Momentum analysis of complex time-periodic flows

Several methods have been proposed to characterize the complex interactions in turbulent wakes, especially for flows with strong cyclic dynamics. This paper introduces the concept of Fourier-averaged Navier-Stokes (FANS) equations as a framework to obtain direct insights into the dynamics of complex coherent wake interactions. The method simplifies the interpretations of flow physics by identifying terms contributing to momentum transport at different time scales. The method also allows for direct interpretation of nonlinear interactions of the terms in the Navier-Stokes equations. By analysing well-known cases, the characteristics of FANS are evaluated. Particularly, we focus on physical interpretation of the terms as they relate to the interactions between modes at different time scales. Through comparison with established physics and other methods, FANS is shown to provide insight into the transfer of momentum between modes by extracting information about the contributing pressure, convective and diffusive forces. The FANS equations provide a simply calculated and more directly interpretable set of equations to analyse flow physics by leveraging momentum conservation principles and Fourier analysis. By representing the velocity as a Fourier series in time, for example, the triadic model interactions are apparent from the governing equations. The method is shown to be applicable to flows with complex cyclic waveforms, including broadband spectral energy distributions.