Scaling analysis of linear and non-linear terms of the Ffowcs-Williams and Hawkings equation

The Ffowcs- Williams and Hawkings (FWH) equation, which rules propagation of noise generated by a rigid body in motion is composed of linear and non-linear terms.

The linear terms are associated to the noise produced by the velocity and pressure over the body in motion, the non-linear ones are associated to the wake. Since the Lighthill seminal work, scaling properties of monopole, dipole and quadrupole sources have been exploited, finding dependence with the Mach number, no matter what the fluid dynamic regime is. Such scaling laws apply to the FWH terms decaying as 1/??. We  complete the classical theory, through analysis of scaling of the additional terms (linear and non-linear) decaying as 1/??² and as 1/??³ , tipically not considered in literature. 

The analysis is carried out  in view of perfect/imperfect scaling when data obtained at laboratory scale are used for prediction at the full scale.

We found that in order to recover perfect similarity (i.e. the Mach number has to be the same at the laboratory scale and at the full scale) the speed of sound at a laboratory scale needs to be properly scaled, otherwise imperfect similarity introduces errors in the acoustic response related both to the linear terms and to the non-linear terms, the latter of great

importance when the wake is characterized by organized vorticity and turbulence.

The error associated to the imperfect scaling is quantified considering the case of a marine propeller in uniform single-phase flow condition. The flow field is obtained using Large Eddy Simulation with a wall-layer model.

We find that the near field is nearly unaffected by imperfect scaling because there the thickness term, which scales perfectly being independent on the Mach number, dominates. On the other hand, the intermediate-to far field noise, dominated by the non-linear terms which scale imperfectly may be substantially affected by the scaling procedure.