Unified solver for fluid dynamics and aeroacoustics in isentropic gas flows

Abstract

The propagation of acoustic waves is a physical phenomenon which can only be described taking into account the compressibility of the medium. The high computational cost of solving numerically the fully compressible Navier-Stokes equations, together with the poor performance of most numerical formulations for compressible flow in the low Mach number regime, has led to the necessity for more affordable numerical models for Computational Aeroacoustics. For low Mach number subsonic flows with neither shocks nor thermal coupling, both flow dynamics and wave propagation can be considered isentropic. Therefore, a joint isentropic formulation for flow and aeroacoustics can be devised which avoids the need for segregating flow and acoustic scales. Under these assumptions density and pressure fluctuations are directly proportional, and a two field velocity-pressure compressible formulation can be derived as an extension of an incompressible solver. On the other hand, the linear system of equations which arises from the proposed isentropic formulation is better conditioned than the homologous incompressible one due to the presence of a pressure perturbation term. Similarly to other compressible formulations the prescription of boundary conditions will have to deal with the backscattering of acoustic waves. In this sense, a separated imposition of boundary conditions for flow and acoustic scales which allows the evacuation of waves through Dirichlet boundaries

A fractional step method for computational aeroacoustics using weak imposition of Dirichlet boundary conditions

“…Since the formulation we present aims at accounting for both flow and acoustic scales at once, a special type of condition must be imposed for the pressure field, whose main purpose is to allow the sound waves to leave the external boundaries of the computational domain smoothly. In this section we review the method proposed in [6] for the prescription of boundary conditions for the isentropic problem.…”

“…In Fig. 4 we present the flow pressure contour for a certain time step of the vortex shedding cycle, qualitatively comparing the proposed fractional step scheme with its monolithic version, already validated in [6] and taken as reference. It can be observed that the pressure pulses evolve radially from the cylinder area with time, yet they do not propagate normally to the flow direction since this case is based on an uniform flow.…”

“…flow and acoustics, their approximation errors still need to be properly analyzed, as they assume only a one-way coupling from flow to acoustics and acoustic terms need to be correctly derived. The formulation presented in this work, recently published in [6], aims at combining the simplicity of hybrid methods with the unified scale computation of DNC. We will see that this methodology can be understood as an extension of the incompressible flow equations, since the final problem to be solved requires to compute only velocity and pressure fields, being the thermal problem mathematically uncoupled.…”

“…This is possible due to the fact that the entropy remains constant, which in turn allows one to establish a connection between density and pressure perturbations, assuming a low Mach number flow, with neither shocks nor thermal sources. The computational cost of the present technique is reduced with respect to other methods since, apart from the fact that both acoustic and flow scales are solved altogether and that we get rid of the energy conservation equation, the final system is better conditioned (we refer to [6] for a detailed discussion). For simplicity in the exposition, the ideal gas law has been assumed, yet other equations of state may be possible.…”

“…Figure 4. Aeolian tones: flow pressure contour in the far field comparing the result obtained when the problem is solved using the proposed fractional step scheme and the monolithic approach in[6], taken as reference.…”