Mass-Conserved Solution to the Ffowcs-Williams and Hawkings Equation for Compact Source Regions

Abstract

A mass-conserved formulation for the Ffowcs-Williams–Hawkings (FW–H) integral is proposed to suppress contributions of spurious mass flux to the far-field sound at very low Mach numbers. The far-field condition and compact-source region assumptions are employed. By using higher-order derivatives of Green’s function, an expansion of the integrand in the monopole term is performed. This expansion transforms the mass-flux like monopole term into a series including different orders of velocity moment. At very low Mach numbers, the zero-order term is exactly the contribution from the spurious mass flux. The proposed mass-conserved formulation is confirmed by using an unsteady dipole, a two-dimensional (2D) incompressible convecting vortex, a circular-cylinder flow, and a co-rotating vortex pair. Additional spurious mass flux is added to the unsteady dipole, 2D incompressible convecting vortex, and flows over a circular cylinder; and the spurious mass flux of the co-rotating vortex pair comes from the residual of an incompressible-flow simulation. The far-field sound is found to be sensitive to spurious mass flux in the unsteady dipole and 2D incompressible convecting vortex cases. Then, the computation of the monopole-term expansion with the flow over a circular cylinder is presented. Fast convergence performance was observed, suggesting that the expansion requires little extra computational resources. Finally, FW–H boundary dependence is observed in the co-rotating vortex-pair case and eliminated by using the proposed mass-conserved formulation.