project

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