Aerodynamic Shape Optimization of Laminar Wings A. Hanifi 1,2, O. Amoignon 1 & J. Pralits 1 1 Swedish Defence Research Agency, FOI 2 Linné 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 CO 2 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:

Parameters Geometry parameters : Mean flow: Disturbance energy: Gradient to find: 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). 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) PSE Euler BL Adj. BL Adj. PSE Adj. Euler Optimization AESOP ShapeOpt

Minimize the objective function: J = u E + d C D + L (C L -C L 0 ) 2 + m (C M -C M 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 Fixed nose radius

Low Mach No., 2D airfoil (wing tip) Subsonic 2D airfoil: M ∞ = 0.39 Re ∞ = 13 Mil Constraints: Thickness ≥ 0.12 CL ≥ CL 0 CM ≥ CM 0 J= u E + d C D 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 ∞ = Re ∞ = 12.1 Mil Constraints: Thickness ≥ t 0 CL ≥ CL 0 CM ≥ CM 0 J= u E + d C D Amoignon, Hanifi, Pralits & Chevalier (CESAR) Transition (N=10) moved from x/C=15% to x/C=50% (caused by separation) Initial Intermediate Final

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 H 12

Minimize a new object function where H sp is a large value.

Minimizing H 12 Not so good!

Minimizing H 12 + C D

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