Assessment of methodologies for the solution of the Ffowcs Williams and Hawkings equation using LES of incompressible single-phase flow around a finite-size square cylinder

The acoustic analogy represents a powerful tool for the prediction of noise generated by the interaction between the flow and a moving body. It is based on decoupling the acoustic problem from the fluid dynamic one: the velocity and pressure fields, obtained through a separate numerical simulation, are used as source terms in an inhomogeneous wave equation whose solution reconstructs the noise in the far field. When the method is based on the fundamental Ffowcs Williams and Hawkings (FW-H) equation, different solving methodologies may be adopted. The present work considers the original FW-H equation and gives the advective formulation of the volume integral terms. The results are compared with those obtained with the Curle and porous formulations. To account for volume integrals, the assumption of compact noise source is needed. This assumption is common in literature, however, in the present work, a dimensional analysis is proposed, in order to indicate in a rigorous way the cases in which the compressibility delays can be avoided. The dimensional analysis is tested in the case of an acoustic monopole field. Successively, the FW-H porous formulation is compared with the original FW-H equation in the case of an irrotational advected vortex. This example puts in evidence the different response of the two methods in the case of a vortex crossing the acoustic domain. Then, different solution strategies of the FW-H are evaluated using a fluid dynamic dataset obtained through large eddy simulation of a turbulent flow around a finite-length cylinder with square section. The analysis allows to point out the strengths and drawback of the different techniques and to achieve, through the comparison of the different solutions, an accurate understanding of the noise source mechanisms taking place in the flow. Finally, a mixed procedure, merging the advantages of the porous formulation with the direct evaluation of the volume integral terms is proposed. It may be used in presence of significant time delays. Overall, the present study is oriented to the analysis of very low Mach number flows, although the complete porous method might be applicable in a more general framework. This aspect will require additional research in the future.