Reduced order time spectral model for the ALE formulation of the compressible Navier-Stokes equations

In this paper we address the construction of a time and a frequency domain Reduced Order Model (ROM) for the ALE formulation of the Navier-Stokes equations. The wide variety of fluid dynamics ROMs reported in literature shares the aim of reducing the dimensionality of dynamical systems by performing a projection of the governing equations onto a basis that may be often constructed via the Proper Orthogonal Decomposition (POD). Unfortunately, the applicability of this kind of ROMs in the time domain still remains problematic as non-linear effects induce numerical stability problems, especially for aeroelastic applications. Nevertheless, in many cases, only the periodic steady state is of concern. In this context, spectral methods like the Time Spectral Method (TSM) avoid stability problems by expressing the conservative variables as Fourier series in time with spatially varying coefficients to be computed. In this work, we develop a POD based ROM which aims to reduce the computational cost of the TSM. Numerical tests have been carried out in order to highlight the potentiality of the proposed technique.