Additional terms for the use of Ffowcs Williams and Hawkings surface integrals in turbulent flows

•The use of the FWH surface integrals in turbulent flows is thoroughly analysed.•Additional terms are proposed to use FWH surface integrals in turbulent flows.•The efficacy of FWH additional surface terms is assessed for a turbulent jet. The paper aims to contribute to a clearer understanding of the spurious noise generated in aeroacoustic computations by the flow turbulence passing through a Ffowcs Williams and Hawkings (FW-H) control surface, and to propose additional surface terms that can effectively reduce this spurious noise. To this end, expressions of FW-H surface and volume integrals are recalled as well as the link between both integrals. A thorough analysis of the pressure radiated by a vortex passing through a control surface is then conducted. This analysis confirms that this spurious noise is due to the fact that some volume sources are not taken into account in the calculations limited to surface integrations. It clearly shows the mechanism involved in the spurious radiated pressure. It also confirms that the FW-H formulation is much less sensitive to the flow turbulence on the control surface than the Kirchhoff method. In the second part of the paper, a new expression of additional surface terms, that can represent an approximation of the complementary volume integral, is rigorously developed starting from the original FW-H formulation. The efficacy of these additional flux terms is first evaluated for an isolated vortex before being assessed for a turbulent jet. The comparison of different approaches finally shows the advantages of using closed control surfaces together with these additional surface terms.