Framework for aerodynamic shape optimization applied to wind turbine airfoils

Key aerodynamic innovations such as the use of slender high-lift airfoils in the power producing part of the blade, and thick flatback airfoils near the blade root, have facilitated a reduction in wind turbine blade mass. This has allowed the design of 10+ MW wind turbines with rotor diameters greater than 200 m.

The growth in the size of the wind turbine rotors is expected to continue, powered by the industry’s never-ending quest of lowering the cost of energy. High-lift airfoils with high relative thickness enable slender and lighter blades. Additionally, the use of airfoils with thick trailing edges (flatbacks) have considerably improved the aerodynamic performance in the structurally vital blade root.

Despite their advantages, airfoils with high relative thickness and thick trailing edges have greater adverse pressure gradients on the upper surface resulting in thicker boundary layers, with a greater chance of unsteady flow phenomena such as flow separation and vortex shedding. For optimizing airfoils under such conditions, the robustness of the Computational Fluid Dynamics (CFD) solver is of paramount importance.

The CFD solver often fails to converge due to challenging flow physics caused by the shapes encountered during the optimization. This work involves the development of a gradient-based, two-dimensional airfoil shape optimization framework using a steady-state, incompressible, Reynolds-averaged Navier- Stokes (RANS)-CFD solver.

The thesis, primarily focuses on improving the robustness of the RANS flow solver to allow for deep convergence of the residuals for the various shapes encountered during the optimization. For this purpose, an existing deep-convergence method found in the literature is modified, and its convergence and robustness properties improved for application to steady-state, RANS flow solvers.

The improvements are demonstrated by finding lift and drag curves for airfoils representative of those used in modern wind turbines, for angles of attack spanning 360 degrees. The flow solutions are converged to machine precision at all angles of attack. Furthermore, the ability of the modified deep convergence method to converge solutions for varying flow conditions over a range of Reynolds numbers is also demonstrated.

Next, a discrete adjoint solver is developed and verified for the two-dimensional flow solver to provide accurate flow gradients. In its current implementation, the discrete adjoint solver utilizes a finite-difference based approach to compute the flow gradients, and handles computations in a sparse manner.

Instead of the fixed-point approach, the discrete adjoint system is solved by applying the so-called “Krylov approach”. This involves assembling the full Jacobian for the adjoint system and using the Krylov-based generalized minimizing residual (GMRES) method to solve it. The effect of including the developed deep-convergence method for obtaining a converged flow field, on the adjoint sensitivity is investigated.

Finally, a robust aerodynamic shape optimization framework for airfoils is developed. The framework supplies the gradients based on a mix of finite-difference, analytical, and complex-step methods. The developed shape-optimization framework is verified against a “black-box” finite-difference based alternative.

The framework consists of various components that include geometry deformation, CFD mesh deformation, the RANS-CFD solver and airfoil polar computation. Geometric constraints are provided by an airfoil parametrization library. The various components have been unified using the OpenMDAO multidisciplinary optimization framework.

Various airfoil configurations encountered during wind turbine operations are tested using the developed framework. The airfoils that are tested include the NACA 0018, NACA 0024 and the NACA 0030. The developed shape optimization framework produces near identical optimal designs to the ones produced by the alternative version of the framework with gradients supplied using a “black-box” finite-difference approach.

The robustness imparted to the developed aerodynamic shape optimization framework by the inclusion of the RANS flow solver enhanced with the modified deep convergence method, is tested. The framework is challenged by an optimization case that starts with a bluff-body undergoing massive flow separation.

The flow solution with the standard RANS flow solver for the bluff body is found to stall entering a limit cycle oscillation. Whereas, by using the RANS flow solver enhanced by the developed deep convergence method, we obtain a flow solution that has been converged to an accuracy that is near machine precision.

The same problem formulation as that for a reference NACA 0018 airfoil optimization is utilized to carry out the bluff body optimization. The results show that the bluff body optimization with the standard flow solver does not progress far from its starting point, indicating bad gradients. In contrast, the bluff body optimization with the convergence-enhanced flow solver is able to arrive at an optimal shape of an airfoil.

Furthermore, the optimized shape from the bluff-body optimization is identical to that arrived upon by the reference NACA 0018 airfoil optimization. The result, not only indicates the robustness imparted by the convergence-enhanced flow solver, but also hints at unimodality in the design space for aerodynamic shape optimization of airfoils.

Finally, the results indicate that for a convergent, discrete adjoint system solved using the “Krylov approach”, applying the developed deep convergence method to the RANS flow solver is sufficient to produce accurate gradients.