5.2 The Direct Eulerian-Lagrangian Approach

For transonic flows where , the aeroelasticproblem is

inherently nonlinear, because

1)

Mixed flow with strong m

oving shocks occur.

2)

Shock m

otion is highly nonlinear (even at sm

all am

plitudes)

3)

The thickness effects are significant and enter the governing

equations in a nonlinear manner.

Flow Regim

es:

a)Subcritical: Flow is entirely subsonic throughout.

b)

Supercritical: Mixed subsonic-supersonic flow (em

bedded

supersonic pockets)

1<

<∞

MM

cr

1<

<∞

crM

M

1<

<∞

MM

cr

Supercritical Flow Regim

e

Aeroelastic Phenomena in Transonic Regim

e

a) Flutter boundary for

NACA 0006

model corresponding to fixed mass ratio

of 20, b) corresponding boundary if U_inf

and mass ratio is decreased until flutter

occurs, c) Theodorsen-G

arrick rule for

subsonic flutter.

Aeroelastic Phenomena in Transonic Regim

e

Effect of thickness on the transonic flutter boundary.

Aeroelastic Phenomena in Transonic Regim

e

Experim

ental data showing the effect of angle of attack on the

transonic flutter boundary.

For details of this form

ulation, one may refer to Prof. Oddvar O. Bendiksen’sresearch.

The Direct Eulerian-Lagrangian Approach

FE Lagrangian Code

Lagrangian Mesh

The Direct Eulerian-Lagrangian Approachcont.

•In this approach the

fluid-structure system

is modeled and

integrated as a single dynam

ical system

without introducing

norm

al or assumed m

odes.

•This approach predicts the energy exchange between the fluid

and structure very accurately.

•In this form

ulation time synchronization avoids phase errors

which m

ay lead to assesment of incorrect instability behavior.

•The

fluid-structure boundary is defined accurately and

consistently using the structural finite elem

ent shape functions

and the actual wing deform

ations.

Aerodynam

ic M

odel

•Classical inviscid Euler Equations, using the Direct Eulerian-

Lagrangian form

ulation.

0dV

ds

tΩ

∂Ω

∂+

⋅=

∂∫

∫W

Fn

1 2 3

u u u eρ ρ ρ ρ ρ

=

W

()

()

()

()

()

11

22

33

jj

jj

j

jj

jj

jj

j

jj

iji

uU

uu

U

uu

U

uu

U

eu

Uu

ρ

ρσ

ρσ

ρσ

ρσ

−

−

−

=

−−

−−

−−

F

()

11

2ii

pe

uu

ργ

=−

−

ijij

pσ

δ=−

..

.i

ii

ii

dV

dS

dV

dS

dS

tt

ϕϕ

ϕϕ

ϕΩ

ΩΩ

∂Ω∂Ω

∂∂

+∇

=+

−∇

∂∂

∫∫

∫∫

∫W

FW

Fn

F

Aerodynam

ic M

odel cont.

•Galerkin FE discretization is applied.

•Space-Discretized equations shown in m

atrix form

.

0ij

ji

i

j

d dt

+−

=∑mW

QD

Structural Model

Structural Model cont.

•Based on a specific type of aeroelastic wing construction that has

been used extensively in wind tunnel tests.

•The structural part is a thin flat aluminum alloy plate of the same

planform

as the wing, covered by balsa wood or lightweight foam

to form

the desired airfoil shape.

•Employs shear deform

able plate type finite elem

ents (the Discrete

Shear Triangle, DST, by Batoz and Lardeur).

Structural Modelcont.

Fluid-Structure Boundary Conditions

•Kinem

atic boundary condition of tangent flow at a boundary is

0.

=≡

∇+

∂∂Dt

DB

Bu

tB

Mappings Between Aerodynam

ic and Structural Meshes

Structural Element

Aerodynam

ic faces

on the wing surface

Tetrahedron

aerodynam

ic cell

xy

z

Mappings Between Aerodynam

ic and Structural Meshes

Dynam

ic M

esh M

otion

•The rest of the CFD m

esh is deform

ed using ‘spring analogy’, see

work by Batina for details.

Consistent Aerodynam

ic Load Calculations

•Aerodynam

ic load vector is evaluated using numerical integration

(Gaussian Quadrature), considering the pressures at lower and

upper skins.

Fluid-Structure Tim

e-Marching

•Four-stage Runge-Kutta schem

e, shown below for the fluid.

()

41

WW

n=

+()

[]

()

()

()

()

()

()

[]

13

4

00

14

WD

WQ

tW

mm

W−

∆−

=−

α

()

[]

()

()

()

()

()

()

[]

00

1

00

11

WD

WQ

tW

mm

W−

∆−

=−

α

()

nW

W=

0

•Reduced order structural equations for the Runge-Kutta Schem

e.

[]

elastic

aero

FF

vM

−= v

q=

.

• • •

Numerical Results:

Flutter Calculations

NACA 0012 Typical Section M

odel (2D)

•Nonlinear flutter-divergence interaction at Mach 0.8 and U = 3.6

•Based on the

short term

plot (RHS) one

would incorrectly

conclude that the model is stable and that no flutter occurs.

•The Direct Eulerian-Lagrangian method has been successfully

applied to other w

ing m

odels including fighter and transport type

wings.

•Recently a geometrically nonlinear structural model has been

implemented in this form

ulation leading to very unique results

that has not been computationally observed before, i.e. high

altitude flutter. For further details please see the work by Prof.

Bendiksen and Dr. Seber.