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DESCRIPTION
presentation on roes schemeTRANSCRIPT
2D Unstructured Euler Solver
2D Unstructured Euler Solver
Prapanj Ganeshan Radhamony
Dept of Mechanical EngineeringUniversity of Alabama at Birmingham
Advanced Computational Fluid Dynamics
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
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.
2D Unstructured Euler Solver
Introduction
The Goal
Assumptions on the Domain
Inviscid compressible flow
Euler Equation
Unstructured Grid - To learn the book keeping
2D Unstructured Euler Solver
Introduction
The Goal
Assumptions on the Domain
Inviscid compressible flow
Euler Equation
Unstructured Grid - To learn the book keeping
2D Unstructured Euler Solver
Introduction
The Goal
Assumptions on the Domain
Inviscid compressible flow
Euler Equation
Unstructured Grid - To learn the book keeping
2D Unstructured Euler Solver
Introduction
The Goal
Assumptions on the Domain
Inviscid compressible flow
Euler Equation
Unstructured Grid - To learn the book keeping
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
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
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
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
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
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 !!
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 !!
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|
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
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
2D Unstructured Euler Solver
Development
Post processing
Output Format
Tecplot format
Cell Centered Data
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
2D Unstructured Euler Solver
Test Cases
Expansion over convex corner
Results
Expansion
grid
density
pressure
stream
2D Unstructured Euler Solver
Test Cases
Expansion over convex corner
Results
Expansion
grid
density
pressure
stream
2D Unstructured Euler Solver
Test Cases
Expansion over convex corner
Results
Expansion
grid
density
pressure
stream
2D Unstructured Euler Solver
Test Cases
Expansion over convex corner
Results
Expansion
grid
density
pressure
stream
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
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
2D Unstructured Euler Solver
Test Cases
Flow over 15 degree wedge
Results
Compression
grid
density
pressure
2D Unstructured Euler Solver
Test Cases
Flow over 15 degree wedge
Results
Compression
grid
density
pressure
2D Unstructured Euler Solver
Test Cases
Flow over 15 degree wedge
Results
Compression
grid
density
pressure
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
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Subsonic flowFlow Conditions
M∞ 0.3P∞ 14.0 psiα 0.0Temperature 460 R
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Subsonic
grid
density
pressure
stream
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Subsonic
grid
density
pressure
stream
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Subsonic
grid
density
pressure
stream
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Subsonic
grid
density
pressure
stream
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Transonic flowFlow Conditions
M∞ 0.73P∞ 15.807 psiα 2.31Temperature 460.6 R
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Transonic
contour
density
pressure
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Transonic
contour
density
pressure
2D Unstructured Euler Solver
Test Cases
RAE2822 Aerofoil
Results
Transonic
contour
density
pressure
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
2D Unstructured Euler Solver
Conclusion
Acknowledgements
Shravan- Inspiration for modular code
Joseph- Helping me with the computer
Dr. Roy- For teaching us a lot