a ship or the shock waves from the nose of the F-18 in the image above.

Types of Shock waves:

• Normal or Bow shock waves:

When an airplane is in transonic flight, the shock wave that forms on top of the wing, and eventually on the bottom of the wing, is called a normal shock wave. If the leading edge of the wing is blunted, instead of being rounded or sharp, a normal shock wave will also form in front of the wing during supersonic flight. Normal shock waves form perpendicular to the airstream. The velocity of the air behind a normal shock wave is subsonic, and the static pressure and density of the air are higher. Figure shows a normal shock wave forming on the top of a wing.

• Oblique shock waves:

An airplane that is designed to fly supersonic will have very sharp edged surfaces, in order to have the least amount of drag. When the airplane is in supersonic flight, the sharp leading edge and trailing edge of the wing will have shock waves attach to them. These shock waves are known as oblique shock waves. Behind an oblique shock wave the velocity of the air is lower, but still supersonic, and the static pressure and density are higher. Figure shows an oblique shock wave on the leading and trailing edges of a supersonic airfoil.

• Expansion waves(not a shock waves):

Air at supersonic speed acts like a compressible fluid. For this reason, supersonic air, when given the opportunity, wants to expand outward. When supersonic air is flowing over the top of a wing, and the wing surface turns away from the direction of flow, the air will expand and follow the new direction. At the point where the direction of flow changes, an expansion wave will occur. Behind the expansion wave the velocity and Mach number increases, and the static pressure, temperature and density decrease. An expansion wave is not a shock wave. Figure shows an expansion wave on a supersonic airfoil.

Simulation:

The following Simulation tools will be incorporated:

1.Converge studio:- To pre-process the data that needs to simulate the case. It plays an important role in generating important 3d input files only using GUI. It does not run simulation directly, however it generates inputs files which use to run the simulation.

2. Cygwin:- Cygwin is a collection of GNU and open-source tools that provide the functionality similar to Linux distribution on windows.
3. Converge msmpi 3.0:- To run the simulation using these input files in Cygwin in series or in parallel.
4. Post convert:- To convert all the output files which are processed by converge msmpi into the files which will be readable for post-processing in Paraview.
5. Paraview :- To post-processing to obtain the contour, plots, etc. It is an open-source, multi-platform data analysis and visualization software.

Geometry Setup:

• Import the “stl” file.
• Boundary Flagging

Case Setup/Simulation Setup:

MATERIAL:

Gas Simulation:

• As only air is our working medium hence uncheck Reaction Mechanism.
• Predifined mixture: air

Global Transport Properties

• Terbulent Prandtl number: 0.9
• Terbulent Schmidt number: 0.78

SIMULATION PARAMETER:

Run Parameter:

• Solver: Pressure-based Steady-state
• Simulation Mode: Full Hydrodynamic
• Gas flow Solver: Compressible (as air is working medium)

Simulation Time Parameter:

• End time: 25000 cyc
• Start time: 0cyc
• Initial time step: 1e-09 s
• Minimum time step: 1e-09 s
• Maximum time step: 1 s

Solver Type:

• Pressure based for Mach number below 0.3 (when velocity is less than 100m/s)-Subsonic flow
• Density based solver for Mach number above 0.3 (when velocity is more than 680m/s)-Supersonic flow

BOUNDARY:

• Naming: Select faces and name as inlet, outlet, Elbow wall, Throttle.
• Bounday condition:

Inlet:

• Bounady Type: Inflow
• Pressure BC: Zero normal gradient(Neumann BC)- As we don’t know the value of pressure at a given velocity
• Velocity BC: 100m/s for subsonic flow

680m/s for supersonic flow

• Species Boundary condition: air
• Temperature BC: 286.1k

Outlet:

• Boundary type: OUTFLOW
• Pressure BC: Zero normal gradient(Neumann BC)- As we don’t know the value of pressure at a given velocity. So by assigning a Neumann Boundary condition, the pressure at a given velocity value will be calculated by Converge itself.
• Velocity BC: Zero normal gradient(Neumann BC)- As we don’t know the value of velocity after an Expansion waves as there are infinite number of expansion waves.
• Temperature Backflow: 300K.

Top and Bottom wall:

• Boundary type: Wall
• Temperature BC: Zero normal gradient(Neumann BC)- As we don’t know the value of temperature at the walls at a given velocity and the pressure which depends on the velocity again.

Front:

• Boundary type: TWO_D

Back:

• Boundary type: TWO_D

REGION & INITIALIZATION:

• Click on Region and Initialization and initialize a Region namely (Region 0).
• Assign flow variables as Velocity-100m/s for Subsonic flow and 680m/s for Supersonic flows.
• Pressure- 50,000Pa.
• Temperature-300K.
• Species: Air

Turbulence model:

• RNG k-epsilon

Base grid:

• dx=0.8m; dy=0.8m; dz=0.8m.

Enable Adaptive Mesh Refinement- (to capture the small change in flow by automatically refinement of the grid at only where there is a small change in flow):

• Select Region 0 in Active Regions under AMR group 1.
• Maximum cells: 200000 (the maximum number of cells possible in the mesh).
• Enable Temperature parameter.
• Embed type: Subgrid-scale(SGS)- SGS is use to show the small change in flow parameter (using subgrid criteria) by scale it down, depends on Max embedding level.
• Max embedding level: 2
• Formula for subgrid : Grid size after subgrid(smallest cell size after subgrid)=Base grid size/(2^Max embedding level)=0.8/(2^2)=0.8/4=0.2metre.
• Timing control type:
• Start: 5000 cycles
• End: 9e9 cycles

Post-variable selection:

Output files:

Time interval for writing 3d output data files=250cyc

Time interval for writing restarting output=250cyc

Export input files and Running the Simulation:

Now we have successfully setup the Case and need to export the input files to run the Simulation.

Run the Simulation in Cygwin using parallel processor.

Post Converting:

Now we need to convert all the output files which are processed by converge msmpi into the files which will be readable for post-processing in Paraview

Paraview:

We need to Post-process the output files to obtain the contour.

Results:

CASE-(1)For Supersonic flow:

Air at supersonic speed acts as a compressible fluid. For this reason, supersonic air, when given the opportunity, wants to expand outward. At the point where the direction of flow changes, an expansion wave will occur. Behind the expansion wave the velocity and Mach number increases, and the static pressure, temperature, and density decrease. An expansion wave is not a shock wave.

Pressure plot:

For Subgrid scale 0.05k For subgrid-scale 0.1k

As the subsonic flow is expanding from a sharp corner, there is a decrease in the pressure behind the expansion wave. Hence the pressure at the outlet is less than the inlet pressure.

Pressure Contour:

The area below the expansion waves belongs to the wake formation as there is very little pressure behind the expansion waves.

Velocity plots:

Velocity Contour:

From the above velocity plots and contour, we can see that the velocity at outlet is maximum than the inlet velocity. This is due to when the flow crosses the Expansion waves it accelerates suddenly.

Temperature plot:

Temperature Contour:

As the supersonic flow crosses the expansion waves, the temperature decreases.

Density plot:

Density contour:

As the supersonic flow crosses the expansion waves, the temperature decreases.

Total cell:

The cell counts for low SGS criteria (0.05k) is more due to more refinement than high SGS criteria (0.1k).

Mach number plot:

As the supersonic flow crosses the expansion waves, the Mach number increases.

CASE-(2)For Subsonic flow:

As the shock Waves or Expansion waves only occur in the supersonic flow because the subsonic flow does not have a high enough speed to resist the speed of the sound and hence there is no collision that means no shock or expansion waves. Hence all the flow parameters like velocity, Mach number, density, pressure, Temperature do not change suddenly as a result of shock or expansion wave, rather they change due to normal change in the flow parameter.

Pressure plot:

Pressure Contour:

The pressure at the inlet is 0.82 Pa and at the outlet is 0.84 Pa. Hence it is an approximately constant pressure.

Velocity plots:

Velocity Contour:

The velocity at the inlet is 100m/s and at the outlet is 91m/s.

Temperature plot:

Temperature Contour:

The temperature at the inlet is 290k and at the outlet is 280k.

Density plot:

There is no change in density between inlet and outlet.

Total cell:

The cell count is 4400 approximately.

Mach number plot:

The Mach number is decreasing slightly due to slight change in velocity between inlet and outlet.

Animation:

For Subgrid scale 0.05k For subgrid-scale of 0.1k

Effect of SGS parameter on shock location:

The Subgrid criteria are only applicable for Supersonic flow as there is no shock waves or expansion waves that occur in subsonic flow. From the above contours, we can see that, for the large Subgrid criteria 0.1k, the Adaptive Mesh Refinement(AMR) is not assigning over a large distance of the Expansion wave. On the other hand for low SGS criteria, there is good capture in physics at a large distance of expansion wave.

Conclusion:

• For Supersonic Flow: The Mach number and velocity increase behind the Expansion waves whereas pressure, temperature, and density decrease.
• For Subsonic Flow: The Mach number, velocity decreases slightly, however, pressure, temperature, and density remains almost constant.
• The Subgrid criteria are only applicable for Supersonic flow as there is no shock waves or expansion waves that occur in subsonic flow.