Aerodynamic Shape Optimization of
Laminar Wings
A. Hanifi1,2, O. Amoignon1 & J. Pralits1
1Swedish
Defence Research Agency, FOI
2Linné 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.
instability waves
Prevent/delay transition by
suppressing the growth
of small perturbations.
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 x.
J can be disturbance growth, drag, …
x can be wall suction, geometry, …
Problem to solve:
J
?
x
Parameters
Geometry parameters :
yi
Mean flow:
Q
Disturbance energy:
Gradient to find:
E
NLF:
y i
E
HLFC:
Q
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).
Pe
E
E
Q



yi
Q
Pe
yi
Adjoint
Stability
equations
Adjoint
Boundary-layer
equations
Adjoint
Euler
equations
Solution procedure
• 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
Euler
BL
PSE
E
E
Optimization
AESOP
Adj.
Euler
Adj.
BL
Adj.
PSE
ShapeOpt
*ShapeOpt is a KTH-FOI software (NOLOT/PSE was developed by FOI and DLR)
Problem formulation
Minimize the objective function:
J = luE + ldCD + lL(CL-CL0)2 + lm(CM-CM0)2
can be replaced by constraints
Accuracy of gradient
Fixed nose radius


J  E   u 2  v2  w2 dx dy
Comparison between gradient obtained from solution of adjoint
equations and finite differences. (Here, control parameters are the
surface nodes)
Low Mach No., 2D airfoil (wing tip)
Subsonic 2D airfoil:
•
M∞ = 0.39
•
Re∞ = 13 Mil
J= luE + ldCD
Constraints:
•
Thickness ≥ 0.12
•
CL ≥ CL0
•
CM ≥ CM0
Transition (N=10) moved from x/C=22% to x/C=55%
Amoignon, Hanifi, Pralits & Chevalier (CESAR)
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
J= luE + ldCD
Initial
Intermediate
Final
Constraints:
•
Thickness ≥ t0
•
CL ≥ CL0
•
CM ≥ CM0
Transition (N=10) moved from x/C=15% to x/C=50% (caused by separation)
Amoignon, Hanifi, Pralits & Chevalier (CESAR)
Low Mach No., 2D airfoil (wing root)
RANS computations with transition
prescribed at:
N=10 or Separation
Separation
at high AoA
Need to account
for separation.
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
xsp
J   H12 dx 
0
where Hsp is a large value.
xTE
H
xsp
sp
dx
Minimizing H12
Not so good!
Minimizing H12 + CD
xsp
J   H12 dx 
0
xTE
H
xsp
sp
dx  CD
Include a measure of wall friction directly into the object function:
J
xTE
c
f
dx
0
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.
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)