2.1. Pressure field in the frequency domain

Consider the problem of leading-edge noise radiation from a stationary rotor ingesting turbulence convected by a uniform flow with axial Mach number $M_z$. The observer is stationary with respect to the rotor hub. Further consider a fixed Cartesian coordinate system positioned at the centre of the hub. The $z$-axis is aligned with the uniform flow and points upstream. The $x$-axis points vertically upwards in the rotor plane and the $y$-axis is such that the coordinate system is right-handed. Figure 1 shows the geometry of the blades in the source cylindrical coordinate system $(r,\gamma,z)$, where the azimuthal direction $\gamma$ has been unwrapped. Both lean and sweep are neglected. The cross-sections of the blades are assumed to be flat plates of chord $C$. The pitch change axis is located at the leading edge and the pitch angle, $\alpha$, is assumed to vary in the radial direction such that blade cross-sections are at zero angle of attack relative to the incoming flow

(2.1)\begin{equation} \alpha=\text{arctan} \left(\frac{U_z}{\varOmega r}\right)\!, \end{equation}

where $U_z$ is the axial flow velocity and $\varOmega$ is the rotor angular velocity. The blade-fixed coordinate system $(X,Y,Z)$ is centred mid-chord with the $X$-axis in the downstream direction and the $Z$-axis in the wall-normal direction. Two consecutive blades are separated by a distance of $2 {\rm \pi}r /B$ in the unwrapped azimuthal direction, where $B$ is the number of blades. The azimuthal angle of a source in the $m$th blade planform, where $m \in [0,B-1]$, is given by $\gamma =\varOmega \tau - \psi + \phi _m$, where $\tau$ is the emission time and $\phi _m=2 {\rm \pi}m/B$. For clarity, let us define the Fourier transform pair $(\,f,\tilde {f})$ used throughout this paper:

(2.2a,b)\begin{equation} \tilde{f}(\omega) = \frac{1}{2 {\rm \pi}} \int^{+\infty}_{-\infty} f(t) {\rm e}^{-{\rm i}\omega t} \,{\rm d}t, \quad f(t)=\int^{+\infty}_{-\infty} \tilde{f}(\omega) {\rm e}^{{\rm i} \omega t} \,{\rm d} \omega. \end{equation}

Figure 1. Blade geometry and source position in cylindrical coordinate system $(r,\gamma =\varOmega \tau + \psi + 2 {\rm \pi}m/B,z)$. Figure adapted from Sinayoko et al. (Reference Sinayoko, Kingan and Agarwal2013). (a) Three-dimensional view of the blade geometry showing the cross-blade spacing in the unwrapped blade path. (b) Zeroth blade geometry and blade-fixed coordinate system $(X,Y,Z)$ in the plane $(z,\gamma )$.

We note that the notation of this section has largely been adopted from Sinayoko et al. (Reference Sinayoko, Kingan and Agarwal2013). The pressure field generated by the $m$th blade due to interaction with the incoming turbulent flow is given by (Sinayoko et al. Reference Sinayoko, Kingan and Agarwal2013)

(2.3)\begin{align} \tilde{p}^{(m)}(\boldsymbol{x}_o,\omega)&=2 {\rm \pi}{\rm i} \tilde{g} \sum^{+\infty}_{n=-\infty} \exp({-{\rm i}n \left( \gamma_0- {\rm \pi}/2 \right)}) \int_{\varSigma} k_L {\rm J}_n(k_r r) \tilde{L}^{(m)}(\boldsymbol{x},\omega-n \varOmega) \nonumber\\ &\quad \times \exp({-{\rm i} [ k_C (X+C/2)- n \phi_m) ]})\,{\rm d}\varSigma , \end{align}

where

(2.4)\begin{equation} \tilde{g}(k,R_e,\theta_e)=\frac{{\rm e}^{-{\rm i} k R_e}}{8 {\rm \pi}^2 R_e (1-M_z \cos \theta_e)} , \end{equation}

and with the integration performed over the blade planform $\varSigma$. We use $\tilde {L}$ to denote the unsteady lift force per unit area and ${\rm J}_n$ to represent the $n$th Bessel function of the first kind. The vectors $\boldsymbol {x}$ and $\boldsymbol {x}_o$ denote, respectively, the current source position and the observer position in the hub-fixed Cartesian coordinate system. The vector $\boldsymbol {X}$ denotes the current source position in the blade-fixed Cartesian coordinate system. We have also introduced $(R_e,\theta _e,\gamma _0)$ to denote the observer position in spherical emission coordinates (Sinayoko et al. Reference Sinayoko, Kingan and Agarwal2013), given by

(2.5a)$$\begin{gather} x_o= R_e \sin \theta_e \cos \gamma_0 , \end{gather}$$

(2.5b)$$\begin{gather}y_o= R_e \sin \theta_e \sin \gamma_0 , \end{gather}$$

(2.5c)$$\begin{gather}z_o= R_e \left( \cos \theta_e - M_z \right) . \end{gather}$$

The acoustic wave frequency and wavenumber are given by $\omega$ and $k=\omega /c_0$, respectively, where $c_0$ is the speed of sound. The modal wavenumbers in the blade-fixed coordinate system are given by

(2.6a,b)\begin{equation} k_L=k_z \cos \alpha - (n/r) \sin \alpha, \quad k_C=k_z \sin \alpha + (n/r) \cos \alpha, \end{equation}

where

(2.7a,b)\begin{equation} k_z=k \cos \theta_e / (1-M_z \cos \theta_e) , \quad k_r=k \sin \theta_e / (1-M_z \cos \theta_e) . \end{equation}

2.2. Aeroacoustic transfer function

The far-field pressure expression presented in § 2.1 remains a function of the lift forces acting on the blade. We proceed to calculate the unsteady lift generated by the blades as a consequence of turbulence ingestion by the rotor. Let us consider a two-dimensional turbulence gust impinging on a blade cross-section. Its upwash velocity is given by

(2.8)\begin{equation} w_g(X,t)=\tilde{w}_g (k_X) \exp({{\rm i} k_X ( U_r t-X)}) , \end{equation}

where $U_r=\sqrt {U_z^2+U_\gamma ^2}$ is the blade relative velocity, and $U_\gamma =\varOmega r$ is the tangential velocity. The streamwise wavenumber is given by $k_X=\omega /U_r$. Taylor's frozen turbulence hypothesis has been adopted here. We can introduce the aerofoil response function, $g$, by writing the unsteady lift produced by the aerofoil interaction with a single sinusoidal gust as (Amiet Reference Amiet1975)

(2.9)\begin{equation} L(X,t)=2 {\rm \pi}\rho U_r g(X,k_X) \tilde{w}_g (k_X) {\rm e}^{{\rm i} \omega t} , \end{equation}

where $\rho$ denotes the far-field uniform flow density. Note that the oscillatory behaviour in the $X$ direction has been absorbed into the aerofoil response function ($g$). The equivalent frequency-domain result is obtained by integrating the contributions of all wavenumbers and taking the Fourier transform. Its final form is (Amiet Reference Amiet1975)

(2.10)\begin{equation} \tilde{L}(X,\omega)= 2 {\rm \pi}\rho U_r g(X,k_X) \tilde{w}_g (\omega) , \end{equation}

where we have used $\tilde {w}_g (k_X)=U_r \tilde {w}_g (\omega )$. A detailed derivation of $g(X,k_X)$ in the high-frequency regime, following Amiet's theory (Amiet Reference Amiet1976a), can be found in Appendix C.1 of Christophe (Reference Christophe2011). Substituting (2.10) in the pressure field generated by the $m$th blade (2.3) yields

(2.11)\begin{align} \tilde{p}^{(m)}(\boldsymbol{x}_o,\omega) &=\frac{{\rm i} \rho \exp({-{\rm i} k R_e})}{2 R_e (1-M_z \cos \theta_e)} \sum^{+\infty}_{n=-\infty} \int^{r_{tip}}_{r_{hub}} k_L {\rm J}_n(k_r r) U_r \tilde{w}_g^{(m)} (r,\omega-n \varOmega)\nonumber\\ &\quad \times \exp({{\rm i} n ( - \gamma_0 + {\rm \pi}/2 + \phi_m )}) \exp({-{\rm i} k_C C/2}) \int^{C/2}_{-C/2} g(X,{\rm K}_X) \nonumber\\ &\quad \times\exp({-{\rm i} k_C X})\,{\rm d}X \,{\rm d}r , \end{align}

where ${\rm K}_X= (\omega -n \varOmega )/U_r$. The aeroacoustic transfer function is defined as

(2.12)\begin{equation} \mathcal{L} (k_X,k_C) = \frac{2}{C} \int^{C/2}_{-C/2} g(X,k_X) \exp({-{\rm i} k_C X})\,{\rm d}X . \end{equation}

The definition of the aeroacoustic transfer function (2.12) can finally be introduced in (2.11), yielding

(2.13)\begin{align} \tilde{p}(\boldsymbol{x}_o,\omega) &=\frac{{\rm i} C \rho \exp({-{\rm i} k R_e})}{4 R_e (1-M_z \cos \theta_e)} \sum^{B-1}_{m=0} \sum^{+\infty}_{n=-\infty} \int^{r_{tip}}_{r_{hub}} k_L {\rm J}_n(k_r r) U_r \tilde{w}_g^{(m)} (r,\omega-n \varOmega)\nonumber\\ & \quad\times \exp({{\rm i} n( - \gamma_0 + {\rm \pi}/2 + \phi_m )- {\rm i} k_C C/2}) \mathcal{L} ({\rm K}_X,k_C) \,{\rm d}r , \end{align}

where we have summed the contributions of each blade to obtain the total far-field pressure $\tilde {p}(\boldsymbol {x}_o,\omega )$.

The aeroacoustic transfer function defined in (2.12) is a more general transfer function than typically found in the literature because $k_C$ has not been substituted by a specific value. The first-order term in Amiet's high-frequency successive approximation method is given by (Amiet Reference Amiet1976a)

(2.14)\begin{equation} \mathcal{L}_1(k_X,k_C)=-\frac{{\rm i} {\rm e}^{{\rm i} \bar{k}_C}}{{\rm \pi} \sqrt{ \bar{k}_X+\beta_r^2 \kappa} \sqrt{\kappa-\bar{k}^{\prime}_X M_r^2 +\bar{k}_C}} \text{erf} \left[ \left(1+{\rm i} \right) \sqrt{\kappa-\bar{k}_X^{\prime} M_r^2 + \bar{k}_C}\right] , \end{equation}

where $k_X^{\prime }=k_X/\beta _r^2$, $\beta _r=\sqrt {1-M_r^2}$, $\kappa =\bar {k}_X^{\prime } M_r$, $M_r=U_r/c_0$ and the overbar denotes quantities made non-dimensional with the length scale $C/2$. The error function of complex argument, $\text {erf}(z)$, is defined as

(2.15)\begin{equation} \text{erf}(z)=\frac{2}{\sqrt{\rm \pi}} \int^z_0 {\rm e}^{-t^2} \,{\rm d}t . \end{equation}

The back-scattering correction is given by

(2.16)\begin{align} \mathcal{L}_2(k_X,k_C)&= \frac{\exp({{\rm i} ({\rm \pi}/4+\bar{k}_C)})}{\theta_1 {\rm \pi}\sqrt{2 {\rm \pi}\left( \bar{k}_X+\beta_r^2 \kappa \right)}}\left[ \vphantom{\sqrt{\frac{2 \kappa}{2 \kappa - \theta_1}}} \left( 1- \exp({-2{\rm i}\theta_1})\right) + (1+{\rm i})\right.\nonumber\\ & \quad \times \left.\left[ - E^*(4 \kappa) + \sqrt{\frac{2 \kappa}{2 \kappa - \theta_1}} \exp({-2{\rm i} \theta_1}) E^* \left( 2 (2 \kappa -\theta_1)\right)\right] \right] , \end{align}

where $\theta _1= \kappa - \bar {k}_X^{\prime } M_r^2 + \bar {k}_C$, and the Fresnel integral is defined as

(2.17)\begin{equation} E^*(z)= \int^{z}_0 \frac{{\rm e}^{-{\rm i} t}}{\sqrt{2 {\rm \pi}t}} \,{\rm d}t . \end{equation}

The aeroacoustic transfer function is thus given by $\mathcal {L}=\mathcal {L}_1+\mathcal {L}_2$ or $\mathcal {L}=\mathcal {L}_1$, with and without trailing edge back-scattering effects, respectively.

For frequencies where the blade chord is compact we require another solution that is valid in the low-frequency regime, i.e. for $\omega C / 2 c_0 (1-M_r^2)<{\rm \pi} /4$ (Amiet Reference Amiet1975). The pressure jump in this regime was studied by Amiet (Reference Amiet1974) and the associated normalised aerofoil response function is given by (Amiet Reference Amiet1989)

(2.18)\begin{equation} g(\bar{X},k_X)=\frac{1}{\beta_r {\rm \pi}} \sqrt{\frac{1-\bar{X}}{1+\bar{X}}} S(\bar{k}_x^{\prime}) \exp({{\rm i} \bar{k}_X^{\prime} ( M_r^2 \bar{X}+f(M_r) )}) , \end{equation}

where $f(M_r)=(1-\beta _r) \ln (M_r)+\beta _r \ln (1+\beta _r)-\ln (2)$, and $S(\bar {k}_X^{\prime })$ is the Sears function given by

(2.19)\begin{equation} S(\bar{k}_X^{\prime}) = \frac{2}{{\rm \pi} \bar{k}_X^{\prime} \left( H_0^{(1)} ( \bar{k}_X^{\prime}) +{\rm i} H_1^{(1)} ( \bar{k}_X^{\prime}) \right) } , \end{equation}

where $H_n^{(1)}$ is the Hankel function of order $n$ of the first kind. Note that the Sears response function in (2.18) includes a compressibility correction. The interested reader is referred to Amiet (Reference Amiet1974, Reference Amiet1993) for details. The aeroacoustic transfer function follows immediately

(2.20)\begin{equation} \mathcal{L}(k_X,k_C)=\frac{1}{\beta_r } S(\bar{k}_X^{\prime}) \exp({{\rm i} \bar{k}_X^{\prime} f(M_r)}) \left[ {\rm J}_0(\bar{k}_x^{\prime} M_r^2 - \bar{k}_C ) - {\rm i} {\rm J}_1(\bar{k}_X^{\prime} M_r^2 - \bar{k}_C ) \right] . \end{equation}

2.3. Far-field noise spectrum

The pressure field generated by the propeller was linked to the incoming turbulence in § 2.2 through an aeroacoustic transfer function for both low- and high-frequency gusts. We proceed to calculate the power spectral density by adopting the following definition

(2.21)\begin{equation} S_{pp}(\boldsymbol{x}_o,\omega)= \frac{\rm \pi}{T} E \left[ \tilde{p}(\boldsymbol{x}_o,\omega) \tilde{p}^{^*}(\boldsymbol{x}_o,\omega)\right] , \end{equation}

where $E[\ldots ]$ denotes the expected value. Using (2.13) in this definition yields

(2.22)\begin{align} S_{pp}(\boldsymbol{x}_o,\omega)&= \frac{C^2 \rho^2}{16 R_e^2 (1-M_z \cos \theta_e)^2} \sum^{B-1}_{m=0} \sum^{B-1}_{k=0} \sum^{+\infty}_{n=-\infty} \int^{r_{tip}}_{r_{hub}} \int^{r_{tip}}_{r_{hub}} k_L k_L' {\rm J}_n(k_r r) {\rm J}_n'(k_r' r') U_r U_r' \nonumber\\ &\quad \times S_{ww}^{(m,k)}(r,r',\omega-n \varOmega) \nonumber\\ &\quad \times\exp\left({{\rm i} \frac{2 {\rm \pi}n}{B} (m-k)-{\rm i} (k_C-k_C')C/2}\right) \mathcal{L}({\rm K}_X,k_C) \mathcal{L}^*({\rm K}_X',k_C') \,{\rm d}r\, {\rm d}r' , \end{align}

where

(2.23)\begin{equation} S_{ww}^{(m,k)}(r,r',\omega)= \frac{\rm \pi}{T} E \left[ \tilde{w}_g^{(m)}(r,\omega) \tilde{w}^{{(k)}^*}_g(r',\omega) \right] . \end{equation}

The problem is thus reduced to calculating the blade-normal unsteady velocity cross-spectrum, $S_{ww}^{(m,k)}$. Following Glegg et al. (Reference Glegg, Morton and Devenport2012), the Fourier transform of the gust velocity component normal to the blade is given by

(2.24)\begin{equation} \tilde{w}_g^{(m)}(r,\omega)=\frac{1}{2 {\rm \pi}} \int^{T}_{-T} \boldsymbol{n}_i^{(m)}(r,\tau) \boldsymbol{v}_i(\boldsymbol{x}_{LE}^{(m)}(\tau),\tau) {\rm e}^{-{\rm i} \omega \tau} \,{\rm d} \tau , \end{equation}

where $\boldsymbol {v}(\boldsymbol {x}_{LE}^{(m)}(\tau ),\tau )$ is the gust velocity vector in the hub-fixed coordinate system, which is a function of the $m$th blade's leading-edge position. The gust velocity is projected onto the blade-normal direction by taking the inner product with the unit normal vector. These are given by

(2.25a)$$\begin{gather} \boldsymbol{x}_{LE}^{(m)}(\tau) = \left[ r \cos \gamma_{LE} , r \sin \gamma_{LE},0\right]^{\rm T}, \end{gather}$$

(2.25b)$$\begin{gather}\boldsymbol{n}^{(m)}(r,\tau)=\left[ -\sin \alpha \sin \gamma_{LE},\sin \alpha \cos \gamma_{LE}, - \cos \alpha \right]^{\rm T}, \end{gather}$$

where $\gamma _{LE}=\varOmega \tau + \phi _m$. Substituting (2.24) in (2.23) and defining the cross-correlation as $R_{ij}(\boldsymbol {x}_{LE}^{(m)},\boldsymbol {x}_{LE}^{(k)},\tau -\tau ')=E [ \boldsymbol {v}_i \boldsymbol {v}_j^* ]$ yields

(2.26)\begin{align} S_{ww}^{(m,k)}(r,r',\omega)&= \frac{1}{4 {\rm \pi}T} \int^{T}_{-T} \int^{T}_{-T} \boldsymbol{n}_i^{(m)}(r,\tau) \boldsymbol{n}_j^{(k)}(r',\tau') R_{ij}(\boldsymbol{x}_{LE}^{(m)},\boldsymbol{x}_{LE}^{(k)},\tau-\tau') \nonumber\\ &\quad\times\exp({{\rm i} \omega(\tau'- \tau)}) \,{\rm d} \tau \,{\rm d} \tau ' . \end{align}

This expression can be used to calculate the upwash velocity cross-spectrum from the knowledge of the two-point velocity correlation in the plane of the rotor. This has the advantage of accounting for the distortion of turbulence on approach to the rotor, bypassing the modelling of this phenomena. It is particularly advantageous in cases where no wavenumber spectrum is readily available or when one has access to direct measurements of the velocity cross-correlation, see Glegg et al. (Reference Glegg, Devenport and Alexander2015) as an example.

2.4. Turbulent upwash velocity cross-spectrum

In § 2.3, we derived an expression for the noise power spectral density which requires knowledge of the turbulent upwash velocity cross-spectrum in the blade-fixed coordinate system as an input. The upwash velocity cross-spectrum can be calculated directly with (2.26) from time-domain measurements of the turbulence velocity in the rotor plane, for example. However, in general it is more useful to express the upwash velocity cross-spectrum as a function of the impinging turbulent velocity spectrum in the hub-fixed coordinate system. Following Glegg et al. (Reference Glegg, Morton and Devenport2012), let us introduce the correlation–spectrum pair of a homogeneous turbulence field as

(2.27)\begin{equation} R_{ij}(\boldsymbol{x}_{LE}^{(m)},\boldsymbol{x}_{LE}^{(k)},\tau-\tau')= \int_{-\infty}^{+\infty} \varPhi_{ij} (\boldsymbol{k}) \exp({{\rm i} \boldsymbol{k} \boldsymbol{\cdot} (\boldsymbol{x}_{LE}^{(m)}-\boldsymbol{x}_{LE}^{(k)}) + {\rm i} U_z k_z (\tau-\tau')})\,{\rm d} \boldsymbol{k} , \end{equation}

where $\boldsymbol {k}=[k_x,k_y,k_z]^{\rm T}$ is the wavenumber vector, and $\varPhi _{ij} (\boldsymbol {k})$ is the turbulent velocity spectrum. Substituting this in (2.26), we obtain

(2.28)\begin{align} S_{ww}^{(m,k)}(r,r',\omega)&= \frac{1}{4 {\rm \pi}T} \int^{+\infty}_{-\infty} \varPhi_{ij} (\boldsymbol{k}) \int^{T}_{-T} \int^{T}_{-T} \boldsymbol{n}_i^{(m)}(r,\tau) \boldsymbol{n}_j^{(k)}(r',\tau')\nonumber\\ & \quad \times \exp({{\rm i} \omega(\tau'- \tau) + {\rm i} \boldsymbol{k} \boldsymbol{\cdot} (\boldsymbol{x}_{LE}^{(m)}(\tau)-\boldsymbol{x}_{LE}^{(k)}(\tau'))+ {\rm i} U_z k_z (\tau-\tau')})\,{\rm d} \tau \,{\rm d} \tau' \,{\rm d} \boldsymbol{k} . \end{align}

We introduce the Jacobi–Anger expansion of $\exp ({ {\rm i} \boldsymbol {k} \boldsymbol {\cdot } \boldsymbol {x}_{LE}^{(m)}(\tau ) })$, to obtain

(2.29)\begin{align} \exp({ {\rm i} \boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{x}_{LE}^{(m)}(\tau) })& = \exp({{\rm i} k_R r \cos ( \varOmega \tau + \phi_m - k_\gamma )}) \nonumber\\ & = \sum^{+\infty}_{p=-\infty} i^p {\rm J}_p(k_R r) \exp({{\rm i} p ( \varOmega \tau +\phi_m - k_\gamma )}) , \end{align}

where $k_R^2=k_x^2+k_y^2$ and $k_\gamma =\text {arctan}(k_y/k_x)$. The unit normal vector can be rewritten in terms of complex exponentials as

(2.30)\begin{align} \boldsymbol{n}^{(m)}(r,\tau)= \left[ -\sin \alpha \frac{{\rm e}^{{\rm i} (\varOmega \tau + \phi_m)}-{\rm e}^{-{\rm i} (\varOmega \tau + \phi_m)}}{2{\rm i}}, \sin \alpha \frac{{\rm e}^{{\rm i} (\varOmega \tau + \phi_m)}+{\rm e}^{-{\rm i} (\varOmega \tau + \phi_m)}}{2}, -\cos \alpha \right]^{\rm T}. \end{align}

Using the expansion introduced in (2.29) for the $m$th and $k$th blades, and (2.30) in (2.28), we obtain the full cross-spectrum of the blade-normal gust velocity component; see Appendix A. Those expressions can be considerably simplified if we assume that the blades are untwisted, i.e. that the blade pitch angle is small $\alpha \ll 1$, leading to $\sin \alpha \approx 0$ and $\cos \alpha \approx 1$. This assumption is not uniformly valid along the blade radial direction. However, it can be argued that because turbulence ingestion noise scales with the fourth power of the blade relative velocity (Glegg & Devenport Reference Glegg and Devenport2017), most of the far-field noise originates in the vicinity of the blade tip. In this region, assuming that the blades operate at low angles of attack, the pitch angle should be small due to the large ratio between the propeller azimuthal velocity and the uniform flow velocity. In this paper, we have made the simplifying assumption that the blades operate at zero angle of attack, and therefore the pitch angle is given by (2.1). At the blade tip, i.e. for $r=r_{tip}$, (2.1) can be written in terms of the advance ratio, $J = U_z {\rm \pi}/ \varOmega r_{tip}$, as $\alpha _{tip}=\text {arctan} ( J/{\rm \pi} )$. The blade tip pitch angle is thus smallest when the rotor is operating at lower advance ratios. Under this assumption, together with the argument that the majority of far-field noise originates in the blade tip region, the untwisted blade approximation should be valid. This is investigated in § 4 for a model cooling fan, an open propeller and a wind turbine. Introducing the approximations $\sin \alpha \approx 0$ and $\cos \alpha \approx 1$ in (A2) and substituting the result in (A1) yields

(2.31)\begin{align} S_{ww}^{(m,k)}(r,r',\omega)&= \frac{1}{U_z} \int^{2 {\rm \pi}}_{0}\int^{+\infty}_{0} k_R \sum^{+\infty}_{p=-\infty} \varPhi_{33} (k_z, k_R, k_\gamma) \nonumber\\ &\quad \times \exp({{\rm i} p ( \phi_m-\phi_k ) }) {\rm J}_p(k_R r) {\rm J}_p(k_R r') \,{\rm d} k_R \,{\rm d} k_\gamma . \end{align}

This approximate expression, or the complete version in (A1), can be used in the result obtained in § 2.3 for the far-field noise spectrum (2.22) and, together with the aeroacoustic transfer functions in § 2.2, they define a complete model of turbulence ingestion noise. The exact and approximate models are named ‘rotational model’ and ‘rotational model with untwisted blade approximation’, respectively. The approximate formulation is of interest as it can speed up far-field noise predictions.

Hereafter, we consider the particular case of homogeneous isotropic turbulence. This type of idealised turbulence is advantageous because there are well-known analytical expressions for the spectrum tensor $\varPhi _{ij}$. The simplicity of the model, with only two parameters, also enables straightforward comparisons between this turbulence ingestion noise model and Amiet's model described in § 3. It does not add unnecessary layers of complexity, when the goal of the comparison is to study other assumptions and approximations made throughout the derivations. Note, however, that in practical applications the assumption of homogeneous isotropic turbulence is not always valid. The turbulent length scales most relevant to turbulence ingestion noise generation are proportional to the radius of the rotor, and inversely proportional to the frequency of interest. The relevant wavenumber range is thus bounded by the rotor radius at low wavenumbers and the blade thickness at high wavenumbers (Simonich et al. Reference Simonich, Amiet, Schlinker and Greitzer1986). Therefore, for wind turbines, for example, noise arises from very large-scale atmospheric turbulence which is unlikely to be homogeneous and isotropic. Conversely, for a cooling fan, a propeller and even a helicopter rotor, the upstream turbulence can be modelled as locally homogeneous and isotropic (Simonich et al. Reference Simonich, Amiet, Schlinker and Greitzer1986). Furthermore, even in cases where the upstream turbulence is homogeneous and isotropic, rotating blades are known to distort turbulence in the streamtube on approach to the rotor disc, particularly in static conditions; see, for example, Robison & Peake (Reference Robison and Peake2014) and Graham (Reference Graham2017).

2.5. Far-field pressure spectrum for homogeneous isotropic turbulence ingestion

As shown earlier, in order to evaluate the turbulent velocity upwash cross-spectrum, given by (A1), one needs prior knowledge of the turbulent velocity energy spectrum. In the case of homogeneous isotropic turbulence, the energy spectrum is given by (see e.g. Christophe Reference Christophe2011)

(2.32)\begin{equation} \varPhi_{ij}(k_1,k_2,k_3)=A\frac{\vert \hat{\boldsymbol{k}} \vert^2 \delta_{ij} - \hat{k}_i \hat{k}_j} {\left(1+ \hat{k}_1^2+\hat{k}_2^2+\hat{k}_3^2\right)^{17/6}} , \quad i,j=\{1,2,3\} , \end{equation}

with

(2.33a,b)\begin{equation} A=\frac{55 \varGamma(5/6) \overline{u^2}}{36 {\rm \pi}\sqrt{\rm \pi} \varGamma(1/3) k_e^3}, \quad k_e=\frac{\sqrt{\rm \pi} \varGamma(5/6)}{\varLambda \varGamma(1/3)}, \end{equation}

where $\varGamma$ is the gamma function, $\delta _{ij}$ is the Kronecker delta and the hat denotes variables made non-dimensional with $k_e$. Note that $(k_1,k_2,k_3)=(k_x,k_y,k_z)$. The energy spectrum is fully characterised by the mean square of the turbulent velocity fluctuations, $\overline {u^2}$, and the turbulence integral length scale, $\varLambda$.

The energy spectrum in (2.32) can be used directly in (A1) to calculate the cross-spectrum of the blade-normal gust velocity component. Alternatively, a simpler result that is faster to compute can be obtained by substituting the energy spectrum of the streamwise turbulent velocity fluctuations given by (2.32) in (2.31) and integrating over the azimuthal wavenumber $k_\gamma$, yielding

(2.34)\begin{align} S_{ww}^{(m,k)}(r,r',\omega)&= \frac{2 {\rm \pi}A}{U_z} \int^{+\infty}_{0} \sum^{+\infty}_{p=-\infty} k_R \frac{\hat{k}_R^2}{\left(1+ \hat{k}_R^2+\hat{k}_z^2\right)^{17/6}} {\rm J}_p(k_R r) {\rm J}_p(k_R r')\nonumber\\ & \quad \times \exp({{\rm i} p ( \phi_m-\phi_k) })\,{\rm d} k_R . \end{align}

Finally, the upwash velocity cross-spectrum given in (2.34) can be substituted in (2.22) to obtain the far-field pressure spectrum at the observer location $\boldsymbol {x}_o$ as

(2.35)\begin{align} S_{pp}(\boldsymbol{x}_o,\omega)&= \frac{C^2 \rho^2 2 {\rm \pi}A}{U_z 16 R_e^2 (1-M_z \cos \theta_e)^2} \sum^{B-1}_{m=0} \sum^{B-1}_{k=0} \sum^{+\infty}_{n=-\infty} \sum^{+\infty}_{p=-\infty} \nonumber\\ &\quad \times \int^{+\infty}_0 \frac{\hat{k}_R^2}{ \left( 1+\hat{k}_R^2+\left( \dfrac{\omega -(\,p+n) \varOmega}{U_\infty k_e}\right)^2 \right)^{17/6}}k_R \exp\left({ \frac{{\rm i} 2 {\rm \pi}(m-k) (\,p+n)}{B}}\right)\nonumber\\ &\quad \times \left| \int^{r_{tip}}_{r_{hub}} k_L U_r {\rm J}_n(k_r r) {\rm J}_p(k_R r) \mathcal{L} ({\rm K}_X,k_C) \exp({-{\rm i} \bar{k}_C })\,{\rm d}r \right|^2 \,{\rm d}k_R . \end{align}

Note that the dependence on the blade numbers $m$ and $k$ appears in the form $(m-k)$ and that, therefore, we need only perform the calculation for every $(m-k)$ combination. Physically, this means that the far-field pressure cross-spectrum of the noise radiated by two blades of the rotor is independent of the particular blades in question, but rather depends on their relative azimuthal position. This reduces the computational time to evaluate (2.35) significantly as only $2B-1$ terms of the double sum over the blade numbers must be computed, instead of $B^2$ terms.

3.1. Noise radiated by aerofoil cascade in rectilinear motion

Consider a cascade of aerofoils moving in rectilinear motion with velocity $U_r$ at zero angle of attack, with a constant and equal offset between adjacent blades as shown in figure 2. The coordinate system $(X,Y,Z)$ is centred mid-chord and mid-span of the zeroth aerofoil, with $X$ in the chordwise downstream direction, $Y$ in the spanwise direction and $Z$ in the wall-normal direction. Let us consider a three-dimensional turbulence gust impinging on the cascade of aerofoils with its upwash velocity given by

(3.1)\begin{equation} w_g(X,Y,t)=\tilde{w}_g (k_X,k_Y) \exp({{\rm i} [k_X ( U_r t-X) + k_Y Y ]}) . \end{equation}

The gust–aerofoil interaction generates unsteady pressure disturbances at the surface of the aerofoil, which, in turn, radiates noise. The far-field pressure power spectral density is given by (Amiet Reference Amiet1989)

(3.2)\begin{align} S_{pp}^{(m)}(\boldsymbol{X}_o,\omega)&=U_r \left( \frac{k \rho Z_o b}{s_o^2} \right)^2 \int^{+\infty}_{-\infty} \int^{+\infty}_{-\infty} \varPhi_{ww} (k_X,k_Y,k_Z) \vert \mathcal{L}_a(k_X,k_Y) \vert^2 \nonumber\\ & \quad \times \exp({- {\rm i} m k_Z s_n }) \frac{\sin^2 \left( d \left( \dfrac{k Y_o}{s_o} - k_Y \right)\right) }{\left( \dfrac{k Y_o}{s_o}-k_Y \right)^2 } {\rm d}k_Y\,{\rm d}k_Z , \end{align}

where $\boldsymbol {X}_o$ denotes the observer position in the blade-fixed coordinate system, and $b$ and $d$ denote the aerofoil semi-chord and semi-span, respectively. The turbulence gust streamwise wavenumber is given by $k_X=\omega /U_r$, and $k_Y$ and $k_Z$ denote the spanwise and wall-normal wavenumbers respectively, with all three components defined in the blade-fixed coordinate system. The amplitude radius, $s_o$, is approximated to zeroth order by

(3.3)\begin{equation} s_o =X_o^2+\beta_r^2 \left( Y_o^2 + Z_o^2 \right) , \end{equation}

where $\beta _r=\sqrt {1-M_r^2}$. We have further introduced the turbulent upwash velocity spectrum in the blade-fixed coordinate system, $\varPhi _{ww}(k_X,k_Y,k_Z)$, and the blade-normal distance between two consecutive blades in the unwrapped cascade, $s_n$, as shown in figure 2. The aeroacoustic transfer function is given by

(3.4)\begin{equation} \mathcal{L}_a (k_X,k_Y) = \frac{2}{C} \int^{C/2}_{-C/2} g(X,k_X,k_Y) \exp({-{\rm i} \kappa ( M_r-X_0/s_o) X}) \,{\rm d}X . \end{equation}

Equation (3.4) differs from that introduced in § 2.2 in that it accounts for three-dimensional turbulence gusts. A detailed derivation of the aeroacoustic transfer function in the high-frequency regime can be found in Appendix C.1 of Christophe (Reference Christophe2011). For frequencies where the blade chord is compact we require another solution that is valid in the low-frequency regime, i.e. for $\omega C / 2 c_0 (1-M_r^2)<{\rm \pi} /4$ (Amiet Reference Amiet1975). In this case, we use the aerofoil transfer function given by Amiet (Reference Amiet1989).

Figure 2. Blade cascade diagram. Turbulent eddy path in the blade-fixed coordinate system highlighted with dashed red line. Figure adapted from Karve et al. (Reference Karve, Angland and Nodé-Langlois2018).

The far-field noise power spectral density given by (3.2) is further simplified if we consider an infinite-span aerofoil, in which case

(3.5)\begin{align} S_{pp}^{(m)}(\boldsymbol{X}_o,\omega)&=\left( \frac{k \rho Z_o b}{s_o^2} \right)^2 {\rm \pi}\,{\rm d} U_r \int^{+\infty}_{-\infty} \varPhi_{ww} (k_X,k_Y,k_Z) \vert \mathcal{L}_a(k_X,k_Y) \vert^2\nonumber\\ &\quad\times \exp({ - {\rm i} m k_Z s_n })\,{\rm d}k_Z, \end{align}

where the spanwise wavenumber is now given by $k_Y=k Y_o/s_o$.

3.2. Blade-to-blade correlation

In § 3.1, we considered the noise generated by a single blade in rectilinear motion. In this section, we incorporate the effects of noise source correlation between blades in the far-field noise prediction. These effects are more prominent when consecutive blades chop the same turbulent eddy as it travels through the rotor. This equates to satisfying the condition $B \varOmega \varLambda / U_z \gg 1$ (Glegg et al. Reference Glegg, Morton and Devenport2012; Glegg & Devenport Reference Glegg and Devenport2017).

3.2.1. Time between blade chops

Of particular importance to the model developed by Amiet (Reference Amiet1989) is the time, as seen by the observer, between blade chops. Figure 2 is particularly useful to a physical understanding of the quantities involved. The blade-normal distance between two blades is given by

(3.6)\begin{equation} s_n=s_b \sin \alpha . \end{equation}

The reader is reminded that $U_r=\sqrt {U_z^2+U_\gamma ^2}$ is the blade relative velocity, where $U_z$ is the axial flow velocity and $U_\gamma =\varOmega r$ is the tangential velocity. The distance between two consecutive blades in the unwrapped azimuthal direction can be expressed in terms of the velocity in this direction, $U_\gamma$, and the blade-passage time $T_B=2 {\rm \pi}/ B \varOmega$, as

(3.7)\begin{equation} s_b= U_{\gamma} T_B . \end{equation}

Substituting (3.7) in (3.6) and knowing that $\sin \alpha = U_z/U_r$ yields

(3.8)\begin{equation} s_n = \frac{U_\gamma U_z T_B}{U_r} . \end{equation}

The time between eddy chops is the time it takes for the wavefront of an eddy to travel from the leading edge of the first blade to the leading edge of the second blade. From figure 2, this distance is given by $s_b \cos \alpha$. Moreover, in the blade-fixed coordinate system the turbulent eddy travels in the downstream direction with velocity $U_r$. The time between eddy chops is thus given by

(3.9)\begin{equation} T_C= \frac{s_b \cos \alpha}{U_r} , \end{equation}

which, using (3.7), equates to

(3.10)\begin{equation} T_C= \frac{U_\gamma^2 T_B}{U_r^2} . \end{equation}

We must now calculate the time between blade chops from the perspective of the observer. Let us denote by $\tau ^{(0)}$ and $\tau ^{(-1)}$ the time taken for the acoustic waves to travel from blade 0 and blade $-1$ to the observer, respectively. The time between blade chops as seen by the observer, $T_O$, can then be expressed as

(3.11)\begin{equation} T_O=T_C+\tau^{(-1)}-\tau^{(0)} . \end{equation}

The time for sound waves to travel from the zeroth blade to the observer is, by definition, given by the phase radius divided by the speed of sound (Garrick & Watkins Reference Garrick and Watkins1953). Approximating the phase radius by retaining the zeroth-order terms of its Taylor expansion about the origin of the coordinate system yields

(3.12a,b)\begin{equation} \tau^{(0)}=\frac{s_o-M_r X_o}{\beta_r^2 c_0}, \quad \tau^{(-1)}=\frac{s_o'-M_r X_o'}{\beta_r^2 c_0}, \end{equation}

where the dash indicates observer coordinates measured in the coordinate system $(X',Y',Z')$ centred mid-chord and mid-span of blade $-1$. The relationship between the two sets of coordinates, expressed in vectorial form as $\boldsymbol {X}_o$ and $\boldsymbol {X}_o'$, is expressed as

(3.13)\begin{equation} \boldsymbol{X}_o=\boldsymbol{X}_o'+\boldsymbol{X}_{BB} , \end{equation}

where the separation between the two blades, as seen from figure 2, is given by

(3.14)\begin{equation} \boldsymbol{X}_{BB}=\left[ \sqrt{s_b^2-s_n^2},0,s_n\right]^{\rm T} . \end{equation}

We now turn to the calculation of $\tau ^{(-1)}$. The first step is to develop the amplitude radius $s_o'$, given by

(3.15)\begin{equation} s_o'= \sqrt{X_o'^2+\beta_r^2 \left( Y_o'^2+Z_o'^2 \right)} , \end{equation}

which after change of variables using (3.13) and Taylor expansions, retaining terms up to first order, yields

(3.16)\begin{equation} s_o'=s_o - \frac{U_\gamma T_B}{U_r s_o} \left( X_o U_\gamma + \beta_r^2 U_z Z_o \right) . \end{equation}

This result allows us to readily calculate $(\tau ^{(-1)}-\tau ^{(0)})$ from (3.12) and (3.13) as

(3.17)\begin{equation} \tau^{(-1)}-\tau^{(0)}= \frac{U_\gamma T_B}{U_r s_o \beta_r^2 c_0} \left[ U_\gamma \left( M_r s_o-X_o \right) - \beta_r^2 U_z Z_o \right] . \end{equation}

The time between blade chops as seen by the observer, $T_O$, can finally be evaluated by (3.11), together with (3.10) and (3.17). This is one of the key contributions of this paper.

3.2.2. Comparison with previous works

Let us compare (3.17) with the equivalent results of Paterson & Amiet (Reference Paterson and Amiet1979), Amiet (Reference Amiet1989) and Karve et al. (Reference Karve, Angland and Nodé-Langlois2018). Paterson & Amiet (Reference Paterson and Amiet1979) concluded that the time between blade chops as seen by the observer is given by $T_O=T_C$ and, thus, neglected the time difference that the acoustic waves take to propagate from each blade to the observer given here by (3.17). Amiet (Reference Amiet1989) extended the model of Paterson & Amiet (Reference Paterson and Amiet1979) to consider this aspect of the problem. In that report, the authors arrived at the following result for the acoustic wave propagation time difference

(3.18)\begin{equation} \tau^{(-1)}-\tau^{(0)}=\frac{\vert \Delta \boldsymbol{X}_o \vert}{\beta_r^2 c_0 s_o} \left(s_o M_r-X_o \right) , \end{equation}

where $\vert \Delta \boldsymbol {X}_o \vert =U_z^2 U_\gamma T_B / U_r^2$. This is in disagreement with our own calculation in (3.17). No clear derivation was presented, although it appears the authors have sought to consider the assumed rectilinear movement of the blades relative to the observer during the time between blade chops. In our derivation, this would manifest as an additional term in (3.13). Although in the first instance this seems to be a reasonable approach, it conflicts with the implicit assumption made in § 3.1 that the observer is fixed with respect to the cascade of aerofoils. The relative motion between the two is only addressed in § 3.3.

More recently Karve et al. (Reference Karve, Angland and Nodé-Langlois2018) presented their own result for the time between blade chops as heard by the observer. However, no detailed derivation was presented. Their result for the acoustic wave propagation time difference is given by

(3.19)\begin{equation} \tau^{(-1)}-\tau^{(0)}=\frac{-U_\gamma U_z T_B Z_o}{U_r c_0 s_o} , \end{equation}

which is equal to the second term of the result presented in this paper in (3.17). The authors appear to have only accounted for one of the components in the blade offset vector $\boldsymbol {X}_{BB}$. However, their overall approach of considering a fixed observer with respect to the cascade of aerofoils is in agreement with the present approach, while contrasting with that of Amiet (Reference Amiet1989). In § 4, we show numerical evidence that both their result and that of Amiet (Reference Amiet1989) lead to incorrect far-field pressure power spectral densities, failing to predict haystacking tones which are a prominent feature of turbulence ingestion noise spectra.

3.2.3. Far-field pressure spectrum

Paterson & Amiet (Reference Paterson and Amiet1979) argued that the far-field autocorrelation function consists of a series of peaks, where the $m$th peak represents the correlation between the noise originating in the zeroth blade and the noise originating in the $m$th blade. Mathematically, this is represented as

(3.20)\begin{equation} R_{pp}(\boldsymbol{X}_o,t)=\sum^{+\infty}_{m=-\infty} R_{pp}^{(m)} (\boldsymbol{X}_o,t+m T_O) , \end{equation}

where $R_{pp}^{(m)}$ denotes the cross-correlation between the far-field pressure generated by the zeroth and the $m$th blade, and $T_O$ is the time between blade chops as heard by the observer given by (3.11). The far-field pressure spectrum is obtained by taking the Fourier transform, yielding

(3.21)\begin{equation} S_{pp}(\boldsymbol{X}_o,\omega)=\sum^{+\infty}_{m=-\infty} S_{pp}^{(m)}(\boldsymbol{X}_o,\omega) \exp({{\rm i} m \omega T_O}) , \end{equation}

where $S_{pp}^{(m)}(\boldsymbol {X}_o,\omega )$ is given by (3.5) or (3.2), with and without an infinite-span wing assumption, respectively. Substituting (3.5) in (3.21), we obtain

(3.22)\begin{align} S_{pp}(\boldsymbol{X}_o,\omega)&= \sum^{+\infty}_{m=-\infty} \left( \frac{k \rho Z_o b}{s_o^2} \right)^2 {\rm \pi}U_r d \int^{+\infty}_{-\infty} \varPhi_{ww} (k_X,k_Y,k_Z) \vert \mathcal{L}_a(k_X,k_Y) \vert^2\nonumber\\ &\quad \times \exp({ {\rm i} m (-k_Z s_n + \omega T_O ) })\,{\rm d} k_Z . \end{align}

Using the delta Dirac train identity

(3.23)\begin{equation} \sum^{+\infty}_{m=-\infty} \exp({{\rm i} 2 {\rm \pi}m a}) = \sum^{+\infty}_{m=-\infty} \delta(m-a) , \end{equation}

we obtain

(3.24)\begin{equation} S_{pp}(\boldsymbol{X}_o,\omega)= \left( \frac{k \rho Z_o b}{s_o^2} \right)^2 \frac{2 {\rm \pi}^2 U_r d}{s_n} \vert \mathcal{L}_a(k_X,k_Y) \vert^2 \sum^{+\infty}_{m=-\infty} \varPhi_{ww} (k_X,k_Y,k_Z) , \end{equation}

where $k_X=\omega /U_r$, $k_Y=k Y_o/s_o$ and $k_Z= (\omega T_O - 2 {\rm \pi}m)/ s_n$. Alternatively, if blade-to- blade correlation effects are negligible, i.e. if the condition $(B \varOmega \varLambda / U_z) \gg 1$ is not respected (Glegg et al. Reference Glegg, Morton and Devenport2012; Glegg & Devenport Reference Glegg and Devenport2017), the far-field power spectral density is simply given by (3.5) with $m=0$.

For homogeneous isotropic turbulence, the full energy spectrum is given by (2.32). The turbulence upwash velocity spectrum in the blade-fixed coordinate system is readily obtained by considering that isotropic turbulence has invariant properties under rotation,

(3.25)\begin{equation} \varPhi_{ww}(k_X,k_Y,k_Z)= A\frac{\hat{k}_X^2+\hat{k}_Y^2}{\left(1+ \hat{k}_X^2+\hat{k}_Y^2+\hat{k}_Z^2\right)^{17/6}} , \end{equation}

where $A$ is given by (2.33).

3.3. Noise radiated by rotating blades

The far-field pressure spectrum in (3.24) corresponds to the noise generated by a cascade of aerofoils in rectilinear motion. In the context of Amiet's theory (Amiet Reference Amiet1989), thoroughly and clearly described in Sinayoko et al. (Reference Sinayoko, Kingan and Agarwal2013), it represents an approximation to the instantaneous power spectral density generated by a strip of rotating blades at a given azimuthal angle. Amiet argued that this is a suitable approximation when the source frequency is much larger than the rotor angular velocity, i.e. $\omega \gg \varOmega$. Obtaining the total far-field pressure spectrum requires:

1. (i) applying a Doppler effect correction to account for the movement of the observer relative to the blade;

2. (i1) averaging the instantaneous power spectral density over all azimuthal angles;

3. (iii) summing the contributions of each blade strip.

Sinayoko et al. (Reference Sinayoko, Kingan and Agarwal2013) provides a complete description of steps (i) and (ii), which we will not reproduce here. A summary of the key results is given instead. Step (iii) alludes to the use of strip theory, where rotor blades are divided into small strips and the total far-field power spectral density is given by the sum of the individual, uncorrelated contributions of each strip.

The time-averaged power spectral density from the $j$th strip element of rotating blades is given by (Sinayoko et al. Reference Sinayoko, Kingan and Agarwal2013)

(3.26)\begin{equation} S_{pp}^{(\,j)}(\boldsymbol{x}_o,\omega)=\frac{1}{2 {\rm \pi}} \int^{2 {\rm \pi}}_{0} \left( \frac{\omega'}{\omega} \right)^2 S_{pp}^{(\,j)}(\boldsymbol{X}_o,\omega',\gamma) \,{\rm d} \gamma , \end{equation}

where $S_{pp}^{(\,j)}(\boldsymbol {X}_o,\omega ',\gamma )$ is the power spectral density given by (3.24) or (3.5), for cases with or without blade-to-blade correlation effects, respectively. We have now made the power spectral density an explicit function of the azimuth angle and the blade strip number; the latter serves to acknowledge the dependence of the results obtained in § 3.2.3 on the radial position as a result of varying geometric properties and flow conditions. The factor $\omega '/\omega$ is the Doppler-shift correction for each azimuth angle and each blade strip, as given by

(3.27)\begin{equation} \frac{\omega}{\omega'}=1+\frac{\boldsymbol{M}_{BO} \boldsymbol{\cdot} \widehat{\boldsymbol{CO}}}{1+\left(\boldsymbol{M}_{FO}-\boldsymbol{M}_{BO} \right) \boldsymbol{\cdot} \widehat{\boldsymbol{CO}}} , \end{equation}

where $\boldsymbol {M}_{BO}$ and $\boldsymbol {M}_{FO}$ are the blade and fluid Mach number vectors relative to the observer,

(3.28a,b)\begin{equation} \boldsymbol{M}_{BO}=\left[ - M_\gamma \sin \gamma ,M_\gamma \cos \gamma,0\right]^{\rm T}, \quad \boldsymbol{M}_{FO}=\left[ 0,0,-M_z\right]^{\rm T}, \end{equation}

where $M_\gamma =U_\gamma /c_o$, and $\widehat {\boldsymbol {CO}}$ is the unit vector from the convected source position, $\boldsymbol {x}_c$, to the observer position, $\boldsymbol {x}_o$,

(3.29)\begin{equation} \widehat{\boldsymbol{CO}} = \frac{\boldsymbol{x}_o-\boldsymbol{x}_c}{ \vert \boldsymbol{x}_o-\boldsymbol{x}_c \vert } , \end{equation}

with $\boldsymbol {x}_c=\boldsymbol {M}_{FO} R_p$. The propagation distance, $R_p$, is, in turn, given by

(3.30)\begin{equation} R_p=\frac{\vert \boldsymbol{x}_o \vert \left(-\vert \boldsymbol{M}_{FO} \vert \cos \varTheta + \sqrt{1-\vert \boldsymbol{M}_{FO} \vert^2 \sin^2 \varTheta} \right)}{1-\vert \boldsymbol{M}_{FO} \vert^2} , \end{equation}

where $\varTheta$ denotes the angle between $\boldsymbol {M}_{FO}$ and $\boldsymbol {x}_o$. It remains to calculate the observer position in the blade-fixed coordinate system $\boldsymbol {X}_o$. From Sinayoko et al. (Reference Sinayoko, Kingan and Agarwal2013), we know that

(3.31)\begin{equation} \boldsymbol{X}_o=\underline{\textit{R}}_y(\alpha) \underline{\textit{R}}_z({\rm \pi}/2-\gamma) \left(\boldsymbol{x}_o-\boldsymbol{x}_p \right) , \end{equation}

where $\underline {\textit {R}}_y$ and $\underline {\textit {R}}_z$ are rotation matrices given by

(3.32a,b)\begin{equation} \underline{\textit{R}}_z(\theta)= \begin{pmatrix} \cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0 & 0 & 1 \end{pmatrix} , \quad \underline{\textit{R}}_y(\theta)= \begin{pmatrix} \cos \theta & 0 & -\sin \theta\\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \end{pmatrix} , \end{equation}

and $\boldsymbol {x}_p$ is the present source position, approximated by

(3.33)\begin{equation} \boldsymbol{x}_p=\boldsymbol{M}_{BO} R_p . \end{equation}

We now have all the information to calculate (3.26). It remains to sum up the power spectral density contributions of each blade strip. The blades are divided into $N_s$ segments following a logarithmic distribution with clustering in the blade tips. The flow and geometric quantities of each strip needed to calculate the far-field pressure spectrum are given by their values at the centre of the strip. The total power spectral density is then given by the sum of the individual contributions in (3.26), as

(3.34)\begin{equation} S_{pp}(\boldsymbol{x}_o,\omega)= B \sum^{N_s}_{j=1} S_{pp}^{(\,j)}(\boldsymbol{x}_o,\omega) . \end{equation}