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