doru caraenidoru caraeni cd-adapco, usa cfd futures conference, august 6-8, 2012 why i did...

Doru Caraeni

CD-adapco, USA

CFD Futures Conference, August 6-8, 2012

Why I did Residual-based schemes research ?

- (1996) Leading the CFD/CAE group (Centrifugal Compressors) at

COMOTI Bucharest

- Challenge: to perform LES of turbulence inside high-PR CC

- Write a new CFD code (together with Aerospace Department at

“Polytechnica” Institute Bucharest) for industrial LES

- Found a few papers about early Residual-Distribution schemes

- Learned more about these scheme at a VKI Advanced CFD course

- Went to Lund Institute to learn LES of turbulence and develop a

(hopefully) best-in-class LES algorithm, for industry (Dec.1997)

Multidimensional Upwind Residual Distribution scheme :

- Fluctuation-Splitting (RDS) proposed in 1986 by professor Phil Roe

- Developed by professors and students at Michigan University (Roe), VKI (Deconinck), Bordeaux University (Abgrall), Polytechnica di Bari, Lund (Caraeni), Univ. of Leeds (Hubbard) etc.

- Compact matrix distribution schemes for steady Euler and Navier-Stokes equations (E.van der Weide, H. Paillere), 1996.

- Second order RD scheme for LES of turbulence using a residual-property preserving, dual time-step approach (Caraeni, 1999).

A short early history of MU-RDS

Multidimensional Upwind Residual Distribution scheme:

- Second order space-time RD scheme for unsteady simulations (2000, VKI)

(using space-time integration/residual-distribution to achieve accuracy)

- Third order RD scheme for steady inviscid flow simulations (2000, LTH)

(node gradient-recovery for quadratic solution representation)

- Third order RD scheme for the unsteady turbulent flow simulations (2001, LTH). (node gradient-recovery and residual-property satisfying)

- Third order results with above gradient-recovery idea reported by Rad and Nishikawa (2002, MU).

- High-order (>3) RD scheme for scalar transport equations (2002, BU & MU).

(sub-mesh reconstruction for high-order solution representation)

“ Third-order non-oscillatory fluctuation schemes for steady scalar

conservation laws ” M. Hubbard, 2008.

A short early history of MU-RDS (cont.)

Ricchiuto, CEMRACS, 2012

A 3rd Order Residual-Distribution scheme

for Navier-Stokes simulations

(Residual-property satisfying formulation)

(A Third Order Residual-Distribution Method for Steady/Unsteady

Simulations: Formulation and Benchmarking, including LES,

Caraeni, VKI, 2005)

jjjijijjt

jijijjiti

jjt

Tuhue

puuu

u

,,,,,

,,,,

,,

)()()()(

)()(

0)(

V

jj

C

jjt FFU ,,,

hu

puu

puu

puu

u

F

j

jj

jj

jj

j

C

j

33

22

11

jjjj

j

j

j

V

j

Tuuu

F

,332211

3

2

1

0

teuuuU ),,,,( 321

t

v

jj

c

jj UFFU ,,,,

dvUFF

dvU

tj

v

j

c

j ])[( ,,

,

Jameson dual-time

algorithm

High-order RD scheme for Navier-Stokes equations

dvT

c

j

c

T F ., Convection flux cell-residual

kn

TiT

uns

T

v

T

c

T

T

i

kn

i

kn

i BUUi

V

,1

,

,11,1)(

Update scheme for steady/unsteady simulations (Caraeni):

T

iB Upwind matrix residual

Distribution coefficient (bounded)

dvT

v

j

v

T F ., Diffusion flux cell-residual

dvUT

t

uns

T ., Unsteady term cell-residual

Ti

T

i IB

High-order RD scheme (cont.) for Navier-Stokes equations

(conservativity)

])(2

1

4

1[ 1,

j

ji

LWT

iKKIB

Distribution schemes (for preconditioned system):

Low Diffusion A (LDA)

Lax-Wendroff (LW)

1,)(

j

ji

LDAT

i KKB

1

1

1

,,

3

1

3

1

3

1

3

1

iiii

iiii

iiiijUji

RRK

RRK

RRnFK

4

1

ii

c

T UK

Computes the convective

cell residual with second

order accuracy (linear data)

High-order RD scheme (cont.) for Navier-Stokes equations

High-order RD scheme (cont.)

How to construct a 3rd-order RDS (Ph.D. 2000, LTH):

1. Use (upwind or upwind-biased) uniformly bounded residual-distribution

coefficients (linearity/accuracy preserving RD scheme), and apply to total cell-

residual

2. Compute the total cell residual (convective + diffusive + unsteady terms)

with the required accuracy:

- we used condition-1 + linear solution, second order accurate integration for

2nd order RDS

- we need to use condition-1 + use quadratic reconstruction, 3rd order

accurate integration for 3rd order RDS

The idea is to use the same accuracy-preserving RD scheme, as for second

order schemes, but compute the total cell residual with 3rd order accuracy.

)(8

)(

2

01

1010

,,)( ii

i

j

i

j

ii

mid

rrZZZZ

Z

T

cc

T

j

c

T dSFdvF ..,

j

jj

jj

jj

j

c

ZZ

pZZ

pZZ

pZZ

ZZ

F

4

33

22

11

0

Use parameter variable Z and

assume a quadratic variation

over the tetrahedral cell.

)2

(1

2

3

2

2

2

140

ZZZZZp

),,,,1( 321 HuuuZ

2

2

ip

uTcH

High-order RD scheme (cont.) Convection residual discretization, 3rd order.

Cell-residual in integral form: Z,j computed with 2nd order accuracy

(multi-step algorithm)

High-order RD scheme (cont.) Convection residual discretization, 3rd order.

4

1 ,

,})({.

i face ij

ijc

j

T

cc

T

i

dn

nFdSF

i

i

i

i

i

i

face

jij

face

jjij

face

jjij

face

jjij

face

jij

face

ijc

j

dZZn

dpZZn

dpZZn

dpZZn

dZZn

dnF

)(

)(

)(

)(

)(

)(

4,

33,

22,

11,

0,

,

i

i

kJ

face

jk

face

ZZdZZI )(

6

1,

)()(

10

1010 }{

)(

ii face

iii

j

i

k

face

jk

i

i

dHH

dZZ

ZZ

kjHHface

face

kj IAdHHi

i

,

)()( ][

kjHHI ,][ Pre-computed matrix

dsFdvFT

v

T

v

j

v

T .,

k

k

face

v

T

vv

T nFdsFk

..4

1

)(

])()[( ,,

,

ii

i

uuu

Assuming a quadratic variation of the Z variables over the cell, the

diffusive flux vector integral can be computed over the cell-face.

Use the values of the Z variable and its gradients,

defined in the nodes of the high-order FEM tetrahedral-cell.

High-order RD scheme (cont.) Diffusion residual discretization, 3rd order.

High-order RD scheme (cont.) Unsteady residual discretization , 3rd Order.

t

UUUU

nnkn

t

.2

43 1,1

,

T

t

T

t

v

T dvUQdvU9

0

)(

,

)(

, ..

dvQUT

t

v

T .)(9

0

)(

,

9,..,4;20.0

3,..,0;05.0

;.)(

T

T

T

VI

VI

dvQI

2nd order discretization in time,

and 3rd order in space:

High-order RD scheme (cont.)

Monotone shock capturing

1. Shock detection or or

2. Blending between the high-order scheme and

a first order positive RD scheme (the N-scheme)

,..),(:

.).1(

Pfwhere

N

i

LDA

i

T

i

= 0 for a smooth flow

= 1 (discontinuity detected)

)(22

avp

ph

nodesN

j

#

1

).

(2

2

V

V

- Uses a Multi-D Upwind Residual-Distribution scheme

- Formulated for fully unstructured grids (tetrahedrons),

- Compact scheme, highly efficient parallel algorithm.

- Implicit time integration (dual time-stepping algorithm).

- 3rd - order accuracy in space (using FEM integration)

- 2nd - order time discretization (BDF2 scheme)

- Acceleration techniques: preconditioning, point-

implicit relaxation, geometric multi-grid, etc.

Summary of this 3rd order RD algorithm

- Steady inviscid flows

a. Sine-bump channel flow inlet Mach 0.5.

- Steady viscous flows

c. Laminar flat-plate boundary layer, Reynolds 2000.

- Unsteady inviscid flows

b. Vortex transport by uniform flow Mach 0.04.

- Shock capturing

d. Shock vortex interaction.

- Large Eddy Simulation

e. LES of turbulent channel flow.

Results

Inviscid sine bump channel flow:

- Inlet Mach number 0.5

Steady Euler

LDA 2nd LDA 3rd

Asymptotic accuracy

Steady Euler

0.0001

0.001

0.01

0.1

0.1 1

h

Err

or

Classical

Num3rd

RDS Solution on 32x8 grid

Maximum entropy production:

- 2nd order scheme E=2.3 e-4

- 3rd order scheme E=5.1 e-6

- Cell centered FVM(2ndO) E=4.2 e-4

Laminar viscous flow over a flat plate:

- Infinite Mach # 0.5, Re 2000

- Grid0 of 32x18 grid points

Laminar flat plate boundary layer

Re=2000

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5

U/Uinf

y*s

qrt

(Uin

f/m

iu/x

)

Blasius-U/Uinf

Num.3rdO Lev0

Num.3rdO Lev1

Num.3rdO Lev2

Num.3rdO Lev3

Laminar flat plate boundary layer

Re=2000

0

2

4

6

8

10

12

14

16

18

0 0.01 0.02 0.03

V/Uinf

y*s

qrt

(Uin

f/m

iu/x

)

Blasius-V/Uinf

Num.3rdO Lev0

Num.3rdO Lev1

Num.3rdO Lev2

Num.3rdO Lev3

Steady Navier-Stokes

Asymptotic accuracy

Laminar flat plate

0.0001

0.001

0.01

0.1

0.1 1

h

Err

or

Num3rd

Unsteady Euler

Vortex transport by Inviscid flow.

- Uniform flow Mach = 0.04

Third order results, grid 64x64



Unsteady Euler (cont.)

Vorticity X=0.05, T=5

-10

-5

0

5

10

15

20

25

30

0 0.025 0.05 0.075 0.1

Y

Om

eg

aZ

3rdO Sol. (32x32x2)

Sol. (32x32x2) T=0

3rdO Sol. (64x64x4)

Sol. (64x64x4) T=0

2ndO Sol. (32x32x2)

2ndO Sol. (64x64x4)

Vorticity Y=0.05 , T=5

-10

-5

0

5

10

15

20

25

30

0 0.025 0.05 0.075 0.1

X

Om

eg

aZ

3rdO Sol. (32x32x2)

Sol.at T=0 (32x32x2)

3rdO Sol. (64x64x4)

Sol. (64x64x4) T=0

2ndO Sol.(32x32x2)

2ndO Sol. (64x64x4)

Asymptotic accuracy

Vortical Euler flow

0.00001

0.0001

0.001

0.01

0.1

0.1 1

h

Err

or

Num3rd

Unsteady Euler (cont.)



Third order results, grid 64x64

T= 0 T = 12 periods T = 24 periods

Shock vortex interaction

Shock-vortex interaction:

- Steady shock in mid channel

- Vortex moves from left to right

Note: vortex preserving strength,

before and after crossing shock

Shock vortex interaction

LES of turbulent channel flow

Turbulent channel flow:

- Reynolds # = 5400

- Ret= 344

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.25 0.5 0.75 1

y/h

U/u

t

DNS(Kim et al.)

2ndO LES

3rdO LES

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5y/h

u' r

ms/u

t

Exp. (Kreplin)

DNS (Kim et al.)

2ndO LES

3rdO LES

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5y/h

v' r

ms/u

t

Exp.

DNS(Kim et al.)

2ndO LES

3rdO LES

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.1 0.2 0.3 0.4 0.5

y/h

w' rm

s/u

t

Exp. (Kreplin)

DNS(Kim et al.)

2ndO LES

3rdO LES

LES Smagorinsky ut/Ub Uc/Ub

3rd O MDU 0.0640 | 1.164

2nd O MDU 0.0627 | 1.186

-------------------------------------------

DNS (Kim et al.) 0.0643 | 1.162

Multidimensional Residual-Distribution

Solving for flow and “optimal” mesh

(Grids and solutions from Residual Minimization,

Nishikawa, Rad, Roe, 2001)

Solving for flow and solution using RDS

Main ideas:

- Use multidimensional RDS to compute solution at vertices,

- There are 5-6 times less vertices than cells in the tetrahedral-

cells mesh …

- Use the extra “conditions” (cell-residual must be driven to

zero) to define mesh motion equations, using an LSQ approach,

- Algorithm computes an improved solution on a “optimized”

mesh, which minimizes the overall error in a specific norm.

(Nishikawa, 2001)


Flow over Joukowsky airfoil

(known theoretical solution)

Original mesh Adapted mesh

(Nishikawa, 2001)


Flow over Joukowsky airfoil

(known theoretical solution)

Original mesh solution Cp, o Adapted mesh Cp, o

(Nishikawa, 2001)

Comparison with theoretical solution ----

- Resolves better real complex multidimensional physics (!)

- It is much more accurate that 2nd order Finite Volume method,

- It is capable of handling complex geometry (formulated tetrahedrons),

- Has a compact stencil algorithm, at every step (which leads to very

efficient parallelization),

- It is relatively to easy to extend to high order accuracy (at least from

2nd to 3rd order), and 3rd order results are significantly more accurate,

- Can be used to solve for flow and node location - using the

combined RDS/LSQ approach - for an optimal solution, on a given

mesh topology.

Why using Multi-D Residual-Distribution schemes ?

Backup slides

High-order Residual Distribution Scheme

for Scalar Transport Equations

on Triangular Meshes

From “High-order fluctuations schemes on triangular meshes”

R.Abgrall and Phil Roe, 2002

High-order RD scheme for scalar equations

)(3

2

2

IIIIII

IVH


From (Hubbard, J. Computational Physics 2007)

“Status of Multidimensional Residual Distribution Schemes and

Applications in Aeronautics”,

Deconinck et al. AIAA 2000-2328.

Space-time Residual Distribution

schemes for unsteady simulations

“Construction of 2nd order monotone and stable residual distribution

schemes: the unsteady case”,

Abgrall et al. VKI 2002

Space-time RD for unsteady simulations

t

ttxu

t

ttxutxu nnnnh

)()(),( 11

1

.)(.

n

n

t

ti

i

h

tx dtdxx

uA

t

u

i

n

i

n

ii

s

n

i

n

i

t

i

i

st

itx

uuKt

uu

)(2

)(3

1

1

,

)(2

)(2

)(3

0

111

1,

,

nn

ii

nn

ii

n

i

n

i

N

ni

N

ni

uuKt

uuKt

uuV

i

n

jj

i

i

n

i

n

jj

i

i

n

uKKu

uKKu

1

1

11

“Upwind-in-time”


s

i

ij

t

ji

t

ii

LDA

ni

LDA

ni

)12

1(

)6

1(

0

1,

,

1)(

j

jii KK

LDA space-time scheme: LDA+N space-time scheme:

j

N

ni

tx

LDA

ni

N

ni

B

ni

B

ni

l

ll

1,

,

1,1,1,

,

)1(

0

“Status of Multidimensional Residual Distribution Schemes and Applications in Aeronautics”

Deconinck et al. 2000


Reflection of a planar

shock from a ramp (density plot)

Shock reflection on a forward facing

step (density plot)

Shock-vortex interaction

From “Construction of 2nd order monotone and stable residual distribution

schemes: the unsteady case”, Abgrall et al. 2002


Convection of vortex

- Periodic BC’s

- One revolution simulated

- 2nd and 3rd order ST-RDS

compared

- Pressure contours displayed

From (Nadege Villedieu ,

VKI Ph.D., 2009)

doru caraenidoru caraeni cd-adapco, usa cfd futures conference, august 6-8, 2012 why i did...

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