Week 5: Prandtl Meyer Shock problem
AIM:
To simulate a supersonic flow through a duct having a sharp corner in order to observe Prandtl Meyer expansion fan over a shock wave.
To observe the variation in flow for different initial conditions and also for various SGS temperatures.
THEORY:
PRANDTL MEYER EXPANSION WAVE
Shock was is a propagating disturbance which moves faster than speed of sound. In the current problem we are dealing with an expansion wave where the properties change across the waves. These are a set of Mach waves where the velocity of the fluid increase while the pressure density and temperature decrease. These Mach waves are weak and fan out as they move away from corner in normal direction. As per Prandtl and Meyer all the expansion waves are isentropic flows where the entropy is constant across the shock.
The Prandtl Meyer function is given as
When we substitute the gamma and M1 we arrive at f1. Now using the below equation, we can find f2 as per angle of inclination of the surface. Now the M2 can be calculated by trial and error method. All the other properties like pressure density and temperature can be found by using isentropic flow equations.
In the given problem we are considering the Neumann bc’s due to the supersonic flow at the outlet and properties are flowing along the normal to the boundary. We use a Neumann condition so that the flow itself provides the results at the outlet as per the inlet conditions (Dirichlet BC’s) by just providing a derivative that is, we just specify a derivative or a trend and as per inlet conditions the flow takes place. Where the speed of fluid is way fast than sound which makes the information flow very hard hence we define such problems only as per inlet conditions and initial region conditions.
BOUNDARY CONDITIONS
- Dirichlet Boundary Condition
This BC’s are specified as a real value on the boundaries directly. When a PDE or ODE is given and the values at the extremes of the domain are directly specified in the terms of real values then it is known as Dirichlet Boundary Condition.
ODE
y'' + y = 0
bc's
y(a) = 0
y(b) = 1
- Neumann Boundary Condition
This is also known as second type boundary condition. When the conditions specified are the derivatives of the dependent variable in the normal direction of a ODE or PDE within the domain then it is known as Neumann Boundary Condition.
It is also necessary that the boundary has to be sufficiently smooth in order to apply Neumann BC’s because they require a normal vector of the boundary.
y'' + y = 0
bc's
y'(a) = 0 ; y'(b) = 1
- Robin Boundary Condition
This is also known as third type boundary condition. When a ODE or a PDE is subjected to bc’s where the conditions posses a linear combination of values of functions and values of its derivatives on the boundary of the domain.
These are also known as convection boundary conditions.
- Mixed Boundary Condition
In a mixed boundary condition problem, the solution has to satisfy the Dirichlet or Neumann BC’s in a mutually exclusive ways on the disjoint parts of the boundary. This differs from the Robin BC’s in that the latter requires the linear combination possibly with pointwise variable coefficient of the Dirichlet and the Neumann boundary value conditions to be satisfied on the boundaries.
- Cauchy Boundary Equation
A Cauchy Boundary condition specifies value of the function and also the normal derivative on the boundaries of the domain. These are highly famous in second order ODE’s.
SETUP
The model was designed and imported into CONVERGE studio for post processing setup. Initially the design was converted into milli meters from meters, then the boundaries are flagged properly.
Then the normals are checked and pointed inwards and next step is to start case setup.
1. Materials
check gas simulation and species, and predefined mixture as air.
Add oxygen and nitrogen as species and set gas simulation and global transport parameters to default and apply.
2. Simulation Parameters
Set run parameters as steady state pressure solver.
In simulation time parameters
start cyc = 0
end cyc = 25000
dt = 1e-7s
solver parameters default
3. Boundary Conditions
Inlet
Dirichlet Boundary Condition
type = inflow
velocity = 680m/s
Temperature = 286.1k
Outlet
Neumann Boundary Condition
Type = outflow
Pressure = zero normal gradient
velocity = zero normal gradient
Front
type = 2D wall
Back
Type = 2D wall
Top and Bottom
Type = wall
Velocity = slip ;since inviscid flow
4. Regions and Initialization
Velocity = 680m/s and 100m/s
Temperature = 300k
Pressure = 1bar
5. Turbulence modeling
Type = RNG k-e
6. Grid Control
check adaptive mesh refinement
Base Grid
mesh size = 0.8m
Adaptive Mesh Refinement
Maximum cells = 200000
set region into active AMR region then check temperature box
max embedding level = 2
Sub grid criterion = 0.01,0.05,0.1.
start = 500cyc
end = 999999999cyc
7. Output/Post-Processing
Post variable selection to default
In Output files--> Writing time Intervals--> Time interval for writing 3D output data files=100cyc.
Here we end at the case setup and we postprocess the files in CYGwin and observe the simulation in Paraview.
RESULTS
Case 1 for 100m/s of region velocity
For SGC = 0.01k
Mesh:
link: https://youtu.be/N1ZWHM7wUXc
Cell Count:
Mass flow rate:
Total Pressure:
Mach:
Temperature:
link: https://youtu.be/e6hQK3SHgTk
Velocity:
For SGC 0.05k
Mesh:
link: https://youtu.be/nE0pxDKivyc
Cell count:
Mass flow rate:
Total Pressure:
Mach:
Temperature:
Link: https://youtu.be/DtgSyeI4l7E
Velocity:
For SGC 0.1
Mesh:
Link: https://youtu.be/u4Ytrql4bhA
Cell count
Mass flow rate:
Total Pressure:
Mach:
Temperature:
link: https://youtu.be/8xsgZOZC2l8
Velocity:
Understanding from case 1:
Cell count is gradually reducing due higher factor of SGC whic makes solution not so precise when compared with low SGC value. When we compare all the other quantities, we observe that their is a very minute difference which can be neglected.
We can observe that the higher SGC effects only at the sharp corner that is at the start of expansion fan.
The results are perfectly alright as per theoretical data that only the mach number increases while the other quantities decrease across the expansion fan.
Case 2 for 680m/s region velocity:
For SGC = 0.01k
Mesh
Cell count:
Mass flow rate:
Total Pressure:
Mach:
Temperature:
Link: https://youtu.be/3k1R5BNmxTQ
Velocity:
For SGC = 0.05k
Mesh:
link: https://youtu.be/vP3eAnDfffs
Cell count:
Mass flow rate:
Total Pressure:
Mach:
Temperature:
link: https://youtu.be/a-AIK5FJhFY
For SGC = 0.1k
Mesh:
link: https://youtu.be/wmy1JYEkTl4
Cell count:
Mass flow rate:
Total Pressure:
Mach:
Temperature:
link: https://youtu.be/Bjlkl1WntKA
Velocity:
Understanding from case 2:
Cell count is gradually reducing due higher factor of SGC whic makes solution not so precise when compared with low SGC value. When we compare all the other quantities, we observe that their is a very minute difference which can be neglected.
We can observe a trend in the temperature and mach number that as the SGC is increasing we observe that the flow is having an oscillatory output due to unrefined solution which is not vissible with SGC of 0.01k.
CONCLUSION
Here we have 2 different cases one with subsonic initial conditon for region and another with super sonic initial condition for region.
We come up to a conclusion that the difference is very minite which can be observed only in the initial cycles of flow where the first set of fluid inside the doman flows outside the domain. Since the solution is steady state simulation we do not observe great difference in the results for different initial conditions.