Conservative integral form of the incompressible Navier–Stokes equations for a rapidly pitching airfoil

This study provides a simple moving-grid scheme which is based on a modified conservative form of the incompressible Navier–Stokes equations for flow around a moving rigid body. The modified integral form is conservative and seeks the solution of the absolute velocity. This approach is different from previous conservative differential forms [1], [2] and [3] whose reference frame is not inertial. Keeping the reference frame being inertial results in simpler mathematical derivation to the governing equation which includes one dyadic product of velocity vectors in the convective term, whereas the previous [2] and [3] needs to obtain the time derivative with respect to non-inertial frames causing an additional dyadic product in the convective term. The scheme is implemented in a second-order accurate Navier–Stokes solver and maintains the order of the accuracy. After this verification, the scheme is validated for a pitching airfoil with very high frequencies. The simulation results match very well with the experimental results [4] and [5], including vorticity fields and a net thrust force. This airfoil simulation also provides detailed vortical structures near the trailing edge and time-evolving aerodynamic forces that are used to investigate the mechanism of the thrust force generation and the effects of the trailing edge shape. The developed moving-grid scheme demonstrates its validity for a rapid oscillating motion.