## Hydro-acoustic Simulations using OpenFOAM

Hydroacoustics focuses on the study of propagating sound in water. Sound may occur because of a wide variety of reasons. Theoretical basis for the analysis of sound generated by a body moving in a fluid is represented by the Ffowcs Williams-Hawkings (FWH) equation, which can be derived from basic conservation laws of mass and momentum written in terms of generalized functions. In the scope of this project turbulent incompressible flow around cavitated propellers will be investigated to predict the influence of cavitation on the noise waveforms. Therefore we aim to implement a transient multiphase hydroacoustics solver based on the FWH equation to determine the noise generated by ship propellers as well as to compute the influence of cavitation occurring. The nonlinear terms are included in the mathematical model because the results from recent studies show that neglecting the quadrupole terms does not yield sufficient enough results. The software of which this study is based is OpenFOAM, an Open Source object-oriented library for numerical simulations in continuum mechanics written in the C++ programming language. OpenFOAM framework is selected as the basis library for code development because of its flexibility in the development of customized numerical solvers.

FWH equation provides a description of sound generated by a solid body moving in a fluid and it can be written as follows:

$D^{2} p'(x,y) = + \frac{\partial}{\partial t} \left[ \rho_{0} \nu_{N} + \rho \left( u_{N} - \nu_{N} \right) \delta(f) \right] - \frac{\partial}{\partial x_{i}} \left[ \Delta P_{ij} n_{j} + \rho u_{i} \left( u_{N} - \nu_{N} \right) \delta(f) \right] + \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} \left[ T_{ij} H(f) \right]$

where D is D'Alembert operator:

$D^{2} = \frac{1}{c_{0}^{2}} \frac{\partial^{2}}{\partial^{2} t^{2}} - \nabla^{2}$

and P is compressible stress tensor:

$P_{ij} = p \delta_{ij} + \mu \left( - \frac{\partial u_{i}}{\partial x_{j}} - \frac{\partial u_{j}}{\partial x_{i}} + \frac{2}{3} \frac{\partial u_{k}}{\partial x_{k}} \right) \delta_{ij}$

FWH equation consists of 3 sources because according to Williams and Hawkings (J.E.F. Williams and D.L. Hawkings, Sound Generation by Turbulence and Surfaces in Arbitrary Motion - 1968) when both the bounding surfaces and the turbulence are compact relative to the radiated length scales, the turbulence is acoustically equivalent to a volume distribution of moving quadrupoles and the surfaces to dipole and monopole distributions. One significant difference of FWH model considers time dependent control surfaces, while Curle assumes they are fixed over the time.

We can investigate each source term of the FWH equation separately. First term on the RHS is called the monopole term:

$+ \frac{\partial}{\partial t} \left[ \rho_{0} \nu_{N} + \rho \left( u_{N} - \nu_{N} \right) \delta(f) \right]$

This term represents volume displacement effects when the surfaces are moving.

The monopole term is supplemented by surface distributions of acoustic dipoles of strength density Pij , which correspond to the second term on the RHS:

$- \frac{\partial}{\partial x_{i}} \left[ \Delta P_{ij} n_{j} + \rho u_{i} \left( u_{N} - \nu_{N} \right) \delta(f) \right]$

where "u_{N}" is velocity component normal to the surface and "\nu_{N}" is the surface velocity component normal to the surface.

The quadrupole term in the FWH equation is the very last term on the RHS and it corresponds to the volume displacement related sound generation as it is aforementioned. In aeroacoustics, it is crucial to consider this nonlinear term only if the flow is in the high transonic or supersonic regime. Even though we are far from being inside these high speed flow regimes for a rotating ship propeller, we still might have to take into account the quadrupole term because of the different nature of underwater physics. A recent study by Ianniello et al (S. Ianniello and R. Muscari and A. DiMascio, Hydroacoustic Characterization of a Marine Propeller Through Acoustics Analogy - 2012) claims that it is vital to consider the nonlinear term for hydro-acoustic applications to capture the sound generation due to the shocks which occur because of the cavity bubbles appearing even for propellers with a relatively low rotational speed.

As an initial step of this work, we reduced the above equation by considering the flow is time independent and incompressible:

$\frac{1}{c_{0}^{2}} \frac{\partial^{2} p'}{\partial^{2} t^{2}} - \nabla^{2} p' = \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} \left( \rho u_{i} u_{j} \right)$

An OpenFOAM solver based on the above equation is implemented. Velocity components on the RHS are computed by the flow solver and pressure fluctuations are resolved using these fields that come from CFD. Adding the acoustic pressure disturbance to the pressure solutions from CFD for each control volume gives us the total pressure in the system.