Prandtl Mayer Shock Problem
INTRODUCTION: This project contains a report in Prandtl Mayer shock problem. An aerodynamic simulation was performed on a given geomtry for two cases:
- Sub-sonic
- Super-sonic
The case was setup in converge. Cygwin and paraview were used to post process the findings.
THEORY: When an object moves in a medium, aerodynamic forces acts on it. The ratio of speed of object in the medium to the speed of sound in that same medium is called Mach number(Ma). Mach number is an important parameter that can be used to determine the boundary conditions as well as study the characteristics of the flow.
For external aerodynamic simulations, following rough classifications can be made to determine the flow charateristics:-
For internal aerodynamics, following rough classifications can be made to determine the flow characteristics:
- Ma<1.0: subsonic
- Ma>1.0: supersonic.
Basically sound can be treated as a disturbance in air pressure and particles. When an object moves in air, the air moves around the object as shown in the figure. This happens because a pressure wave is created at the leading edge of the object which directs the air around the object.
This pressure wave is created by air molecules and particles reflecting from the surface of the object. During the sub-sonic case, the molecule reflect and travel faster than the body does. Due to this the origin of the pressure wave is ahead of the body. This pressure waves travel with the speed of sound.
Notice that the pressure waves in subsonic case is propagating and no intereference takes place.
But when the object moves with speed greater than the speed of sound, the particles reflect and the waves get stacked up since object is moving greater than the speed of sound. It can be compared to many pressure waves combining into a single wave. This single wave is called as shock wave. The centre of the shock waves lies inside the object. The constructive intereference of many such shock waves gives us a conical profile.
The half angle of the cone is called the angle of the pressure wave. This angle is greatest at Mach1 (i,e. 90 degrees) and as the objects speed increases, the angle decreases.
Fig:Object travelling at Mach 1
Fig:Object travelling at speed >Mach 1
These shock waves cause tremendous amount of stress on the object. Hence objects must be designed to widthstand these shocks. These shocks cause maximum stress at the origin since shock waves are concentrated at a small region. Thus air shuttles have blunt nose which produces bow shock instead of oblique shock and the shocks are not localised in a small region.
NEUMANN BOUNDARY CONDITIONS: In neumann boundary condition the derivitive of the solution is applied on the boundary of domain. This type of boundary condition is used for heat flux problems in which heat flux from the surface would serve as the boundary condition.
CASE SETUP: The geometry was imported into the converge. Boundary was scaled for the problem to required limit. Next step was to flag the boundary and asign the surface meaningful names. Geometry was then checked for any errors. The next step was to do the case setup.
APPLICATION: General flow
MATERIAL: Air
SIMULATION PARAMETERS: density based steady state solver, simulation parameters: 25000cycles,
initial time step=max time step=1e-07
BOUNDARY CONDITION: inlet=inflow type(velocity = 680 m/s, temp=286K), outlet=outflow(neumann boundary confitions), front 2D and Back 2D are TWO D walls, Top-bottom walls are slip type walls.
REGIONS AND INITIALIZATION: all booundaries are assignes region 0 and velocity= 680m/s, temp=300K, pressure=50000 Pa
PHYSICAL MODELLING: default, RNG-ke method
GRID CONTROL: Base grid dx=dy=dz=0.8, Adaptive mesh refinement: Temperature based :- max embedding level=2, start time is 5000 cycle, timing control type= SEQUENTIAL.
three cases for sub-grid criteria were performed:sub-grid criterion
CASE 1: sgs=.05
CASE 2: sgs= .1
CASE 3: sgs=.01
OUTPUT/POST PROCESSING: Default
USER DEFINED FUNCTIONS: Default
POST PROCESSING:
1. Base grid:
CASE 1:
CASE 2:
CASE 3:
As the SGS parameter is decreased, the number of cells in the simulation increases. This leads to increased computation time and the results are more well defined. The location in which the shock occur is also dependent on the SGS. The changes taking place in temperature over large area are captured as the smaller temperature difference is now used to create a cell as the SGS creiteria is decreased. The decrease in parameters of the SGS thus the increases the region in which the shock-fan is observed.
The location of the shock in all the cases with sgs parameter remains same at the point where geometry expands.
2. DENSITY:
CASE 1:
CASE 2:
CASE 3:
3. TEMPERATURE:
CASE 1:
CASE 2:
CASE 3:
Due to the shock created the temperature rises in the system before the shock occurrence. This is due to pressure waves. The pressure waves create a high pressure region in the system before the shock location and hence the temperature rises because so many molecules are being forced together in the confined spaces. The work is done on the the system and its balanced by the gain in heat energy of the region( red in color) that increases the temperature.
4. VELOCITY:
CASE 1:
CASE 2:
CASE 3:
The low velocity before the shock is due to the relative high pressure at the origin of the shock.
The density, temperature and velocity distribution are well defined as the SGS parameter decreases giving us sharper results.
5. ANIMATION:
CASE 1:
CASE 2:
CASE 3:
The above results clearly tell us that as the SGS parameter is decresed, the sharpness of the results also increases.
SUB-SONIC:
In the super sub sonic flow the initial conditions used were different than the super sonic case. For subsonic case the velocity used is 100 m/s. Which gives the Mach number <0.3, thus the flow is incompressible and the density effects are negligible. Thus a density based solver must be used for the analysis of the sub-sonic flow.
A case was setup in converge using the pressure based solver and following result of cell count was obtained:
From the above graph we can see that the solution blows up after 5000 cycles and never reaches convergence. This suggests that the pressure based solver is not fit for the sub-sonic cases.
When the above setup was solved using a density based solver following results were obtained:
1. Cell count
The total cell count converged at the begining of the simulation and did not deter till the end. Thus the density based solver is to be used for the sub sonic case as discussed above.
2. Density
3. Velocity
Like Cell count, other physical parameters like density, pressure, temperature and velocity all are steady in sub sonic flow. This happens because the speed of object is less than the speed of sound in that medium and hence there is not any formation of pressure waves in the system. The properties are fairly governed by normal flowing fluid laws.
Temperature animation of sub-sonic flow is shown below, we can see that there is no change in the temperature during the flow.