Abstract
The adjoint approach in gradient-based optimization combined with computational fluid dynamics is commonly applied in various engineering fields. In this work, the gradients are used for the design of a two-dimensional airfoil shape, where the aim is a change in lift and drag coefficient, respectively, to a given target value. The optimizations use the unconstrained quasi-Newton method with an approximation of the Hessian. The flow field is computed with a finite-volume solver where the continuous adjoint approach is implemented. A common assumption in this approach is the use of the same turbulent viscosity in the adjoint diffusion term as for the primal flow field. The effect of this so-called “frozen turbulence” assumption is compared to the results using adjoints to the Spalart–Allmaras turbulence model. The comparison is done at a Reynolds number of R e = 2 × 10 6 for two different airfoils at different angles of attack.
Highlights
The design of aerodynamic shapes is increasingly based on Computational Fluid Dynamics (CFD).Since CFD is computationally more expensive than the coupling of potential flow theory with boundary layer corrections, a computationally inexpensive optimization technique is preferable
This results in another set of Partial Differential Equations (PDEs), the adjoint equations, in which each adjoint variable refers to a variable of the flow field
Solving the adjoint equations has a comparable cost to solving the primal state equations [3] and the gradients can be computed from primal and adjoint fields with minor additional calculations compared to the CFD iterations
Summary
The design of aerodynamic shapes is increasingly based on Computational Fluid Dynamics (CFD). Osusky et al [13] used discrete adjoints for drag reduction of wings in compressible flows They compared the results by optimizations based on Euler and Reynolds-averaged Navier–Stokes (RANS) equations using the Spalart–Allmaras model and obtained inferior designs by the inviscid flow analysis. Dwight and Brezillon [14] investigated the effect of frozen turbulence, there called the “constant eddy-viscosity assumption” They used discrete adjoints in compressible flows and optimized airfoils at small angles of attack without any separation. Use discrete adjoints with OpenFOAM, more often the continuous approach is followed when using this flow solver [21,22] As this code is very suitable for the implementation of analytical equations, as well as the expected lower memory costs, continuous adjoints are used in our work.
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