Finite element hybrid and direct computational aeroacoustics at low Mach numbers in slow time-dependent domains

In this work, we address the finite element computation of flow noise in the presence of arbitrarily slowly moving rigid bodies at low Mach numbers, by means of hybrid and direct computational aeroacoustics (CAA) strategies. As regards the former, the problem could be dealt with by means of the Ffowcs Williams–Hawkings acoustic analogy. That reduces to Curle's analogy for a static body, which is analogous to a problem of diffraction of sound waves generated by flow eddies in the vicinity of the body. Acoustic analogies in CAA first demand computing the flow motion to extract an acoustic source term from it and then use the latter to calculate the acoustic pressure field. However, in the case of low Mach number flows, the Ffowcs Williams–Hawkings and Curle analogies present a problem as they require knowing the total pressure distribution on the body's boundary (i.e. the aerodynamic pressure plus the acoustic one). As incompressible computational fluid dynamics (CFD) simulations are usually performed to determine the flow motion, the acoustic pressure distribution on the body surface is unavoidably missing, which can yield acoustic analogies inaccurate. In a recent work, it was proposed to tackle that problem for static and rigid surfaces, by keeping the incompressible CFD and then splitting the acoustic pressure into direct and diffracted components. Two separate wave equations were solved for them, in the framework of the finite element method (FEM). In this article, we extend that work to compute the aerodynamic sound generated by a flow interacting with a slowly moving rigid body. The incompressible Navier–Stokes equations are first solved in an arbitrary Lagrangian–Eulerian (ALE) frame of reference to obtain the acoustic source term. Advantage is then taken from the same computational run to separately solve two acoustic ALE wave equations in mixed form for the incident and diffracted acoustic pressure components. For validation of the total acoustic pressure field, an ALE formulation of a direct CAA approach consisting of a unified solver for a compressible isentropic flow in primitive variables is considered. The performance of the exposed methods is illustrated for the aeroacoustics of flow past a slowly oscillating two-dimensional NACA airfoil and for flow exiting a duct with a moving teeth-shaped obstacle at its termination.