Ghost-cell Immersed Boundary Method for Inviscid

8/9/2019 Ghost-cell Immersed Boundary Method for Inviscid

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Overview

Overview

Introduction and background

Potential flows

Euler flows

Conclusions

B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 2 / 22

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Introduction

Outline

1 Introduction

2 Potential flow

3 Euler flow

4 Conclusions


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Introduction

Motivation

‘Vision 2020’: studying and minimizing the aviation industry’s impact on

the global climate. Reductions in:Fuel consumptionNoiseEmissions


I d i

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Introduction

Motivation

‘Vision 2020’: studying and minimizing the aviation industry’s impact on

the global climate. Reductions in:Fuel consumptionNoiseEmissions

Development of unconventional aircraft configurations


I t d ti

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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

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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

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Potential flow

Outline

1 Introduction

2 Potential flow

3 Euler flow

4 Conclusions


Potential flow

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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

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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 = φ

m.

1 1 1 1

1 1 1

1

1

0

−1000

0

(c)

1

2

3

4

φi

φm

i

b

m nn

(d)B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 9 / 22

Potential flow

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Bilinear interpolation - case I

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

1 x1 y1 x1y11 x2 y2 x2y21 x3 y3 x3y31 x4 y4 x4y4

abcd

=

φ1φ2φ3φ4

.

value of φm follows from:

φm = a + bxm + cym + dxmym.

1

23

4

i

m

Case I


Potential flow

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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

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Bilinear interpolation - case II

1 x1 y1 x1y11 x2 y2 x2y21 x3 y3 x3y30∂x∂n

b1

∂y∂n

b1

xb1∂y∂n

b1

+ yb1∂x∂n

b1

abcd

=φ1φ2φ30

.

1

3

2

ib1

m

Case II


Potential flow

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Bilinear interpolation - case III

1 x1 y1 x1y11 x2 y2 x2y20∂x∂n

b1

∂y∂n

b1

xb1 ∂y∂n

b1

+ yb1∂x∂n

b1

0∂x∂n

b2

∂y∂n

b2

xb2 ∂y∂n

b2

+ yb2∂x∂n

b2

abcd

=φ1φ200

.

12

ib1

b2

m

Case III


Potential flow

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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

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Results

0 0.2 0.4 0.6 0.8 11

8

6

4

2

0

2

4

6

8

1

IBM 0.005

IBM 0.01

IBM 0.02

IBM 0.04


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Euler flow


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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

∆y j +

Gi,j−1/2 −Gi,j+1/2

∆xi.

MUSCL with minmod limiter to reconstruct ql

i+1/2,j and qr

i+1/2,j .Fluxes with HLL or exact Riemann solver.

RK3 time discretization (TVD).


Euler flow

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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

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Ghost-cell method


1 un = 0 −→ un,i = −un,m,2

∂p

∂n =

ρu2tR

b

−→ pi = pm + 2∆n

ρu2tR

b

,

1

23

4

i

m

Case I


Euler flow

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Ghost-cell method


1 un = 0 −→ un,i = −un,m,2

∂p

∂n =

ρu2tR

b

−→ pi = pm + 2∆n

ρu2tR

b

,

3∂s

∂n

= 0 −→ ρi = ρm pi

pm

1/γ

,

1

23

4

i

m

Case I


Euler flow

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Ghost-cell method


1 un = 0 −→ un,i = −un,m,2

∂p

∂n =

ρu2tR

b

−→ pi = pm + 2∆n

ρu2tR

b

,

3∂s

∂n = 0 −→ ρi = ρm

pi pm

1/γ

,

4∂H

∂n = 0 −→ u2t,i = u

2t,m +

2γ

γ − 1

pmρm

−

piρi

.

1

23

4

i

m

Case I


Euler flow

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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

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3


Euler flow

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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

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Unsteady supersonic flow over a cylinder at M=2

6

3


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Conclusions

C l


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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

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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.

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Ghost-cell Immersed Boundary Method for Inviscid

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