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Centre for Computational Fluid Dynamics
Jacek Rokicki, Robert Wieteska
Institute of Aeronautics and Applied Mechanics
Warsaw University of Technology
ECOMASS 2007
Egmond aan Zee, 05-08.09.2006
High-order WENO schemes onunstructured tetrahedral meshes
Centre for Computational Fluid Dynamics
Typical problem in practical aerodynamics
Evaluate CD, CM
Centre for Computational Fluid Dynamics
MOTIVATION
High accuracy estimations of integral coefficients are required by industry (e.g., 1 drag count accuracy)
Yet the unstructured 3D computational meshes are notsufficiently refined or have a poor quality
� highly distorted mesh cells are common in the „Navier-Stokes” meshes
� During adaptation, shock-waves leaving the refined mesh area, tend to crash the code
Centre for Computational Fluid Dynamics
OBJECTIVES
1. Achieving higher accuracy via higher-order disretisation within the FV WENO scheme(Extension of linear reconstruction to the quadratic)
2. Smaller sensitivity to poor quality mesh – esp. to high aspect ratio cells
Centre for Computational Fluid Dynamics
HIGHER-ORDER 3D RECONSTRUCTION
Centre for Computational Fluid Dynamics
The Euler Model of Fluid
Equation of state:
0)( =⋅∇+∂∂
UFUt
++⊗=
T
T
VpE
pVV
V
r
rr
r
)(
)(
ρρ
ρIUF
=E
V
ρρρr
U
( )
−−=2
1VV
EkpTrr
ρ
Centre for Computational Fluid Dynamics
Finite Volume approachWENO reconstruction – 2D/3D
The reconstruction function for control volume is defined as a weighted average ∑
=
∇⋅=∇m
iiio UU
1
ω
Centre for Computational Fluid Dynamics
Reconstruction within the computational cell
Linear)()()( 2hOUUU jjj +−⋅∇+= xxx
)()()()()( 32T21 hOUUUU jjjjjj +−⋅∇⋅−+−⋅∇+= xxxxxxx
Gradient and Hessian approximated basing on data from neighbouring cells
)( 1hOUj+∇
)( 12 hOUj+∇)( 2hOU
j+∇
A.
B.
A.
B.
Centre for Computational Fluid Dynamics
Is basic stencil system sufficient ?
Does not allow to increase accuracy to h2
Centre for Computational Fluid Dynamics
Extended Stencil 2D/3D
Extended stencil
Centre for Computational Fluid Dynamics
Linear reconstruction
mϕϕϕϕ ,...,,, 210
( ) ( )01
0
1,
p
p
m
pppp ww
rGG =−= ∑
=
ϕϕϕ
under/over-determined linear systemSolved by the sequence of Householder transformations
002
000 ),( rrrr −=+∇=− ppTpp hOϕϕϕ
pp ϕ,r
00 ,ϕr
Known values:
( ) )(1
00 hOwm
p
Tppp∑
=
+∇= ϕϕ rGG
I=
Centre for Computational Fluid Dynamics
3rd order reconstruction (2nd order gradient)
)(2
1 300
20000 hOp
Tp
Tpp +∇+∇=− rrr ϕϕϕϕ
0,1
00 =∈=∀ ∑=
×m
pp
Tppp
nnT w rErGEE R
under/over-determined linear system:
( ) ( )01
0
1,
p
p
m
pppp ww
rGG =−=∑
=
ϕϕϕ
( ) 01
02
01
00 2
1p
m
p
Tppp
m
p
Tppp rrGrGG ∑∑
==
∇+∇= ϕωϕωϕ
vvvm
p
Tppp
n =∈∀ ∑=1
0, rG ωR
Centre for Computational Fluid Dynamics
3rd order reconstruction(2nd order gradient)
0,1
00 =∈=∀ ∑=
×m
pp
Tppp
nnT w rErGEE R
vvwvm
p
Tppp
n =∈∀ ∑=1
0, rGR
==
=++39
25
2
)1(n
nnnn
underdetermined linear system -solved by the sequence of Householder transformations
Solution minimises vector coefficients Gp
( ) ( )01
0
1,
p
p
m
pppp ww
rGG =−=∑
=
ϕϕϕm
nnn <++
2
)1(
Centre for Computational Fluid Dynamics
3rd order reconstruction (1st order Hessian)
ErErGEE =∈=∀ ∑=
×m
pp
Tppp
nnT w1
00,R
0,1
0 =∈∀ ∑=
m
p
Tppp
n vwv rGR
==
=++39
25
2
)1(n
nnnn
underdetermined linear system -solved by the sequence of Householder transformations
Solution minimises matrix coefficients Hp
( ) ( ) 2
01
0 ,−
=
=−=∑ pp
m
pppp ww rHH ϕϕϕ
Centre for Computational Fluid Dynamics
Numerical 3D test
)422sin(),,( 222)( 222
+++++= ++− yzzxyyxezyxu zyx
X
-1
-0.5
0
0.5
1
Y
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Centre for Computational Fluid Dynamics
Coefficients
Basic stencil Full stencil
Centre for Computational Fluid Dynamics
HIGHER-ORDER RECONSTRUCTION ON DISTORTED/ANIZOTROPIC 3D MESHES
Centre for Computational Fluid Dynamics
Anizotropic test meshes
)422sin(),,( 222)( 222
+++++= ++− yzzxyyxezyxu zyx
K,8
1,
4
1,
2
1
,
00
010
001
=
⋅
=
η
η
η rr
Centre for Computational Fluid Dynamics
Anizotropic meshes
)422sin(
),,(222
)( 222
+++++×= ++−
yzzxyyx
ezyxu zyx
rr ⋅
=η
η
00
010
001
η = 1/4
η=η=η=η=
90.71/64
51.61/32
14.71/16
2.51/8
01/4
01/2
% of failureηηηη
Centre for Computational Fluid Dynamics
Anizotropic meshes
η=η=η=η=
L1
Centre for Computational Fluid Dynamics
Anizotropic meshes
Reasons of failure:
1. ill conditioning of the matrix ~h-3
2. weights should include „directional” information
( ) ( )01
0
1,
p
p
m
pppp ww
rGG =−=∑
=
ϕϕϕ
Centre for Computational Fluid Dynamics
How to measure the local anisotropy ?
CppC rrr −=
pCr
[ ] ∑=
⋅==m
ppcpcm
0
def
0 ,..., TrrrrMM
0>⋅⋅ rMrT1.
2. IM α≈ For spherical symmetry of the cloud
Centre for Computational Fluid Dynamics
Local transformation
[ ] ∑=
⋅==m
ppcpcm
0
def
0 ,..., TrrrrMM
0>⋅⋅ rMrT
pcpc rMr ⋅= − 2/1*pCr
∗pCr
[ ] IrrMM == **0
* ,..., m
*2/1 GMG ⋅= −
2/1*
2/1 −− ⋅⋅= MHMH∗HG* ,
IM α≈∗
Centre for Computational Fluid Dynamics
Anizotropic test meshes
Local transformation
η=η=η=η=
η=η=η=η=
Centre for Computational Fluid Dynamics
3D FLOW EXAMPLES
Centre for Computational Fluid Dynamics
3D sinusoidal bump in the channel
3 meshes:A – 2620 cellsB – 21915 cellsC – 160958 cells
Centre for Computational Fluid Dynamics
3D sinusoidal bump – Ma=0.5
mesh C – 160958 cells
Standard 2nd order WENO . . . . . .
3rd order WENO ________
Centre for Computational Fluid Dynamics
3D sinusoidal bump – Ma=0.5
Pressure loss at the boundary 2nd vs. 3rd order
3 meshes:A – 2620 cellsB – 21915 cellsC – 160958 cellsBA
C
C
Centre for Computational Fluid Dynamics
3D sinusoidal bump – Ma=0.5
Pressure loss at the boundary 3rd order vs. RDS LDA
X
1-P
t/Ptin
f
-0.5 0 0.5 1 1.5-0.02
-0.01
0
0.01
0.02
0.03
0.04mesh A - third ordermesh B - third ordermesh C - third ordermesh A - LDAmesh B - LDAmesh C - LDA
X
1-P
t/Ptin
f
-0.5 0 0.5 1 1.5-0.01
-0.005
0
0.005
0.01mesh A - third ordermesh B - third ordermesh C - third ordermesh A - LDAmesh B - LDAmesh C - LDA
3 meshes:A – 2620 cellsB – 21915 cellsC – 160958 cells
Centre for Computational Fluid Dynamics
TRANSONIC CASE3D SINUSOIDAL BUMP
Centre for Computational Fluid Dynamics
Nonlinear WENO Weighting
Extended stencil
3rd Order on single stencil
2nd Order WENO
Centre for Computational Fluid Dynamics
Full Nonlinear 3rd Order WENO Weighting
Extended stencil
Central stencil
3-4 biased stencils
+
Centre for Computational Fluid Dynamics
3D sinusoidal bump – Ma=0.55
mesh B – 21915 cells
0.55
0.50
0.59
0.6 9
0.83
1.02
0.64
0.55
0.50
0.59
0.831.
21
Centre for Computational Fluid Dynamics
Summary
• Reconstruction scheme extended to higher-order of accuracy
• Special procedure proposed for highly distorted meshes based on additional transformation
• Less entropy produced by Higher order-schemes for the 3D subsonic case
• Full WENO weighting is effective also for the third-order scheme
• High-order scheme is expensive but this can be balanced by the increased effort in the linear subiterations
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