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=

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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

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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≈∗

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Anizotropic test meshes

Local transformation

η=η=η=η=

η=η=η=η=

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3D FLOW EXAMPLES

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3D sinusoidal bump in the channel

3 meshes:A – 2620 cellsB – 21915 cellsC – 160958 cells

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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

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TRANSONIC CASE3D SINUSOIDAL BUMP

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Nonlinear WENO Weighting

Extended stencil

3rd Order on single stencil

2nd Order WENO

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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