A non-linear multigrid method for the steady Euler equations

Higher-order accurate Euler-flow solutions are presented for some airfoil test cases. Second-order accurate solutions are computed by an Iterative Defect Correction process. For two test cases even higher accuracy is obtained by the additional use of a ~xtrapolation technique. Finite volume Osher-type discretizations are applied throughout. Two interpolation schemes (one with and one w~hout a flux limiter) are used for the computation of the second-order defect. In each Defect Correction cycle, the solution is computed by non-linear mu~igrid iteration, in which Collective Symmetric Gauss-Seidel relaxation is used as the smoothing procedure. The computational method does not require tuning of parameters. The solutions show a good resolution of discontinuities, and they are obtained at low computational costs. The rate of convergence seems to be grid-independent

Adjoint Optimization of a Wing Using the CSRT Method

This paper will demonstrate the potential of the Class-Shape-Refinement-Transformation (CSRT) method for aerodynamically optimizing three-dimensional surfaces. The CSRT method was coupled to an in-house Euler solver and this combination was used in an optimization framework to optimize the ONERA M6 wing in transonic conditions. The gradients of the flow variables with respect to the design parameters were computed using an adjoint solver integrated in the Euler code. The optimization was performed by a trust region reflective algorithm. A two-step approach was used to optimize the wing. First, a “general” optimization was done using the Bernstein coefficients of the shape function. Second, a “regional” refinement was performed using the B-spline coefficients of the refinement function. It was shown that using this strategy a considerable improvement of the lift-to-drag ratio of 22 % could be achieved. This result proves that an adjoint optimization using the CSRT method is an effective way to improve transonic wing performance.Aerodynamics, Wind Energy & PropulsionAerospace Engineerin