Prandtl Meyer Shock problem
AIM : To setup the simulation for a flow over a supersonic inlet and observe how the shock wave is generated around a sharp corner
SIMULATION-SETUP :
Case setup : CONVERGE CFD STUDIO
Commanad editor : Cygwin
Post-processor : Paraview
THEORY OF SHOCK WAVES :
What is a shock wave ??
Consider a body travelling at the speed (v) greater than the speed (c) of sound, ie,, v > c at this instant of time, the wave fronts lag behind the source in a cone-shaped region with the source at the vertex, in other words the high pressure fonts are generated with large magnitude known as shock waves
Physics behind the shock wave :
Shock wave plays an important role in case of body travelling at supersonic speed to reduce the velocity of flow in the combustion chamber and to increase the pressure of the flow, for producing the maximum amount of thrust.
When the speed of the moving object in a medium becomes equal to the velocity of sound(340 m/s) in the medium, the wavefronts cannot escape from the source and pile up. This results in a large amplitude sound barrier. 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.
BC’s for shock problems
Since we are setting up a steady state simulation over a supersonic inlet appropriate BC’s are needed to setup the entire simulations.
Different types of boundary conditions that are used for shock wave problems are, Dirichlet boundary condition, Neumann boundary condition and mixed boundary condition
DIRICHLET BC :
When using a Dirichlet boundary condition, the value of variable is prescribed at the boundary.
NEUMANN BC :
When a Neumann boundary condition, is applied to the PDE that has to be solved ie,,,, the Navier stokes equation, the condition specifies the values in which the derivative of a solution is applied within the boundary of the domain.
Here for the pressure there is Dirichlet BC either at the inlet or outlet at all because, the flow is supersonic and the static pressure plays no role here and also as the outlet says that there is no DI boundary conditions at all, this means that the value of the pressure are derived from the initial conditions.
Prandtl Meyer shock waves :
fig 1 (expansion waves), flow is made to turn ' away from itself '
fig 2 (oblique shock wave), flow is made to turn ' towards itself '
Consider the supersonic flow as shown in the above figure 1 & 2
Let ‘θ’ be the wedge inclination
‘M1 and M2’ be the inlet and exit mach number
‘P1, T1, ρ1’ and ‘P2, T2, ρ2’ be the inlet and outlet pressure, temperatue and density
When a supersonic flow is introduced over a wedge as shown in the fig (1), expansion waves are generated. This is because the wedge is inclined at an angle –θ to the flow and the flow is made to turn ‘away from itself ’ expansion waves are generated.
In this scenario since the expansion waves expands and also the flow coming through it expands the exit mach number is greater than the inlet mach number. ie,,,,, M2 > M1
Similarly if the flow is made to turn ‘towards itself ’ , oblique shock waves are generated, this is because since the flow is made to turn towards itself, compression of pressure fonts occurs and the flow gets compressed and results in oblique shock waves.
Across oblique shock waves the velocity at the exit decrease the pressure, temperature, and density increases. ie,,,,,,
M2 < M1
P2 > P1, T2 > T1 and ρ2 > ρ1
The increase in pressure here plays an important role in case of combustion process in ram jet and scram jet engines. As these oblique shock waves are responsible for the increase in pressure and efficient burning of exhaust gases at high pressure, once they are made pass over the nozzle resulting in high exhaust velocity of the gases and producing enormous amount of thrust, to achieve a mach of M > 3
Prandtl-Meyer function ( Prandtl-Meyer angle )
The physical interpretation of the Prandtl-Meyer function is that it is the angle through which you must expand a sonic (M=1) flow to obtain a given Mach number. The value of the Prandtl-Meyer function is therefore called the Prandtl-Meyer angle
Prandtl-Meyer function therefore helps in determining the angle of the wedge inclined or declined for a supersonic flow, such that the inlet mach number and specific heat ratio ( γ ) of the continuum flowing is defined.
THE CASE SETUP
The case setup was done using the converge software.
The geometry was imported to the converge and the was cleaned to check if there were any voids or gap in it.
The diagnosis was done and confirmed that there are no errors.
Boundary flagging :
Boundary flagging was done and the geometry was divided into 4 different regions, namely inlet, outlet, top-bottom (walls) and front and back 2D.
Since the top and bottom surface have no role to play in this simulation they were assigned just as walls, and as we are interested in what is happening in the inside of geometry the front surface and back surface of the geometry were grouped as front and back 2d individually.
These boundaries were grouped into a volumetric region and were assigned individually to those regions
Boundary condtions :
1) Inlet: Inflow type; At a temperature of 286.1 Kelvin, velocity is 340 m/s, velocity along x-axis was kept 680 m/s making the inlet Mach number was assigned to 2 .
2) Outlet: Outflow type. Neumann Boundary Condition was used which is dependent on the initial conditions of the flow.
3) Top & Bottom: Wall boundary type with 'Slip' condition. The slip condition causes the fluid to flow only along a wall instead of stopping at the wall, which typically occurs along a wall. Fluid is prevented from flowing through the wall, however.
4) Front & Back: Since the case was setup for 2-D flow, 'TWO-D' conditions were assigned to front and back surfaces of a supersonic inlet.
Simulation Parameters :
The simulation that we are interested here is steady-state simulation as were are interested in looking the final results and development of shock waves at the supersonic inlet.
The simulation was ran for 25000 cycles with an initial time step of 1e-07 and for a maximum of 1sec.
The simulation mode was set to full-hydrodynamic. This means that in the post processing result not only mesh will be generated also all parameters like velocity, pressure, temperature etc.. will be solved.
TURBULENCE-MODELLING :
The turbulence model taken here is RNG K-Ɛ.
MESHING :
A base grid of dx = 0.8m, dy = 0.8m, and dz = 0.8m size was applied to the entire geometry and there were around 43000 cells created without AMR. Since the development of shock wave is very crucial and has to be observed carefully, it is necessary to define Adaptive Mesh Refinement ( AMR ) where ever the shock occurs to monitor the changer or curvature in temperature.
Mesh with edges for all SGS of 0.01 K , 0.05 K, and 0.1 K.
Converge uses an octry data method to define the adaptive mesh refinement ie,,, whenever the the curvature in temperature is greater than the SGS ( sub-grid scale temperature) converge refines the mesh. This is nothing but the place where shock occurs, mesh is refined.
The simulation was done for three different SGS ie,,, 0.05 K, 0.01 K and 0.1 K. So whenever the value of > 0.05 the mesh is being refined or generated in the region of shock wave.
2 layers of embedding layers has been made. The size of cells in each layer can be determined by the formula,
Grid size = (base grid)/2^(embedding level) Therefore for first level of embedding grid size is
= 0.8/2^1 = 0.4 m.Above figure shows a SGS of 0.05 K.
Similarly for second level of embedding grid size is
= 0.8/2^2 = 0.2 m.Above figure shows a SGS of 0.1 K.
Therefore the smallest cell refined in our entire simulation is 0.2m.
OUTPUT FILES :
For every 200 cyc a 3d output file is written, since we are running the simulation for 25000 cyc, therefore the total number of 3d output files that are dumped in the output folder for the post-processing in paraview is
=25000/200 = 125 3d output files. Hence there are 125 time steps or cycles for the animation.Post Processing the 3d output files
So when all the 3d output files are getting dumped into the output files by the cygwin, it is important to convert it into format that the paraview can understand like .case format or .vtm format using cygwin.
Once the post processed files are ready paraview can read those files and the desired contour of the different parameters can be obtained and the animation can be done.
CONTOUR PLOTS
MACH NUMBER (M) :
Above figure shows the inlet mach number (M1)
Above figure shows the exitmach number (M2)
Above figures shows the plot of mach number v/s number of cycles
The inlet mach number was set to supersonic condition ie,, 680 m/s or M = 2 ( M = velocity of the object / speed of sound) in the boundary condition itself.
M=680/340
M=2
When a supersonic flow encounters a supersonic inlet shock waves are generated. These may be oblique shock waves or expansion waves depending on the type of supersonic inlet used (sharp or blunt)
In our case once the flow at a velocity of mach 2 encounters a supersonic inlet, there is a sharp corner in the x-axis of the geometry where the flow interacts and leads to the development of expansion waves.
As said earlier in the downstream of the expansion waves ie,,, behind the shock wave as the flow expands it’s velocity increase and thus the exit mach number obtained from the simulation is M = 2.15, ie,,, the flow is leaving at a velocity of 731 m/s which is higher than that of the velocity of the flow upstream of the shock wave (680 m/s).
Total Cell Count :
Total cell-count for SGS of 0.05 K.
Total cell-count for SGS of 0.1 K.
Total cell-count for SGS of 0.01 K.
The total cell count must be a constant value. Since we are defining a sub-grid scale (SGS) criterion, the total cell count in this simulation varies accordingly.
SGS 0.01 K :
A lower value of sub-grid scale (SGS) means that the more number of cells or elements are being refined in the region of shock waves by monitoring the curvature in temperature ie,,, in other words converge refines the mesh (increase in cell count in shock region) whenever the value of
> 0.01 K (SGS), since SGS value of 0.01 K occurs first when compare to SGS of 0.05 K and 0.1 K, more number of cells are clearly refined in this case.
The shock waves observed in this case are much finer and accurate.
A total of 15500 cells are refined here.
SGS 0.05 K :
A SGS scale of 0.05 K refines the number of cells or elements in lesser number when compared to the SGS scale of 0.01 K. Hence here much lesser number of cells are refined than compared to the previous case.
Converge refines the mesh (increase in cell count in shock region) whenever the value of
> 0.05 K
The result obtained is not much finer as in the previous cases and also the shock wave are not well-defined for a higher value of SGS.
A total of 7090 cells are refined here.
SGS 0.1 K :
A higer value of SGS means lesser number of cells are refined, when compared to previous two cases.
Converge refines the mesh (increase in cell count in shock region) whenever the value of
> 0.05 K
A total of 5170 cells are refined here.
Therefore it is clear that with the increase in SGS criterion, the cells refined are lesser and the results tends to deviate from the analytical solution and convergence decreases.
MASS FLOW-RATE :
mass-flow rate for all SGS criteria of 0.01 K, 0.05 K and 0.1 K.
Mass-flow rate involves the terms like density, area and velocity of the flow. It is the quantity of fluid flowing through the given geometry in unit time interval in kg/s.
Since the velocity in the downstream of the shock wave is greater than the velocity in the upstream of the shock wave ie,,,, V2 > V1 the mass-flow rate through the boundary id 2 ie,,, at the exit will be larger than the mass-flow rate at the inlet.
Mass-flow increase with the increase in velocity of the flow in the downstream of the shock wave as shown in the above figures.
ANIMATION LINKS :
SGS of 0.05 K.
1) Pressure
2) Temperature
SGS of 0.1 K.
1) Pressure
2) Temperature