Application of a variational approach to the computation of forces around a wing

In this paper we present an application of the variational approach, introduced by Quartapelle and Napolitano (AIAA J 21:911–913, 1983) and developed further by Protas et al. (J Comput Phys 159:231–245, 2000), which requires only the velocity fields and its derivatives to determine forces acting on a body. First, the approach is presented and adapted to our 3D test problem. The obtained expression for the hydrodynamic force involves a harmonic function η$$\eta$$ whose determination is also presented. Then, numerical flow fields obtained with LES are used in order to evaluate the influence of different parameters on the forces and to offer a validation of the proposed approach. This allows us to assess the accuracy of the method and its advantages. Next, a comparison of the proposed variational approach with the momentum equation approach presented by David et al. (2009) is discussed. The momentum equation approach offers a non-intrusive method to determine forces, but requires the pressure field around the object. On the other hand, the variational approach requires the determination of the vorticity field on the surface of the wing, which is not always trivial to obtain with sufficient accuracy, but the computation of the pressure field can be avoided. This paper aims to compare both methods in a practical setting and show their relative advantages and sensitivities to different parameters for a 3D numerical/experimental study of a flow field around a NACA0015 airfoil. To the best of our knowledge, this is the first application of the variational approach in an experimental setting.