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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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 2 / 22
Introduction
Outline
1 Introduction
2 Potential flow
3 Euler flow
4 Conclusions
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 3 / 22
Introduction
Motivation
‘Vision 2020’: studying and minimizing the aviation industry’s impact onthe global climate. Reductions in:
Fuel consumptionNoiseEmissions
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 4 / 22
Introduction
Motivation
‘Vision 2020’: studying and minimizing the aviation industry’s impact onthe global climate. Reductions in:
Fuel consumptionNoiseEmissions
Development of unconventional aircraft configurations
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 4 / 22
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 5 / 22
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 6 / 22
Potential flow
Outline
1 Introduction
2 Potential flow
3 Euler flow
4 Conclusions
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 7 / 22
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.
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 8 / 22
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)
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 9 / 22
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 10 / 22
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 11 / 22
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 11 / 22
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 12 / 22
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.
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 13 / 22
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 13 / 22
Euler flow
Outline
1 Introduction
2 Potential flow
3 Euler flow
4 Conclusions
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 14 / 22
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).
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 15 / 22
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 16 / 22
Euler flow
Ghost-cell method
Four boundary conditions are needed (Dadone & Grossman, 2004):
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 16 / 22
Euler flow
Ghost-cell method
Four boundary conditions are needed (Dadone & Grossman, 2004):
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 16 / 22
Euler flow
Ghost-cell method
Four boundary conditions are needed (Dadone & Grossman, 2004):
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 16 / 22
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 17 / 22
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
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 18 / 22
Euler flow
Unsteady supersonic flow over a cylinder at M=2
6
3
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 19 / 22
Conclusions
Outline
1 Introduction
2 Potential flow
3 Euler flow
4 Conclusions
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 20 / 22
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.
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 21 / 22
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.
B. Sanderse, B. Koren (CWI, TU Delft) Ghost-cell IBM for compressible flows June 17, 2009 22 / 22