Aerodynamic Shape Optimization of Laminar Wings

A. Hanifi1,2, O. Amoignon1 & J. Pralits1

1Swedish Defence Research Agency, FOI2Linné Flow Centre, Mechanics, KTH

Co-workers: M. Chevalier, M. Berggren, D. Henningson

Why laminar flow? Environmental issues!

A Vision for European Aeronautics in 2020:

”A 50% cut in CO2 emissions per passenger kilometre (which means a 50% cut in fuel consumption in the new aircraft of 2020) and an 80% cut in nitrogen oxide emissions.”

”A reduction in perceived noise to one half of current average levels.”

Advisory Council for Aeronautics Research in Europe

Drag breakdown

G. Schrauf, AIAA 2008

Friction drag reduction

Possible area for Laminar Flow Control:

Laminar wings, tail, fin and nacelles -> 15% lower fuel consumption

Transition control

Transition is caused by

breakdown of growing

disturbances inside the

boundary layer.

Prevent/delay transition by

suppressing the growth

of small perturbations.

instability waves

Control parameters

Growth of perturbations can be controlled through e.g.:

• Wall suction/blowing

• Wall heating/cooling

• Roughness elements

• Pressure gradient (geometry)

} active control

} passive control

Theory

We use a gradient-based optimization algorithm to minimize a given objective function J for a set of control parameters .

J can be disturbance growth, drag, …

can be wall suction, geometry, …

Problem to solve:?

J

Parameters

Geometry parameters :

Mean flow:

Disturbance energy:

Gradient to find:

iy

Q

iy

E

Q

E

NLF: HLFC:

Gradients

Gradients can be obtained by :

• Finite differences : one set of

calculations for each control

parameter (expensive when no.

control parameters is large),

• Adjoint methods : gradient for all

control parameters can be found by

only one set of calculations including

the adjoint equations (efficient for

large no. control parameters).

i

e

ei y

P

P

Q

Q

E

y

E

Adjoint Stability

equations

Adjoint Boundary-layer

equations

Adjoint Euler

equations

• Solve Euler, BL and stability equations for a given geometry,

• Solve the adjoint equations,

• Evaluate the gradients,

• Use an optimization scheme to update geometry

• Repeat the loop until convergence

Solution procedure

*ShapeOpt is a KTH-FOI software (NOLOT/PSE was developed by FOI and DLR)

PSEEuler BL

Adj.BL

Adj. PSE

Adj.Euler

Optimization

EE

AESOP ShapeOpt

Minimize the objective function:

J = uE + dCD + L(CL-CL0)2 + m(CM-CM

0)2

can be replaced by constraints

Problem formulation

Comparison between gradient obtained from solution of adjoint equations and finite differences. (Here, control parameters are the surface nodes)

Accuracy of gradient

dydxwvuEJ 222

Fixed nose radius

Low Mach No., 2D airfoil (wing tip)

Subsonic 2D airfoil:

• M∞ = 0.39

• Re∞ = 13 Mil

Constraints:

• Thickness ≥ 0.12

• CL ≥ CL0

• CM ≥ CM0

J= uE + dCD

Amoignon, Hanifi, Pralits & Chevalier (CESAR)

Transition (N=10) moved from x/C=22% to x/C=55%

Low Mach No., 2D airfoil

Optimisation history

Low Mach No., 2D airfoil (wing root)

Subsonic 2D airfoil:

• NASA TP 1786

• M∞ = 0.374

• Re∞ = 12.1 Mil

Constraints:

• Thickness ≥ t0

• CL ≥ CL0

• CM ≥ CM0

J= uE + dCD

Amoignon, Hanifi, Pralits & Chevalier (CESAR)

Transition (N=10) moved from x/C=15% to x/C=50% (caused by separation)

InitialIntermediateFinal

Low Mach No., 2D airfoil (wing root)

RANS computations with transition prescribed at:

N=10 or Separation

Need to account for separation.

Separation at high AoA

Amoignon, Hanifi, Pralits & Chevalier (CESAR)

Low Mach No., 2D airfoil (wing root)

Optimization of upper and lower surface for laminar flow

Amoignon, Hanifi, Pralits & Chevalier (CESAR)

The boundary-layer computations stop at point of separation:

No stability analyses possible behind that point.

Force point of separation to move downstream:

Minimize integral of shape factor H12

Minimize a new object function

where Hsp is a large value.

dxHdxHJTE

sp

sp x

x

sp

x

0

12

Minimizing H12

Not so good!

Minimizing H12 + CD

D

x

x

sp

x

CdxHdxHJTE

sp

sp

0

12

Include a measure of wall friction directly into the object function:

cf is evaluated based on BL computations.

Turbulent computations downstream of separation point if no turbulent separation occurs.

Gradient of J is easily computed if transition point is fixed.

Difficulty: to compute transition point wrt to control parameters.

TEx

f dxcJ0

3D geometry

Extension to 3D geometry:

Simultaneous optimization of several cross-sections

Important issues:

• quality of surface mesh (preferably structured)

• extrapolation of gradient values

• paramerization of the geometry

2D constant-chord wing

Structured grid(medium)

Unstructured grid(medium)

Unstructured grid(fine)

2D constant-chord wing

Structured grid(medium)

Unstructured grid(medium)

Unstructured grid(fine)