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Multi-objective aerodynamic shape optimization for unsteady viscous flows

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Abstract

This paper presents an adjoint method for the multi-objective aerodynamic shape optimization of unsteady viscous flows. The goal is to introduce a Mach number variation into the Non-Linear Frequency Domain (NLFD) method and implement a novel approach to present a time-varying cost function through a multi-objective adjoint boundary condition. The paper presents the complete formulation of the time dependent optimal design problem. The approach is firstly demonstrated for the redesign of a helicopter rotor blade in two-dimensional flow and in three-dimensional viscous flow, the technique is employed to validate and redesign the NASA Rectangular Supercritical Wing (RSW).

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Abbreviations

b :

Boundary velocity component

c :

Airfoil chord length

c a :

Axial force coefficient

c n :

Normal force coefficient

D,d:

Artificial dissipation flux

f :

Flux vector

f v :

Viscous flux vector

F :

Numerical flux vector

F v :

Numerical viscous flux vector

\({\mathcal{G}}\) :

Gradient

\(\bar{\mathcal{G}}\) :

Smoothed gradient

H :

Enthalpy

i,j:

Cell indices

I :

Cost function

k :

Wave number

ℒ:

Objective function

M :

Mach number

M ∞ :

Freestream Mach number

ℳ:

Objective function at final time

N :

Number of Fourier modes

p :

Pressure

p ∞ :

Freestream pressure

R :

Residual

\(\hat{R}\) :

Fourier coefficient of residual

s :

Surface arc length

\({\mathcal{S}}\) :

Shape function

S :

Face areas of computational cell

T :

Time period

t :

Time

t * :

Pseudo time

u :

Velocity (physical domain)

V :

Cell volume

w :

State vector

\(\hat{w}\) :

Fourier coefficient of state vector

x :

Coordinates (physical domain)

α :

Angle of attack

α m :

Maximum angle of attack

α o :

Mean angle of attack

θ :

Coefficient for implicit smoothing technique

κ :

Coefficient of thermal conductivity

Λ:

Spectral radius of the flux Jacobian

λ :

Second coefficient of viscosity

μ :

Coefficient of viscosity

ν :

Coefficient for artificial dissipation flux

ξ :

Coordinates (computational domain)

ρ :

Density

σ :

Viscous stress

τ :

Pseudo time

ψ :

Lagrange multiplier

\(\hat{\psi}\) :

Fourier coefficient of Lagrange multiplier

ω r :

Reduced frequency

ϖ :

Weights for cost functions

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Authors and Affiliations

  1. CFD Lab, Department of Mechanical Engineering, McGill University, 688 Sherbrooke Street West, Room 711, Montreal, Quebec, H3A 2S6, Canada

    Siva K. Nadarajah & Charles Tatossian

Authors
  1. Siva K. Nadarajah
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  2. Charles Tatossian
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Correspondence to Siva K. Nadarajah.

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Nadarajah, S.K., Tatossian, C. Multi-objective aerodynamic shape optimization for unsteady viscous flows. Optim Eng 11, 67–106 (2010). https://doi.org/10.1007/s11081-008-9036-4

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