An azimuthal Fourier domain formulation of the Ffowcs Williams and Hawkings equation

Abstract

The acoustic field of high-speed turbulent jets is dominated by a small number of low-wavenumber azimuthal Fourier modes. Accordingly, it is of interest to directly obtain individual azimuthal modes of the acoustic field from simulation data or models of the jet near-field. To this end, we manipulate the Ffowcs Williams and Hawkings (FW–H) equation to obtain a new formulation that lives entirely in the azimuthal Fourier domain—it delivers individual azimuthal modes of the acoustic field as a function of the same azimuthal modes of the near-field FW–H source term or, upon linearization of the source terms, as a function of the same azimuthal modes of the near-field flow variables. As an added benefit, all surface integrals are converted into line integrals in the streamwise–radial plane. After verifying and validating our formulation using a monopole problem with an exact solution and large-eddy simulation data, respectively, we show how our method can be used to efficiently and naturally compute the acoustic field associated with resolvent modes of a Mach 1.5 jet, thus avoiding the need to compute the modes on a large computational domain to capture their acoustic radiation.