Propeller tip-vortex cavitation and its broadband noise

*Johan Bosschers is a PhD student in the Engineering Fluid Dynamics Research Group his supervisors are prof.dr.ir. H.W.M. Hoeijmakers from the faculty Engineering Technology and prof.dr.ir. T.J.C. van Terwisga from the Delft Technical University.*

Comfort on board has become an important aspect in the design of ships over the last decades, especially comfort for passengers on cruise vessels and for owners of yachts. Noise and vibration need to be minimized, which poses design constraints on the propeller with respect to cavitation. Cavitation on propellers for ships requiring high levels of comfort should be minimized, leaving often only a tip-vortex cavity. However, tip-vortex cavitation is known to be a cause of broadband pressure fluctuations on the hull above the propeller, typically in a frequency range from 30 to 100 Hz, which has led to vibration problems for some ships. Broadband noise of cavitation is also of relevance for the underwater radiated noise emitted by the ship, not only from the perspective of the acoustic signature of military ships, but also because of the impact of noise on fish and marine mammals.

The objective of the present PhD thesis was to develop prediction methods for broadband noise, including hull-pressure fluctuations, by developed propeller tip-vortex cavitation. However, developed tip-vortex cavitation and its broadband noise are far from understood, because most research on vortex cavitation has been focussed on its inception. Therefore, fundamental aspects of a cavitating vortex were first investigated, aiming at understanding the mechanisms involved in the generation of broadband noise. This has been achieved by theoretical and computational studies of the kinematics, dynamics and acoustics of vortex cavitation and the analysis of experimental data. The knowledge and formulations obtained were used for the development of a semi-empirical prediction method. The formulations were also used to develop a novel methodology for model tests to correct for the Reynolds-number scale effect on the radius of the vortex-cavity and its broadband noise.

The kinematics of the flow around a vortex cavity was investigated by deriving an analytical solution for the radial distribution of the azimuthal velocity and pressure from Navier-Stokes equations for axisymmetric incompressible flow. After extending this vortex model with a semi-empirical formulation to account for vorticity roll-up, published experimental data could be matched well. The results show that the --by approximation zero-shear-stress-- boundary condition at the cavity interface is realistic. Remarkable was that the measured relation between cavity radius and cavitation number could only be reproduced if the measured increase in viscous core radius in cavitating flow was not taken into account. It was also found that the analytical vortex model becomes independent of vortex strength and Reynolds number when the ratio of cavity radius and viscous core radius is presented as a function of the ratio of cavitation number and cavitation inception number. The vortex model suggests that the effect of viscosity on cavity radius reduces with increasing cavity radius.

The dynamics of a vortex cavity was first studied by numerically solving the Navier-Stokes equations for axisymmetric, unsteady, and incompressible flow. The results show that the collapse of the cavity is inertia driven and show the presence of a resonance frequency. The dynamics in 3-D flow were investigated by analysing the dispersion relation that describes the propagation of waves on the vortex-cavity interface. The results of the analytical formulation for potential flow with an ad-hoc correction for viscosity show acceptable agreement with published experimental data. Several criteria for resonance have been proposed. However, resonance has not been demonstrated because all criteria correspond to neutrally stable waves.

The acoustics of a vortex cavity was investigated by studying the analytical formulation for the radiated noise by structural vibrations of a cylinder of finite length. The far-field formulation is similar to that of a monopole, but at the distance where the hull-pressure fluctuations are measured, the situation is slightly more complicated. An analytical formulation was developed for the effect on the hull-pressure spectrum of the variation, from one blade passage to the other, of amplitude and phase of the pressure signal.

The mechanisms of broadband noise generated by vortex-cavitation were analysed and reviewed using the analytical formulations derived in the fundamental studies. It was shown that the transient oscillatory dynamics of the vortex cavity, in combination with variability between blade passages, generates the broadband hump. An important excitation source of a vortex cavity is the shedding of the sheet cavity into the vortex cavity. It was also shown that the collapse of the closure-vortex cavity, generated by the side-entrant jet of the sheet, produces broadband noise.

A semi-empirical prediction method for broadband noise by propeller tip-vortex cavitation was developed by combining the results of the fundamental studies with experimental data obtained from model tests and sea trials. This new method predicts broadband hull-pressure fluctuations as well as underwater radiated noise. It makes use of results of a boundary element method that computes the flow on the propeller operating in a ship's wake field. Despite its simplicity, the method gives good results, also for ships that were not used to determine the empirical parameters from fits.

A new methodology for model tests was developed and evaluated to correct for the viscous-scale effect on the radius of the vortex cavity and, thereby, on its broadband noise. The methodology makes use of a formulation for a vortex model in which only the cavitation number and cavitation inception number are required to evaluate this scale effect. A semi-empirical relation was used to relate the ratio of the cavity radius on model-scale and that on full-scale to a difference in noise level. The first results of the methodology are encouraging but more detailed validation studies are required.

In general, a basic understanding of developed vortex cavitation and its associated broadband noise has been obtained, but several details need to be further investigated.