project
DESCRIPTION
presentation on roes schemeTRANSCRIPT
![Page 1: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/1.jpg)
2D Unstructured Euler Solver
2D Unstructured Euler Solver
Prapanj Ganeshan Radhamony
Dept of Mechanical EngineeringUniversity of Alabama at Birmingham
Advanced Computational Fluid Dynamics
![Page 2: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/2.jpg)
2D Unstructured Euler Solver
Table of contents
1 IntroductionThe GoalGoverning Equation
2 DevelopmentPre-ProcessingCore ComputationsPost processing
3 Test CasesExpansion over convex cornerFlow over 15 degree wedgeRAE2822 Aerofoil
4 Conclusion
![Page 3: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/3.jpg)
2D Unstructured Euler Solver
Introduction
Motivation
Inspiration
Fascination for fluid flow. The feeling of greatness in simulatingnatural phenomenon. We usually see the mathematics of theproblem extablished, and then we see the colorful results. I wantedto see the bridge between the two.
![Page 4: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/4.jpg)
2D Unstructured Euler Solver
Introduction
The Goal
Assumptions on the Domain
Inviscid compressible flow
Euler Equation
Unstructured Grid - To learn the book keeping
![Page 5: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/5.jpg)
2D Unstructured Euler Solver
Introduction
The Goal
Assumptions on the Domain
Inviscid compressible flow
Euler Equation
Unstructured Grid - To learn the book keeping
![Page 6: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/6.jpg)
2D Unstructured Euler Solver
Introduction
The Goal
Assumptions on the Domain
Inviscid compressible flow
Euler Equation
Unstructured Grid - To learn the book keeping
![Page 7: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/7.jpg)
2D Unstructured Euler Solver
Introduction
The Goal
Assumptions on the Domain
Inviscid compressible flow
Euler Equation
Unstructured Grid - To learn the book keeping
![Page 8: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/8.jpg)
2D Unstructured Euler Solver
Introduction
The Goal
Problem to Solve
Internal and External flows
Two kinds of boundaries - Farfield and Wall
Flow regime - Subsonic to Supersonic
Steady State problems
![Page 9: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/9.jpg)
2D Unstructured Euler Solver
Introduction
Governing Equation
Euler Equation
The PDE
∂U
∂t+∂Fx
∂x+∂Fy
∂y+∂Fz
∂z= 0
The discretized form
∆U
∆tV + Σ~F · n̂ds = 0
![Page 10: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/10.jpg)
2D Unstructured Euler Solver
Development
Pre-Processing
Grid Generation
1 Gridgen - Unstructured grid
2 Cobalt Format
3 Edge to cell information
4 Extract cells to node information
![Page 11: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/11.jpg)
2D Unstructured Euler Solver
Development
Pre-Processing
Grid Generation
1 Gridgen - Unstructured grid
2 Cobalt Format
3 Edge to cell information
4 Extract cells to node information
![Page 12: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/12.jpg)
2D Unstructured Euler Solver
Development
Pre-Processing
Grid Generation
1 Gridgen - Unstructured grid
2 Cobalt Format
3 Edge to cell information
4 Extract cells to node information
![Page 13: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/13.jpg)
2D Unstructured Euler Solver
Development
Pre-Processing
Gometric Variables
1 Edge normals
2 Cell Areas
3 Direction of rotation
4 This code works only for a negative area !!
![Page 14: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/14.jpg)
2D Unstructured Euler Solver
Development
Pre-Processing
Gometric Variables
1 Edge normals
2 Cell Areas
3 Direction of rotation
4 This code works only for a negative area !!
![Page 15: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/15.jpg)
2D Unstructured Euler Solver
Development
Core Computations
Intercell fluxes -Roe Scheme
Finteface =1
2(FL + FR − |A|(QR − QL))
where
|A|(QR − QL) = |∆F1|+ |∆F23|+ |∆F4|
![Page 16: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/16.jpg)
2D Unstructured Euler Solver
Development
Core Computations
Boundary Fluxes
Farfield Boundaries
The free stream values are substituted in the farfield boundaryghost cells
Wall boundary
Finterface =
0
Pnx
Pny
0
![Page 17: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/17.jpg)
2D Unstructured Euler Solver
Development
Core Computations
Time Stepping
Euler time stepping is used in this code
Ut+∆ti = Ut
i −∆t
ARi
where R(i) is the Residual
![Page 18: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/18.jpg)
2D Unstructured Euler Solver
Development
Post processing
Output Format
Tecplot format
Cell Centered Data
![Page 19: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/19.jpg)
2D Unstructured Euler Solver
Test Cases
Expansion over convex corner
Prandtl Meyer ExpansionFlow conditions
M∞ 2.5P∞ 12.0 psiρ∞ 0.94347 kg/m3
θ 15 degrees
![Page 20: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/20.jpg)
2D Unstructured Euler Solver
Test Cases
Expansion over convex corner
Results
Expansion
grid
density
pressure
stream
![Page 21: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/21.jpg)
2D Unstructured Euler Solver
Test Cases
Expansion over convex corner
Results
Expansion
grid
density
pressure
stream
![Page 22: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/22.jpg)
2D Unstructured Euler Solver
Test Cases
Expansion over convex corner
Results
Expansion
grid
density
pressure
stream
![Page 23: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/23.jpg)
2D Unstructured Euler Solver
Test Cases
Expansion over convex corner
Results
Expansion
grid
density
pressure
stream
![Page 24: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/24.jpg)
2D Unstructured Euler Solver
Test Cases
Expansion over convex corner
Comparison of Results
. Exact Present Solver
M2 3.23 3.74P2P1
0.3274 0.3295ρ2ρ1
0.4505 0.4408
![Page 25: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/25.jpg)
2D Unstructured Euler Solver
Test Cases
Flow over 15 degree wedge
Flow Conditions
M∞ 2.5P∞ 14.7 psiρ∞ 1.2222 kg/m3
θ 15 degrees
![Page 26: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/26.jpg)
2D Unstructured Euler Solver
Test Cases
Flow over 15 degree wedge
Results
Compression
grid
density
pressure
![Page 27: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/27.jpg)
2D Unstructured Euler Solver
Test Cases
Flow over 15 degree wedge
Results
Compression
grid
density
pressure
![Page 28: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/28.jpg)
2D Unstructured Euler Solver
Test Cases
Flow over 15 degree wedge
Results
Compression
grid
density
pressure
![Page 29: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/29.jpg)
2D Unstructured Euler Solver
Test Cases
Flow over 15 degree wedge
Comparison of Results
. Exact Present Solver
Angle of oblique shock 36.9449 36.88
M2 1.87 2.12P2P1
2.4675 2.4682ρ2ρ1
1.8665 1.856502
![Page 30: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/30.jpg)
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Subsonic flowFlow Conditions
M∞ 0.3P∞ 14.0 psiα 0.0Temperature 460 R
![Page 31: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/31.jpg)
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Subsonic
grid
density
pressure
stream
![Page 32: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/32.jpg)
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Subsonic
grid
density
pressure
stream
![Page 33: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/33.jpg)
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Subsonic
grid
density
pressure
stream
![Page 34: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/34.jpg)
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Subsonic
grid
density
pressure
stream
![Page 35: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/35.jpg)
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Transonic flowFlow Conditions
M∞ 0.73P∞ 15.807 psiα 2.31Temperature 460.6 R
![Page 36: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/36.jpg)
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Transonic
contour
density
pressure
![Page 37: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/37.jpg)
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Transonic
contour
density
pressure
![Page 38: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/38.jpg)
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Transonic
contour
density
pressure
![Page 39: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/39.jpg)
2D Unstructured Euler Solver
Conclusion
Future Works
Fix bugs regarding coordinates
Incorporate more boundary conditions
Run more test cases
Incorporate Roe Entropy fix
Incorporating more Schemes
Higher Order time-stepping
![Page 40: Project](https://reader031.vdocuments.mx/reader031/viewer/2022020306/5464c359b4af9f2e588b499c/html5/thumbnails/40.jpg)
2D Unstructured Euler Solver
Conclusion
Acknowledgements
Shravan- Inspiration for modular code
Joseph- Helping me with the computer
Dr. Roy- For teaching us a lot