Airfoil lift finite element computations using H1 (Ω) and H(div,Ω) approximation spaces

Resumo

Computing the lift generated by an airfoil is crucial in aircraft design, especially for aircraft performance prediction, aircraft stability/control, and optimization of airfoil shape for improving efficiency. The problem of incompressible flow around an airfoil is studied herein as a weakly irrotational flow, resulting a div-curl problem in a 2D double-connected domain. The original div-curl problem is ill-posed because it is defined over a multiply-connected domain. Traditionally, a cut in the original domain and additional constrain for the velocity potential and velocity potential gradient at the cut are required for the numerical solution, which applies a rotational correction method based on Helmholtz decomposition of vector fields. The lift coefficient can be finally computed from the numerical solution of the problem. In the present work, the lift of an airfoil is computed from finite element solutions using different approximation spaces: The conventional H1 (Ω) potential-field space and a special class of H(div,Ω) velocity-field space, i.e., the divergence-free space. Accurate analysis using both approximations is performed with h-refinement strategies. The lift coefficient computed from the analysis is compared against available reference solutions. The computational performance and accuracy of the lift computation using the two different approaches are discussed. Finally, the difference in the velocity solution obtained using H1 (Ω) and H(div,Ω) spaces is presented, which can be interpreted as a new posteriori error estimator for the problem.