AIAA Paper 2006-0697, 01/10/2006

Numerical Simulation of 3-D Wing Flutter with

Fully Coupled Fluid-Structural Interaction

Xiangying Chen, Ge-Cheng Zha

Dept. of Mechanical and Aerospace Engineering

University of Miami

Coral Gables, Florida 33124

Ming-Ta Yang

Discipline Engineering - Structures

Pratt & Whitney

400 Main Street, M/S 163-07

East Hartford, CT 06108

1

Background

• Important Issue: Aircraft Wing and Turbomachinery Blade

Flutter

• Fully coupled fluid-structure model is necessary to capture the

nonlinear flow phenomena

Objective

• Develop High Fidelity Predicting Tool for Wing/Blade Flutter

2

Numerical Strategy

• 3D Time Accurate RANS

• Dual Time Stepping, Implicit Gauss-Seidel Iteration

• Low Diffusion Upwind Scheme

• 2nd Order Accuracy in Space and Time

• Baldwin-Lomax Turbulence Model

• Enforce Geometry Conservation Law

3

Governing Equations: 3D RANS

∂Q′

∂τ+∂Q′

∂t+∂E′

∂ξ+∂F′

∂η+∂G′

∂ζ=

1

Re

(

∂E′v

∂ξ+∂F′

v

∂η+∂G′

v

∂ζ

)

(1)

Q′ =Q

JE′ =

1

JE, (2)

4

Q =

ρ

ρu

ρv

ρw

ρe

, E =

ρU

ρuU + ξxp

ρvU + ξy p

ρwU + ξz p

ρeU + pU

,

U = ξt + ξxu+ ξy v + ξzw, U = U − ξt

Ev =

0

τxx − ρu′′u′′

τxy − ρu′′v′′

τxz − ρu′′w′′

Qx

5

τij = −2

3µ∂uk∂xk

δij + µ(∂ui∂xj

+∂uj∂xi

) (3)

Qi = uj(τij − ρu′′u′′)− (qi + CpρT ′′u′′i ) (4)

qi = −µ

(γ − 1)Pr

∂a2

∂xi(5)

• Molecular viscosity µ = µ(T )

• Total energy:

ρe =p

(γ − 1)+

1

2ρ(u2 + v2 + w2) + k (6)

• Turbulent: Baldwin-Lomax model

6

Time Marching Scheme:

Implicit Gauss-Seidel Relaxation, Dual Time Stepping

[

(

1

∆τ+

1.5

∆t

)

I −

(

∂R

∂Q

)n+1,m]

δQn+1,m+1 = Rn+1,m−3Qn+1,m − 4Qn +Qn−1

2∆t

(7)

R = −1

V

∫

s

[(F − Fv)i+ (G−Gv)j+ (H −Hv)k]·ds (8)

7

Roe’s Riemann Solver on Moving Grid System

E′i+ 1

2

=1

2[E′′(QL)+E

′′(QR)+QLξtL+QRξtR−|A|(QR−QL)]i+ 12

(9)

A = TΛT−1 (10)

(U + C, U − C, U , U , U) (11)

U = ξt + ξxu+ ξy v + ξzw (12)

C = c√

ξ2x + ξ2y + ξ2z (13)

ξt = (ξtL + ξtR√

ρR/ρL)/(1 +√

ρR/ρL) (14)

8

Boundary Conditions

• Upstream: All variables specified except pressure extrapolated

from interior

• Downstream: All variables extrapolated except pressure specified

• Solid wall boundary conditions: Non-slip condition

u0 = 2xb − u1, v0 = 2yb − v1 (15)

and adiabatic and the inviscid normal momentum equation

∂T

∂η= 0,

∂p

∂η= −

(

ρ

η2x + η2y

)

(ηxxb + ηy yb) (16)

9

Geometric Conservation Law

S = Q

[

∂J−1

∂t+

(

ξtJ

)

ξ

+(ηtJ

)

η+

(

ζtJ

)

ζ

]

(17)

Sn+1 = Sn +∂S

∂Q∆Qn+1 (18)

10

Structural Model

Governing equation:

Md2u

dt2+C

du

dt+Ku = f (19)

u =

u1...

ui

...

uN

, f =

f1...

fi...

fN

, ui =

uix

uiy

uiz

, fi =

fix

fiy

fiz

.

11

Modal Approach

u(t) =∑

j

aj(t)φj = Φa (20)

Mode shape matrix: Φ = [φ1, · · · , φj , · · · , φ3N ].

Modal Coordinates: a = [a1, a2, a3, · · · ]T

d2ajdt2

+ 2ζjωj

dajdt

+ ω2jaj =

φTj f

mj

(21)

12

State form:

[M]∂{S}

∂t+ [K]{S} = q (22)

where

S =

aj

aj

, M = [I], K =

0 −1(

ωjωα

)2

2ζ(

ωjωα

)

,

q =

0

φ∗jT f∗V ∗

(

bsL

)2 mv∗

.

Time marching: same as flow solver

(

1

∆τI+

1.5

∆tM+K

)

δSn+1,m+1 = −M3Sn+1,m − 4Sn + Sn−1

2∆t−KSn+1,m+qn+1,m+1

(23)

13

Fully Coupled Fluid-Structural Interaction Procedure

CFD and structuralresiduals < ε ?

No Yes

Nex

tPse

udo

Tim

eS

tep

Sn-

1 =S

n

Sn =

Sn+

1

Qn-

1 =Q

n

Qn =

Qn+

1

Qn,

m=

Qn,

m+

1

Sn,

m=

Sn,

m+

1

Sn+1,m+1

Qn+1,m+1

Stop if n≥nmaxN

extP

hysi

calT

ime

Ste

p

Initial flow field and structuralsolutions, Qn, Sn

Aerodynamic forces

Structural displacementby solving structuralequations

CFD moving and deformingmesh

CFD flow field by solvingflow governing equations

14

Results: ONERA M6 Wing Mesh M=0.84, Re=19.7×106

15

ONERA M6 Wing Surface Pressure

ExperimentComputation

X/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.0

-0.5

0.0

0.5

1.0

1.5

Section 7 (z/b = 0.99)

X/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.0

-0.5

0.0

0.5

1.0

1.5

Section 4 (z/b = 0.8)

X/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.0

-0.5

0.0

0.5

1.0

1.5

Section 3 (z/b = 0.65)

X/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.0

-0.5

0.0

0.5

1.0

1.5

Section 2 (z/b = 0.44)X/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.0

-0.5

0.0

0.5

1.0

1.5

Section 6 (z/b = 0.95)

X/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.0

-0.5

0.0

0.5

1.0

1.5

Section 5 (z/b = 0.9)X/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.0

-0.5

0.0

0.5

1.0

1.5

Section 1 (z/b = 0.2)

16

Structural Solver Validation: Plate Wing

node 491

node 510

17

Structural Solver Validation: Plate Wing

5 modes, Present computationfull model, ANSYS5 modes, ANSYS

Node #: 491

Time

Uy

0 1 2 3 4 5-0.1

-0.05

0

0.05

0.1

Time

Ux

0 1 2 3 4 5-0.08

-0.04

0

0.04

0.08

Time

Uz

0 1 2 3 4 5-8

-4

0

4

8

18

AGARD Wing 445.6, Damped response M∞ = 0.96 and V ∗

= 0.28, 1st 5 Modes

Dimensionless time

Gen

eral

ized

disp

lace

men

t

0 50 100 150 200 250-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015Mode 1Mode 2Mode 3

19

AGARD Wing 445.6, Neutrally stable response. M∞ =

0.96 and V ∗ = 0.29, 1st 5 Modes

Dimensionless time

Gen

eral

ized

disp

lace

men

t

0 50 100 150 200 250-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015Mode 1Mode 2Mode 3

20

AGARD Wing 445.6, Diverging response M∞ = 0.96 and V ∗

= 0.315, 1st 5 Modes

Dimensionless time

Gen

eral

ized

disp

lace

men

t

0 50 100 150 200 250-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015Mode 1Mode 2Mode 3

21

Pressure Contours, Uppermost position M∞ = 0.96 and V ∗

= 0.29

0.68

0.69

0.70

0.72

0.74

0.76

0.78

0.70

0.72

0.74

0.76

0.78

0.68

0.80

0.80

x

z

1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

22

Pressure Contours, Neutral position M∞ = 0.96 and V ∗ =

0.29

0.69

0.71

0.74

0.75

0.77

0.79

0.81

0.70

0.68

0.71 0.

74 0.76

0.78

0.80

x

z

1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

23

Pressure Contours, Lowermost position M∞ = 0.96 and V ∗

= 0.29

0.71

0.72

0.74

0.76

0.78

0.80

0.81

0.72

0.71

0.72 0.

74

0.76 0.78

0.80

0.80

0.71

x

z

1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

24

Computed AGARD Wing 445.6 Flutter Boundary

Mach number

Flu

tter

spee

din

dex

0.2 0.4 0.6 0.8 1 1.2 1.40.2

0.3

0.4

0.5

0.6

Present computationExperiment

25

Conclusions

• A fully coupled 3D fluid-structural interaction methodology for

flutter prediction is developed.

• A dual time stepping implicit unfactored Gauss-Seidel iteration

with Roe scheme is employed.

• The structural response using modal approach with 1st 5 modes

agrees excellently with finite element solution.

• The predicted AGARD Wing 445.6 flutter boundary agrees well

with experiment.

26