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Design optimization of unsteady airfoils with continuous adjoint method

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Abstract

In this paper, a new unsteady aerodynamic design method is presented based on the Navier-Stokes equations and a continuous adjoint approach. A basic framework of time-accurate unsteady airfoil optimization which adopts time-averaged aerodynamic coefficients as objective functions is presented. The time-accurate continuous adjoint equation and its boundary conditions are derived. The flow field and the adjoint equation are simulated numerically by the finite volume method (FVM). Feasibility and accuracy of the approach are perfectly validated by the design optimization results of the plunging NACA0012 airfoil.

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Abbreviations

w :

state vector

f ci :

inviscid flux vector

F ci :

Cartesian inviscid flux vector

f vi :

viscous flux vector

F vi :

Cartesian viscous flux vector

u i :

velocity components of the fluid

u bi :

velocity components of the boundary

k :

coefficient of thermal conductivity

E :

total energy

p :

pressure

σ ij :

viscous stresses

ξ i :

computational coordinates

R :

residual

S :

shape function

T :

time period

t :

time

t f :

final time

t n :

nondimensional time

V :

cell volume

P r :

Prandtl number

δ ij :

Kronecker delta

i var :

the sequence number of design variables

K ij :

transformation metrics

C L :

lift coefficient

C D :

drag coefficient

I :

objective function

Ψ :

Lagrange multiplier

G :

local gradient

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Correspondence to Xiaowei Li.

Additional information

Project supported by the Shanghai Municipal Natural Science Foundation (No. 13ZR1415700)

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Fang, L., Li, X. Design optimization of unsteady airfoils with continuous adjoint method. Appl. Math. Mech.-Engl. Ed. 36, 1329–1336 (2015). https://doi.org/10.1007/s10483-015-2010-9

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  • DOI: https://doi.org/10.1007/s10483-015-2010-9

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

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