2.1. Wake modelling

The unsteady loading on the downstream propeller blades at radius, r, is calculated using an equivalent two-dimensional problem in which the wakes from an upstream cascade of blades interact with the blades of a downstream cascade. This situation is illustrated in figure 2. The upstream and downstream blades translate vertically downwards and upwards at Mach numbers $\varOmega _1r/c_0$ and $\varOmega _2r/c_0$, respectively. At time $\tau = 0$ the pitch-change axis of the front rotor reference blade is aligned with the pitch-change axis of the rear rotor reference blade at $y = 0$ and the vertical spacing between the mid-chord positions of the blades on each cascade is equal to $2{\rm \pi} r/B$. The blades are modelled as infinitely thin flat plates which are aligned with the local flow direction but otherwise have identical characteristics (chord length, sweep, lean and drag coefficient) to the actual rotor blade at that particular radius. Also, the effect of the flow induced by the rotors is neglected such that the stagger angle, $\alpha $, of each blade is defined by $\tan \alpha = zM_T/M_x$, where $z = 2r/D$ and $M_T = \varOmega D/2c_0$. The Mach number of the blades relative to the air flow is $M_r = \sqrt {M_x^2 + z^2M_T^2 } $ with corresponding relative velocity $U_r = M_rc_0$.

Figure 2. Schematic of the equivalent two-dimensional cascade problem (a). Sweep definition and velocity triangle (b).

The reference blade of the upstream cascade produces a wake with mean deficit velocity ${\bar{u}}^{\prime}$ aligned with the negative $X_1$-direction (the chordwise direction – see figure 2) which will be modelled using the two-dimensional Gaussian model defined in Parry (Reference Parry1988, Reference Parry1997):

(2.1)\begin{equation}{\bar{u}}^{\prime}(r,Y_1) = 2U_{r_1}\sqrt {\displaystyle{{\ln 2} \over {\rm \pi}}} \sqrt {\displaystyle{{c_1C_{D_1}} \over {L_1}}} \; \exp \left\{ {-\displaystyle{{\ln 2} \over {b_{1/2}^2 }}Y_1^2 } \right\},\end{equation}

where $Y_1$ is the chord-normal coordinate of the upstream rotor blade, $L_1 = [gM_{r_1}/M_x-s_1 + s_LM_{r_1}/M_{r_2}]$ is the length (or convection distance) of the wake which is defined as the distance along the helical path which the upstream blades trace through the air measured from the mid-chord of the upstream reference blade to the axial location of the leading edge of the downstream propeller blades, $s_L = s_2-c_2/2$ is the leading-edge sweep of the downstream propeller blade (see figure 2) and $b_{1/2}$ is the wake half-width which is defined as $b_{1/2} = {\textstyle{1 \over 4}}\sqrt {C_{D_1}c_1L_1} $.

Because the upstream blades are identical and evenly spaced, the periodic velocity deficit of the upstream cascade of blades in the unwrapped cascade representation is given by

(2.2)\begin{equation}{\bar{v}}^{\prime} = \mathop \sum \limits_{n = {-}\infty }^\infty {\bar{u}}^{\prime}\left( {r,Y_1 + \displaystyle{{2{\rm \pi} r\cos \alpha_1} \over {B_1}}} \right).\end{equation}

Because ${\bar{v}}^{\prime}$ is periodic in $Y_1$, Poisson's summation formula can be used to convert this expression to a sum of convected harmonic gusts:

(2.3)\begin{equation}{\bar{v}}^{\prime} = \displaystyle{{B_1U_{r_1}C_{D_1}c_1G_{n_1}} \over {4{\rm \pi} r\cos \alpha _1}}\mathop \sum \limits_{n = {-}\infty }^\infty \exp \left\{ {{\rm i}\displaystyle{{nB_1} \over {r\cos \alpha_1}}Y_1} \right\},\end{equation}

where

(2.4)\begin{equation}G_{n_1} = \exp \left\{ {-{\left[ {\displaystyle{{n_1B_1b_{1/2}M_{r_1}} \over {zDM_x\sqrt {\ln 2} }}} \right]}^2} \right\}.\end{equation}

In order to calculate the unsteady loading on the downstream blade row, the velocity deficit incident on the chord line of the reference blade on the downstream cascade should be expressed in terms of the chordwise coordinate of that blade $X_2$ (shown in figure 2). The $X_2$ coordinate is related to the $Y_1$ coordinate by the following expression:

(2.5)\begin{equation}Y_1 = (\varOmega _1 + \varOmega _2)r\tau \cos \alpha _1-g\sin \alpha _1-(X_2 + s_L)\sin (\alpha _1 + \alpha _2).\end{equation}

The upstream rotor wake deficit velocity is aligned with the ${-}X_1$ coordinate and therefore the mean upwash velocity (which is the component of velocity in the $Y_2$ direction) onto the downstream reference blade is given by $w = {-}\sin (\alpha _1 + \alpha _2){\bar{v}}^{\prime}$. Substituting (2.5) into (2.4) yields

(2.6)\begin{equation}w = \mathop \sum \limits_{n_1 = {-}\infty }^\infty w_{n_1}\exp \{ {\rm i}k_X(U_{r_2}\tau -X_2)\} ,\end{equation}

where

(2.7)\begin{equation}w_{n_1} = {-}\; \displaystyle{{B_1C_{D_1}c_1U_{r_1}(M_{T_1} + M_{T_2})G_{n_1}} \over {2{\rm \pi} DM_{r_2}}}\exp \left\{ {-{\rm i}k_Xs_L-{\rm i}n_1B_1\displaystyle{{2g} \over D}\displaystyle{{M_{T_1}} \over {M_x}}} \right\},\end{equation}

and

(2.8)\begin{equation}k_X = \displaystyle{{2n_1B_1} \over {DM_{r_2}}}(M_{T_1} + M_{T_2}).\end{equation}

As mentioned above, the approach adopted here assumes that the convection of the wakes from the upstream propeller is not affected by the swirl and induced axial and radial velocity between the propellers. Clearly, the wakes produced by a practical propeller will in fact deform due to the induced velocities. It is, however, not our intention to address these additional wake deformation effects in this paper. Rather, the purpose of this paper is to present a framework for analysing the noise generated by blade-wake interactions and to show how the geometry of the downstream blades can be used to alter the timing of the blade-wake interactions at different radii and thus the level of noise which is produced. It is also important to note that, from an aerodynamic point of view, our approach follows the helicoidal surface theory of Hanson (Reference Hanson1980, Reference Hanson1983, Reference Hanson1985) in which the effects of induced flow – both axial and circumferential – are neglected. That approach has been shown to produce good agreement with measured data (see, for example, Parry & Crighton Reference Parry and Crighton1989; Parry Reference Parry1997). The effects of induced velocity are straightforward to include and methods similar to the one presented here, but which include these effects, have been validated against experimental and numerical data in Kingan et al. (Reference Kingan, Ekoule, Parry and Britchford2014) and Ekoule et al. (Reference Ekoule, McAlpine, Parry, Kingan and Sohoni2017).

2.2. Unsteady loading

The total unsteady lift force per unit area acting on the chord line of the reference blade can be expressed as the sum of the unsteady ‘responses’ of the blade to each upwash harmonic, i.e.

(2.9)\begin{equation}\Delta p = \mathop \sum \limits_{n_1 = 1}^\infty 2\; \Re \{ \Delta p_{n_1}\} ,\end{equation}

where ${{\rm \Delta} }p_{n_1}$ is the response of the reference blade to a harmonic gust of the form $w_{n_1}\exp \{ {\rm i}k_X(U_{r_2}\tau -X_2)\} $ which is given by

(2.10)\begin{equation}\Delta p_{n_1} = 2{\rm \pi} \rho _0U_{r_2}w_{n_1}{\mathbb S}(\sigma _2,M_{r_2},\bar{X}_2,\varLambda )\exp \left\{ {{\rm i}2{\rm \pi} n_1\displaystyle{\tau \over T}} \right\},\; \end{equation}

where $T = 2{\rm \pi} /[B_1(\varOmega _1 + \varOmega _2)]$ is the period of the loading, $\bar{X}_2 = 2X_2/c_2$ is a dimensionless chordwise coordinate, $\sigma _2 = k_Xc_2/2$ is the reduced frequency and ${\mathbb S}$ is a non-dimensional response function. In this paper we follow the approach used in Kingan & Parry (Reference Kingan and Parry2019a) and use the high-frequency response function of Adamcyzk (Reference Adamczyk1974a,Reference Adamczykb). In this approach it is assumed that the response of the reference blade to a gust of the form $w_{n_1}\exp \{ {\rm i}k_X(U_{r_2}\tau -X_2)\} $ is equal to that of a two-dimensional, semi-infinite flat plate immersed in a flow of velocity $U_{r_2}$ with a leading-edge sweep angle set such that the spanwise trace velocity of each gust along the leading edge is equal to the spanwise trace velocity of the wake centreline along the leading edge of the reference blade of the rotor. The equivalent problem is shown in figure 3. Because the gust convects with the flow, the spanwise ($Z_2$) component of the trace Mach number of the gust along the leading edge is given by $M_{r_2}\cot \varLambda $.

Figure 3. Swept flat-plate coordinate definitions.

At radius r, the time $\tau = \tau _0(r) \gt 0$ at which the wake centreline from the reference blade on the upstream cascade, defined by $Y_1 = 0$, first impinges on the leading edge of the downstream reference blade, located at $X_2 = 0$, can be determined by setting $X_2 = Y_1 = 0$ in (2.5) and rearranging to yield

(2.11)\begin{equation}\tau _0(r) = \displaystyle{1 \over {c_0}}\left[ {\displaystyle{{gM_{T_1}} \over {M_x(M_{T_1} + M_{T_2})}} + S_L(r)} \right]\; ,\end{equation}

where, for convenience, we have introduced $S_L(r) = s_L/M_{r_2}$. The spanwise trace Mach number of this impingement point can be evaluated by taking the derivative of $\tau _0(r)$ with respect to r, rearranging and making use of the inverse function theorem to yield

(2.12)\begin{equation}M_t = \displaystyle{1 \over {c_0}}\displaystyle{{{\rm d}r} \over {{\rm d}\tau _0}} = \displaystyle{1 \over {S^{\prime}_L}}.\end{equation}

The blade leading-edge sweep angle in the equivalent flat-plate response problem, $\varLambda $, is set such that $M_t = M_{r_2}\cot \varLambda $ and thus

(2.13)\begin{equation}\tan \varLambda = M_{r_2}{S}^{\prime}_L.\end{equation}

Kingan & Parry (Reference Kingan and Parry2019a) express Adamczyk's high-frequency, isolated aerofoil response function in the following forms:

(2.14)\begin{equation}{\mathbb S}(\sigma _2,M_{r_2},\bar{X}_2,\varLambda ) = \displaystyle{{\exp \left\{ {-{\rm i}{\rm \pi} /4 + {\rm i}\sigma_2\displaystyle{{\left[ {(M_{r_2}^2 -{\tan }^2\varLambda )-\sqrt {M_{r_2}^2 -{\tan }^2\varLambda } } \right]} \over {[1-(M_{r_2}^2 -{\tan }^2\varLambda )]}}{\bar{X}}_2} \right\}} \over {{\rm \pi} \sqrt {{\rm \pi} \sigma _2{\bar{X}}_2} \sqrt {1 + \sqrt {M_{r_2}^2 -{\tan }^2\varLambda } } }}\; ,\; \end{equation}

for $M_t \gt 1\; $ (so-called super-critical gusts), and

(2.15)\begin{align} {\mathbb S}(\sigma _2,M_{r_2},{\bar{X}}_2,\varLambda )& = \exp \left\{ {-{\rm i}\displaystyle{{\rm \pi} \over 4} + \displaystyle{{\rm i} \over 2}{\tan }^{{-}1}\sqrt {{\tan }^2\varLambda -M_{r_2}^2 } } \right\} \cr &\quad \times \displaystyle{{\exp \left\{ {\sigma_2\displaystyle{{\left[ {-{\rm i}({\tan }^2\varLambda -M_{r_2}^2 )-\sqrt {{\tan }^2\varLambda -M_{r_2}^2 } } \right]} \over {(1 + {\tan }^2\varLambda -M_{r_2}^2 )}}{\bar{X}}_2} \right\}} \over {{\rm \pi} \sqrt {{\rm \pi} \sigma _2{\bar{X}}_2} {[1 + ({\tan }^2\varLambda -M_{r_2}^2 )]}^{1/4}}}\; , \end{align}

for $M_t \lt 1$ (so-called sub-critical gusts).

The accuracy of this response function in relation to the problem considered here is discussed in detail in appendix A of Kingan & Parry (Reference Kingan and Parry2019b). That paper considered a set of almost identical contra-rotating propellers to those considered here (the only significant difference being the blade numbers and a slight change in the tip Mach numbers). It was shown that the effect of including a correction to enforce the unsteady Kutta condition at the trailing edge had a negligible effect on the total unsteady loading on the propeller blade. Nevertheless, it is acknowledged that the response function used here is approximate and will not model the pressure jump on the blade surface close to the trailing edge accurately. This fact should be borne in mind when interpreting results presented later in the paper which depend on (2.15) for the chordwise distribution of loading (namely figures 6 and 8). Kingan & Parry (Reference Kingan and Parry2019b), following the analysis of Amiet (Reference Amiet1976), also showed that the high-frequency response function was expected to produce inaccurate results when the dimensionless parameter $|\kappa |\lt {\rm \pi}/4$, where

(2.16)\begin{equation}\kappa \equiv \displaystyle{{\sigma _2\cos \varLambda \sqrt {M_{r_2}^2 {\cos }^2\varLambda -{\sin }^2\varLambda } } \over {(1-M_{r_2}^2 {\cos }^2\varLambda )}}.\end{equation}

This criterion is only satisfied in the examples presented in this paper for the case presented in figure 15, for which the gust trace velocity is sonic across the blade leading edge.

The effect of the spanwise trace Mach number on the sound field radiated from the leading edge of an aerofoil due to its interaction with a convected harmonic gust is well known. This phenomenon has been studied extensively in the context of sound radiated from the leading edge of an aerofoil immersed in a turbulent flow. For example, Amiet (Reference Amiet1975) presented formulae for calculating the sound radiated from a flat-plate aerofoil with infinite span. In this case, the radiated sound field only depends on gusts with supersonic wavenumbers. However, Amiet noted that the surface pressure spectrum on the aerofoil is dependent on both the sub- and supercritical gusts. Later studies, for example by Roger & Serafini (Reference Roger and Serafini2005) and Roger & Moreau (Reference Roger and Moreau2010), have extended Amiet's work to include the effect of finite span and the contribution of the subcritical gusts. Note, however, that all of these papers addressed interactions on non-rotating aerofoils.

2.3. Acoustic radiation

The far-field acoustic pressure at location $\boldsymbol{x}$ and time t produced by a thin rotating blade immersed in a flow can be calculated using a slightly modified form of (3.19) in Najafi-Yazdi et al. (Reference Najafi-Yazdi, Brés and Mongeau2011):

(2.17)\begin{equation}p(\boldsymbol{x},t)\cong \displaystyle{1 \over {4{\rm \pi} c_0}}\displaystyle{\partial \over {\partial t}}\int_{R_h}^{R_t} {\int_0^{c_2} {\left[ {\displaystyle{{{{\rm \Delta} }p(X_2,r,\tau )\tilde{\boldsymbol{R}}\boldsymbol{\cdot }\boldsymbol{n}} \over {R^\ast (1-\boldsymbol{M}\boldsymbol{\cdot }\tilde{\boldsymbol{R}})}}} \right]} } _{\tau = t-R/c_0}{\rm d}X_2{\rm d}r.\end{equation}

This model assumes that the observer and rotating blade are immersed in a flow of Mach number $M_x$ in the negative $x_1$ direction where $R = [R^\ast{ + } M_x(x_1-y_1)]/\; (1-M_x^2 )$ is described by Najafi-Yazdi et al. as the acoustic distance between the source and receiver positions, $R^\ast{ = } \sqrt {{(x_1-y_1)}^2 + (1-M_x^2 )[{(x_2-y_2)}^2 + {(x_3-y_3)}^2]} $ and the radiation vector $\tilde{\boldsymbol{R}} = \tilde{R}_1\boldsymbol{e}_1 + \tilde{R}_2\boldsymbol{e}_2 + \tilde{R}_3\boldsymbol{e}_3$, where

(2.18a–c)\begin{equation}\tilde{R}_1 = \displaystyle{{(x_1-y_1) + M_xR^\ast } \over {R^\ast (1-M_x^2 )}},\quad \tilde{R}_2 = \displaystyle{{(x_2-y_2)} \over {R^\ast }},\quad \tilde{R}_3 = \displaystyle{{(x_3-y_3)} \over {R^\ast }},\end{equation}

and $\boldsymbol{e}_1$, $\boldsymbol{e}_2$ and $\boldsymbol{e}_3$ represent unit vectors in the $x_1$, $x_2$ and $x_3$ directions, respectively. The source position is defined in Cartesian coordinates as

(2.19a–c)\begin{equation}y_1 = {-}\displaystyle{{M_x} \over {M_{r_2}}}(s_L + X_2),\quad y_2 = r\cos \varPhi ,\quad y_3 = r\sin \varPhi ,\end{equation}

where $\varPhi = \varOmega \tau -(2M_{T_2}/DM_{r_2})(s_L + X_2)$.

The receiver positions are defined in Cartesian coordinates as $x_1 = R_r\cos \theta _r$, $x_2 = R_r\sin \theta _r\cos \phi $, and $x_3 = R_r\sin \theta _r\sin \phi $, where $R_r$ is the reception distance, $\theta _r$ is the reception polar angle and $\phi $ is the azimuthal angle of the observer.

The unit vector aligned with the local force exerted by the blade on the fluid is defined as

(2.20)\begin{equation}\boldsymbol{n} = {-}\displaystyle{{zM_{T_2}} \over {M_{r_2}}}\boldsymbol{e}_1-\displaystyle{{M_x} \over {M_{r_2}}}\sin \varPhi \boldsymbol{e}_2 + \displaystyle{{M_x} \over {M_{r_2}}}\cos {\Phi }\boldsymbol{e}_3,\end{equation}

and the Mach number of the source is defined as

(2.21)\begin{equation}\boldsymbol{M} = {-}\boldsymbol{e}_2zM_{T_2}\sin \varPhi + \boldsymbol{e}_3zM_{T_2}\cos \varPhi .\end{equation}

Each of the terms in the square brackets within the integrand in (2.17) is evaluated at the source (or retarded) time $\tau = t-R/c_0$.

In this paper, we will also make use of ‘emission coordinates’ to define the position of the observer. These emission coordinates, defined by the radiation distance $R_e$ and polar angle $\theta _e$, are related to the ‘reception coordinates’ via $R_r\cos \theta _r = R_e(\cos \theta _e-M_x)$ and $R_r\sin \theta _r = R_e\sin \theta _e$. In particular, the ‘emission radius’, $R_e$, is equal to the ‘radiation distance’, R, for a source located at the origin of the coordinate system at the ‘emission time’ (the time at which sound is emitted). The ‘emission polar angle’, $\theta _e$, is equal to the angle between the radiation vector, $\tilde{\boldsymbol{R}}$, and the $x$-axis and corresponds to the polar angle at which a ‘ray’, emitted from the origin and travelling at sonic velocity relative to the fluid, will reach the observer position.

3.1. Wake interaction and blade response

For a blade with zero leading-edge sweep ($s_L = 0$, $\varLambda = 0$ along the blade span), the trace Mach number of the gust across the leading edge is infinite ($M_t\to \infty $) as the wake centreline impinges on the leading edge at the same instant at all points across the blade span. For such a case, the aerofoil response is identical to Landahl's (Reference Landahl1961) two-dimensional, high-frequency, isolated aerofoil response function which can be expressed as

(3.1)\begin{equation}{\rm \Delta} p_{n_1} = \displaystyle{{\rho _0c_0^2 B_1C_{D_1}c_1M_{r_1}(M_{T_1} + M_{T_2})G_{n_1}} \over {{\rm \pi} D{[{\rm \pi} k_X^{(n_1)} (1 + M_{r_2})X_2]}^{0.5}}}\exp \left\{ {{\rm i}k_X^{(n_1)} U_{r_2}(\tau -\tau^\ast )-{\rm i}\displaystyle{{\rm \pi} \over 4}} \right\},\end{equation}

where $\tau ^\ast (X_2) = \tau _0 + X_2/(c_0 + U_{r_2})$, which can be interpreted as follows: after the upstream rotor reference blade wake centreline impinges on the leading edge of the downstream blade at time $\tau _0$, consistent with the two-dimensional response assumption, a pressure pulse is generated which moves downstream along the surface of the blade at speed $c_0 + U_{r_2}$. Thus $\tau ^\ast (X_2)$ represents the time at which this pulse reaches the chordwise position $X_2$.

3.2. Acoustic pressure radiated from a single radius

In order to interpret physically the pressure field radiated from the rotor blade, (2.17) is rewritten as a radial integral of a pressure per unit span function $P(r;\boldsymbol{x},t)$, i.e.

(3.2)\begin{equation}p(\boldsymbol{x},t)\cong \int_{R_h}^{R_t} {P(r;\boldsymbol{x},t)\,{\rm d}r} ,\end{equation}

where

(3.3)\begin{equation}P(r;\boldsymbol{x},t) = \displaystyle{1 \over {4{\rm \pi} c_0}}\displaystyle{\partial \over {\partial t}}\int_0^{c_2} {{\left[ {\displaystyle{{{{\rm \Delta} }p(X_2,r,\tau )\tilde{\boldsymbol{R}}\boldsymbol{\cdot }\boldsymbol{n}} \over {R^\ast (1-\boldsymbol{M}\boldsymbol{\cdot }\tilde{\boldsymbol{R}})}}} \right]}_{\tau = t-R/c_0}} \,{\rm d}X_2.\end{equation}

Figure 5(a) plots $P(r;\boldsymbol{x},t)$, radiated from radius $r = 0.7R_t$ against time for this case (zero leading-edge sweep). The figure also indicates the pressure at the observer times

(3.4)\begin{equation}t_n(r) = \tau _n + \displaystyle{{R_L(r,\tau _n)} \over {c_0}},\quad \tau _n = \tau _0 + nT,\quad n\in {\mathbb Z},\end{equation}

which are the times where the sound emitted from the leading edge of the blade, at the instant the centreline of the nth wake impinges on the leading edge at that radius, arrives at the observer location. In (3.4), $R_L(r,\tau _n)$ is the radiation distance R (defined in § 2.3) from the leading edge (located at $X_2 = 0$) at radius r to the observer position at source time $\tau _n$. From figure 5 it is observed that the times $t_n$ correspond well with the peaks in the pressure time history. Figure 5(b) also plots the wake and blade locations at different values of $\tau _n$, where it should be noted that $\tau _n$ is constant along the blade span for a blade with no lean or leading-edge sweep. Several interesting features are evident in the pressure time history:

1. (i) The time spacing between peaks is strongly dependent on the location of the rotor blade relative to the observer when the noise is produced. When the blade is moving towards the observer, the peaks are more closely spaced together, whilst the peaks are more spread out when the blade is moving away.

2. (ii) The sign of the peak pressure is positive when the blade moves towards the observer (when the pressure surface faces the observer) and negative when it moves away (when the suction surface is facing the observer).

3. (iii) The amplitudes of the peaks are affected by convective amplification and are strongly dependent on the location of the rotor blade relative to the observer. When the blade is travelling towards the observer, the peaks are of significantly higher amplitude. Conversely, when the blade is moving away from the observer the peak amplitude is significantly lower.

All of these features represent differences in the acoustic field dependent on the direction of blade motion relative to the observer. For rectilinear motion they would be referred to simply as Doppler effects but, as they relate to variations in blade motion relative to the observer, we can use terminology common in the field of radar and describe them as micro Doppler effects which have been discussed by, for example, Van Bladel (Reference Van Bladel1976), Chen et al. (Reference Chen, Li, Ho and Wechsler2006) and Chen (Reference Chen2019). However, to the authors’ knowledge, there is little in the literature on these effects as they relate to turbomachinery or propeller noise.

The integrand in (3.3), which defines $P(r;\boldsymbol{x},t)$, contains the product of ${{\rm \Delta} }p(X_2,r,\tau )$, which varies impulsively with time due to the interaction of the rear blade with each of the narrow wakes from the front propeller (with this interaction taking place over a very short timescale), and the remaining terms which vary relatively slowly (with a period of one revolution of the blade). Thus we can reasonably approximate this expression by moving the partial derivative with respect to t inside the integral (see Farassat & Succi Reference Farassat and Succi1983) and applying it only to ${{\rm \Delta} }p$ yields

(3.5)\begin{equation}P(r;\boldsymbol{x},t)\approx \displaystyle{1 \over {4{\rm \pi} c_0}}\int_0^{c_2} {{[G(X_2,r,\tau;\boldsymbol{x}){{\rm \Delta} }\dot{p}({X_2,r,\tau } )]}_{\tau = t-R/c_0}} \,{\rm d}X_2,\end{equation}

where ${{\rm \Delta} }\dot{p}$ is the partial derivative with respect to $\tau $ of ${{\rm \Delta} }p$ and

(3.6)\begin{equation}G(X_2,r,\tau;\boldsymbol{x}) = \displaystyle{{\tilde{\boldsymbol{R}}\boldsymbol{\cdot }\boldsymbol{n}} \over {R^\ast {(1-\boldsymbol{M}\boldsymbol{\cdot }\tilde{\boldsymbol{R}})}^2}}\; .\end{equation}

Figure 6 plots contours of constant ${{\rm \Delta} }\dot{p}(X_2,r,\tau )$ versus $2X_2/c_2$ (vertical axis) and $(\tau -\tau _n)/T$ (horizontal axis) at $r = 0.7R_t$. Close to the leading edge ($X_2\to 0^ + $) ${{\rm \Delta} }\dot{p}$ has impulsive peaks at times $\tau _n = \tau _0 + nT$ (where $n\in {\mathbb Z}$) which is the time at which a wake centreline crosses the leading edge. Downstream of the leading edge ($X_2 \gt 0$) ${{\rm \Delta} }\dot{p}$ has a peak value close to $\tau _n^\ast{ = } \tau _n + X_2/(c_0 + U_{r_2})$ which represents the time at which the pulse generated at the leading edge at time $\tau _n$ reaches the chordwise position $X_2$ and which is shown by the red line superimposed on the plot.

Figure 6. Contours of constant ${{\rm \Delta} }\dot{p}(X_2,r,\tau )$ plotted against $\bar{X}_2$ (vertical axis) and $(\tau -\tau _n)/T$ (horizontal axis) at $r = 0.7R_t$. Note that ${{\rm \Delta} }\dot{p}(X_2,r,\tau )$ is singular at $X_2 = 0$ and the function is only plotted for $2X_2/c_2 \ge 0.01$. The red dashed line denotes $\tau _n^\ast $.

Figure 7 plots $G(X_2,r,\tau;\boldsymbol{x})$ versus normalised chordwise distance $2X_2/c_2$ (vertical axis) and normalised source time $(\tau -\tau _0)/T$ (horizontal axis) at $r = 0.7R_t$. As expected, the function G varies slowly over one blade revolution and has maximum magnitude when the blade is moving towards the observer ($(t-t_0)/T\approx 10$) due in part to the micro Doppler amplification term $(1-\boldsymbol{M}\boldsymbol{\cdot }\tilde{\boldsymbol{R}})^{{-}2}$ and also to the directivity term $\tilde{\boldsymbol{R}}\boldsymbol{\cdot }\boldsymbol{n}$ which is positive when the net loading exerted by the blade on the air points towards the observer, is negative when it points away and is zero when they are orthogonal. Clearly, the variation in magnitude of the micro Doppler amplification term over a blade revolution will increase as the propeller tip Mach number increases. For propellers with high subsonic tip Mach numbers, the micro Doppler amplification will significantly increase the magnitude of impulses emitted when the blade moves towards the observer and significantly reduce the magnitude of the impulses when the blade moves away from the observer. In the far field, the term $R^\ast{\sim} R_e(1-M_x\cos \theta _e)$ which remains constant.

Figure 7. Contours of constant $G(X_2,r,\tau;\boldsymbol{x})$ plotted against $2X_2/c_2$ (vertical axis) and $(\tau -\tau _0)/T$ (horizontal axis) at $r = 0.7R_t$.

Figure 8(a) plots the integrand in (3.5), $[G(X_2,r,\tau;\boldsymbol{x}){{\rm \Delta} }\dot{p}(X_2,r,\tau )]_{\tau = t-R/c_0}$, versus $2X_2/c_2$ (vertical axis) and normalised observer time $(t-t_0)/T$ (horizontal axis) at $r = 0.7R_t$ for one blade revolution. Figure 8(b) is a replica of figure 8(a) but with red curves indicating $t_n^\ast = \tau _n^\ast{ + } R/c_0$ superimposed. It can be seen that these curves match closely the locations of the local peak values of the integrand. The function G modulates the impulsive ${{\rm \Delta} }\dot{p}$ function so that the peak magnitude of the integrand is large and positive as the rear blade moves toward the observer ($(t-t_0)/T = 10$) and is relatively smaller and negative as the rear blade moves away from the observer ($(t-t_0)/T = 0$ or $20$). Unlike figures 6 and 7, the horizontal axis in figure 8 is the normalised time at the observer position. Due to the micro Doppler effect, the pulses are spaced more closely together in time as the blade moves towards the observer and are more spread out as the blade moves away. Here, an infinitesimally small period of time at the observer position, $\delta t$, is related to an infinitesimally small period of time at the source position, $\delta \tau $, by the expression $\delta t = (1-\boldsymbol{M}\boldsymbol{\cdot }\tilde{\boldsymbol{R}})\delta \tau $. Clearly, the significance of this effect is enhanced as the propeller tip Mach number increases. The impulsive nature of ${{\rm \Delta} }\dot{p}$ at the leading edge means that there is a singularity in ${{\rm \Delta} }\dot{p}$ as $X_2\to 0^ + $ and we thus expect that sound generated in that vicinity will dominate. Nonetheless, contributions from the entire blade chord remain important. In particular, when any of the observer time ($t_n^\ast $) curves in figure 8(b) are vertical, the peak sound generated at each chordwise position arrives at the observer position at the same (observer) time leading to high peak noise levels there, due to constructive interference. For observer time ($t_n^\ast $) curves that are not vertical, the peak sound pressures from different chordwise positions arrive at the observer position at different times; the smaller the gradient the larger the interval over which the arrival times are spread. The result is less constructive interference and lower peak noise levels radiated from a given radius. For the particular case considered here, the observer time ($t_n^\ast $) curve has a smaller slope for sound emitted when the blade is moving towards the observer, compared to that for the sound emitted when the blade is moving away from the observer.

Figure 8. Plot of contours of constant $[G(X_2,r,\tau;\boldsymbol{x}){{\rm \Delta} }\dot{p}(X_2,r,\tau )]_{\tau = t-R/c_0}$ plotted against $2X_2/c_2$ (vertical axis) and $(t-t_0)/T$ (horizontal axis) at $r = 0.7R_t$ for one blade revolution. Panel (b) is a replica of (a) with red dashed lines indicating $t_n^{\ast} = \tau _n^{\ast} + R/c_0$. Note that ${{\rm \Delta} }\dot{p}(X_2,r,\tau )$ is singular at $X_2 = 0$ and thus only $2X_2/c_2 \ge 0.01$ are plotted.

3.3. Acoustic pressure radiated from the entire blade span

Thus far it has been shown that the sound pressure per unit span produced at a particular radius consists of a series of impulsive sounds and that the peak of these sounds corresponds closely with the times $t_n(r)$. We hypothesise that if the function $t_n(r)$ is stationary at a particular radius, then sounds emitted close to that radius will also interfere constructively to produce a net impulse of higher amplitude. Moreover, if the observer time function $t_n(r) = \tau _n + R_L(r,\tau _n)/c_0$ is constant across the blade span, then the ‘impulses’ from all radii will coalesce at the observer location producing an even higher amplitude impulse in the pressure time history produced by the entire blade. Conversely, it is hypothesised that all the high-amplitude, and sharp, impulses would be replaced by lower amplitude, and smooth, impulses if the observer time function $t_n(r)$ contained no stationary points over the blade span. Further, it is hypothesised that the peak levels of the resultant smooth impulses would continue to fall as the variation in the observer time function $t_n(r)$ increased over the blade span (i.e. the peak noise levels would be minimised if the average gradient of observer time, over the blade span, were maximised). The fall in peak amplitudes is due to the increased variation in reception times from the various spanwise contributions that, in turn, minimises the amount of constructive interference of the impulses across the span.

For the case where the observer is located in the far-field ($R_r\to \infty $) it is straightforward to show that the gradient of $t_n(r)$ is given by

(3.7)\begin{equation}{t}^{\prime}_n(r)\cong{-}\displaystyle{{\sin \theta _r} \over {c_0\sqrt {1-M_x^2 {\sin }^2\theta _r} }}\cos (\phi _n-\phi ) = {-}\displaystyle{{\sin \theta _e} \over {c_0(1-M_x\cos \theta _e)}}\cos (\phi _n-\phi ),\; \end{equation}

where

(3.8)\begin{equation}\phi _n = \varOmega _2(\tau _0 + nT),\quad \; \phi _n\;{\rm modulo}\;2{\rm \pi} ,\end{equation}

is the azimuthal angle at which the centreline of the nth wake impinges on the leading edge of the downstream blade. (All azimuthal angles are defined as having values between 0 and $2{\rm \pi} $ radians. Formulae which allow azimuthal angles with values outside this range, such as (3.8), should be interpreted as modulo $2{\rm \pi} $.)

Thus, $t_n(r)$ is stationary when the observer is located on-axis ($\theta _e = 0$ or ${\rm \pi} $) or when the observer is located at the azimuthal angle

(3.9)\begin{equation}\phi = \phi _n \pm \displaystyle{{\rm \pi} \over 2}.\end{equation}

When (3.9) is satisfied, the sound pressure emitted from the leading edge at all radial locations at source time $\tau _n(r)$ will arrive at the observer at the same time. This situation occurs when the observer is located at an azimuthal angle such that a line from source to the observer is perpendicular to the blade leading edge (at angles ${\pm} {\rm \pi}/2$ radians from the impingement angle) as, in the acoustic far-field, the sound travels the same acoustic distance from any radius. Conversely, the radial gradient of ${t}^{\prime}_n(r)$ in (3.7) is maximised at azimuthal angles corresponding to the impingement angle ($\phi = \phi _n$), or its direct opposite ($\phi = \phi _n + {\rm \pi}$ radians) because the acoustic distance the sound travels varies most rapidly with radius at these locations. This rapid variation in propagation distance will result in less constructive interference between the sound emitted from different radii and a lower impulsive peak in the sound at the observer location.

For the case considered here, the upstream propeller has 10 blades ($B_1 = 10$), the propellers rotate at the same speed ($\varOmega _1 = \varOmega _2$) and the observer is located at azimuthal angle $\phi = \varOmega _2\tau _0(R_h)-{\rm \pi} /2$. Then (3.9) is satisfied for $n = 0$, 10 and 20 ($0 \le n \le 20$). Thus, we expect maximum constructive interference when the sound, from the rear blade's interaction with the 0th, 10th and 20th wakes, arrives at the observer location. The gradient of the observer time function ${t}^{\prime}_n(r)$, defined by (3.7), is maximised for $n = 5$ and 15 and thus we expect a minimum of constructive interference to occur when the sound, from the rear blade's interaction with the fifth and 15th wakes, arrives at the observer location.

Figure 9(a) plots the pressure time history at the observer location and contains an impulsive peak close to non-dimensional observer time $(t-t_0(R_h))/T = 10$. There are also significant pressure impulses with peaks close to non-dimensional times of 0 and 20 with the pressure being relatively constant for non-dimensional times from 4 to 8 and from 12 to 16. In order to aid interpretation of these results, contours of the integrand of (3.2), $P(r;\boldsymbol{x},t)$, are also plotted in figure 9(b,c) against non-dimensional radius $2r/D$ (vertical axis) and non-dimensional observer time (horizontal axis). Figure 9(c) is a replica of the figure 9(b) but with red curves superimposed to indicate $t_n(r) = \tau _n + R_L(r,\tau _n)/c_0$. These curves closely match the location of the peak values of the integrand. It can be seen that ${t}^{\prime}_n(r) = 0$ (and the red curves are vertical) at non-dimensional times of 0, 10 and 20 when the high-amplitude impulsive sounds occur and which correspond to the source times at which the centrelines of the $n = 0,\;10$ and 20 wakes impinge on the rear blade leading edge. We also see that, away from these peaks, the sound pressure is relatively constant at the intermediate times where the red curves are shallowest, corresponding to the points at which ${t}^{\prime}_n(r)$ is maximum. These maxima occur at the source times at which the centreline of the $n = 5$ and 15 wakes impinge on the edge such that the sound arrives at the observer location at non-dimensional times satisfying, approximately, $5.9 \le (t-t_0)/T \le 7.2$ and $12.8 \le (t-t_0)/T \le 14.1$, respectively. The impulsive sound is thus strongly dependent on the amount of constructive interference of the signals emitted from different radii. However, other effects are also important, as observed by the differences between the impulses at non-dimensional times of 0 (when the blade is moving away from the observer) and 10 (when the blade is moving towards the observer). These differences are due to micro Doppler and source directivity effects as discussed in § 3.2.

Figure 9. Plot of pressure versus non-dimensional time (a). Plot of contours of constant $P(r;\boldsymbol{x},t)$ plotted against non-dimensional radius (vertical axis) and non-dimensional time (horizontal axis) (b,c). Panel (c) shows superimposed red dashed curves of $t_n(r) = \tau _n + R_L(r,\tau _n)/c_0$. Note that panel (a) showing pressure versus time was also reproduced using the frequency-domain method described in Kingan & Parry (Reference Kingan and Parry2019a).

Having considered the observed acoustic pressure over a complete rotation period, it is important to consider the effects local to the impulsive peaks. Thus, in order to obtain a better understanding of the underlying physics at, and near to, the impulsive pressure that is produced at the observer time $(t-t_0)/T = 10$, we will zoom in to Figure 9 there. Figure 10 thus represents a magnified version of figure 9 that shows just the non-dimensional time period $9.5 \le (t-t_0)/T \le 10.2$. Figure 10(a) plots the far-field acoustic pressure against non-dimensional observer time and figure 10(b) plots contours of constant $P(r;\boldsymbol{x},t)$ against non-dimensional observer time and non-dimensional radius, where $P(r;\boldsymbol{x},t)$ is defined by (3.3). In order to aid our interpretation of the observed acoustic pressure, $P(r;\boldsymbol{x},t)$ is also plotted against non-dimensional radius $2r/D$ at fixed non-dimensional observer times 9.6, 9.65, 9.7 and 9.75 in figure 11 and 9.95 and 10 in figure 12. The former group represents times when the observer receives the sound generated by the 9th ($n = 9$) wake interaction (for which ${t}^{\prime}_n(r)\ne 0$), and the latter group represents times at which the observer receives the sound from the 10th ($n = 10$) wake interaction (for which ${t}^{\prime}_n(r) = 0$). These times are also indicated by red circles in figure 10(a) and by the red lines in figure 10(b).

Figure 10. Plot of acoustic pressure against non-dimensional time (a). Contours of constant $P(r;\boldsymbol{x},t)$ plotted against non-dimensional radius (vertical axis) and non-dimensional time (horizontal axis) (b).

Figure 11. Plot of $P(r;\boldsymbol{x},t)$ against non-dimensional radius at four different observer times, corresponding to the interaction of the $n = 9$ wake with the rear propeller blade. (a) $(t - t_0)/T = 9.6$; (b) $(t - t_0)/T = 9.65$; (c) $(t - t_0)/T = 9.7$; (d) $(t - t_0) /T = 9.75$.

Figure 12. Plot of $P(r;\boldsymbol{x},t)$ against non-dimensional radius at two different observer times, corresponding to the interaction of the $n = 10$ wake with the rear propeller blade. (a) $(t - t_0)/T = 9.95$; (b) $(t - t_0)/T = 10$.

At a non-dimensional time of $(t-t_0)/T = 9.6$ most of the sound is generated from the narrow range of non-dimensional radii centred on $2r/D = 0.88$, where the $n = 9$ wake impinges on the propeller blade. This narrow band of radii is observed as the ‘hump’ in panel (a) of P (the acoustic pressure per unit span) in figure 11. Prior to the times shown in figure 11, the band is located further inboard and moves outboard as time increases. The shape of the hump changes slowly with radius, due to spanwise differences in wake shape, relative air velocity, blade angle, etc. so the integral of P (and thus the radiated pressure) along the blade radius also changes slowly whilst the hump is contained entirely on the blade span. As the wake interaction moves over the blade tip, and the hump moves off the blade tip, the value of the integral changes more rapidly producing a mildly impulsive sound close to $(t-t_0)/T = 9.7$, as can be seen in figure 10. Although the results are not shown here we add, for completeness, that similar behaviour occurs at the blade hub close to times $(t-t_0)/T = 9.3$, where this $n = 9$ wake first impinges on the downstream blade at the hub, before moving outboard. Such mildly impulsive behaviour is typical of interactions in which ${t}^{\prime}_n(r)$ is large, with the result that the acoustic pressure remains relatively constant whilst the wake interaction occurs over the entire span of the downstream propeller blade but fluctuates more rapidly as the wake passes over the hub or tip regions of the downstream blade. Figure 11(a–d) shows the change in the radiated pressure per unit span, P, as the interaction of the $n = 9$ wake with the downstream blade moves outboard to the tip.

At the second group of non-dimensional times, $(t-t_0)/T = 9.95$ and $10.0$, the sound at the observer location corresponds to that generated by the $n = 10$ wake interacting with the downstream propeller blade. Here the observer time function ${t}^{\prime}_n(r)$ is close to or equal to zero, which means that sound generated at all radii arrives at the observer location at (almost) the same reception time. The pressure per unit span, P, varies with radius, but remains positive along the blade span with the result that the radial integral of P is large (see figure 12a,b). The acoustic pressure peaks as the observer receives the noise generated when the wake centreline impinges on the rear propeller leading edge. The sound associated with this interaction is highly impulsive because the noise from wake interactions over a range of radii arrives at the observer location over a very narrow time period. This impulsive behaviour is typical of interactions in which ${t}^{\prime}_n(r)$ is small, such that there is constructive interference between the impulses emitted from each radii producing a strong net pressure impulse that has a high peak value and which occurs over a relatively short time.

It is also interesting to consider the case of an observer located on the propeller axis at $\theta _e = 0{\rm ^\circ }$ (with all other variables held constant). Figure 13(a) plots the pressure time history there and it is observed to be highly impulsive and, due to the rotational symmetry of the problem, perfectly periodic. In order to aid interpretation of these results, contours of pressure per unit span $P(r;\boldsymbol{x},t)$ are plotted in figure 13(b) against non-dimensional radius (vertical axis) and non-dimensional observer time (horizontal axis). Red lines indicating $t_n(r) = \tau _n + R_L(r,\tau _n)/c_0$ are superimposed on the lower plot and it can be seen that ${t}^{\prime}_n(r) = 0$ for all wake interactions. This result means that, for all radii, the observer receives, simultaneously, the sound from the interactions of the wake centrelines with the leading edge of the downstream propeller blade. The net sound field at the observer location, on the axis, is thus impulsive and of high amplitude.

Figure 13. Plot of acoustic pressure against non-dimensional time on the propeller axis $(\theta _e = 0^\circ )$ (a). Contours of constant $P(r;\boldsymbol{x},t)$ for this case plotted against non-dimensional radius (vertical axis) and non-dimensional time (horizontal axis) with red dashed lines indicating $t_n(r) = \tau _n + R_L(r,\tau _n)/c_0$ superimposed (b).

The results obtained here for $\theta _e = 0^\circ $, in the time domain, echo the high blade number asymptotic solutions for contra-rotating propellers of Kingan & Parry (Reference Kingan and Parry2019a) and Parry & Kingan (Reference Parry and Kingan2019), which were derived in the frequency domain. They showed that solutions of different asymptotic orders were possible, dependent on the blade sweep and the observer location; indeed, noise could be increased for a swept propeller, relative to a straight-bladed propeller, if the leading-edge sweep was ‘critical’. However, the noisiest possible solution occurred when the blades were straight and the observer was positioned on the engine axis.

In this paper, we are only interested in the sound radiated from a single downstream propeller blade. However, the sound field on axis due to radiation from multiple downstream propeller blades is a relatively simple case to consider. This is because the impulse produced by each blade-wake interaction is identical, regardless of the location of that interaction. For a downstream propeller with a blade count equal to an integer factor or multiple of the blade count of the upstream propeller, interactions will occur simultaneously. This will result in constructive addition of the pressure signals from each interaction at the observer location and very high levels of impulsive noise. Alternative blade count combinations, which produce non-simultaneous interactions, will produce a total pressure signal which consists of a series of impulses with lower amplitudes spaced at time intervals corresponding to the time-interval between subsequent interactions (which could be a relatively small period of time).