un approccio ``libero'' alla moderna aeroelasticità ... coupling of structural and...

POLITECNICO DI MILANO

Corso di Laurea in Ingegneria Aeronautica

Un approccio “libero” alla modernaAeroelasticita Computazionale

Relatore: Prof. Paolo Mantegazza

Tesi di Laurea di:

Giulio Romanelli matr. 679778

Elisa Serioli matr. 679777

Anno Accademico 2007/2008

Un approccio “libero” alla moderna Aeroelasticita Computazionale

Introduction Structural system Aerodynamic system Coupling procedure Aeroelastic system Conclusions

Introduction

General framework:

academic and industrial aeroelastic problems

transonic regime

Computational Aeroelasticity (CA)

Up to now:

coupling of structural and aerodynamic commercial software

MSC.Nastran, FEMAP

FLUENT, GAMBIT, Tecplot

Target

To build a toolbox of free software for aeroelastic analysis in the transonic regime



Block diagram

Post-processorPre/Post-processor Mesh generator

ParaViewGmshSalome

Aeroelastic Solver

Aerodynamic SolverStructural Solver

Aeroelastic Interface

OpenFOAMCode Aster

NAEMO MASSA

Linear interpolation

Build [Ham(k, M∞) ] Build V∞ − ω, V∞ − g



Free software

Definition:

Fundamental freedoms by R. Stallmanand Free Software Foundation (1986):

1 run the program for any purpose

2 study and modify the program

3 copy the program

4 improve the program and releasethe improvements to the public,so the whole community benefits

source code available

freedom not price

copyleft licences GNU GPL and LGPL



Structural solver

Reference: Structural analysis commercial software MSC.Nastran by MSC

Requirements:

complete finite element library (beam, plate, 3D, . . .)

complete set of materials (iso/ortho/aniso-tropic, visco-elasto-plastic σ − ε law)

complete set of boundary and load conditions

static and dynamic, linear and non-linear analysis; modal analysis

possibility to add/develop new functionalities/modules

efficient pre/post-processing (mesh handling, numerical results visualization)

Structural analysis free software Code Aster by EDF

Fortran 77, Python, C, CAL programming languages

GNU General Public License (GPL)

detailed documentation and test problems (in French)



Code Aster

· · ·

DEBUT( )

FIN( )

List of commands JDC

Catalogue

Superviseur

Jeveux

CalculCommand #1

Command #2

Command #N

syntax control

commands translation

execution control

execute all operations

memory control



Beam (1)

Beam modal and static analysis:

Geometry: R = 0.15 m, e = 0.05 m, L = 10 m

Material: steel, ρ = 7800 kg/m3, E = 2.1 · 1011 Pa, ν = 0.3

Boundary conditions: simply supported (bending), fixed (torsion)

Loads: F = 2500 N , M = 5000 Nm

FEM dicretization: 20 beam (POU D E) elements

comparison with exact solution

e

Rx

y

y

z z

L

m, I, EJ, GJ

MF

=

=

sx = 0

sy = 0

sz = 0

θx = 0

sx = 0

sy = 0

sz = 0

θx = 0



Beam (2)

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8 10

z1 [−

]

x [m]

¯fz1

= 7.828 Hz

× 10−2

Code Aster

Exact

zexact1 (x) =

√

2

mLsin(πx)

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

ϑ1 [−

]

x [m]

¯fϑ1

= 161.062 Hz

× 10−1

Code Aster

Exact

θexact1 (x) =

√

2

ILsin(πx)

0

2

4

6

8

10

12

0 2 4 6 8 10

z [

m]

x [ m ]

¯

× 10−4

Code Aster

Exact

zexact(x) =

F

12EJ

(

3

4L

2x− x

3

)

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8 10

ϑ [

°]

x [ m ]

¯

× 10−4

Code Aster

Exact

θexact(x) =M

2GJx



Wing-like structure

x

y

z

bxbx

by

by=

=

=

=

F1 = 13200 N

F2 = 13200 N

F3 = 13200 N

F4 = 6600 N

1600mm

600 mm

200 mm200 mm

300

mm

300 mm 25

mm

25

mm

25 mm20 mm

3.2

mm

2.5 mm2.5 mm2.5 mm

2.5 mm

1.5 mm 1.5 mm

2.0

mm

3.0

mm

Wing-like structure static analysis:

Material: alluminium, ρ = 2800 kg/m3,E = 7.1 · 1010 Pa, ν = 0.3125

FEM discretization: 224 beam (POU D E)and 1440 plate (DKT) elements

comparison with MSC.Nastran numericalresults (RBE3 distributed loads)



Aerodynamic problem (1)

Supersonic bubble

Shock wave

Curved shock wave

M∞ = M∞, cr + ε

M∞, cr < M∞ < 1

M∞ > 1

M >1

M >1

M >1

M <1

M <1

M <1

Transonic regime:

modern airplanes designed to fly at M∞ ∈ (0.7, 1.3)

subsonic and supersonic regions simultaneously present

non-reversible procecesses such as shock waves

strong non-linearities

possible shock wave interaction with boundary layer(buffeting) or structural displacement (aileron buzz)

Computational Aeroelasticity

classical linearized mathematical models (smalldisturbances theory) efficient but not accurate

Computational Fluid Dynamics (CFD)



Aerodynamic problem (2)

Mathematical models: compromise between accuracy and efficiency

1 Full potential equations (1 unknown per cell)

omoentropic irrotational flow but OK up to M∞ = 1.3÷ 1.4

linearized wake models for lifting bodies

2 Euler equations (Nd + 2 unknowns per cell)

high Re, thin boundary layer, moderate α

accurate prediction of pressure distribution

3 Reynolds Averaged Navier Stokes equations (Nd + 2 + Nt unknowns per cell)

accurate prediction of aerodynamic lift and drag loads

no free software to create boundary layer meshes



Aerodynamic solver

Reference: Aerodynamic analysis commercial software FLUENT by ANSYS

Requirements:

complete finite volume library (cell-centered or node-centered)

complete set of materials (PIG, real reacting gas mixture, . . .)

complete set of boundary conditions (Riemann, slip, . . .)

steady and unsteady, implicit and explicit, segregated and coupled solvers

possibility to add/develop new functionalities/libraries

efficient pre/post-processing (mesh handling, numerical results visualization)

Aerodynamic analysis free software OpenFOAM by Wikki

C++, Bash programming languages

GNU General Public License (GPL)

uncomplete documentation and test problems



OpenFOAM

OpenFOAM-1.4.1

applications

solvers

utilities

Applications

Solvers

Utilities

src

finiteVolume

meshTools

OpenFOAM

PStream

thermophysicalModels

wmake

Libraries

Numerical methods

Numerical grid

Kernel

Parallel computing

Physical models

Compiler



Evaluation of existing solvers (1)

L = 4.17 m

h=

1m

α1 α2 =

θθ

v1v2

v2

v3

1©

2©

3©M1, P1, T1

M2, P2, T2

M3, P3, T3

2D Oblique shock reflection

M∞ = 2.9, α1 = 29

comparison with exact solution

PC AMD64 2.2 Ghz, 1 Gbyte RAM

(A) Ne = 40 × 10 = 400 ∆t = 4·10−5 (B) N

e = 80 × 20 = 1600 ∆t = 2·10−5

(C) Ne = 160 × 40 = 6400 ∆t = 1·10−5 (D) N

e = 320 × 80 = 25600 ∆t = 5·10−6




0.1

1

10

0.001 0.01 0.1 1

||eh||

L1 [−

]

h [ m ]

O(h)

O(h2)

× 10−2

rhoSonicFoam

rhoSonicFoam solver

density based

cell-centered

segregated algorithm

conservative variables

0.01

0.1

1

10

100

100 1000 10000 100000

CPU

tim

e [

s]

Ne

O(Ne)

O(Ne

2)

× 10−1

rhoSonicFoam

Figure: Thermodynamic pressure P contourand density ρ isolines




0.1

1

10

0.001 0.01 0.1 1

||eh||

L1 [−

]

h [ m ]

O(h)

O(h2)

× 10−2

rhopSonicFoam

rhoSonicFoam

rhopSonicFoam solver

density-pressure based

cell-centered


conservative variables

PISO correction loop

0.01

0.1

1

10

100

100 1000 10000 100000

CPU

tim

e [

s]

Ne

O(Ne)

O(Ne

2)

× 10−1

rhoSonicFoam

rhopSonicFoam





0.1

1

10

0.001 0.01 0.1 1

||eh||

L1 [−

]

h [ m ]

O(h)

O(h2)

× 10−2

sonicFoam

rhopSonicFoam

rhoSonicFoam

sonicFoam solver

density-pressure based

cell-centered


nonconservative variables

PISO correction loop

0.01

0.1

1

10

100

100 1000 10000 100000

CPU

tim

e [

s]

Ne

O(Ne)

O(Ne

2)

× 10−1

rhoSonicFoam

rhopSonicFoam

sonicFoam




Euler equations

Governing equations for time-dependent, compressible, ideal (inviscid µ = 0 andnon-conducting κ = 0) fluid flows in coupled, integral, conservative form:

V ⊆ RNd

S = ∂V ⊆ RNd−1

β

Sinflow ⊆ S | β · n ≤ 0

n

n

ddt

∫V

udV +

∮S

f(u) · n dS = 0

u(x, 0) = u0(x)

u(x ∈ Sinflow, t) = ub(t)

Conservative variables vector and inviscid flux function tensor:

u =

ρ

m

Et

=

ρ

ρ v

ρe + 12ρ|v|2

f =

m

mρ ⊗m + P [ I ]

mρ (Et + P )

Polytropic Ideal Gas (PIG) thermodynamic model: γ = Cp/Cv, R = Ru/M



FV Framework

Sh = ∂Vh

Ωi Ωj

Γij nij

On each cell: averaged cons. variables vector

Ui(t) =1

|Ωi|

∫Ωi

u(x, t) dV

On each interface: numerical fluxes vector

Fij(t) =1

|Γij |

∫Γij

f(u) · n(x, t) dS

Cell-centered FVM: spatially discretized Euler equations ODE system

dUi

dt+

1

|Ωi|

Nf∑j=1

|Γij |Fij = 0 Targets:

Monotone and sharp solution

near discontinuities

2nd order of accuracy in space

in smooth flow regions?



Monotone numerical fluxesPSfrag

Γij

Ωi

x, u

y, v

z, w

X, vX

Y, vY

Z, vZ

R(nG

ij)

nij

G

L

Several Godunov-type monotone 1st orderexpressions for the numerical fluxes vectorpresented in Literature

FIij = Fij(Ui, Uj)

Implemented in AeroFoam:

Approximate Riemann Solver (ARS)

Convective Upwind and Split Pressure (CUSP)

Harten-Lax-vanLeer (HLL/C)

Osher-Solomon (OS)



High resolution numerical fluxes

Idea: combine a monotone 1st order numerical flux FIij (works fine near shocks)

and a 2nd order numerical flux FIIij (works fine in smooth flow regions) by means

of a flux-limiter function Φ

FHRij = FI

ij + Φ (FIIij − FI

ij ) = FIij + Aij ,

Implemented in AeroFoam

Lax-Wendroff (LW)

Jameson-Schmidt-Turkel (JST)

Remark

To build the antidissipative numerical fluxes vector Aij = Aij(Ui, Uj ; Ui∗ , Uj∗)numerical solutions Ui∗ and Uj∗ on extended cells Ωi∗ and Ωj∗ are also needed



Extended cells connectivity

Idea: continue Ωi and Ωj cells along nij , e.g. Ωj∗ : Ωq ∈ B(Pj) = Ωq|Pj ∈ Ωqsuch that ∆⊥ = ‖(xq − xij)− (xq − xij) · nij nij‖ is minimum

Extended cells connectivity data structures initialized in the pre-processing stage

Ωi ΩjΩi∗ Ωj∗Pi

Pj

Γijnij

A. Incremental search algorithm

initial guess xA = sA bnij

Ωj∗ : Ωq such that xA ∈ Ωq

if Ωj∗ ≡ Ωj update sA = 2 sA

B. Nonincremental search algorithm

xB = sB bnij where sB = 4|Ωj |/|Γij |Ωj∗ : Ωq such that xB ∈ Ωq



Time discretization

0

0.5

1

1.5

2

2.5

3

-3 -2.5 -2 -1.5 -1 -0.5 0

RK2

RK3

RK4

Re

Im

Explicit Runge-Kutta method of orderp = 2, 3, 4 as a compromise between:

computational efficiency

memory requirements

CFL stability condition

order of accuracy in time

Local Timestepping (LT)

speed up convergence to steady state

on each cell ∆ti so that Coi≡Comax



Boundary conditions

At each boundary interface Γij ∈ Sh = ∂Vh suitable numerical solutions UGCj , UGC

j∗must be set on the fictitious or ghost cells ΩGC

j , ΩGCj∗

Ωi

ΩGC

j

Ωi∗

ΩGC

j∗

Γij

nij

vij

Sh = ∂Vh

uij



Riemann boundary conditions

At each asymptotic/external boundary interface Γij ∈ S∞

Boundary type Flow type NbcAssignedvariables

Extrapolatedvariables

Sinflow

vij · bnij < 0

Supersonic flow

vij · bnij > cij

(∼SupersonicInlet)

Nd + 2 T, v, P −

Subsonic flow

vij · bnij < cij

(∼Inlet)

Nd + 1 T, v P

Soutflow

vij · bnij > 0

Supersonic flow

vij · bnij > cij

(∼ExtrapolatedOutlet)

0 − T, v, P

Subsonic flow

vij · bnij < cij

(∼Outlet)

1 T v, P



Slip boundary conditions

At each solid/impermeable boundary interface Γij ∈ Sb

linear extrapolation of numerical solutions UGCj , UGC

j∗ on ghost cells ΩGCj , ΩGC

j∗

set to zero normal velocity component and update conservative variables

vj = vj − (vj · nij) nij Etj = Et

j −1

2ρj |vj |2 +

1

2ρj |vj |2

Generalization: transpiration boundary conditions

The geometric and kinematic effects of any given body displacement law s(x, t)(rigid and deformative) can be simulated by means of a transpiration velocity:

vj = vj − (vj · nij) nij + Vn nij

Mesh is not deformed runtime (expensive) but only in the post-processing stage



Comparison with OF solvers (1)

(C) Ne = 160 × 40 = 6400 ∆t = 1·10−5

2D Oblique shock reflection

M∞ = 2.9, α1 = 29

(C) mesh Ne = 160× 40 = 6400

rhoSonicFoam rhopSonicFoam

sonicFoam AeroFoam



Comparison with OF solvers (2)

0.1

1

10

0.001 0.01 0.1 1

||eh||

L1 [−

]

h [ m ]

O(h)

O(h2)

× 10−2

AeroFoam

sonicFoam

rhopSonicFoam

rhoSonicFoam

Least Squares fit for ‖eh‖L1 = A hp

Solver A p Rh

rhoSonicFoam 0.267 0.543 0.998

rhopSonicFoam 0.425 0.765 0.998

sonicFoam 0.048 0.045 0.221

AeroFoam 1.601 1.382 0.997

0.01

0.1

1

10

100

100 1000 10000 100000

CPU

tim

e [

s]

Ne

O(Ne)

O(Ne

2)

× 10−1

rhoSonicFoam

rhopSonicFoam

sonicFoam

AeroFoam

0

1

2

3

4

5

6

7

100 1000 10000 100000

Speedup

[−]

Ne

rhoSonicFoam

rhopSonicFoam

sonicFoam



Incompressible limit

2D Fixed cylinder

M∞ = 0.1 !

15K triangular mesh (Gmsh)

comparison with exact solution(potential theory)

single iteration CPUtime = 0.09 s

-1

0

1

2

3

-1 -0.5 0 0.5 1

−C

p [−

]

x/R [−]

CpExact = 1 − 4 sin

2ϑ

AeroFoam

Exact



NACA 0012 airfoil

2D NACA 0012 airfoil

M∞ = 0.75, α = 4


comparison with FLUENT andexperimental data (AGARD 138)


-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

−C

p [−

]

x/c [−]

Upper Surface

Lower Surface

WT

FLUENT

AeroFoam



Thin airfoil

2D Thin airfoil

t/c = 0.01, M∞ = 0.5, 0.7, 1.0, 1.2,vertical step gust vg/V∞ = tan(1)


comparison with exact solution(small disturbances theory)


0.4

0.6

0.8

1

1.2

1.4

1.6

0 2 4 6 8 10 12 14 16

CL

/α/2

π [−

]

τ [−]

M∞

=0.5

M∞

=0.7

Exact

AeroFoam

0.4

0.6

0.8

1

1.2

1.4

1.6

0 2 4 6 8 10 12 14 16

CL

/α/2

π [−

]

τ [−]

M∞

=1.2

M∞

=1.0

Exact

AeroFoam



ONERA M6 wing (1)

b = 1.196 m

c r=

0.8

06

m

c t=

0.5

09

m

c

1

2

3

4

5

67

ΛLE = 30

ΛTE = 15.8

M∞ = 0.84

0.4

4c r

x

y

3D ONERA M6 wing

M∞ = 0.84, α = 3.06

350K tetrahedral mesh (GAMBIT)





ONERA M6 wing (2)

-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−C

p [−

]

x/c [−]

Upper Surface

Lower Surface

y/b = 0.20

WT

FLUENT

AeroFoam-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−C

p [−

]

x/c [−]

Upper Surface

Lower Surface

y/b = 0.95

WT

FLUENT

AeroFoam



RAE A wing + body (1)

b = 0.457 m

cr = 0.228 m

ct = 0.076 m

L = 1.928 m

xw = 0.609 m

xb =0.760 m

c

Λc/2 = 30

x

Rb =Ro/2

ϕ

R(x) Ro = 0.076 m

xo = 0.508 m

1

2

3

4

5

6

3D RAE A wing + body

M∞ = 0.9, α = 1






RAE A wing + body (2)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−C

p [−

]

x/L [−]

ϕ = +15°

ϕ = −15°

Body

WT

FLUENT

AeroFoam

-1

-0.5

0

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−C

p [−

]

x/c [−]

Upper Surface

Lower Surface

y/b = 0.925

Wing

WT

FLUENT

AeroFoam



Aeroelastic interface

Target: closed loop connection between structural and aerodynamic sub-systems

Requirements:

connection between topologically different domains and non-conformal meshes

exact treatment of rigid motions

conservation of momentum and energy transfer (Lyapunov energetic stability)

ua = [ I ] usPrinciple of virtual work←−−−−−−−−−−−−→ F a

s = [ I ]T F aa

Reference test problem:

AGARD 445.6 wing

Structural mesh Ns = 121 nodes

Aerodynamic mesh Nba = 5506 triangles



Linear interpolation

Ωj

xs, j1

xs, j2

xs, j3

s, j1

us, j2

us, j3

x⊥

a, i

ua, i

nj

n0, j

∆nj

us, j(x)

Algorithm:

project i-th aerodynamic node ontothe structural mesh and find parentstructural element Ωj

x⊥a, i = xa, i − (xa, i · bnj) bnj

piece-wise linear intepolation function

eus, j(x) =3X

k=1

us, jk φk(x)

Lagrangian basis functions

φk(x) = ak + bk x + ck y

set (i, jk) interpolation matrix element

[ I|i, jk ] = φk(x⊥a, i)



Verification

Mode n 1 (bending) Mode n 2 (torsion)

Mode n 3 (bending) Mode n 4 (torsion)



Transpiration velocity (1)

Idea:

simulate geometric and kinematic effects of a given body displacement law s(x, t)

mesh is not deformed runtime (computationally expensive)

coherent with local linearization of aerodynamic loads to build aeroelastic ROM

Linearized formulation

parametric description of body deformed configuration x(ξ, η, t), velocity v(ξ, η, t)

x = x0 − V ∞ t + s v ≡ dx

dt= −V ∞ + s

first order expression for the deformed normal versor bn(ξ, η, t)

bn ≡ x/ξ × x/η

‖x/ξ × x/η‖= bn0 + ∆bn +O(s2)

transpiration velocity

Vn = −V ∞ · ∆bn + s · bn0 = [ A ] q + [ B ] q



Transpiration velocity (2)

Finite Differences (FD) linearized formulation

non-linear (numerically linearized) normal versor deformation ∆bnj by FDM

∆bnj = bnj − bn0, j

uniform on each parent structural element ∆bni = ∆bnj ∀ x⊥a, i ∈ Ωj

superposition principle (geometric and kinematic inputs can be splitted)

Finite Differences (FD) non-linear formulation

mixed geometric-kinematic non-linear (numerically linearized) contribution retained

Vn = −vj · ∆bnj + sj · bn0, j + sj · ∆bnj

wider (M∞, α) validity interval, generality

perhaps not fully coherent with aerodynamic loads linearization



Pitching NACA 64A010 airfoil

2D NACA 64A010 airfoil

M∞ = 0.796, ∆α = 0 ± 1.01,k = ωLa/V∞ = 0.202, La = c/2

3.5K triangular mesh (Gmsh)

comparison with Flo3xx andexperimental data (AGARD 702)


-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-1 -0.5 0 0.5 1

CL

[−]

α [ ° ]

Cycling to limit cycle

WT

Flo3xx

AeroFoam



Aeroelastic problemreplacemen

E

A

FEM

Modal base

Hybrid base

Full potential

Euler

RANS

vg

a

ua , ua

us , us

F a

a

F a

s

[I ][I ]T



Linearized aerodynamic loads

Classic aeroelasticity:

frequency domain

efficient Reduced Order Model (ROM) for the locally linear(ized) unsteadyGeneralized Aerodynamic Forces (GAF) due to small structural displacements

aerodynamic transfer function matrix

Qa(s) = q∞ [ Ham(k, M∞) ] q(s)

Transonic regime

strongly non-linear flow field compatible with GAF local linearization?

fully non-linear trimmed steady solution

unsteady GAF satisfactorily linear with the small structural displacements



How to build [ Ham(k, M∞) ]

Idea: for each M∞ numerical simulation conceptually identical to wind tunnelexperimental procedure to identify aerodynamic loads ROM

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10

q(τ)

/Aq [−

]

τ [−]

τq

Algorithm:

1 non-linear trimmed steady solution

2 store Qa(τ) time history due to i-thblended step generalized displacement

qi(τ) =

8<:Aq

2[1− cos(kq τ)] τ < τq

Aq τ ≥ τq

3 build i-th column of [ Ham(k) ] with theFast Fourier Transform (FFT) algorithm

[ Ham(k)|i ] =Fk( Qa(τ) )

Fk( qi(τ) )



Block diagram

i = 1 i = 2 i = Nsi = Ns − 1

· · ·

· · ·

TRIM

GAFGAF GAFGAF

FFTFFT FFTFFT

ASSEMBLY COLUMNS

[ Ham(k) ]



Structural model

AGARD 445.6 wing

laminated mahogany weakened model n3

Ground Vibration Test (GVT) data

[rmr] = 1 kg [rcr] ' 0 [rkr] = [rω20,ir]

only modes n1÷ 4 considered

FEM model set-up with a genetic optimizer(SciLab + Code Aster)

offset between experim. and numerical data

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0

0.01

0.02

0.03

0.04

0.05

0.06

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Error mode n

1 Error mode n

2 Error mode n

3 Error mode n

4

ef1= 0.3% ef2

= 2.9% ef3= 4.1% ef4

= 2.1%



Aerodynamic model

b

cr

ct

c

1

2

3

4

Λc/4

M∞

cr/4

x

y

NACA 65A004

AGARD 445.6 wing

M∞ ∈ [ 0.678, 1.141 ], α0 = 0

La = c = 0.463 m





Trim @ M∞ = 0.960

Trim

symmetric airfoil, α0 = 0

trimmed structural solution undeformed

only aerodynamic steady state solution

qualitatively comparison with CFL3D

0 0.2 0.4 0.6 0.8 1 1.2

0

0.2

0.4

0.6

0.8

-0.4

-0.2

0

0.2

0.4

0.6

Cp



Dynamic linearity @ M∞ = 0.960

Dynamic linearity check

time response to 1st modal step input

doubled input q(2)1 = 2 q

(1)1

time response ratio H = Q(1)11 /Q

(2)11

µH = 0.5052, σH = 0.0172

0

1

2

3

4

0 0.5 1 1.5 2 2.5 3

q1 [m

]

τ [−]

× 10−3

(1)

(2)-2

-1.5

-1

-0.5

0

0 0.5 1 1.5 2 2.5 3

Q11

[m

2]

τ [−]

× 10−1

(1)

(2)



GAF(1) @ M∞ = 0.960

-4

-3

-2

-1

0

1

0 1 2 3 4 5

Q11

[m

2]

τ [−]

a

× 10−1

EDGE

AeroFoam

-4

-3

-2

-1

0

1

0 1 2 3 4 5

Q21

[m

2]

τ [−]

a

× 10−1

0

0.5

1

1.5

2

0 1 2 3 4 5

q1 [m

]

τ [−]

kmax = 10

ε = tan(1°)

× 10−1



GAF(2) @ M∞ = 0.960

-10

-5

0

5

10

15

0 1 2 3 4 5

Q12

[m

2]

τ [−]

a

× 10−1

EDGE

AeroFoam

0

0.5

1

1.5

2

0 1 2 3 4 5

q2 [m

]

τ [−]

kmax = 10

ε = tan(1°)

× 10−1

-10

-5

0

5

10

15

0 1 2 3 4 5

Q22

[m

2]

τ [−]

a

× 10−1



[ Ham(k) ] @ M∞ = 0.960

-4

-3

-2

-1

0

1

0 0.2 0.4 0.6 0.8 1

Ham

,11

[m

]

k [−]

-10

-5

0

5

10

0 0.2 0.4 0.6 0.8 1

Ham

,12

[m

]

k [−]

EDGE

Literature

AeroFoam

Re

Im

-4

-3

-2

-1

0

1

0 0.2 0.4 0.6 0.8 1

Ham

,21

[m

]

k [−]

-10

-5

0

5

10

0 0.2 0.4 0.6 0.8 1

Ham

,22

[m

]

k [−]



Flutter analysis (1)

0

20

40

60

80

100

200 250 300 350 400

f [

Hz

]

V∞

[ m/s ]

VF

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

200 250 300 350 400

g [−

]

V∞

[ m/s ]

VF

Mode n° 1

Mode n° 2

Mode n° 3

Mode n° 4

V∞−ω−g diagrams @ M∞ =0.960

MASSA post-processor

Rogers’ method with ‖eI‖max ≤ 5%

frequency coalescence

modes n1− 2 dynamically important

mode n1 unstable

flutter point

(fF = 14.176 Hz

VF = 306.538 m/s



Flutter analysis (2)

0.3

0.4

0.5

0.6

0.7

0.8

0.6 0.7 0.8 0.9 1 1.1 1.2

Iω [−

]

M∞

[−]

Experimental

CFL3D

EDGE

FLUENT

AeroFoam

0.2

0.3

0.4

0.5

0.6

0.7

0.6 0.7 0.8 0.9 1 1.1 1.2

Iv [−

]

M∞

[−]

Experimental

CFL3D

EDGE

FLUENT

AeroFoam

Frequency ratio and flutter index

frequency ratio Iω = ωF /ωα

flutter index Iv = VF /Lαωα√

µ

comparison with NASA Langley TDTexp. data and CFL3D, EDGE, FLUENT

good agreement for M∞ < 1

overpredicted flutter point for M∞ > 1

all the reference solvers but EDGE (ona finer mesh) show the same behaviour

a local adaptive refinement may help



Conclusions and future work

Conclusions:

a toolbox of free software (Code Aster+OpenFOAM) for CA has been developed

satisfactory agreement with similar toolbox of commercial software (NAEMO CFD)

computational cost still too high: very difficult to implement a fully implicit timeintegration scheme in OpenFOAM (lduMatrix class, maybe in OpenFOAM 1.5)

. . . and future work:

aeroservoelastic applications: flutter suppression active control design (BACT wing)

runtime mesh deformation: new face swapping moving algorithm

tackle industrial aeroelastic 3D problems (e.g. Piaggio P-180)


un approccio ``libero'' alla moderna aeroelasticità ... coupling of structural and...

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