2 The FWH equation and the standard solving formulations

The FWH equation represents a rearrangement of the fundamental conservation laws of mass and momentum into an inhomogeneous wave equation, written in terms of generalized functions

(2.1) $$\begin{eqnarray}\displaystyle \bar{\Box }^{2}p^{\prime }(\boldsymbol{x},t) & = & \displaystyle \frac{\bar{\partial }}{\partial t}\{[{\it\rho}_{0}v_{n}+{\it\rho}(u_{n}-v_{n})]{\it\delta}(f)\}-\frac{\bar{\partial }}{\partial x_{i}}\{[{\rm\Delta}\unicode[STIX]{x1D617}_{ij}\hat{n}_{j}+{\it\rho}u_{i}(u_{n}-v_{n})t]{\it\delta}(f)\}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\bar{\partial }^{2}}{\partial x_{i}\partial x_{j}}\{\unicode[STIX]{x1D61B}_{ij}H(f)\},\end{eqnarray}$$

where $p^{\prime }=c_{0}^{2}\tilde{{\it\rho}}$ , $\tilde{{\it\rho}}={\it\rho}-{\it\rho}_{0}$ being the density perturbation and ${\it\rho}_{0}$ , $c_{0}$ the density and sound speed respectively in the undisturbed medium; under the assumption of small perturbations ( $\tilde{{\it\rho}}/{\it\rho}_{0}\ll 1$ ), $p^{\prime }$ represents the acoustic pressure. The equation $f=0$ (where $f$ is the support of the Dirac ${\it\delta}(f)$ and the Heaviside $H(f)$ functions) indicates an arbitrary surface, whose choice heavily affects the physical meaning of the different terms. The D’Alembert operator is given by $\bar{\Box }^{2}=[1/c_{0}^{2}\bar{\partial }^{2}/\partial t^{2}]-\bar{{\rm\nabla}}^{2}$ , the Lighthill tensor is $\unicode[STIX]{x1D61B}_{ij}={\it\rho}u_{i}u_{j}+\unicode[STIX]{x1D617}_{ij}-c_{0}^{2}\tilde{{\it\rho}}{\it\delta}_{ij}$ and $\unicode[STIX]{x1D617}_{ij}$ represents the compressive stress tensor. The fluid and surface velocity components are indicated by $u_{i}$ and $v_{i}$ respectively, while $n$ denotes the projection along the outward normal to the surface. The presence of the Dirac and Heaviside functions points out the different nature of the source terms: two surface terms directly related to the effects of the discontinuity at $f=0$ in the flow field and a volume term accounting for all sources acting outside it. When $f=0$ coincides with the body surface $S$ , the impermeability condition $u_{n}=v_{n}$ simplifies (2.1), and the use of the Green function approach results in several different integral formulations, suitable for numerical implementation. Among these and for a long time, the formulation 1A developed by Farassat (Reference Farassat1981), has been the standard approach in most computer codes devoted to the aeroacoustic (linear) analysis of rotating blades. By accounting for an inertial frame of reference and under the assumptions of (i) fluid at rest, (ii) inviscid flow (which reduces the stress tensor to the scalar pressure field on the blade surface, $\unicode[STIX]{x1D617}_{ij}=(p-p_{0}){\it\delta}_{ij}=\tilde{p}{\it\delta}_{ij}$ ) and (iii) isentropic transformations (for which the pressure–density relationship can be approximated by the linear term of its series expansion, i.e. $p^{\prime }=c_{0}^{2}\tilde{{\it\rho}}$ ), the surface terms of (2.1) can be written in the following form:

(2.2) $$\begin{eqnarray}\displaystyle 4{\rm\pi}p_{T}(\boldsymbol{x},t) & = & \displaystyle \int _{S}\left[\frac{{\it\rho}_{0}\dot{(v_{n})}}{r|1-M_{r}|^{2}}+\frac{{\it\rho}_{0}v_{n}(\dot{M_{i}}\hat{r}_{i})}{r|1-M_{r}|^{3}}\right]_{{\it\tau}}\,\text{d}S\nonumber\\ \displaystyle & & \displaystyle +\,\int _{S}\left[\frac{{\it\rho}_{0}v_{n}c_{0}(M_{r}-M^{2})}{r^{2}|1-M_{r}|^{3}}\right]_{{\it\tau}}\,\text{d}S,\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle 4{\rm\pi}p_{L}(\boldsymbol{x},t) & = & \displaystyle \int _{S}\left[\frac{\dot{\tilde{p}}\cos {\it\theta}+\tilde{p}\dot{\hat{n}}_{i}\hat{r}_{i}}{c_{0}r|1-M_{r}|^{2}}+\frac{\tilde{p}\cos {\it\theta}(\dot{M_{i}}\hat{r}_{i})}{c_{0}r|1-M_{r}|^{3}}\right]_{{\it\tau}}\text{d}S\nonumber\\ \displaystyle & & \displaystyle +\,\int _{S}\left[\frac{\tilde{p}\cos {\it\theta}-\tilde{p}M_{n}}{r^{2}|1-M_{r}|^{2}}+\frac{\tilde{p}\cos {\it\theta}\left(M_{r}-M^{2}\right)}{r^{2}|1-M_{r}|^{3}}\right]_{{\it\tau}}\,\text{d}S,\end{eqnarray}$$

where, as already said, $S$ represents the surface of the body. In this case (and only in this case), the surface terms can be identified as the thickness and loading noise components, with their physical meaning related to the fluid displacement and surface loads due to the body motion, and are usually labelled as the ‘linear’ terms of the FWH equation. Here, $\tilde{p}$ is the gauge pressure on $S$ , $\hat{n}_{i}$ and $\hat{r}_{i}$ represent the components of the unit vectors along the outward normal (with respect to $S$ ) and the radiation directions respectively, $\cos {\it\theta}=\hat{\boldsymbol{n}}\boldsymbol{\cdot }\hat{\boldsymbol{r}}$ , the subscript $r$ indicates the projection along $\boldsymbol{r}$ , $M=|\boldsymbol{v}|/c_{0}$ is the source Mach number and the dot over symbols indicates the time derivative. It is very important to note that all of the kernels are computed at the emission (retarded) time ${\it\tau}=t-r/c_{0}$ . The difference between ${\it\tau}$ and the corresponding observer time $t$ is called the compressibility delay: it represents a essential feature of the propagation mechanisms and points out that sound propagates in the medium at a finite speed. For our purposes, it is also important to note that each of the integral kernels exhibits a far-field and a near-field component, formally depending on the inverse and the inverse-squared of the source-observer distance $r=|\boldsymbol{x}(t)-\boldsymbol{y}({\it\tau})|$ . Henceforth, the phrase ‘FWH surface terms’ (unless otherwise specified) will explicitly refer to the two terms given by (2.2) and (2.3).

3 Thinking about monopoles, dipoles and rotating sources

As already said, the FWH thickness and loading noise components are often associated with some distribution of monopoles and dipoles respectively. Although theoretically correct, however, this correlation is certainly not the best way to describe the behaviour of the surface terms (2.2) and (2.3) when the body motion is rotational, and a different, likewise simple and more appropriate model could be used. As is well known, a monopole is a source radiating sound equally in all directions. It is often represented by a sphere pulsating at a fixed frequency $f_{0}$ , whose radius alternatively expands and contracts, thus generating a mass/volume flux and a spherical omnidirectional wave. Close to the sphere surface (in the near field), the fluid is accelerated from rest and its velocity is in phase quadrature with the pressure wave; very soon, however, this shift disappears and all around (in the far field) the noise decreases following the $1/r$ law ( $r$ being the source-observer distance). This means a $6$ dB reduction at any doubling of the distance. A dipole is usually represented by two close-set monopoles of equal strength but with opposite sign, or, alternatively, a sphere oscillating along a prescribed axis. While one source expands, the other contracts, so that the net flux of mass is zero and the sound is generated by the net force on the fluid, that is a momentum flux. In this case, the sound field has a well-defined directionality (along the dipole axis), but the mutual interactions of the monopoles make the noise decay more pronounced in the proximity of the source; thus, in the near field the sound propagates according to the inverse-squared law $1/r^{2}$ , while maintaining the $1/r$ dependence in the far field. In other words, we can think of the near field like a sort of transition region where the different pressure disturbances progressively mix together, to achieve a single stabilized front radiating thereafter (in the far field) as a ‘plane’ wave. Qualitatively speaking, the near-field spatial dimension is estimated by the wavelength ${\it\lambda}$ of the resulting noise,

(3.1) $$\begin{eqnarray}O_{L_{nf}}=O({\it\lambda})=O(c_{0}T),\end{eqnarray}$$

where $T$ is the signal period. A sketch of these models is shown in figure 1. Figure 1(a) points out the omnidirectionality of the monopole source, while figure 1(b) and figure 1(c) give a schematic representation of the dipole near and far field respectively, as perceived at a fixed measurement point $P$ .

Figure 1.

Schematic representation of the monopole and dipole elemental sources: (a) monopole, (b) dipole near field, (c) dipole far field.

Despite the usual and ideal modelling of the FWH surface terms through these elemental sources, the generating and propagating noise mechanisms concerning a rotating body can be, indeed, very different. The underlying problem is that the theoretical structure of the aforementioned monopole and dipole models is not sufficient to provide an immediate exhaustive description of the acoustic behaviour of a rotating source, and the missing, although essential, elements are the kinematic variables characterizing the rotational motion. First of all, it is interesting to note that even by accounting for a single rotating source, a sort of auto-interaction of kinematic nature occurs and gives rise to a transition region (that is, a near field) in the proximity of the source. In fact, at each position along the circular trajectory, the source induces a pressure disturbance in the surrounding fluid, but while the whole revolution is completed these disturbances interact with each other in a way strictly dependent on the rotational speed ${\it\omega}$ . Intuitively, the interaction is more pronounced at smaller values of ${\it\omega}$ , since the shift between the subsequent disturbances increases. The period appearing in the qualitative assessment of near-field size now corresponds to the revolution period $T=2{\rm\pi}/{\it\omega}$ , and (3.1) can be written as

(3.2) $$\begin{eqnarray}O_{L_{nf}}=O\left(\frac{2{\rm\pi}c_{0}}{{\it\omega}}\right)=O\left(\frac{2{\rm\pi}R}{M_{R}}\right).\end{eqnarray}$$

This relation states that, for a fixed value $R$ of the radius of motion, a source rotating at $M_{R}=0.01$ can give rise to a near field 100 times larger than the near field of the same source rotating at sonic speed. How does the noise behave in this (eventually very large) region? Some suggestions are provided in figure 2, depicting an observer at different distances from the source. The far field (figure 2 a) can be still recognized as the region where the noise decreases according to the $1/r$ law and the observer perceives the rotating source as a point (compact) source. The distance $d$ between this observer and the rotation axis is larger than $2R$ (the size of the circular trajectory), but the far-field size strongly depends on ${\it\omega}$ : the higher the rotational speed, the more compact the source and, consequently, the closer to the source the far field is. When $d$ has a value comparable to $2R$ (figure 2 b), the observer is in a region characterized by the interactions between the pressure disturbances induced by the motion; then, it is reasonable to identify this region as the near field and, like a dipole, to expect a noise decay governed by the inverse-squared law $1/r^{2}$ . Nevertheless, a further reduction of the distance $d$ (figure 2 c) gives rise to a rather anomalous situation.

Figure 2.

Noise propagation mechanisms related to a rotating source: (a)  $d\gg 2R$ , far field, (b) $d\approx O(2R)$ , near field, (c) $d_{min}<2R$ , very near field.

Within a revolution period, in fact, the observer experiences both $d_{min}<2R$ and $d_{max}=d_{min}+2R>2R$ , thus very close to the source (which, moreover, corresponds to the numerical singularity $r=0$ ) and in a position previously identified as a near field. Still, the way in which the sound reaches the observer in such a very near field is strictly related to the rotational motion and, as for the near field of a dipole, we could expect this region to widen with decreasing rate. This configuration, however, points out very well that the ‘structure’ of a rotating source is deeply different with respect to the simple monopole and dipole models depicted in figure 1, and suggests that the whole aforementioned transition region may extend much further out. Furthermore, according to our description of the near field, it is quite hard to believe that, approaching the blade, the noise could always decrease following the $1/r^{2}$ law for any value of ${\it\omega}$ . Thus, the question is how the noise actually behaves in the near field and how this behaviour is affected by the rotational speed. Unfortunately, in the past, research was always focused on the effects of an aeronautical high $M_{R}$ in the far field, while it substantially ignored the effects of a marine low $M_{R}$ in the near field. The literature offers many papers concerning possible analytic solutions for a rotating source, in both the time and frequency domains (Morfey & Tanna Reference Morfey and Tanna1971; Tanna & Morfey Reference Tanna and Morfey1971; Poletti Reference Poletti2010; Poletti & Teal Reference Poletti and Teal2011; Tang et al. Reference Tang, Qi and Mao2013). Most of them, however, refer to a monopole source (equipped with an auto-pulsating frequency $f_{0}$ ), are focused on high values of the rotational speed and, above all, are based on rather complex mathematical manipulations, which make the physics of the underlying phenomena difficult to understand. For this reason, our investigations will be carried out through a practical approach, by directly investigating the behaviour of the FWH surface terms for a single rotating source.

4 The rotpole

In order to characterize the acoustic behaviour of a rotating source and to analyse its own features through the Farassat formulation 1A, it is useful to think of a new elemental source: the rotpole. Like the well-known models concerning the elemental sources, the rotpole ( $\mathscr{R}$ ) can be represented by a small sphere of radius $a$ , located at a distance $R\gg a$ from the $z$ axis of a Cartesian frame of reference $O(xyz)$ and rotating counterclockwise around it (that is, in the $xy$ disk plane) at a fixed rotational speed ${\it\omega}$ . The rotpole, however, is rigid: it has no pulsating motion and thus is able to generate sound only by virtue of its own rotation (then through the displacement of the surrounding fluid) and/or an associated force rotating with it. A sketch of this simple model is shown in figure 3. The main aim of our analysis is to understand how the kinematics of the source affects the sound propagation and, then, the acoustic integrals of (2.2) and (2.3). In particular, the interest is here focused on noise decay (moving far from the source) as a function of the rotational speed.

Figure 3.

The rotpole $\mathscr{R}$ .

It is worth mentioning that this kind of analysis could be performed by modelling the rotpole by a spherical mesh and computing, at different points, the thickness and (with $\tilde{p}$ available) loading noise components. Nevertheless, in order to get to the heart of the problem, we identify the rotpole with a point source and, for each surface term, analyse the behaviour of a kernel function accounting for the only dependence of the corresponding integral on kinematic parameters. It is fundamental to stress that, in this way, the calculations are carried out analytically: they do not require any numerical validation and are free from any link to the shape of the body source or the corresponding aero/hydrodynamic loads. In the following, unless otherwise specified, the calculations will refer to water; thus, the sound speed $c_{0}$ is set to $1500~\text{m}~\text{s}^{-1}$ and the reference pressure for the computation of the sound pressure level (SPL) in dB is $p_{ref}=10^{-6}$ Pa (the usual reference value underwater).

4.1 The rotpole kernel functions

The rotpole kernel functions, indicated by $k(\cdot )$ , may be considered as a practical way to describe the acoustic behaviour of a rotating source and, at the same time, to achieve a deep understanding of the general behaviour of the FWH surface terms. First of all, as we refer to a point source, the coordinates of the rotpole and their subsequent time derivatives (see figure 3) can be simply written in the following form:

(4.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle \mathscr{R}\equiv (R\cos {\it\omega}t,R\sin {\it\omega}t,0), & \displaystyle\end{eqnarray}$$

(4.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}\equiv (-{\it\omega}R\sin {\it\omega}t,+{\it\omega}R\cos {\it\omega}t,0), & \displaystyle\end{eqnarray}$$

(4.1c ) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\boldsymbol{v}}\equiv (-{\it\omega}^{2}R\cos {\it\omega}t,-{\it\omega}^{2}R\sin {\it\omega}t,0). & \displaystyle\end{eqnarray}$$

For our purposes, we remove the integral sign from (2.2) and (2.3) and identify the kernel functions $k(\cdot )$ with the corresponding integral kernels, rid of all terms not directly accounting for the kinematic variables of interest. In particular, we assume here that the strength of the source terms (expressed by $({\it\rho}_{0}\dot{v_{n}})$ , ${\it\rho}_{0}v_{n}$ for thickness and $\dot{\tilde{p}}$ , $\tilde{p}$ for loading components) is responsible of the SPL value at any point, but it does not affect the relationship between the source kinematics and noise propagation at all. Thus, in our definition of the kernel functions, the abovementioned strength is simply ignored (or, in other words, is set to 1). The author is well aware of the poor mathematical strictness of this approach, but its practical advantages will soon be clear. On the other hand, the attention here is not focused on the prediction of noise (it would be quite bizarre to compute a thickness noise from a point source), but rather on the numerical trend of the acoustic integrals, as ruled by the rotational motion. In any case, it will be shown later that the analytical results provided by such a practical approach exactly match the numerical solutions provided by formulation 1A. With this explanation and the previously mentioned assumption concerning $({\it\rho}_{0}\dot{v_{n}})$ , ${\it\rho}_{0}v_{n},\dot{\tilde{p}}$ and $\tilde{p}$ , the rotpole kernel functions can be written in the following form:

(4.2) $$\begin{eqnarray}k_{T}({\it\omega},r,c_{0})=k_{T}^{F}+k_{T}^{N}=\left\{\frac{1}{r}\left[\frac{{\dot{M}}_{i}\hat{r}_{i}}{|1-M_{r}|^{3}}\right]\right\}^{F}+\left\{\frac{1}{r^{2}}\left[\frac{c_{0}\left(M_{r}-M^{2}\right)}{|1-M_{r}|^{3}}\right]\right\}^{N},\end{eqnarray}$$

(4.3) $$\begin{eqnarray}\displaystyle k_{L}({\it\omega},r,c_{0},\hat{\boldsymbol{n}}) & = & \displaystyle k_{L}^{F}+k_{L}^{N}=\left\{\frac{1}{r}\left[\frac{\cos {\it\theta}}{c_{0}|1-M_{r}|^{2}}+\frac{\dot{\hat{n}}_{i}\hat{r}_{i}}{c_{0}|1-M_{r}|^{2}}+\frac{{\dot{M}}_{i}\hat{r}_{i}\cos {\it\theta}}{c_{0}|1-M_{r}|^{3}}\right]\right\}^{F}\nonumber\\ \displaystyle & & \displaystyle +\left\{\frac{1}{r^{2}}\left[\frac{\cos {\it\theta}-M_{n}}{|1-M_{r}|^{2}}+\frac{\cos {\it\theta}\left(M_{r}-M^{2}\right)}{|1-M_{r}|^{3}}\right]\right\}^{N},\end{eqnarray}$$

where the superscripts $F$ and $N$ emphasize the distinction between the far- and near-field components, depending on $1/r$ and $1/r^{2}$ respectively. First of all, note the absence of the term $\dot{(v_{n})}$ of (2.2) in the definition (4.2) of $k_{T}^{F}$ . Actually, many of the integral kernels exhibit the unit vector $\hat{\boldsymbol{n}}$ , locally defining the outward normal direction with respect to the integration domain and rigidly attached to it. By assuming a generic triad of components in a frame of reference in-built with the point source $\hat{\boldsymbol{n}}|^{{\it\eta}}=(n_{1},n_{2},n_{3})$ , satisfying the condition $(n_{1}^{2}+n_{2}^{2}+n_{3}^{3})^{1/2}=1$ , the quantities $\hat{n}_{i}$ appearing in the kernel functions may be obtained by the simple relation

(4.4) $$\begin{eqnarray}\hat{\boldsymbol{n}}=\left[\begin{array}{@{}c@{}}n_{x}\\ n_{y}\\ n_{z}\end{array}\right]=\unicode[STIX]{x1D64F}\;\hat{\boldsymbol{n}}|^{{\it\eta}}=\left[\begin{array}{@{}ccc@{}}\cos {\it\omega}t & -\sin {\it\omega}t & 0\\ \sin {\it\omega}t & \cos {\it\omega}t & 0\\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{@{}c@{}}n_{1}\\ n_{2}\\ n_{3}\end{array}\right],\end{eqnarray}$$

that is

(4.5) $$\begin{eqnarray}\hat{\boldsymbol{n}}=\left(n_{1}\cos {\it\omega}t-n_{2}\sin {\it\omega}t,n_{1}\sin {\it\omega}t+n_{2}\cos {\it\omega}t,n_{3}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D64F}$ is the counterclockwise rotation matrix. In our analysis concerning a point source, as well as in the case of a rigid body for which the only time dependence of the unit vector $\hat{\boldsymbol{n}}$ is related to the rotational motion, (4.5) gives

(4.6) $$\begin{eqnarray}\dot{\hat{\boldsymbol{n}}}=\left[-{\it\omega}\left(n_{1}\sin {\it\omega}t+n_{2}\cos {\it\omega}t\right),{\it\omega}\left(n_{1}\cos {\it\omega}t-n_{2}\sin {\it\omega}t\right),0\right],\end{eqnarray}$$

and it is easy to verify (by (4.1b ), (4.1c ), (4.5) and (4.6)) that $\dot{(v_{n})}=\dot{v}_{i}\hat{n}_{i}+v_{i}\dot{\hat{n}}_{i}$ is equal to zero. In other words, the local shape of a (rigid) body characterizes the strength of the thickness component, but it does not affect how the thickness noise propagates in the field at all. On the contrary, the presence of $\cos {\it\theta}$ , $\dot{\hat{n}}_{i}\hat{r}_{i}$ and $M_{n}$ forces inclusion of the unit normal vector in the list of the independent variables for $k_{L}$ and suggests an actual dependence of the loading noise propagation on the local shape of the body. The effects of $\hat{\boldsymbol{n}}$ on the rotpole loading kernel function will be discussed in § 4.3.2. Other interesting features of $k_{L}$ lie in the presence of the $1/c_{0}$ shrinking factor affecting all terms of $k_{L}^{F}$ and, above all, the nature of the first two ratios, exhibiting the same denominator, but intentionally written in a separate way. The first term (with $\cos {\it\theta}$ in the numerator), in fact, is the only one corresponding to a different source strength ( $\dot{\tilde{p}}$ , as opposed to $\tilde{p}$ ); then, this term does not scale as all the other ones and can significantly alter the weight of the far-field component in the assessment of the overall behaviour of $k_{L}$ . This unsteady part of the loading kernel function could be simply removed by assuming a steady-state condition for the rotpole (like a point of a helicopter rotor in hover or a marine propeller in a uniform flow, for which $\dot{\tilde{p}}=0$ ), but it is important to take it into account since it is known (at least in aeroacoustics) that its contribution can be dominant just at low blade rotational velocities. Then, for completeness, we preserve it in the definition of (4.3) and will deal with its possible effects in the following. At this point, it is fundamental to note that, for fixed $c_{0}$ , ${\it\omega}$ and a measurement point $P(x_{P},y_{P},z_{P})$ , the expressions

(4.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle r=|\boldsymbol{r}|=\sqrt{\left(x_{P}-R\cos {\it\omega}t\right)^{2}+\left(y_{P}-R\sin {\it\omega}t\right)^{2}+z_{P}^{2}}, & \displaystyle\end{eqnarray}$$

(4.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle M_{r}=M_{i}\hat{r}_{i}=\frac{v_{i}r_{i}}{c_{0}r}=\frac{{\it\omega}R}{c_{0}r}\left[-x_{P}\sin {\it\omega}t+y_{P}\cos {\it\omega}t\right], & \displaystyle\end{eqnarray}$$

(4.7c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\dot{M}}_{i}\hat{r}_{i}=\frac{\dot{v}_{i}r_{i}}{c_{0}r}=\frac{1}{c_{0}r}\left[{\it\omega}^{2}R^{2}-{\it\omega}^{2}R\left(x_{P}\cos {\it\omega}t+y_{P}\sin {\it\omega}t\right)\right], & \displaystyle\end{eqnarray}$$

(4.7d ) $$\begin{eqnarray}\displaystyle & \displaystyle M_{n}=\left(\frac{{\it\omega}R}{c_{0}}n_{2}\right), & \displaystyle\end{eqnarray}$$

(4.7e ) $$\begin{eqnarray}\displaystyle & \displaystyle M^{2}=\left(\frac{{\it\omega}R}{c_{0}}\right)^{2}, & \displaystyle\end{eqnarray}$$

(4.7f ) $$\begin{eqnarray}\displaystyle & \displaystyle \cos {\it\theta}=\hat{n}_{i}\hat{r}_{i}=\frac{1}{r}\left[\left(x_{P}n_{1}+y_{P}n_{2}\right)\cos {\it\omega}t+\left(x_{P}n_{2}-y_{P}n_{1}\right)\sin {\it\omega}t-Rn_{1}+z_{P}n_{3}\right], & \displaystyle\end{eqnarray}$$

(4.7g ) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{\hat{n}}_{i}\hat{r}_{i}=\frac{1}{r}\left[-{\it\omega}\left(x_{P}n_{1}+y_{P}n_{2}\right)\sin {\it\omega}t-{\it\omega}\left(x_{P}n_{2}-y_{P}n_{1}\right)\cos {\it\omega}t-{\it\omega}Rn_{2}\right] & \displaystyle\end{eqnarray}$$

enable the evaluation of the far- and near-field components of each kernel function in an analytical form. In this way, they can be easily computed over a revolution period $T=2{\rm\pi}/{\it\omega}$ and turned into the frequency domain by a discrete Fourier transform (DFT) to evaluate the corresponding SPL at $P$ . Then, by changing the measurement point, the SPL decay for $k_{T}$ and $k_{L}$ can be determined as a function of $r$ . In the following, the decay curves will always be compared with the $1/r^{n}$ ‘reference laws’, expressed by the geometric relationship

(4.8) $$\begin{eqnarray}\text{SPL}=\text{SPL}_{0}-20\log \left(\frac{d}{d_{0}}\right)^{n},\end{eqnarray}$$

where SPL and $\text{SPL}_{0}$ represent the pressure levels at two (aligned) points located at distances $d$ and $d_{0}$ respectively from the rotation axis. In particular, $n=1$ and $n=2$ define the well-known inverse ( $1/r$ ) and inverse-squared ( $1/r^{2}$ ) laws, corresponding to an SPL decrease of $6$ dB and $12$ dB at doubled ( $d=2d_{0}$ ) distances.

4.2 Very-near-field effects

Before starting the analysis of noise decay, it is interesting to show how the behaviour of the acoustic integrals may strictly depend on the rotational motion and, hence, require a good deal of caution. As already mentioned in the previous section, a measurement point close to the rotpole (in a region we called the very near field) experiences a remarkable variation of $r$ within the revolution period. Figure 4 shows the rotpole $1/r$ (a) and $1/r^{2}$ (b) curves, computed by (4.7a ) in one revolution period, at 13 subsequent points $P_{i}$ located along the $x$ axis (see figure 3); $R$ is set to $1~\text{m}$ and the distance from the rotation axis starts from $x_{P_{1}}=2~\text{m}$ , increases by $1~\text{m}$ up to the tenth point ( $x_{P_{i}}=x_{P_{i-1}}+1,i=2,3\ldots 10$ ) and then is set to 16, 32 and 64 m. The greater the distance, the more pressed towards the $x$ axis the curves are, especially, of course, for $n=2$ .

Figure 4.

The rotpole (a) $1/r$ and (b) $(1/r^{2})$ functions within a revolution period, at 13 points along the $x$ axis. The point $P_{1}$ is located at only 2 m from the origin.

However, at points closer to the source, the functions exhibit a notable variation during the revolution; in particular, $P_{1}$ experiences a minimum distance from the rotpole $r_{min}=1~\text{m}$ and a maximum one $r_{max}=3~\text{m}$ . The question is: what is, in this case, the behaviour (in dB) of the $1/r$ and $1/r^{2}$ functions? As is well known, the inverse law $1/r$ provides a decrease of $6$ dB at each doubling of the distance. Nevertheless, this is true for a fixed distance, while, in our case, $r$ is a function of time, and the more the point approaches the source, the more it ‘suffers’ this time dependence. Figure 5 shows the $1/r$ and $1/r^{2}$ functions of the rotpole, as computed in dB at the prescribed measurement points (with $z_{P}=0$ ). Both of the curves exhibit an evident discrepancy with respect to the reference decay laws, and in the region between points $P_{1}$ and $P_{2}$ the SPL drop appears to be steeper. From a numerical point of view, the reason for this behaviour is clear: when $O(x_{P},y_{P})\approx O(R)$ , the two terms in brackets of (4.7a ) give rise to a non-negligible fluctuation of $r$ and the curves do not follow the standard decay. By increasing the distance (and then $O(x_{P},y_{P})$ ), the weight of the oscillating terms $R\cos {\it\omega}t$ and $R\sin {\it\omega}t$ in (4.7a ) reduces and the curves soon align to the reference laws. At a first glance, the very-near-field effects can appear to not be so relevant: the gap between the $1/r$ curves soon settles on approximately $1.8$ dB and even reduces by moving out of the disk plane, where the fluctuating terms are also contrasted by the coordinate $z_{P}\neq 0$ ; nevertheless, the gap rises up to $6.2$ dB for $1/r^{2}$ and can become significant (as will be shown later) for higher values of $n$ . In any case, these results reveal in advance how the behaviour of the acoustic kernels may be intimately related to the kinematics of the rotational motion.

Figure 5.

The rotpole $1/r$ and $1/r^{2}$ functions in dB, given by (4.7a ), highlight the discrepancy with respect to the reference lines, corresponding to relation (4.8).

4.3 Noise decay

At this stage, let us start our analysis on the relationship between the rotpole rotational speed and the kernel function decay in the field. To this aim, two measurement lines will be taken into account throughout this section: line $A$ corresponds to the $x$ axis, so as to assess the noise propagation in the plane of motion, while line $B$ is orthogonal to the disk plane and allows us to evaluate the behaviour of the numerical solution in a very important direction (downstream of the propeller, where the blade tip vortex spreads underwater). The rotpole is always located at a distance $R=1~\text{m}$ from the rotation axis, corresponding to the $z$ axis of the reference. Each line includes 10 ordered points, starting from the disk plane points $A_{1}\equiv B_{1}\equiv (2,0,0)$ ; along line $A$ , the distance of each point from the origin is doubled, while along line $B$ , the doubling refers to the distance from the disk plane. The coordinates of all measurement points are listed in table 1, for clarity, with $y_{A_{i}}=z_{A_{i}}=y_{B_{i}}=0$ and $x_{B_{i}}=2$ for $i=1,2\ldots 10$ , and a sketch of the measurement lines is depicted in figure 6.

Table 1.

Coordinates of measurement points along lines $A$ and $B$ .

Figure 6.

Sketch of the measurement lines $A$ and $B$ .

4.3.1 Thickness kernel function

The thickness kernel functions in dB at points $A_{i}$ for a rotpole at rotational Mach numbers of $M_{R}=0.8$ and $M_{R}=0.4$ are shown in figure 7. Each panel shows the far- and near-field components separately, the overall SPL for $k_{T}$ and the reference decay laws of interest. At $M_{R}=0.8$ (figure 7 a), the far-field component is dominant: it fully overlaps the overall SPL and is very close to the $1/r$ reference law, although exhibiting a small gap due to the very-near-field effects, here affecting the region between points $A_{1}$ and $A_{2}$ . In practice, at this value of the rotational speed, the near-field component follows, as expected, the $1/r^{2}$ law, but it does not contribute at all to the resulting SPL, and the noise level decays according to $1/r$ everywhere in the field. By halving the rotational speed ( $M_{R}=0.4$ , figure 7 b), the near-field component becomes significant in the proximity of the source. There, in fact, the overall SPL at $A_{1}$ is dictated by the $k_{T}^{N}$ component and the resulting decay curve clearly appears as the envelope of the two contributions. Nevertheless, the far-field component soon gets the upper hand, and at point $A_{3}$ (at only $8~\text{m}$ from the origin) the overall decay is already realigned to the $1/r$ reference line. Then, in spite of a notable drop of the noise level due to the lower rotational speed (compared with $M_{R}=0.8$ , at $A_{1}$ the overall SPL reduces from 194 to 168 dB), there is no remarkable qualitative change of the acoustic field, which remains deeply characterized by the $1/r$ law. It is worth pointing out that the rotational Mach numbers taken into account here represent two standard values for aeronautical devices, and our analysis of noise propagation in the field, based on the kernel functions, does not add any new information on the well-known behaviour of the thickness noise component: as expected, within this range of the rotational velocity and in the plane of motion, this surface term behaves in a way very similar to a monopole. Nevertheless, by decreasing the blade rate, the general trend of the decay curves progressively changes.

Figure 7.

Rotpole thickness kernel function decay in the disk plane (measurement line $A$ ) at $M_{R}=0.8$ (a) and $M_{R}=0.4$ (b).

Figure 8(a) shows the rotpole $k_{T}$ function at $M_{R}=0.08$ , that is ten times slower with respect to the first test case. At this value of the velocity, as suggested by relation (3.2), the near field is more extended and the $k_{T}^{N}$ component dominates up to point $A_{4}$ , while the overall level starts to follow the inverse law only at point $A_{6}$ , at a distance of 64 m from the origin. This situation corresponds to a more considerable change of the acoustic field, with a constant gap of approximately $20$ dB between the reference curve $1/r$ and the thickness kernel function in the far field. However, at $M_{R}=0.01$ , finally representing a more realistic value for a marine propeller, the SPL drop becomes really significant. The results are shown in figure 8(b). Here, the dominance of the near-field component covers a distance from the origin up to 128 m (point $A_{7}$ ), and the governing $1/r^{2}$ decay law postpones the alignment of the overall SPL to the $1/r$ law at point $A_{9}$ , located half a kilometre from the rotpole. Furthermore, at such a long distance from the source, the noise level has already experienced a drop of approximately $40$ dB with respect to the value dictated by (4.8) with $n=1$ .

Figure 8.

Rotpole thickness kernel function decay in the disk plane (measurement line $A$ ) at $M_{R}=0.08$ (a) and $M_{R}=0.01$ (b).

Due to the way in which we have defined the kernel functions, this fall of the SPL represents an intrinsic feature of the rotpole behaviour, just ruled by the kinematics of the rotational motion and, in particular, the low value of $M_{R}$ . Moving out of the disk plane ( $z_{P}\neq 0$ ), the behaviour of the kernel function becomes more interesting.

Figure 9.

Plot of $M_{r}$ at points $A_{i}$ (disk plane, (a)) and $B_{i}$ (out of the disk plane, (b)).

Figure 10.

Rotpole thickness kernel function decay out of the disk plane (measurement line $B$ ) at $M_{R}=0.8$ (a) and $M_{R}=0.4$ (b).

The far- and near-field components of (4.2) include the terms ${\dot{M}}_{i}\hat{r}_{i}$ and $M_{r}$ (expressions (4.7b ) and (4.7c )). Within the disk plane, these terms exhibit a dependence on $r$ strictly related to the rotational motion, and the $r$ appearing in the denominator of (4.7a ) (where now $z_{P}=0$ ) is fully balanced by the terms in brackets in the corresponding numerator; thus, both ${\dot{M}}_{i}\hat{r}_{i}$ and $M_{r}$ obviously exhibit a fluctuating waveform in the revolution period, but do not change with $r$ along the measurement line $A$ .

Figure 11.

Rotpole thickness kernel function decay out of the disk plane (measurement line $B$ ) at $M_{R}=0.08$ (a) and $M_{R}=0.01$ (b).

On the contrary, out of the disk plane $z_{P}$ is not zero and expressions (4.7b ) and (4.7c ) reveal, due to $r$ in the denominator, a dependence on the inverse law for ${\dot{M}}_{i}\hat{r}_{i}$ and $M_{r}$ . Such a behaviour, of course, comes from the two-dimensional nature of the rotational motion of the source and can be easily verified by numerically solving the previously mentioned equations. This solution is depicted in figure 9, showing the $M_{r}$ term at points $A_{i}$ (a) and $B_{i}$ (b), by increasing $i=1,2\ldots \,$ in one period. The additional $1/r$ dependence of ${\dot{M}}_{i}\hat{r}_{i}$ and $M_{r}$ suggests that (4.2) should be rewritten in the form

(4.9) $$\begin{eqnarray}k_{T}({\it\omega},r,c_{0})=\left\{\frac{1}{r^{2}}\left[\frac{{\dot{M}}_{i}r_{i}-c_{0}M^{2}}{|1-M_{r}|^{3}}\right]\right\}^{F}+\left\{\frac{1}{r^{3}}\left[\frac{v_{i}r_{i}}{|1-M_{r}|^{3}}\right]\right\}^{N},\end{eqnarray}$$

so as to point out that the kernel function $k_{T}$ is now characterized by a far-field term depending on $1/r^{2}$ and a near-field one depending on $1/r^{3}$ . In other words, out of the rotational plane the acoustic integrals combine in a different way to provide the far- and near-field components, and the thickness source term never behaves like a monopole (ruled by the inverse law), not even in the far field. This fact has important consequences, especially at a low rotational speed. (It should be noted that $M_{r}$ also appears in the denominator of all integral kernels, but in a way that does not alter the $r$ dependence of the kernel itself. The Doppler factor $|1-M_{r}|$ , in fact, oscillates as in figure 9(a) in the plane of motion, while out of that plane it simply tends to 1, when $M_{r}\rightarrow 0$ by increasing $r$ .) Figure 10 shows the thickness kernel functions on measurement line $B$ for a rotpole at $M_{R}=0.8$ (a) and $M_{R}=0.4$ (b). The curves confirm the new dependence of $k_{T}^{F}$ and $k_{T}^{N}$ on $r$ and that, as for points $A_{i}$ at such values of the rotational speed, the near-field component does not provide any significant contribution to the overall SPL. In essence, the $k_{T}$ function varies as $1/r^{2}$ in the whole field, thus confirming, as expected, that the thickness term decreases more rapidly than in the disk plane. Nevertheless, once again, by reducing the rotational speed to more realistic values for marine devices, the situation changes significantly. Figure 11 depicts the $k_{T}$ decay curves along line $B$ , at $M_{R}=0.08$ (a) and $M_{R}=0.01$ (b). At these values of the Mach number, the near-field size progressively increases and the overall SPL decay is dominated by $k_{T}^{N}$ in a large part of the field. In particular, at $M_{R}=0.08$ the near-field component governs the overall SPL (thus imposing a decay ruled by the $1/r^{3}$ law) up to point $B_{5}$ , while at $M_{R}=0.01$ the far-field component regains a relevant role only at point $B_{8}$ , at 128 m from the disk plane. However, at such a blade rate, by the time the overall SPL realigns to the $1/r^{2}$ law, the kernel function $k_{T}$ is rapidly going to zero.

4.3.2 Loading kernel function

The behaviour of the kernel function for the loading source term exhibits many analogies with $k_{T}$ , since the parameters governing the relationship between the noise decay and the source kinematics are essentially the same. Nevertheless, $k_{L}$ includes a strict dependence on the unit normal vector $\hat{\boldsymbol{n}}$ and the additional elements $M_{n}=v_{i}\hat{n}_{i}/c_{0}$ , $\cos {\it\theta}=\hat{n}_{i}\hat{r}_{i}$ and $\dot{\hat{n}}_{i}\hat{r}_{i}$ , which make the situation a little more complicated. The projection of the Mach number along the normal direction $M_{n}$ does not include any reference to the source-observer distance; it only depends on the local shape of the body source and the rotational speed, so it does not affect the $r$ dependence of the kernel function on its own. On the contrary, $\cos {\it\theta}$ and $\dot{\hat{n}}_{i}\hat{r}_{i}$ can behave in a manner very similar to ${\dot{M}}_{i}\hat{r}_{i}$ and $M_{r}$ , and, then, alter the $k_{L}$ decay with the $r$ distance. In order to analyse the possible relationships and to assess the behaviour of $k_{L}$ , we have to fix an arbitrary unit vector $\hat{\boldsymbol{n}}$ (and then the triad $\hat{\boldsymbol{n}}|^{{\it\eta}}=(n_{1},n_{2},n_{3})$ appearing in (4.4)), which, theoretically speaking, corresponds to defining an infinitesimal ‘oriented’ surface around the point source. Without loss of generality, we will account for three limiting cases: (i)  $\hat{\boldsymbol{n}}|_{1}^{{\it\eta}}=(1,0,0)$ , corresponding to a unit normal vector in the disk plane, well representing a point at the tip of a rotating blade; (ii)  $\hat{\boldsymbol{n}}|_{2}^{{\it\eta}}=(0,1,0)$ , a vector still located in the rotational plane, but parallel to the rotpole velocity, simulating a point at blade leading edge; (iii)  $\hat{\boldsymbol{n}}|_{3}^{{\it\eta}}=(0,0,1)$ , that is a vector orthogonal to the disk and corresponding, more or less, to a generic point on the suction side of the blade. A sketch of these vectors is shown in figure 12, with the corresponding $O(xyz)$ components to be used in the computation of the $k_{L}$ function. By using (4.1b ), (4.7a ), (4.5) and (4.6) it is possible to write down an analytical form for each of the abovementioned parameters and to analyse their behaviour as a function of $r$ . Nevertheless, this result can also be achieved by simply reasoning about the orientation of the vector $\hat{\boldsymbol{n}}$ and the 2D nature of the rotational motion.

Figure 12.

The unit normal vectors $\hat{\boldsymbol{n}}$ used to analyse the behaviour of the $k_{L}$ function.

As already said, if we assume the rotpole to be a point of a rigid body, the unit normal vector is a time function just by virtue of the rotation; thus, the 2D nature of the rotational motion ( $v_{z}=0$ ) always cancels the corresponding (third) component of $\dot{\hat{\boldsymbol{n}}}$ , independently of $\hat{\boldsymbol{n}}$ (see (4.6)). This means that, like ${\dot{M}}_{i}\hat{r}_{i}$ and $M_{r}$ , the term $\dot{\hat{n}}_{i}\hat{r}_{i}$ (given by (4.7g )) does not change with $r$ in the disk plane ( $z_{P}=0$ ), while it introduces an additional $1/r$ dependence on the kernel function out of the plane of motion ( $z_{P}\neq 0$ ). Concerning $\cos {\it\theta}=\hat{n}_{i}\hat{r}_{i}=(\hat{\boldsymbol{n}}\boldsymbol{\cdot }\boldsymbol{r})/r$ , given by (4.7f ), at measurement points located in the plane of rotation ( $z_{P}=0$ ), the scalar product in the numerator always balances the $r$ appearing in the denominator; thus, in the disk plane, also $\cos {\it\theta}$ does not change with $r$ . On the contrary, out of that plane the behaviour depends on $\hat{\boldsymbol{n}}$ , and two different cases may occur: (i) if the unit normal vector lies in the disk plane ( $n_{3}=0$ , as for $\hat{\boldsymbol{n}}_{1}$ and $\hat{\boldsymbol{n}}_{2}$ ), the scalar product cancels the $z_{P}$ term in the numerator and $\cos {\it\theta}$ introduces a further $1/r$ dependence in $k_{L}$ ; (ii) if the unit normal vector does not lie in the disk plane ( $n_{3}\neq 0$ , as for $\hat{\boldsymbol{n}}_{3}$ ), the numerator fully balances the source-observer distance in the denominator and $\cos {\it\theta}$ will provide a constant value moving further and further away from the source. For clarity, some examples are explicitly shown in figure 13, where the arrows indicate the increasing order ( $i=1,2,\ldots \,$ ) of measurement points, as listed in table 1. Figure 13(a,b) shows the function $\dot{\hat{n}}_{i}\hat{r}_{i}$ as determined at measurement points $A_{i}$ located in the plane of motion (a) and at points $B_{i}$ out of the disk plane (b), by setting $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{1}$ . The function, of course, oscillates in the revolution period because of the rotation, but in the first case this fluctuation is not altered by the value of the source-observer distance, while in the second case it tends to vanish moving away from the source. Analogously, the function $\cos {\it\theta}=\hat{n}_{i}\hat{r}_{i}$ never changes with $r$ in the disk plane (as shown in figure 13(c,d), referred to $\hat{\boldsymbol{n}}_{1}$ in (c) and $\hat{\boldsymbol{n}}_{2}$ in (d)).

Figure 13.

Plots of $\dot{\hat{n}}_{i}\hat{r}_{i}$ and $\cos {\it\theta}$ at points $A_{i}$ and $B_{i}$ (see table 1), for different $\hat{\boldsymbol{n}}$ : (a) $\dot{\hat{n}}_{1}\boldsymbol{\cdot }\hat{\boldsymbol{r}}$ at $A_{i}$ (disk plane), (b) $\dot{\hat{n}}_{1}\boldsymbol{\cdot }\hat{\boldsymbol{r}}$ at $B_{i}$ (out of disk plane), (c) $\cos {\it\theta}=\hat{n}_{1}\boldsymbol{\cdot }\hat{\boldsymbol{r}}$ at $A_{i}$ (disk plane), (d) $\cos {\it\theta}=\hat{n}_{2}\boldsymbol{\cdot }\hat{\boldsymbol{r}}$ at $A_{i}$ (disk plane), (e) $\cos {\it\theta}=\hat{n}_{2}\boldsymbol{\cdot }\hat{\boldsymbol{r}}$ at $B_{i}$ (out of disk plane), (f) $\cos {\it\theta}=\hat{n}_{3}\boldsymbol{\cdot }\hat{\boldsymbol{r}}$ at $B_{i}$ (out of disk plane).

On the contrary, out of the plane of motion the same function exhibits a $1/r$ dependence when the unit normal vector has a $z$ -component equal to zero (as for $\hat{\boldsymbol{n}}_{2}$ , figure 13 e) or, alternatively, it does not change with $r$ when this component is not zero (as for $\hat{\boldsymbol{n}}_{3}$ , figure 13 f). Here, in particular, $\hat{\boldsymbol{n}}_{3}$ is just orthogonal to the disk plane, as the measurement line $B$ ; then, the function ‘feels’ the rotation in the proximity of the source, but at the furthest points $B_{i}$ it tends to assume the value 1, since the vectors $\hat{\boldsymbol{n}}$ and $\hat{\boldsymbol{r}}$ tend to be parallel ( ${\it\theta}$ tends to zero).

Figure 14.

Rotpole loading kernel function decay in the disk plane (line $A$ ) at $M_{R}=0.8$ for $\hat{\boldsymbol{n}}_{1}$ (a) and $\hat{\boldsymbol{n}}_{2}$ (b).

Coming back to the loading kernel function, it is quite clear now that (4.3) represents the $k_{L}$ function just at points located in the plane of rotation, where the loading term actually exhibits a far-field component depending on $1/r$ and a near-field one following the $1/r^{2}$ decay law. Figure 14 depicts the numerical solutions of this equation along the measurement line $A$ (disk plane), for $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{1}$ (a) and $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{2}$ (b) at $M_{R}=0.8$ . At such a high value of the rotational speed, the contribution from the near-field component seems to be non-negligible only for $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{2}$ and very close to the source: in general, the whole acoustic field is dominated by the far-field term and the overall SPL soon aligns to the inverse law $1/r$ . This behaviour is very similar to the function $k_{T}$ shown in figure 7(a), and the notable gap in the corresponding SPL values confirms that, in the plane of motion and at this blade rate, the thickness noise component can be dominant, depending, of course, on the actual operating conditions and the role played by the ‘unsteady’ source strength $\dot{\tilde{p}}$ .

Figure 15.

Rotpole loading kernel function decay in the disk plane (line $A$ ) at $M_{R}=0.01$ for $\hat{\boldsymbol{n}}_{1}$ (a) and $\hat{\boldsymbol{n}}_{2}$ (b).

In any case, by reducing the rotational speed to a realistic value for marine devices ( $M_{R}=0.01$ , figure 15), the general trend of the SPL decay curve changes again: the near-field term becomes dominant for an extended region and gives rise to a much more relevant drop of the overall noise level. It is worth noting that, for both $\hat{\boldsymbol{n}}_{1}$ and $\hat{\boldsymbol{n}}_{2}$ , a full realignment with the $1/r$ law is reached just at point $A_{9}$ , at half a kilometre from the rotpole, and that for $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{3}$ the $k_{L}$ function (in the disk plane) is zero. As for $k_{T}$ , moving out of the plane of motion, the behaviour of $k_{L}$ becomes more intriguing. Whether or not the normal vector $\hat{\boldsymbol{n}}$ lies in the disk plane, both $\cos {\it\theta}$ and $\dot{\hat{n}}_{i}\hat{r}_{i}$ (as well as ${\dot{M}}_{i}\hat{r}_{i}$ and $M_{r}$ ) introduce an additional $1/r$ dependence in the integral kernels on the loading component; then, (4.3) should be (more correctly) written in the following form:

(4.10) $$\begin{eqnarray}\displaystyle k_{L}({\it\omega},r,c_{0},\hat{\boldsymbol{n}}) & = & \displaystyle \left\{\frac{1}{r^{2}}\left[\frac{\hat{n}_{i}r_{i}}{c_{0}|1-M_{r}|^{2}}+\frac{\dot{\hat{n}}_{i}r_{i}-c_{0}M_{n}}{c_{0}|1-M_{r}|^{2}}\right]\right\}^{F}\nonumber\\ \displaystyle & & \displaystyle +\,\left\{\frac{1}{r^{3}}\left[\frac{\hat{n}_{i}r_{i}}{|1-M_{r}|^{2}}+\frac{({\dot{M}}_{i}r_{i})\hat{n}_{i}r_{i}}{c_{0}|1-M_{r}|^{3}}-\frac{\hat{n}_{i}r_{i}M^{2}}{|1-M_{r}|^{3}}\right]\right\}^{N}\nonumber\\ \displaystyle & & \displaystyle +\,\left\{\frac{1}{r^{4}}\left[\frac{\hat{n}_{i}r_{i}(M_{i}r_{i})}{|1-M_{r}|^{3}}\right]\right\}^{VN}.\end{eqnarray}$$

Equation (4.10) states that, moving far from the source and out of the disk plane, the loading noise can propagate along three separate components, which follow the $1/r^{2}$ (far-field), $1/r^{3}$ (near-field) and $1/r^{4}$ (very-near-field) decay laws. These components are shown in figure 16, where, for brevity, only the normal vectors $\hat{\boldsymbol{n}}_{1}$ at $M_{R}=0.8$ (a) and $\hat{\boldsymbol{n}}_{2}$ at $M_{R}=0.01$ (b) are shown. The curves show the perfect alignment of the different terms with the abovementioned reference laws, the negligible contribution always provided by the very-near-field term and still (and above all) that accounting for the lowest rotational speed imposes the near-field component as the dominant one for a large part of the whole acoustic field. In this case, this means a drop of the overall SPL ruled by the very stringent $1/r^{3}$ law, so that when the overall SPL finally realigns to the inverse law (more or less at point $B_{9}$ , at 256 m from the disk plane), the kernel function is very close to zero.

By accounting for $\hat{\boldsymbol{n}}_{3}$ ( $n_{z}\neq 0$ ), $\cos {\it\theta}$ does not change moving far from the rotpole; in this case, the $k_{L}$ function should be written in the further alternative form

(4.11) $$\begin{eqnarray}\displaystyle k_{L}({\it\omega},r,c_{0},\hat{\boldsymbol{n}}) & = & \displaystyle \left\{\frac{1}{r}\left[\frac{\cos {\it\theta}}{c_{0}|1-M_{r}|^{2}}\right]\right\}^{F}\nonumber\\ \displaystyle & & \displaystyle +\,\left\{\frac{1}{r^{2}}\left[\frac{\dot{\hat{n}}_{i}r_{i}+c_{0}\left(\cos {\it\theta}-M_{n}\right)}{c_{0}|1-M_{r}|^{2}}+\frac{({\dot{M}}_{i}r_{i})\cos {\it\theta}}{c_{0}|1-M_{r}|^{3}}-\frac{M^{2}\cos {\it\theta}}{|1-M_{r}|^{3}}\right]\right\}^{N}\nonumber\\ \displaystyle & & \displaystyle +\,\left\{\frac{1}{r^{3}}\left[\frac{\cos {\it\theta}(M_{i}r_{i})}{|1-M_{r}|^{3}}\right]\right\}^{VN},\end{eqnarray}$$

so as to point out the coexistence of three separate components, now following the $1/r$ , $1/r^{2}$ and $1/r^{3}$ decay laws. It should be noted that for $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{3}$ the loading kernel function is zero at the first point $B_{1}$ ( $M_{n}=n_{1}=n_{2}=z_{P}=0$ ), and the decay curves start from point $B_{2}$ , as depicted in figure 17, showing the SPL of the corresponding $k_{L}$ function at $M_{R}=0.8$ (a) and $M_{R}=0.01$ (a). In this case, the near-field component dominates the whole field, independently of the rotational Mach number, while both the very-near- and far-field terms seem to be completely negligible; however, as already mentioned, the role played by the far-field component could be altered significantly by the unsteadiness of the problem and the kernel characterized by the strength $\dot{\tilde{p}}$ . At this stage, regardless of the local orientation of the normal vector and the steady or unsteady nature of the underlying aero/hydrodynamic problem, it is very important to stress the fundamental result achieved by our analysis: in the presence of a rotational motion, the noise propagation mechanisms are ruled by the rotational speed.

Figure 16.

Rotpole loading kernel function decay out of the disk plane (line $B$ ) at $M_{R}=0.8$ for $\hat{\boldsymbol{n}}_{1}$ (a) and $M_{R}=0.01$ for $\hat{\boldsymbol{n}}_{2}$ (b).

This result (i) represents an essential feature of a rotating source, (ii) heavily affects its behaviour in the near field and (iii) is hardly recognizable through the theoretical models of monopoles and dipoles. The relationship between the propagation mechanisms and ${\it\omega}$ can only be of kinematic nature and due to the source auto-interactions occurring within the revolution. In the usual range of velocities for aeronautical devices, these interactions rapidly damp down: the near field is spatially very limited and the wave fronts generated by the body surface (that is, the thickness and loading noise components) soon stabilize and dominate the far field. On the contrary, in the range of velocities for a marine propeller, the near field becomes much larger and the same interactions increase significantly, giving rise to a more pronounced and rapid drop of noise. Such a behaviour is caused by the dominance of noise components dictated by more stringent decay laws compared with $1/r$ , and proves the destructive nature of the abovementioned auto-interactions, as for dipoles and quadrupoles, compared with monopoles. This result suggests the existence of a sharp-cutting borderline between the aeroacoustic and hydroacoustic analysis of rotating blades, especially when (as will be shown in § 5) different blades rotate simultaneously.

Figure 17.

Rotpole loading kernel function decay out of the disk plane (line $B$ ) at $M_{R}=0.8$ (a) and $M_{R}=0.01$ for $\hat{\boldsymbol{n}}_{3}$ (b).

4.4 Kernel function maps

Some additional interesting features of the rotpole kernel functions can be found by computing the maps of the overall SPL and assessing the corresponding directivity in different planes. Figure 18 shows the dB isocurves of the $k_{T}$ SPL in the disk plane, at $M_{R}=0.8$ (a) and $M_{R}=0.01$ (b), determined on a polar mesh of $200\times 80$ measurement points, covering an area of $100R$ . As expected, in this plane the thickness integral kernel always exhibits a radial directivity and, then, a uniform decay in all directions. Nevertheless, the drop of the SPL is very different in the two cases, following the inverse and the inverse-squared laws at the highest and the lowest rotational speed respectively. This can be easily verified through the different range of values (see the legends of figure 18 a,b) or, alternatively, the comparison between figures 7(a) and 8(b). The radial directivity just arises from the symmetry of the motion in its own plane and represents, de facto, the only affinity between the rotpole thickness term and a monopole source.

Figure 18.

Thickness kernel function maps in the $xy$ (disk) plane for a rotpole at $M_{R}=0.8$ (a) and $M_{R}=0.01$ (b).

Figure 19.

Thickness kernel function maps in the $xz$ plane (orthogonal to disk plane) for a rotpole at $M_{R}=0.8$ (a) and $M_{R}=0.01$ (b).

In fact, by moving out of the disk, the $k_{T}$ directivity abruptly changes and exhibits the typical two-lobed structure of a dipole source, as shown in the two $xz$ maps of figure 19, still depicted with a different scale because of the notable variation of the SPL range. It can be noted how, at the highest value of the rotational speed, the $k_{T}$ SPL isocurves are more pressed towards the disk plane, while at $M_{R}=0.01$ the lobes appear to be more circular and uniformly distributed. The SPL maps of the loading kernel function in the disk plane are, in essence, identical to the ones of $k_{T}$ (figure 18) and are not reported. On the contrary, it is interesting to look at the $k_{L}$ maps in the $xz$ plane, depicted in figure 20, still for $M_{R}=0.8$ (a) and $M_{R}=0.01$ (b), but referred to a different choice of the unit normal vector, ( $\hat{\boldsymbol{n}}_{1}$ and $\hat{\boldsymbol{n}}_{3}$ in (a,b) and (c,d) respectively). When the unit vector lies in the plane of motion, the $k_{L}$ contour map at $M_{R}=0.8$ shows the expected two-lobed structure of a dipole, which (as for $k_{T}$ ) becomes more uniform by decreasing the rotational speed. Nevertheless, for a unit vector orthogonal to the plane of motion, the same function at the highest speed exhibits a typical quadrupolar structure, which turns back into a two-lobed shape at $M_{R}=0.01$ with the dipole axis now orthogonal to the disk plane.

Figure 20.

Loading kernel function maps in the $xz$ (out of disk) plane for a rotpole at $M_{R}=0.8$ (a,c) and $M_{R}=0.01$ (b,d) for (a,b) $\hat{\boldsymbol{n}}_{1}$ and (c,d) $\hat{\boldsymbol{n}}_{3}$ .

Looking at these last results, it is evident how both the rotational speed and the normal vector may have a deep influence on the directivity of the loading noise component. For this reason, we accounted for an additional more generic vector $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{4}$ , by using the triad $\hat{\boldsymbol{n}}|^{{\it\eta}}=(\sqrt{2}/4,\sqrt{2}/4,\sqrt{3}/2)$ ; this choice corresponds to a vector making an angle ${\it\beta}=60^{\circ }$ with the $xy$ plane and an angle ${\it\gamma}=45^{\circ }$ with the local ${\it\eta}_{1}$ and ${\it\eta}_{2}$ axes, so that in the main reference $O(xyz)$ its components become $\hat{\boldsymbol{n}}_{4}=(\sqrt{2}/4\cos {\it\omega}t-\sqrt{2}/4\sin {\it\omega}t,\sqrt{2}/4\sin {\it\omega}t+\sqrt{2}/4\cos {\it\omega}t,\sqrt{3}/2)$ . The $k_{L}$ SPL maps corresponding to $\hat{\boldsymbol{n}}_{4}$ are depicted in figure 21, at four different values of the rotational Mach number and now using the same range for SPL values. The main dipole nature of an aeronautical $k_{L}$ out of the disk plane is confirmed by the two lobes occurring at $M_{R}=0.8$ (figure 21 a), although the map shows a slight asymmetry along the ‘dominant’ vertical direction of the fixed vector $\hat{\boldsymbol{n}}_{4}$ . By decreasing the speed to $M_{R}=0.4$ (figure 21 b) this asymmetry becomes more evident, while a further reduction of the Mach number ( $M_{R}=0.08$ , figure 21 c) progressively reduces the two-lobed shape of the SPL contour map. Finally, at $M_{R}=0.01$ , the map confirms the impressive drop of the overall SPL and shows, for a marine $k_{L}$ , a structure similar to an elongated monopole source. It is worth pointing out that these maps just refer to the $k_{L}$ function of a single rotpole, equipped with an artificially imposed $\hat{\boldsymbol{n}}$ , and in a fixed plane; they do not represent the 3D directivity of the loading noise due to a general blade, whose trend is certainly affected by the shape of the body too. Nevertheless, they emphasize very well how the acoustic behaviour of a rotating source strongly depends on the rotational speed, not only in terms of intensity (as previously shown through the noise decay curves), but also in terms of directivity.

Figure 21.

Loading kernel function maps in the $xz$ plane for a rotpole equipped with $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{4}$ , at $M_{R}=0.8$ (a), $M_{R}=0.4$ (b), $M_{R}=0.04$ (c) and $M_{R}=0.01$ (d).

5 Multiple rotpoles

The analysis of the kernel functions for a single rotpole has shown how the SPL decay notably increases at a low (marine) rotational speed, and that this decay is dictated by the mutual interactions of the disturbances generated by the source along its own circular trajectory. For this reason, rather than a generic interaction, we should talk of a destructive interference. Then, a very interesting question arises: what would happen if such an interference were directly imposed by the presence of one (or more) additional rotpole(s)? As is well known, the elemental dipole and quadrupole sources can be represented through a suitable combination of monopoles, and their lower acoustic efficiency is just caused by the mutual interaction of the single elements. The situation, in this case, is very similar, and a definite answer to the above question can be achieved by accounting for rotpoles of $n$ th order, where $n$ (spatially shifted) sources rotate simultaneously. In particular, just like dipoles and quadrupoles, we will assess here the behaviour of a dirotpole and a quadrurotpole, constituted by two and four rotpoles with the same direction of rotation (analogously to what happens in a multibladed propeller), shifted by ${\it\beta}_{d}=j{\rm\pi}$ (with $j=0,1$ ) and ${\it\beta}_{q}=j{\rm\pi}/2$ (with $j=0,1,2,3$ ) respectively. A sketch of these multiple rotating sources is shown in figure 22, where ${\it\psi}_{j}={\it\omega}t+{\it\beta}_{j}$ indicates the azimuth angle of the $j\text{th}$  rotpole.

Figure 22.

The dirotpole (a) and the quadrurotpole (b) sources.

From a numerical point of view, the behaviour of a rotpole of $n$ th order can still be examined through the kernel functions, by adding up the contributions from the single sources; then, (4.2) and (4.3) are replaced by

(5.1) $$\begin{eqnarray}\left\{k_{T}\right\}^{n}({\it\omega},r_{j},c_{0})=\mathop{\sum }_{j=1}^{n}\left\{\frac{1}{r}\left[\frac{{\dot{M}}_{i}\hat{r}_{i}}{|1-M_{r}|^{3}}\right]\right\}_{j}^{F}+\left\{\frac{1}{r^{2}}\left[\frac{c_{0}\left(M_{r}-M^{2}\right)}{|1-M_{r}|^{3}}\right]\right\}_{j}^{N},\end{eqnarray}$$

(5.2) $$\begin{eqnarray}\displaystyle \left\{k_{L}\right\}^{n}({\it\omega},r_{j},c_{0},\hat{\boldsymbol{n}}_{j}) & = & \displaystyle \mathop{\sum }_{j=1}^{n}\left\{\frac{1}{r}\left[\frac{\cos {\it\theta}}{c_{0}|1-M_{r}|^{2}}+\frac{\dot{\hat{n}}_{i}\hat{r}_{i}}{c_{0}|1-M_{r}|^{2}}+\frac{{\dot{M}}_{i}\hat{r}_{i}\cos {\it\theta}}{c_{0}|1-M_{r}|^{3}}\right]\right\}_{j}^{F}\nonumber\\ \displaystyle & & \displaystyle +\,\left\{\frac{1}{r^{2}}\left[\frac{\cos {\it\theta}-M_{n}}{|1-M_{r}|^{2}}+\frac{c_{0}\left(M_{r}-M^{2}\right)}{|1-M_{r}|^{3}}\right]\right\}_{j}^{N},\end{eqnarray}$$

where the coordinates of $j$ th rotpole, of course, are

(5.3) $$\begin{eqnarray}\mathscr{R}_{j}\equiv \{R\cos ({\it\omega}t+{\it\beta}_{j}),R\sin ({\it\omega}t+{\it\beta}_{j}),0\}\end{eqnarray}$$

and (4.7) has to be changed accordingly. It is worth pointing out that, concerning $\{k_{L}\}^{n}$ , the single elements of the multiple source are associated with a unique choice for $\hat{\boldsymbol{n}}$ , and, in the following, the SPL of the kernel functions will be determined by explicitly removing the possible mean value in (5.1), (5.2), so as to point out the effects of the abovementioned destructive interference on the pure fluctuating ‘tonal’ component of the integral terms (which, acoustically speaking, is the only relevant one).

Let us start the analysis by taking into account the typical Mach numbers of an aeronautical device and the thickness kernel function. Figure 23 shows the comparison between the $\{k_{T}\}^{n}$ noise decay curves, determined by (5.1) along measurement line $A$ (disk plane), for a single rotpole ( $n=1$ ), a dirotpole ( $n=2$ ) and a quadrurotpole ( $n=4$ ), at $M_{R}=0.8$ (a) and $M_{R}=0.4$ (b). At the highest value of the velocity, no particular interaction occurs between the single sources: as expected, all the curves soon align to the inverse law $1/r$ , and the presence of additional sources actually gives rise to a slight increase of the overall SPL. At $M_{R}=0.4$ , however, the general trend changes. The rotpole and dirotpole curves, in fact, now perfectly overlap, while the noise decay of the quadrurotpole becomes more pronounced in the proximity of the source (between $A_{1}$ and $A_{2}$ ), where the curve aligns to the $1/r^{3}$ reference law. Then, although the realignment with the $1/r$ line occurs fairly quickly, at a few metres from the origin, the lower speed triggers a destructive interference between the single rotpoles.

Figure 23.

The $\{k_{T}\}^{n}$ function decay for a rotpole, dirotpole and quadrurotpole in the disk plane (line $A$ ), at $M_{R}=0.8$ (a) and $M_{R}=0.4$ (b).

Figure 24.

The $\{k_{T}\}^{n}$ function decay for a rotpole, dirotpole and quadrurotpole in the disk plane (line $A$ ), at $M_{R}=0.08$ (a) and $M_{R}=0.01$ (b).

By reducing the rotational Mach number, this interference increases significantly. At $M_{R}=0.08$ (figure 24 a), the SPL decay of the dirotpole exhibits a starting slope aligned to $1/r^{3}$ , and when the curve assumes the trend of the inverse law (more or less at point $A_{5}$ ), the SPL has already fallen down to $95$ dB and exhibits a gap of more than $30$ dB with respect to the $1/r$ line. The drop suffered by the quadrurotpole $k_{T}$ function is much more pronounced: up to $8~\text{m}$ from the origin (point $A_{3}$ ), the slope of the curve even follows the $1/r^{5}$ law, corresponding to a decrease of $30$ dB at each doubling of distance, so that when the curve regains the trend of the inverse law (still at point $A_{5}$ ), the SPL value has fallen down to $55$ dB. At a more realistic rotational speed for a marine propeller ( $M_{R}=0.01$ ), the drop of the overall SPL for multirotpoles becomes impressive, as shown in figure 24(b). In this case, the alignment of the decay curves of the dirotpole and the quadrurotpole with the $1/r^{3}$ and $1/r^{5}$ laws respectively extends for a large part of the field, up to point $A_{6}$ , located at 64 m from the origin. This means that, at such a value of the rotational Mach number, the thickness term along line $A$ (disk plane) simply collapses in a spatially very limited region, and its contribution to the acoustic field is destined to vanish very rapidly.

Figure 25.

Numerical solution of (5.1) at point $A_{4}$ , for a quadrurotpole, a dirotpole and the single rotpoles at $M_{R}=0.8$ (a) and $M_{R}=0.01$ (b).

Such a shocking drop of the overall SPL from multirotpoles can be easily explained by looking at the corresponding time histories of the $\{k_{T}\}^{n}$ function. Figure 25 shows the numerical solution of (5.1) at the disk plane point $A_{4}$ ( $16~\text{m}$ from the origin) for the dirotpole, the quadrurotpole and each of the single rotpoles, at $M_{R}=0.8$ (a) and $M_{R}=0.01$ (b). At the highest speed, the kernel functions from the separate sources show a fluctuating waveform affecting, more or less, a quarter of a period; thus, by adding the contributions from the single rotpoles, no relevant interaction occurs and the solutions for the multirotpoles are characterized by two or four peak values very similar to the ones from the single elements. On the contrary, by bringing down ${\it\omega}$ , the fluctuation not only reduces in amplitude, but progressively covers a larger part of the revolution period; therefore, by adding the contributions from the single sources, a destructive interference between the time-shifted $k_{T}$ functions results.

Figure 26.

The $\{k_{T}\}^{n}$ function decay for a rotpole, dirotpole and quadrurotpole out of the disk plane (line $B$ ), at $M_{R}=0.8$ (a) and $M_{R}=0.01$ (b).

This result is shown in figure 25(b), referred to $M_{R}=0.01$ : besides the notable variation of the range values, the multirotpoles give rise to a mutual cancellation of the single components, and at only $16~\text{m}$ from the rotation axis (point $A_{4}$ ), the dirotpole $\{k_{T}\}^{2}$ function provides a much lower (approximately $80\,\%$ less) peak value compared with the single source, while the $\{k_{T}\}^{4}$ function of the quadrurotpole is very close to zero. Out of the plane of motion, the destructive interference occurs also at the higher rotational velocities. In this case, in fact, the $k_{T}$ function decreases much more rapidly than in the disk plane (see figure 10 a), and the time histories from each source soon interact with each other, in a way very similar to that shown in figure 25(b). Figure 26 shows the decay curves along line $B$ , out of the disk plane. At $M_{R}=0.8$ (a), the SPL for $\{k_{T}\}^{n}$ follows the $1/r^{2}$ , $1/r^{3}$ and even the $1/r^{5}$ reference laws for the rotpole, the dirotpole and the quadrurotpole respectively; this result confirms (and, somehow, better explains) the lower noise level provided by multibladed propellers and rotors in both marine and aeronautical applications. At $M_{R}=0.01$ (b), however, the SPL drop becomes really devastating. At such a low value of the rotational speed, as already shown in figure 11(b), the $k_{T}$ function (expressed by (4.9)) aligns to the stringent $1/r^{3}$ law for an extended region. Nevertheless, in the presence of multiple sources, the interferences occurring in the near field (up to point $B_{7}$ ) increase and the dirotpole $\{k_{T}\}^{2}$ function follows the $1/r^{5}$ decay law, while the quadrurotpole function even aligns to the $1/r^{9}$ reference line, thus approaching zero at point $B_{4}$ , at only $8~\text{m}$ from the origin. In other words, at $M_{R}=0.01$ and out of the disk plane, the thickness kernel function for a quadrurotpole vanishes in a handful of radii from the rotation axis. The behaviour of the loading kernel function $\{k_{L}\}^{n}$ , provided by (5.2) for multirotpoles, is very similar to $\{k_{T}\}^{n}$ ; here, for brevity, it is reported only at the highest and lowest values of the tested rotational velocities. Figure 27 shows the SPL decay curves along line $A$ (disk plane) for multiple sources rotating at $M_{R}=0.8$ (a,c) and $M_{R}=0.01$ (b,d), corresponding to $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{1}$ (a,b) and $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{2}$ (c,d). As for the thickness term, at the high speed and in the plane of motion, the presence of simultaneously rotating sources slightly increases the overall noise level, and the decay curves soon align to the typical far-field inverse law $1/r$ . On the other

Figure 27.

The $\{k_{L}\}^{n}$ function decay for a rotpole, dirotpole and quadrurotpole in the disk plane (line $A$ ), at $M_{R}=0.8$ (a,c) and $M_{R}=0.01$ (b,d) for $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{1}$ (a,b) and $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{2}$ (c,d).

hand, as the speed is reduced, the destructive interference occurs and rapidly increases. As said above, for a single rotpole at $M_{R}=0.01$ the phenomenon is necessarily auto-induced by the rotational motion, and in a large region (the near field) it causes a drop of the SPL ruled by the inverse-squared law. The presence of additional sources makes the interference more intense and gives rise, in the same region, to a trend of the overall SPL now ruled by the $1/r^{3}$ and $1/r^{5}$ laws for the dirotpole and the quadrurotpole respectively. The behaviour of the $\{k_{L}\}^{n}$ function out of the plane of motion is even more impressive and is shown in figure 28 for $M_{R}=0.8$ (a,c,e) and $M_{R}=0.01$ (b,d,f), now also showing the case $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{3}$ (e,f). These results demonstrate the deep influence of the rotational speed on the integral kernels of the loading source term, along the measurement line $B$ : in a region of approximately $128$ radii from the rotation axis (up to point $B_{8}$ ) and turning $M_{R}$ from $0.8$ to $0.01$ , the $\{k_{L}\}^{n}$ SPL turns from a trend ruled by the $1/r^{2}$ law to $1/r^{3}$ for the single rotpole, from $1/r^{3}$ to $1/r^{5}$ for the dirotpole and even from $1/r^{5}$ to $1/r^{9}$ for the quadrurotpole. It is interesting to note how this behaviour is totally independent of the normal vector: as depicted in figure 25, in fact, the destructive interference arises from the time shift of pressure disturbances spread over the whole revolution period, which is a mechanism not related at all to the selected orientation of $\hat{\boldsymbol{n}}$ . On the contrary, this mechanism could be notably affected by the unsteadiness of the underlying aero/hydrodynamic solution, which implicitly entails different contributions from the single sources and reckons the unknown balance between the $\{k_{L}\}^{n}$ (far-field) components characterized by $\tilde{p}$ and $\dot{\tilde{p}}$ . This issue will be briefly discussed in § 7; nevertheless, all of these results confirm the strict dependence of noise propagation on ${\it\omega}$ , and, above all, the occurrence of an impressive destructive interference in the case of multiple sources at low rotational speed.

Figure 28.

The $\{k_{L}\}^{n}$ function decay for a rotpole, dirotpole and quadrurotpole in the disk plane (line $A$ ), at $M_{R}=0.8$ , (a,c,e) and $M_{R}=0.01$ (b,d,f) for $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{1}$ (a,b), $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{2}$ (c,d) and $\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}_{3}$ (e,f).

6 Marine propeller ‘versus’ helicopter rotor

The validity of the analysis on the kernel functions for a rotating point source, its relationship with the ‘full’ numerical solution of formulation 1A for a three-dimensional body and, above all, its practical consequences can be demonstrated by performing a very simple standard FWH-based prediction of the thickness noise only, thus avoiding any problem related to the availability of aero/hydrodynamic data. To this end, we will solve (2.2) for both a helicopter rotor and a marine propeller model, here suitably scaled in order to have the same radius $R=1~\text{m}$ , at the same operating conditions in air and underwater. Such a test, of course, is just explanatory and also concerns some unrealistic configurations; nevertheless, it enables appreciation of if and how the accounting for different fluids (each with its own values of ${\it\rho}_{0}$ and $c_{0}$ ) and the geometrical differences of the two devices may affect the numerical solution of (2.2). A sketch of the models is shown in figure 29. The marine propeller is the INSEAN E779A, rotating around the $x$ axis. As is usual for this kind of device, the blade exhibits a remarkable variation of chord and twist angle along the span and a rather high solidity, with a thickness decreasing from the 60/70 % of the blade towards the tip. On the other hand, the rotor model corresponds to the BO-105 main rotor and rotates around the $z$ axis. In this case, both the chord and the thickness spanwise distribution are uniform, and the blade planform has a much higher aspect ratio. We assume that the fluid is at rest and both of the models are simply rotating at a fixed $M_{R}$ . The measurement points are located along the measurement lines $A$ and $B$ of figure 6, here suitably oriented to account for the different rotation axes of the two devices. The rotational velocity ${\it\omega}=c_{0}M_{R}/R$ is set according to the speed of sound, being $c_{0}=340~\text{m}~\text{s}^{-1}$ for the rotor and $c_{0}=1500~\text{m}~\text{s}^{-1}$ for the propeller; furthermore, the fluid density ${\it\rho}_{0}$ and the reference pressure for the dB computation are set to $1.234~\text{kg}~\text{m}^{-3}$ and $1000~\text{kg}~\text{m}^{-3}$ , $p_{ref}=2\times 10^{-5}$  Pa and $p_{ref}=10^{-6}$ Pa for air and water respectively. The calculations will concern both the single blade of each device and the alternative two-bladed and four-bladed configurations, in order to check the reliability of the results found for the dirotpole and quadrurotpole sources.

Figure 29.

Three-dimensional sketches of the E779A marine propeller (a) and the BO-105 helicopter main rotor (b) models, used for thickness noise prediction.

Figure 30 shows the thickness noise decay in the disk plane (line $A$ ), at $M_{R}=0.8$ (a) and $M_{R}=0.01$ (b). For clarity, each panel shows the ‘unrealistic’ cases (the marine propeller operating underwater at $M_{R}=0.8$ and the rotor in air at $M_{R}=0.01$ ) in a grey colour. In practice, the decay of thickness noise from the helicopter rotor at $M_{R}=0.8$ (figure 30 a) overlaps the inverse law $1/r$ , thus confirming the well-known aeroacoustic behaviour of this term in the far field. Moreover, at such a high value of the rotational speed, no destructive interference characterizes the multibladed devices, and the thickness noise from the two- or four-bladed rotors is slightly higher compared with the single blade. The marine propeller exhibits a much higher value of the SPL, due to the very large solidity with respect to the aeronautical blade ( $R$ being the same), and, above all, the same trend for all of the noise decay curves. This result is very interesting. It confirms that a helicopter rotor in air and a marine propeller in water basically behave in the same way: the surrounding fluid and the shape of the body source determine the overall noise level, but they do not affect the noise propagation mechanisms, which, in essence, are governed by the rotational Mach number.

Figure 30.

Thickness noise decay of the E779A propeller and the BO-105 rotor models in the disk plane (line $A$ ), at $M_{R}=0.8$ (a) and $M_{R}=0.01$ (b).

The acoustic equivalence of the two devices and the notable reduction of the thickness noise due to the much lower blade rate are confirmed in figure 30(b). As revealed in advance by the kernel function $k_{T}$ , the thickness noise from any blade rotating at $M_{R}=0.01$ decreases more rapidly in the field, following the $1/r^{2}$ law instead of the $1/r$ law in a very extended region (see also figures 7 and 8). Furthermore, the mutual interactions characterizing the behaviour of multiple rotating sources make this drop impressive for any multibladed device. The thickness noise decay curves along line $B$ , at points located downstream the blades, are shown in figure 31, and fully confirm the intrinsic acoustic features of a rotating source/body examined through the kernel functions. In this case also, in fact, the decay of the thickness term from the rotor in air soon aligns to the inverse-squared law, while the corresponding curve from the propeller underwater, at the lowest speed, shows a deeper slope ( $1/r^{3}$ ) in an extended region and gives rise to a more significant and rapid decrease of the thickness contribution.

Figure 31.

Thickness noise decay of the E779A and the BO-105 models out of the disk plane (downstream line $B$ ), at $M_{R}=0.8$ (a) and $M_{R}=0.01$ (b).

It can be noted how the curve for the four-bladed rotor at $M_{R}=0.8$ (figure 31 a) exhibits a very slight slope change at the last point, while the one from the two-bladed device seems not to change at all. This suggests that the absence of the expected realignment to the inverse-squared law falls down out of the range of distances taken into account here, and, in the presence of multiple sources, the interference mechanisms may affect a spatially very extended region, also at a high rotational rate. In any case, these results perfectly match the decay curves depicted in figures 10(a) and 11(b), referred to a single point source and without any term related to the source strength. Table 2 lists the overall SPLs for the four-bladed propeller at $M_{R}=0.01$ (with the distance from the plane of motion) at the first five points along line $B$ ; these values correspond to the decay curve depicted in figure 31(b). At the disk plane point $B_{1}$ the thickness noise level is rather high ( $144$ dB), but at only $8~\text{m}$ from the disk plane (point $B_{4}$ ) it collapses to 32 dB, and goes to zero before $B_{5}$ .

Table 2.

Thickness noise SPLs from the E779A propeller along $B$ , at $M_{R}=0.01$ .