the shock tube problem · 2016-12-22 · conclusions openfoam can simulate the 2d shock tube...
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![Page 1: The Shock Tube Problem · 2016-12-22 · Conclusions OpenFOAM can simulate the 2D shock tube problem solutions match to the analytical ones for resolution of at least 300x300 2D solutions](https://reader030.vdocuments.site/reader030/viewer/2022040507/5e452ed9e3298c391520bd7e/html5/thumbnails/1.jpg)
December 16th, 2016
The Shock Tube Problem
ME702 Final project
Luisa Capannolo
![Page 2: The Shock Tube Problem · 2016-12-22 · Conclusions OpenFOAM can simulate the 2D shock tube problem solutions match to the analytical ones for resolution of at least 300x300 2D solutions](https://reader030.vdocuments.site/reader030/viewer/2022040507/5e452ed9e3298c391520bd7e/html5/thumbnails/2.jpg)
The Shock Tube problem
● Studied by Gary A. Sod in 1978● 1D problem● analytical solutions are known● used to test and validate computational fluid models
![Page 3: The Shock Tube Problem · 2016-12-22 · Conclusions OpenFOAM can simulate the 2D shock tube problem solutions match to the analytical ones for resolution of at least 300x300 2D solutions](https://reader030.vdocuments.site/reader030/viewer/2022040507/5e452ed9e3298c391520bd7e/html5/thumbnails/3.jpg)
x
z
y
The Shock Tube problem
● Studied by Gary A. Sod in 1978● 1D problem● analytical solutions are known● used to test and validate computational fluidmodels
p = 100 kPau = 0 m/sρ = 1.000 kg/m3
p = 10 kPau = 0 m/sρ = 0.125 kg/m3
diaphragm
![Page 4: The Shock Tube Problem · 2016-12-22 · Conclusions OpenFOAM can simulate the 2D shock tube problem solutions match to the analytical ones for resolution of at least 300x300 2D solutions](https://reader030.vdocuments.site/reader030/viewer/2022040507/5e452ed9e3298c391520bd7e/html5/thumbnails/4.jpg)
● Studied by Gary A. Sod in 1978● 1D problem● analytical solutions are known● used to test and validate computational fluid models
diaphragm
shock wave rarefaction wave
contact discontinuity
The Shock Tube problem
![Page 5: The Shock Tube Problem · 2016-12-22 · Conclusions OpenFOAM can simulate the 2D shock tube problem solutions match to the analytical ones for resolution of at least 300x300 2D solutions](https://reader030.vdocuments.site/reader030/viewer/2022040507/5e452ed9e3298c391520bd7e/html5/thumbnails/5.jpg)
Governing Equations
Conservation of mass
Conservation of momentum
Conservation of energy
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Governing Equations
Conservation of mass
Conservation of momentum
Conservation of energy
Assume:● inviscid flows (no viscosity)● no heat transfer
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diaphragm
shock wave rarefaction wave
contact discontinuity
Analytical solutions
![Page 8: The Shock Tube Problem · 2016-12-22 · Conclusions OpenFOAM can simulate the 2D shock tube problem solutions match to the analytical ones for resolution of at least 300x300 2D solutions](https://reader030.vdocuments.site/reader030/viewer/2022040507/5e452ed9e3298c391520bd7e/html5/thumbnails/8.jpg)
OpenFOAM 2D shock tube set-up
0.2 m
x
z
y
10 m
2 m
2
0
4
7
3
6
5
1x
z
y
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OpenFOAM 2D shock tube set-up
0.2 m
x
z
y
10 m
2 m
2
0
4
7
3
6
5
1x
z
y
grid refinement
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OpenFOAM 2D shock tube set-up
0.2 m
x
z
y
10 m
2 m
2
0
4
7
3
6
5
1x
z
y
empty walls
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OpenFOAM 2D shock tube set-up
0.2 m
x
z
y
10 m
2 m
x
z
y
p = 100 kPau = 0 m/sT = 348.432 K
p = 10 kPau = 0 m/sT = 278.746 K
non-uniform internal flow
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OpenFOAM 2D shock tube set-up
0.2 m
x
z
y
10 m
2 m
Δt = 10-6 st = 10 mssampling rate: 0.5 msresolution cases: 20x20x1
30x30x1 50x50x1 100x100x1 300x300x1 400x400x1
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OpenFOAM rhoCentralFoam solver
● Greenshields et al., 2010
● compressible fluids
● finite volumes method
● values are provided at the centroid of the volume cell
● cells are contiguous polyhedral volumes
● volume integrals in divergence and gradientterms are converted to surface integrals
via Gauss’s theorem
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Results: time evolutionline cuts along x-axis
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Results: time evolutionline cuts along x-axis20x20x1
pressure
temperature
speed
density
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Results: time evolutionline cuts along x-axis20x20x1 50x50x1
pressure
temperature
speed
density
![Page 17: The Shock Tube Problem · 2016-12-22 · Conclusions OpenFOAM can simulate the 2D shock tube problem solutions match to the analytical ones for resolution of at least 300x300 2D solutions](https://reader030.vdocuments.site/reader030/viewer/2022040507/5e452ed9e3298c391520bd7e/html5/thumbnails/17.jpg)
Results: time evolution100x100x1 300x300x1line cuts along x-axis
pressure
temperature
speed
density
![Page 18: The Shock Tube Problem · 2016-12-22 · Conclusions OpenFOAM can simulate the 2D shock tube problem solutions match to the analytical ones for resolution of at least 300x300 2D solutions](https://reader030.vdocuments.site/reader030/viewer/2022040507/5e452ed9e3298c391520bd7e/html5/thumbnails/18.jpg)
Results: resolutionline cuts along x-axis
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Results: resolutionline cuts along x-axis
pressure
temperature
speed
density
![Page 20: The Shock Tube Problem · 2016-12-22 · Conclusions OpenFOAM can simulate the 2D shock tube problem solutions match to the analytical ones for resolution of at least 300x300 2D solutions](https://reader030.vdocuments.site/reader030/viewer/2022040507/5e452ed9e3298c391520bd7e/html5/thumbnails/20.jpg)
Resultsline cuts along x-axis for 300x300x1 resolution grid
pressure
temperature
speed
density
![Page 21: The Shock Tube Problem · 2016-12-22 · Conclusions OpenFOAM can simulate the 2D shock tube problem solutions match to the analytical ones for resolution of at least 300x300 2D solutions](https://reader030.vdocuments.site/reader030/viewer/2022040507/5e452ed9e3298c391520bd7e/html5/thumbnails/21.jpg)
Resultsline cuts along x-axis for 300x300x1 resolution grid
pressure
temperature
speed
density
![Page 22: The Shock Tube Problem · 2016-12-22 · Conclusions OpenFOAM can simulate the 2D shock tube problem solutions match to the analytical ones for resolution of at least 300x300 2D solutions](https://reader030.vdocuments.site/reader030/viewer/2022040507/5e452ed9e3298c391520bd7e/html5/thumbnails/22.jpg)
Results: 2D plotsxy plane cut
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Results: 2D plotsxy plane cut
pressure temperature
speed density
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Results: 2D plotsxy plane cut
pressure temperature
speed density
shock wave
shock wave shock wave
shock wave rarefaction wave
rarefaction wave rarefaction wave
rarefaction wave
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Python solvers: Lax-Friedrichs (1-step)
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Python solvers: Lax-Friedrichs (1-step)
pressure
speed
density
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Python solvers: MacCormack (2-step)
pressure
speed
density
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OpenFOAM vs. MacCormack
pressure
speed
density
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Conclusions
● OpenFOAM can simulate the 2D shock tube problem
● solutions match to the analytical ones for resolution of at least 300x300
● 2D solutions are the same for the 1D case
● can extract 1D solutions to 2D and 3D case, if the diaphragm is along x only
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Conclusions
● OpenFOAM can simulate the 2D shock tube problem
● solutions match to the analytical ones for resolution of at least 300x300
● 2D solutions are the same for the 1D case
● can extract 1D solutions to 2D and 3D case, if the diaphragm is along x only
● MacCormack 2-step scheme also approximates well the analytical solutions, but additional artificial viscosity is needed
● MacCormack 2-step scheme has more diffusion
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0.2 m
x
z
y
10 m
2 m
x
z
y
2
0
4
7
3
6
5
1x
z
y