Ghost-cell immersed boundary method for inviscid compressible...

Ghost-cell immersed boundary method for inviscid

compressible flows

B. Sanderse, B. Koren

CWI, TU Delft

June 17, 2009

B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 1 / 22

Overview

Overview

Introduction and background

Potential flows

Euler flows

Conclusions


Introduction

Outline

1 Introduction

2 Potential flow

3 Euler flow

4 Conclusions


Introduction

Motivation

‘Vision 2020’: studying and minimizing the aviation industry’s impact onthe global climate. Reductions in:

Fuel consumptionNoiseEmissions


Introduction

Motivation

‘Vision 2020’: studying and minimizing the aviation industry’s impact onthe global climate. Reductions in:

Fuel consumptionNoiseEmissions

Development of unconventional aircraft configurations


Introduction

Goal

A computational method for preliminary aerodynamic design andoptimization of unconventional aircraft configurations.

Requirements:

Fast (solving and mesh generation)

Accuracy similar/better than codes currently used in optimization

Handle ‘complex’ geometries


Introduction

Immersed boundary method

Ghost-cell method for full-potential equation and Euler equations:

No mesh generation

Straightforward implementation

Fast solution

Able to describe shock waves


Potential flow

Outline

1 Introduction

2 Potential flow

3 Euler flow

4 Conclusions


Potential flow

Full potential equation

Governing equation (conservative):

(ρφx)x + (ρφy)y = 0,

or (non-conservative):

(a2− u2)φxx − 2uvφxy + (a2

− v2)φyy = 0.

Simple test equation for ghost-cell method on compressible flows.

Error in shock jumps small until M ≈ 1.3.

Fast solution with Approximate Factorization algorithm.


Potential flow

Ghost-cell method

Identification ghost cells and fluid cells.Boundary condition:

~u · ~n = 0 →

∂φ

∂n= 0.

Set value in ghost cell:φi = φm.

1 1 1 1

1 1 1

1

1

0

−1000

0

(c)

1

2

3

4

φi

φm

i

b

mnn

(d)


Potential flow

Bilinear interpolation - case I

Assume φ(x, y) = a + bx + cy + dxy. With four surrounding points:

1 x1 y1 x1y1

1 x2 y2 x2y2

1 x3 y3 x3y3

1 x4 y4 x4y4

abcd

=

φ1

φ2

φ3

φ4

.

value of φm follows from:

φm = a + bxm + cym + dxmym.

1

23

4

i

m

Case I


Potential flow

Bilinear interpolation - case II

Use ∂φ∂n = 0 in point b1 by differentiating the bilinear form:

∂φ

∂n= b

∂x

∂n+ c

∂y

∂n+ d

(

xb∂y

∂n+ yb

∂x

∂n

)

= 0,

1

3

2

ib1

m

Case II


Potential flow

Bilinear interpolation - case II

1 x1 y1 x1y1

1 x2 y2 x2y2

1 x3 y3 x3y3

0(

∂x∂n

)

b1

( ∂y∂n

)

b1xb1

( ∂y∂n

)

b1+ yb1

(

∂x∂n

)

b1

abcd

=

φ1

φ2

φ3

0

.

1

3

2

ib1

m

Case II


Potential flow

Bilinear interpolation - case III

1 x1 y1 x1y1

1 x2 y2 x2y2

0(

∂x∂n

)

b1

( ∂y∂n

)

b1xb1

( ∂y∂n

)

b1+ yb1

(

∂x∂n

)

b1

0(

∂x∂n

)

b2

( ∂y∂n

)

b2xb2

( ∂y∂n

)

b2+ yb2

(

∂x∂n

)

b2

abcd

=

φ1

φ2

00

.

12

ib1

b2

m

Case III


Potential flow

Results

NACA 0012; 400 points to describe surface.

M∞ = 0.8, α = 0.

Domain 11 × 5, stretched grid.

x

y

4.5 5 5.5 6 6.50

0.2

0.4

0.6

0.8

1

0.4

0.6

0.8

1

1.2

Figure: Mach contour lines, ∆x = ∆y = 1/200.


Potential flow

Results

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

IBM 0.005

IBM 0.01

IBM 0.02

IBM 0.04


Euler flow

Outline

1 Introduction

2 Potential flow

3 Euler flow

4 Conclusions


Euler flow

Governing equations

Unsteady, 2D inviscid flow:

∂q

∂t+

∂f(q)

∂x+

∂g(q)

∂y= 0 (1)

Finite-volume discretization:

Ωijdq

dt=

(

F i−1/2,j − F i+1/2,j

)

∆yj +(

Gi,j−1/2 − Gi,j+1/2

)

∆xi.

MUSCL with minmod limiter to reconstruct qli+1/2,j and qr

i+1/2,j .

Fluxes with HLL or exact Riemann solver.

RK3 time discretization (TVD).


Euler flow

Ghost-cell method

Four boundary conditions are needed (Dadone & Grossman, 2004):

1 un = 0 −→ un,i = −un,m,

1

23

4

i

m

Case I


Euler flow

Ghost-cell method


1 un = 0 −→ un,i = −un,m,

2∂p

∂n=

(

ρu2t

R

)

b

−→ pi = pm + 2∆n

(

ρu2t

R

)

b

,

1

23

4

i

m

Case I


Euler flow

Ghost-cell method


1 un = 0 −→ un,i = −un,m,

2∂p

∂n=

(

ρu2t

R

)

b

−→ pi = pm + 2∆n

(

ρu2t

R

)

b

,

3∂s

∂n= 0 −→ ρi = ρm

(

pi

pm

)1/γ

,

1

23

4

i

m

Case I


Euler flow

Ghost-cell method


1 un = 0 −→ un,i = −un,m,

2∂p

∂n=

(

ρu2t

R

)

b

−→ pi = pm + 2∆n

(

ρu2t

R

)

b

,

3∂s

∂n= 0 −→ ρi = ρm

(

pi

pm

)1/γ

,

4∂H

∂n= 0 −→ u2

t,i = u2t,m +

2γ

γ − 1

(

pm

ρm−

pi

ρi

)

.

1

23

4

i

m

Case I


Euler flow

Results

NACA 0012; 800 points to describe surface.

M∞ = 0.8, α = 1.25.

Domain 21 × 20, stretched grid.

−0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3


Euler flow

Results

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

IBM 0.04

IBM 0.02

IBM 0.01


Euler flow

Unsteady supersonic flow over a cylinder at M=2

6

3


Conclusions

Outline

1 Introduction

2 Potential flow

3 Euler flow

4 Conclusions


Conclusions

Conclusions

Ghost-cell immersed boundary method for inviscid flows:

Useful method for design and optimization.

General method for compressible flows in complex geometries.

Accuracy similar to body-fitted grids.


Conclusions

Improvements and suggestions

For steady flows: speed-up with multigrid.

Thin body (e.g. trailing edge) treatment.

Local grid refinement.

Comparison with the same discretization on a body-fitted grid.

Research into numerical entropy generation and conservationproperties.

Assess sensitivity needed for optimization.


Ghost-cell immersed boundary method for inviscid compressible...

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