A wing that is simultaneously heaving and pitching may extract energy from an oncoming air flow, thus acting as a turbine. This paper analyzes the unsteady aerodynamics of such a device by means of time-dependent laminar flow simulations performed with a research compressible finite volume Navier–Stokes solver. The study confirms the findings of another independent report that the efficiency of the power extraction of this device can be of the order of 35%, and such an efficient operating condition is characterized by a strong dynamic stall. This study is part of a wider research programme aimed at developing a general-purpose computational framework for unsteady aerodynamic and aeroacoustic wind energy engineering. In view of aeroacoustic applications, the developed flow solver uses the compressible formulation of the Navier–Stokes equations with carefully optimized low-speed preconditioning. To demonstrate the modeling capabilities, the accuracy and the high computational performance of the developed low-speed preconditioning technology, the unsteady aerodynamics of the energy-extracting device is simulated by using a computationally challenging freestream Mach number of 0.001. A mixed preconditioning strategy that maintains both the nominal accuracy and the computational efficiency of the solver also for time-dependent low-speed problems is presented. The study also assesses the impact of a semi-implicit treatment of the unsteady source term associated with the discretization of the physical time-derivative of the governing equations on the numerical stability of the explicit multigrid integration.