2.1. Fluid-dynamic data

Although the study is thoroughly described in the original paper, here, we give brief information about the numerical set-up and main results. The case refers to a laboratory-scale, open-water numerical experiment of a five-blade propeller in pulling conditions. The benchmark propeller is the SVA VP1304, whose complete documentation including geometry, experimental data and numerical results, is available online (https://www.sva-potsdam.de/en/potsdam-propeller-test-case-pptc). The fluid-dynamic field was obtained solving the Navier–Stokes equations for incompressible flows using LES in conjunction with a dynamic Lagrangian model for the closure of the subgrid-scale (SGS) stresses; at the wall we use an equilibrium wall-layer model which allows us to avoid the resolution of the very thin viscous sub-layer developing over the solid surfaces. In this way, we were able to use a grid of approximately 3 million cells, mostly clustered in the wake region (Figure 1). It is worth mentioning that previous studies (Wang Reference Wang1999; Piomelli & Balaras Reference Piomelli and Balaras2002; Wang & Moin Reference Wang and Moin2002; Radhakrishnan & Piomelli Reference Radhakrishnan and Piomelli2008) discussed the performance of the equilibrium wall-layer model, which in non-separated flows, has been proven to give accurate first- and second-order statistics even in the presence of flow complexities, such as inertial unsteadiness, rotation and thermal stratification. The reliability of the data set obtained in the simulation has been verified comparing turbulent quantities with those obtained by other authors using incompressible-flow LES in conjunction with grids as large as two orders of magnitude more. The use of incompressible formulation of the Navier–Stokes equations is justified by the low rotational speed compared with the speed of sound. We obtained the following integral thrust $K_{T}$ and torque $K_{Q}$ coefficients over the propeller, for the value of the advance coefficient $J=1.068$

(2.12a,b)\begin{equation} K_{T}=0.3414,\quad K_{Q} =0.09051, \end{equation}

accurate to within $3.5\,\%$ and $0.5\,\%$, respectively, when compared with experimental data. We made a qualitative comparison with results reported in high-resolution LES (Kumar & Mahesh Reference Kumar and Mahesh2017; Posa et al. Reference Posa, Broglia, Felli, Falchi and Balaras2019) where wall-resolving LES were performed, involving the use of a very large number of grid points. Some quantities such as axial velocity, vorticity magnitude, turbulent kinetic energy, root mean square and turbulent SGS viscosity were plotted at different sections perpendicular to the mean flow. The instability of the tip vortex occurred at approximately three diameters downstream, where larger levels of turbulent kinetic energy are observed. A population of large-scale vorticity was observed in the wake, although in the simulation the mutual induction and interaction between the tip vortex and that emitted at the hub was not observed, basically due to the fact that in pulling conditions the hub is replaced by the shaft. In this sense two main stable vortex structures develop, one being the tip vortex and the other the shaft vortex. This is shown in figure 2(a) where isosurfaces of the quantity $Q D^{2}/U_0^{2} = 80$ ($Q=0.5 (||\varOmega ||^{2}-||S||^{2}$) where $\varOmega$ and $S$ are the instantaneous rotation rate tensor and deformation rate tensor, respectively) are plotted coloured with the vorticity magnitude and the Lighthill term $\partial _{x_i x_j} T_{ij}$. The figure highlights the presence of the main vortical structures, the shaft vortex being persistent up to five diameters downstream. Figure 2(b) shows the same isosurfaces as (a), but coloured using the Lighthill term, which gives an indication of the regions of the wake where noise sources are more intense.

Figure 1.

Computational grid adopted for the LES of the propeller in uniform flow.

Figure 2.

Isosurface $Q = 20\,000$ (non-dimensional $Q D^{2}/U_0^{2} = 80$, optimal value for the visualization of the two vortex structures): (a) contour of non-dimensional vorticity magnitude; (b) contour of Lighthill term.

The acoustic analysis related to this simulation had a clear outcome. Indeed, the authors observed a strong tonal noise at the rotational frequency, whose cause was recognized to be the strong vortex, that rolls up around the shaft. Being caused by a structure present in the wake, this acoustic signal was intercepted by the volume terms. Our results corroborated the findings of Ianniello (Reference Ianniello2016), who stated that ‘the usual assumption of believing the flow non-linear sources to be negligible for blades rotating at low subsonic speed is totally wrong when applied to hydroacoustics’. These results are in some sense quite new, since, in high-speed aeroacoustics, for a long time, it has been believed that the contribution to the noise coming from the wake is negligible compared with the linear terms. This is certainly true for helicopter blades, working in the compressible regime and characterized by very-high-pressure loads over the moving surfaces; however, as shown by other authors (see for example the comparison of measurements with numerical results reported in Ianniello, Muscari & Di Mascio Reference Ianniello, Muscari and Di Mascio2014) and also discussed in the present paper, this is not the rule in other physical configurations, like those studied herein and, in general, in low-Mach-number cases. Another example which is consistent with our results comes from the analysis of the database produced in the huge simulation of Posa et al. (Reference Posa, Broglia, Felli, Falchi and Balaras2019). The authors carried out a simulation of a seven-blade marine propeller, in an open-water pushing condition, adopting a grid of approximately 800 million cells. The acoustic computations carried out using the acoustic analogy revealed a strong broadband low-frequency noise, coming exclusively from the wake (Broglia et al. Reference Broglia, Cianferra, Posa, Felli and Armenio2020). In addition, the contribution of the wake to the far-field noise was shown to be more important than that coming from the linear part. The fact that the signal in Posa et al. (Reference Posa, Broglia, Felli, Falchi and Balaras2019) and Broglia et al. (Reference Broglia, Cianferra, Posa, Felli and Armenio2020) appears more broadband compared with the signal observed in the present work is mainly due to the different operating conditions and geometry of the propeller, namely, to the pulling low-loading condition reproduced in Cianferra et al. (Reference Cianferra, Petronio and Armenio2019b) versus the pushing high-loading conditions of Posa et al. (Reference Posa, Broglia, Felli, Falchi and Balaras2019). Indeed, the shaft located downstream the propeller, as it is in the case herein discussed, gives rise to a stable shaft vortex, identified as a main source of noise; in the work of Posa et al. (Reference Posa, Broglia, Felli, Falchi and Balaras2019) a hub vortex develops, it travels downstream and interacts with the tip vortex causing a broader spectrum.

2.2. The acoustic field

The diameter of the model-scale propeller is $D_m = 0.25 \ \textrm {m}$ and the scaling factor adopted is $\lambda = 24$. The purpose is to use our data of pressure and velocity fields and to apply the FWH equation, first using imperfect similarity and successively using perfect similarity. Although the analysis has a general importance, here, we use hydrodynamic scaling. According to (2.4) and following the previous discussion, in the case of perfect similarity, we consider $c_{0_m}=c_{0_f}/\sqrt {\lambda }$, and in the case of imperfect similarity we consider $c_{0_m}=c_{0_f}$; here $c_{0_f} = c_{w} = 1400\ \textrm {m}\ \textrm {s}^{-1}$, where subscript $w$ denotes water. We calculate the dimensional acoustic pressure contributions associated with the single integrals of the FWH equation, for both procedures. We compute the acoustic pressure through (2.1) which corresponds to the original formulation proposed by Ffowcs-Williams and Hawkings. Over the years, a number of formulations have been developed depending on the problem under investigation. Specifically, efforts have been devoted to the computation of the noise generated by the wake, nowadays understood as to be responsible for the low-frequency part of the spectrum. Among others, the porous formulation (Di Francescantonio Reference Di Francescantonio1997) widely used in recent hydroacoustic noise applications allows us to neglect the volume terms. Recently this problem was carefully investigated by Cianferra et al. (Reference Cianferra, Ianniello and Armenio2019a). The authors found that the direct computation of the volume terms under the assumption $t=\tau$ (thus neglecting the time delays) is accurate in a large number of cases and also provided a criterion for the use of this computationally cheap and effective method. The authors also showed that the direct evaluation of the volume integral provides results more accurate than those obtained using the porous formulation, the latter being too sensitive to the position of the porous surface around the body and the wake. Details on the methodology are given in Cianferra et al. (Reference Cianferra, Ianniello and Armenio2019a) and are not duplicated here. For the evaluation of the volume terms we consider the fluid-dynamic instantaneous LES data contained within a cylindrical domain of volume $W$, diameter equal to $1.16D$ aligned with the propeller axis. For the evaluation of the linear terms we use the pressure over the propeller surface $S$ and its own velocity.

Six microphones are selected, whose positions are reported in table 1 and sketched in figure 3: two (1,2) are chosen on the propeller plane, at increasing distance from the propeller axis; two (3,4) are placed in the wake region at a distance of approximately $14D$ from the centre of the propeller, over lines passing through the centre of the propeller and inclined by different angles with respect to the symmetry axis (respectively $45^{\circ }$ and $15^{\circ }$); two (5,6) are located in the far wake, over the propeller axis. Since the FWH terms decay differently with source–observer distance $r$, their own contribution varies considerably depending on the locations of the microphones. First, we analyse the time records over $t^{*}=4$ of full-scale acoustic pressure evaluated by means of the FWH equation, using the perfect scaling and the imperfect one, respectively. This is carried out both for the linear terms and for the nonlinear terms. Successively, we evaluate the spectra in decibel scale of the acoustic pressure obtained using the two procedures. In the following analysis we refer to full-scale quantities, indicating the FWH terms as $Th$, $L1$, $L2$, $V1$, $V2$ and $V3$.

Figure 3.

Microphones considered for the acoustic analysis, listed in table 1 together with fluid-dynamic computational domain sketch (black box). Figure is not to scale.

Table 1.

Microphones considered for the acoustic analysis; $x$ is the streamwise direction with negative values indicating the downstream direction.

Figure 4 contains the time signals of the full-scale acoustic pressure associated with the linear terms, evaluated using the two methods. Note that, since the first and third terms of (2.10) scale perfectly, the time signals are the same in the two cases, whereas the time signal of the second term of (2.10) changes, increasing by $\sqrt {\lambda }$. For the microphones $1 \text{ and }2$ (a,b) placed on the plane of the propeller, the thickness noise term prevails over the others, whereas the third term ($L2$) is always very small. The second term ($L1$), whose evaluation at the model scale using the speed of sound of water introduces errors, becomes increasingly important with the increase of the distance from the noise source since it decays with $1/r$. However, its own contribution remains substantially smaller than the thickness term. The situation is similar for the microphones $3 \text{ and }4$ placed in the wake zone (c,d), not very far from the propeller. In these cases, the error introduced by imperfect scaling is very small. The situation dramatically changes for the microphones $5 \text{ and }6$ placed in the far wake over the axis of the propeller (e, f) where, for symmetry reasons, the thickness term is zero. The error introduced by imperfect scaling, which affects the second integral, becomes increasingly large with increasing distance from the propeller. This can be attributed to the fact that $L1$ decays with $1/r$ whereas $L2$ decays with $1/r^{2}$.

Figure 4.

Linear terms of FWH equation (2.1): $Th$, red solid line; incorrect $L1$, dashed black line; correct $L1$, solid black line; $L2$, green solid line. The microphones are listed in table 1.

The time records of the full-scale acoustic pressure associated with the FWH nonlinear volume terms are given in figure 5 for the six microphones. Note that, in this case, two out of three terms scale imperfectly, namely $V1$, whose scaling properties were exploited in the seminal research of Lighthill, and $V2$ whose scaling properties are discussed in the present work. At the microphone $1$ which is close to the propeller (a), the term which scales perfectly ($V3$) is dominant. This is reasonable since the microphone is very close to the source and $V3$ contains $1/r^{3}$ in the integral kernel, while $V1$ and $V2$ contain respectively $1/r$ and $1/r^{2}$ in their integral kernels. Moving to microphone $2$ placed far from the source, the contribution of imperfect scaling becomes increasingly important as well as the error in the evaluation of the noise, as shown in figure 5(b). In the wake, out of the propeller disk, (microphones $3 \text{ and }4$) the three volume terms have comparable importance (c,d) and thus the imperfect scaling error is not negligible. In the far wake, over the propeller axis, due to the discussed different decay rate of the kernels of the volume integrals, the error associated with the imperfect scaling becomes increasingly important with increasing distance from the noise source. Some considerations derive from this analysis: first, comparing figures 4(f) and 5(f) it clearly appears that, downstream of the propeller, the nonlinear contribution given by the wake is an order of magnitude larger than that given by the linear terms; second, in the very near field the thickness term prevails over the unsteady loading term, due to the significant contribution given by the rotation of the solid elements, stating the need to consider a more complete form of the FWH equation even in the simplified linear analysis.

Figure 5.

Nonlinear terms of FWH equation (2.1): incorrect $V1$, dashed red line; correct $V1$, solid red line; incorrect $V2$, dashed black line; correct $V2$, solid black line; $V3$, solid green line. The microphones are listed in table 1.

To highlight how imperfect scaling affects the total signal, in figure 6 we show the difference between the spectra, in terms of sound spectrum level (SPL) for the six microphones. Specifically, for each microphone we calculate the entire spectrum given by the sum of the six integral contributions, using the perfect scaling and the imperfect one. The decibel scale is obtained computing $SPL = 20 \log _{10} fft(p(t))/p_{ref}$, with $fft(p(t))$ the Fourier transform of the time signal $p(t)$ expressed in Pascals and $p_{ref} = 1\ \mathrm {\mu } \textrm {Pa}$. The frequency is made non-dimensional with the revolution frequency $f_r = n \ \textrm {Hz}$, $n$ being the number of revolutions per second.

Figure 6.

SPL of FWH signal: imperfect similarity (dashed red lines); perfect similarity (solid blue lines). The microphones are those listed in table 1.

The analysis of the spectra shows that, very close to the propeller (microphone 1, panel a), as expected, the two methods give practically the same result. This occurs because the dominant terms in the near field, namely $Th$ and $V3$, scale perfectly. Further, we may observe that, close to the propeller, two main peaks dominate, one at the rotation frequency $f/f_r = 1$ and the other one at the blade frequency $f/f_r = N$ ($N$ is number of blades). It is worth noting also the appearance of sub-harmonics of the blade frequency, at $f/f_r = 2,3,4$. A specific analysis of the fluid-dynamic versus acoustic field (not shown) suggests that, at microphone 1, which is very close to the propeller, the signal is largely affected by the complex interaction between the various acoustic sources. We observed a rapid decay of the aforementioned sub-harmonics considering other microphones (not shown here) located on the propeller plane, at gradually increasing distances. At microphone $2$ a substantial difference of approximately 20 dB is observed, due exclusively to the difference previously observed in the analysis of nonlinear terms, figure 5(b). Thus, the imperfect scaling is not conservative with respect to the expected maximum noise level, since it underestimates the noise signal at the full scale. At microphones $3$ and $5$ the imperfect scalings of terms $L1$ and $V1$ are not able to produce significant variation in the spectrum, because they are less important than the terms scaling perfectly (namely $Th$ and $V3$) except for a slight drop of the perfect similarity signal, that may explained by a destructive interference between the term $V1$ and $V3$ occurring when $V1$ scale perfectly (figure 5c,e).

The fourth microphone is placed in the wake, with an angle of $15^{\circ }$ with respect to the propeller axis. There, the contribution of $V3$ becomes comparable to that of the other two volume terms, therefore, the error introduced by imperfect scaling is noticeable over the whole spectrum, as shown in figure 6(d). At microphone $6$, located in the far wake, the signal obtained by the perfect scaling, resulting from the composition of solid lines (red, black and green) of figure 5( f), gives rise to a tonal signal at the rotation frequency. Actually, this is mostly given by the $V1$ term, since the $V2$ and $V3$ term signals are in opposition of phase and tend to cancel out. Similarly, the signal obtained with imperfect scaling, composed of the sum of solid green line and dotted red and black lines of figure 5( f), is tonal at the rotation frequency, given exclusively by the dominant term $V3$. Overall, the differences in the spectra are essentially due to the relative importance of the terms which scale imperfectly over those which scale perfectly.

To summarize, the analysis shows that carrying out laboratory-scale experiments using the speed of sound of water introduces a scale effect rendering incorrect the evaluation of the full-scale acoustic pressure starting from experimental data. The error is only partially associated with the noise propagating from the propeller surface (linear terms) and, more significantly, it is due to the imperfect scaling of the noise generated by the fluid-dynamic field in the wake around the propeller. The error is negligible close to the propeller, where the signal is dominated by terms $Th$ and $V3$ which scale perfectly, and, in general, the peak at the rotation frequency is well reproduced even by imperfect scaling. However, far from the acoustic source, in the high-frequency range, considerable differences (up to about 20 dB) arise, between the signal obtained with the perfect scaling and that obtained with the imperfect scaling. Imperfect scaling introduces an underestimation of some noise source terms at the full scale, hence the imperfect scaling procedure is not conservative with respect to the noise level emitted by the full-scale propeller.

3.1. Monopole source near a free surface

Reflection is a crucial aspect of noise propagation in an underwater environment, and is quite a complex topic in the presence of interfaces and salinity/temperature variation in the fluid column (for a discussion see, among others, Athanassoulis et al. Reference Athanassoulis, Belibassakis, Mitsoudis, Kampanis and Dougalis2008, Reference Athanassoulis, Belibassakis, Gerostathis and Prospathopoulos2018). The problem of interfaces is shared by a number of marine engineering systems, among others, ship propellers, marine turbines and wind turbines mounted at sea. However, the presence of archetypal reflecting interfaces may be addressed easily by applying the method of images in case of time-domain wave equation (see for details Rienstra & Hirschberg Reference Rienstra and Hirschberg2004). The main difference with respect to the unbounded domain is that, in the presence of a free surface, two acoustic contributions arrive at the microphone: the direct signal coming from the source, and that reflected by the free surface. A free surface is a sharp interface which reflects the signal in the interior of the field and where $p=0$; it is different from a solid surface which reflects in the interior and where the signal is doubled (see Rienstra & Hirschberg Reference Rienstra and Hirschberg2004). For this reason, in the presence of a free surface the reflection is mimicked by the image of the source, placed anti-symmetrically to the real source with respect to the plane of the free surface, thus having opposite sign with respect to the real source.

A monopole is a pressure field generated by a pulsating sphere. The pressure field has spherical symmetry and may be represented by harmonic waves of angular frequency $\omega = 2{\rm \pi} f$, travelling outward from the source point $\boldsymbol {x}_s$

(3.1)\begin{equation} p(\boldsymbol{x},t) = \frac{p_I}{r} \sin(\omega t - k r), \end{equation}

where $r = |\boldsymbol {x} - \boldsymbol {x}_s|$, $p_I$ is the intensity of the source expressed in $\textrm {Pa}\ \textrm {m}$ and the wavenumber $k = 2 {\rm \pi}/ c_0 T= \omega / c_0$. Note that, here, we consider the real part of the source, given by a sine wave. In the presence of a free surface we must consider the contribution of two monopoles, placed anti-symmetrically with respect to the free surface. One is placed at $\boldsymbol {x}_s = (x,y, z) = (0, 0, 0)$, with the reflection plane located at $y=L$. The imaginary source is located at $\boldsymbol {x}_i=(0, 2L, 0)$. With $r_1 = |\boldsymbol {x} - \boldsymbol {x}_s|$ and $r_2 = |\boldsymbol {x} - \boldsymbol {x}_i|$, at any point $\boldsymbol {x}$, the acoustic pressure is built of the sum of the two contributions

(3.2)\begin{equation} p(\boldsymbol{x},t) = \frac{ p_I}{r_1} \sin(\omega t - k r_1) - \frac{ p_I}{r_2} \sin(\omega t - k r_2). \end{equation}

The imaginary monopole has negative sign, giving $p=0$ at the surface.

Now, we make (3.2) non-dimensional by considering a signal amplitude $p_0$, a time scale $T= 2 {\rm \pi}/\omega$ and a length scale $L$, the distance of the source from the free surface. Introducing the non-dimensional variables $p* = p/p_0$, $t^{*} = t \omega /2{\rm \pi}$, $r_i^{*} = r_i/L$, with $i=1,2$, and considering that $sin \theta = - \sin (\theta + {\rm \pi})$ we obtain

(3.3)\begin{equation} p^{*} = \frac{p^{*}_I}{r^{*}_1} \sin\left(2{\rm \pi} t^{*} - \frac{\omega L}{c_0} r_1^{*}\right) + \frac{ p^{*}_I}{ r_2^{*}} \sin\left(2{\rm \pi} t^{*} - \frac{\omega L}{c_0} r_1^{*} - \frac{\omega L}{c_0} (r_2^{*}-r_1^{*}) + {\rm \pi}\right), \end{equation}

with $p^{*}_I=p_I / p_0 L$. The non-dimensional intensity of the monopole depends on a pressure impulse and the distance from the free surface. Here, we explicitly highlight the phase lag between the two signals. Now, we exploit some properties of this monopole. We observe that the composition of the two signals (direct and reflected) is constructive or destructive depending on the phase lag between the signals. The second sine wave of (3.3) interferes destructively with the first one when ${\omega L}/{c_0} (r_2^{*}-r_1^{*}) \in [2m{\rm \pi} - {\rm \pi}/3, 2m{\rm \pi} + {\rm \pi}/3]$, with $m=0,\pm 1, \pm 2,\ldots$ integer numbers.

At any point of the free surface (where $r^{*}_1=r^{*}_2$) the phase lag between the two sine waves is ${\rm \pi}$ and the signals cancel out. Close to the source location $r^{*}_1 \rightarrow 0$ and $r^{*}_2 \rightarrow 2$; the phase lag is given by $-2\omega L/c_0+{\rm \pi}$, so it tends to ${\rm \pi}$ (destructive interaction) for low frequencies and small $L$; however, for very small $r^{*}_1$, the contribution of the image monopole is negligible compared with that of the real source. At points located over the line passing through the two point sources and orthogonal to the free surface, $r^{*}_2=\pm r^{*}_1 + 2$ with positive sign for points located by the opposite side with respect to the free surface and negative sign for points located between the source and the free surface, with $r^{*}_1 < 1$. In the former case $r^{*}_2=r^{*}_1 + 2$ and the phase lag between the two signals is $-2 \omega L /c_0+{\rm \pi}$. This value gives destructive interaction when $(2m-1/3) {\rm \pi}< 2 \omega L/c_0 < (2m+1/3) {\rm \pi}$, meaning that, for a given distance $L$ from the free surface, there is a range frequencies such that the free surface causes a destructive interaction of the signals, although the total signal cannot be zero because the contribution of the real signal, as in the previous case $r^{*}_1 \rightarrow 0$, is always larger than that of the image one in the near field. In the far field, when $2/r^{*}_1 \sim 0$, $r^{*}_2/r^{*}_1 \sim 1$, the magnitude of the two signals is comparable and the phase lag behaves as discussed above. For points by the side of the free surface $r^{*}_2=-r^{*}_1 + 2$, with $r^{*}_1 < 1$, the two sine waves of (3.3) are in phase when $(1-r^{*}_1) = {\rm \pi}c_0/2 \omega L < 1$ and fully destructive for $r^{*}_1=1$, at the free surface as already discussed. However, also in this case, for $r^{*}_1 \longrightarrow 0$ the intensity of the real signal tends to be larger than that coming from the image source. Considering the points over a line passing through the real source and parallel to the free surface, we are in the case where $r^{*^{2}}_2 = 2^{2} + r^{*^{2}}_1$. Note that the microphones located on the propeller wake fall into this category. Dividing by $r^{*}_1$, we obtain $r^{*^{2}}_2/r^{*^{2}}_1 = 1 + (2/r^{*}_1)^{2}$, which highlights that for $r^{*}_1 < 2$ the contribution of image source becomes small when compared with real source contribution, and for moderate-to-large values $r^{*}_1 > 2$, $r^{*}_2/r^{*}_1 \sim 1$ the two sources have comparable magnitude. For the phase, it is easy to show that destructive interaction requires ${\rm \pi} (2m-1/3)<4 \omega L/ [c_0 (r^{*}_2+r^{*}_1) ] <{\rm \pi} (2m+1/3)$. Finally, in the general cases, the two signals are in phase when $\omega L /c_0 (r^{*}_2 - r^{*}_1) = (2m+1){\rm \pi}$. The condition $r^{*}_1 \sim r^{*}_2$, which ensures a comparable amplitude of the two sine waves, leads to $\omega L/c_0 \gg 1$. Thus, pure doubling of the signal may occur at high frequencies or for large values of $L$.

The discussion holds for a monopole field, or, in general for signals with spherical symmetry (for example airguns); however, when considering sources characterized by strong directivity, the analysis becomes more complex. For example, in figure 7 we show how the directivity of three types of elementary sources, monopole, dipole and quadrupole, changes in the presence of a free surface. We consider the real source located at a distance $L$ from the surface, and the image source placed anti-symmetrically with respect to the surface. The microphones are located at distance $D_c$ from the real source. The monopole was already discussed above. In the cases of the dipole and quadrupole sources, the discussion for the monopole still holds mainly for angle of $270^{\circ }$, where the observer is on the line passing through the two sources. Also, changes in the shape of the directivity are observed under variation of the frequency $\omega$ and of the ratio $L/D_c$. This is the case, among others, of rotors, such as wind turbines or the ship propeller discussed in the previous section, which exhibit a dipole-like field and a quadrupole-like one for the linear and nonlinear parts of the noise signal, respectively. The rotation of the source adds complications, since the image rotates counterclockwise when the real source rotates clockwise. Specifically, in the presence of rotation the phase lag $\phi$ (consider a sinusoidal wave $\sin (\omega (t-\phi ))$ as an example) between the signals has an additional shift when compared with the case in the absence of rotation.

Figure 7.

Directivity of (a) monopole, (b) dipole and (c) quadrupole. Unbounded domain (crosses); free surface (circles).

In the next section the method of images is applied to the marine propeller case and results analysed also in view of the discussion of the present section.

3.2. Rotating complex source near a free surface

Integral solution of a time-domain, non-homogeneous, wave equation in a homogeneous medium, makes use of the Green's function adopted under the assumption of free-space propagation. As for the monopole, the free surface is modelled by considering an additional imaginary source, which works as the real source mirror image with respect to the reflecting plane. As a result, certain boundary conditions are satisfied automatically by the presence of the mirror image. Instead of free-space Green's function, the half-space Green's function is adopted, which is built in such a way to consider both sources, real and imaginary.

The half-space Green's function is therefore the fundamental solution of the problem

(3.4)\begin{equation} \begin{cases} \displaystyle(\partial_{tt} - c^{2} \nabla^{2}) \boldsymbol{G}(\boldsymbol{x},t,\boldsymbol{x}_0,\tau) = \delta (\boldsymbol{x} - \boldsymbol{x}_0, t - \tau) & \boldsymbol{x} \in \mathbb{R}^{+},\\ \displaystyle \boldsymbol{G}(\boldsymbol{x}, t,\boldsymbol{x}_0,\tau) = 0 & \boldsymbol{x} \in \partial \mathbb{R}^{+}, \end{cases} \end{equation}

where the source $\delta (\boldsymbol {x} - \boldsymbol {x}_0, t - \tau )$ is an impulse originating at time $\tau$, at point $\boldsymbol {x}_0$. Note that the boundary $\partial \mathbb {R}^{+}$ of the domain corresponds to the plane $\boldsymbol {x}=0$ and the homogeneous Dirichlet condition is placed to mimic the $p=0$ condition. The Green's function solving (3.4) for the three-dimensional space, has the form

(3.5)\begin{equation} \boldsymbol{G}(\boldsymbol{x},t,\boldsymbol{x}_0,\tau) = \frac{ \delta ( t - \tau - |\boldsymbol{x} - \boldsymbol{x}_0| )}{ 4{\rm \pi} |\boldsymbol{x} - \boldsymbol{x}_0| } - \frac{ \delta( t - \tau - |\boldsymbol{x} - \boldsymbol{x}^{*}_0| )}{ 4{\rm \pi} |\boldsymbol{x} - \boldsymbol{x}^{*}_0|}, \end{equation}

with $\boldsymbol {x}^{*}_0$ being the image source location. Note that the first term, which corresponds to the real source impulse, is non-zero when $t = \tau + |\boldsymbol {x} - \boldsymbol {x}_0|/c_0$, while the impulse coming from the imaginary source arrives at the same point when $t = \tau + |\boldsymbol {x} - \boldsymbol {x}^{*}_0|/c_0$. This time delay gives a phase lag which may amplify or destruct the original signal, as already discussed for the archetypal sinusoidal wave of the monopole case.

The convolution of the half-space Green's function with the source term of the differential FWH wave equation leads to the integral equation describing the case of acoustic propagation in presence of the free surface $\boldsymbol {x} = 0$.

We apply the method of images to the case of the marine propeller discussed in the previous section. Note that, given the peculiarities of the Green's function method, the free surface is approximated as a perfectly smooth plane, not disturbed by the presence of the propeller ($Fn =0$). This does not limit the importance of the present study for a number of reasons: for the specific case of a marine propeller, laboratory tests are usually carried out in the $Fn \longrightarrow 0$ limit; in the general case of a complex source close to an interface (a wind or a marine turbine among others) the case of a flat surface in the most general one. The introduction of a wave system would introduce, at least, two more free parameters. To account for a non-flat free surface, methods different from the Green's function have to be used. This is beyond the scope of the present paper.

As in the previous section, the model-scale propeller has a diameter of $D=0.25\ \textrm {m}$, it is centred at the origin of the frame of reference $\boldsymbol {O} = (0, 0, 0)$ and the wake develops along the negative $x$-axis. The speed of sound for the model-scale propeller is set to $c_0 = c_w /\sqrt {\lambda }$, with $\lambda = 24$, thus considering perfect similarity. Note that, in the present problem,

(3.6)\begin{equation} p^{*}=f(J,V_r,L/D), \end{equation}

with $L/D$ the additional non-dimensional parameter, associated with the presence of the interface, controlling the problem; also, geometric similarity ensures that points behave similarly at the two scales when the ratio $L/D_c$ is the same, where $D_c$ is the distance of the microphone from the source. We first analyse the directivity, obtained considering 100 microphones placed on the $y=0$ plane (a vertical longitudinal plane), at constant distance $D_{c}$ from the source. In the plots, the distance from the centre indicates the value of the root mean square $p_{rms}$ of the acoustic pressure (in Pascals) at the microphone located at the corresponding angle. The analysis is performed for each single term of the FWH equation. The directivity is reported in figures 8 to 11 for an unbounded domain in the presence of the free surface. The directivity is calculated for two distances from the source, respectively $D_c = 10D$ in figures 8 and 10, $D_c = 100D$ in figures 9 and 11. Finally, we consider two values for the distance $L$ of the source from the free surface, $L = 2D$ in figures 8 and 9, $L = 8D$ in figures 10 and 11. In all plots, the downstream direction (the wake) is $180^{\circ }$; further, when considering the presence of the free surface, depending on $D_c$ and $L$ the range of angles at which we can observe the underwater acoustics (and therefore the directivity) changes; for this reason the values are not shown for angles giving microphones outside of the fluid domain.

Figure 8.

Directivity of linear terms (a,c,e) and nonlinear terms (b,d, f), evaluated at a radius of $10D$, on the plane $y=0$. Unbounded domain (blue crosses); free surface (red circle). Free surface at $L=2D$.

Figure 9.

Directivity of linear terms (a,c,e) and nonlinear terms (b,d, f), evaluated at a radius of $100D$, on the plane $y=0$. Unbounded domain (blue crosses); free surface (red circle). Free surface at $L=2D$.

Figure 10.

Directivity of linear terms (a,c,e) and nonlinear terms (b,d, f), evaluated at a radius of $10D$, on the plane $y=0$. Unbounded domain (blue crosses); free surface (red circle). Free surface at $L=8D$.

Figure 11.

Directivity of linear terms (a,c,e) and nonlinear terms (b,d, f), evaluated at a radius of $100D$, on the plane $y=0$. Unbounded domain (blue crosses); free surface (red circle). Free surface at $L=8D$.

As a preliminary test, we checked the cancellation of the acoustic pressure at the free surface, as required by the Dirichlet condition at the boundary. In all cases examined the model gives the expected result. We first observe that, for linear terms (left panels in figures 8 to 11), the intensity of acoustic pressure fluctuations is in general weaker than that produced by the nonlinear terms (right panels in the same figures), showing, once more, the importance of the latter with respect to the former. Note that, when considering the decibel scale with a reference pressure of $1\ \mathrm {\mu } \textrm {Pa}$, the signals below $10^{-6}\ \textrm {Pa}$ correspond to negative values. The directivity of linear terms (dipole-like) remains nearly unchanged in all cases, when the presence of the free surface is considered. In particular, the terms are very small on the propeller axis in all cases, whether or not there is a free surface. The exception is the thickness term, which by symmetry reasons is zero in an unbounded domain and has non-zero values at $0^{\circ }$ and $180^{\circ }$ in the free-surface case. A signal amplification is observed in almost all cases. A verification of this result was carried out by analysing the time history of pressure signals (not reported here). In fact, at distances $D_c=10D$ or $100D$, the linear term signals are characterized by a wide spectrum, maintaining the rotation frequency $f_r = 15\ \textrm {Hz}$ as the dominant one. The signal coming from the imaginary source is comparable in amplitude and frequency content to the signal emitted by the real source, however, the discussion on the interaction properties of the monopole per se cannot explain this signal composition. As observed before, rotation may affect the composition of the signals. The phase lag occurring in the case of reflection and rotation is given by $\phi = L(r^{*}_2 - r^{*}_1)/c_0 + 1/f_r *1/2$, which, in our case, makes the low-frequency signal constructive. In particular, in the case $L/D_c = 0.02$, the interaction between the signals appears constructive up to $f \sim 20\ \textrm {Hz}$. In other words, the low-frequency amplification of the signal is the result of a combination of the phase lag discussed before for the monopole case and the effect of rotation. On the other hand, in the case $D_c=10D$ and $L=8D$ ($L/D_c\sim 1$), the intensity of the real source is much more relevant than the imaginary one, whose contribution is therefore negligible (figure 10).

The directivity of nonlinear terms exhibits a dipole-like shape for small distances from the source and tends to a quadrupole shape with large distances (see figures 9(b,d, f) and 11(b,d, f), respectively). In addition, the relative contribution of the nonlinear terms $V1$, $V2$ and $V3$ appears reversed in the transition from $D_c= 10D$ to $D_c=100D$. This is expected, given the law which rules the decay with distance from the source of the different terms; this justifies the prevalence of the third term when close to the source, case $D_c=10D$ in figure 8, and the increasing importance of the first and second terms at large distances (figure 9). For the cases $L/D_c=0.02$ and $L/D_c=0.08$, reported in figures 9 and 11, respectively, the quadrupole shape appears rotated by $90^{\circ }$ (red circles), so that the interaction of the signals is destructive by the free-surface side, on the propeller axis as well as on the propeller plane. Finally, as observed for linear terms, in the case $L/D_c=0.8$, the presence of the free surface is of minor importance. The results presented in this section show that the presence of the free surface substantially modifies the signal, giving a larger and larger contribution when $L/D_c \ll 1$.