3.1 Precession of the interaction

In this section we consider the interactions between specific blades on the rear rows and the wakes from specific blades of the front row. Here, an interaction occurs when the centreline of the wake from an upstream blade impinges on the leading edge of a downstream blade. In reality, of course, since the wake and blade leading edge are of finite thickness, this interaction occurs over a finite time. Tyler & Sofrin (Reference Tyler and Sofrin1962) refer to these interactions as blade–vane events or, more simply, just as ‘events’. In the general case, the wakes from the front propeller do not all interact simultaneously with the rear propeller blades for the simple reason that the number of wakes, $B_1$, is not the same as the number of blades, $B_2$, on the rear row. In the time domain, the interactions can be viewed as occurring at locations that ‘jump’ around the circumference, sometimes by considerable distances. For rotor–stator interactions, Tyler & Sofrin (Reference Tyler and Sofrin1962) referred to this effect as the ‘firing order’. Here we will suppose that this sequence of events represents the precession of the interaction around the circumference and consider it in detail.

In order to determine the speed at which the individual interactions precess around the circumference, we start by assuming that, in figure 3, the blades are numbered in the positive $\phi$ direction on each row and that the interaction between blade $0$ on the front row and blade $0$ on the rear row occurs at source time $\tau = 0$. For the purposes of the analysis that follows, we assume that the blade numbers $B_1$ and $B_2$ are relatively prime. If $B_1$ and $B_2$ have a common factor then there will be more than one interaction occurring simultaneously and the analysis applies over an appropriate subsection of the annulus, with the results then repeating over each subsection. Of course, the blades chosen as the zeroth blades on each row are arbitrary. The pitch is given by $P_1 = 2{\rm \pi} r/B_1$ and $P_2=2{\rm \pi} r/B_2$ on the front and rear row, respectively, so that the azimuthal gap between blade $j_1$ on the front row and blade $j_2$ on the rear row is

(3.1)\begin{equation} \Delta \phi_{j_1,j_2} = j_1 \frac{P_1}{r} - j_2 \frac{P_2}{r} = 2 {\rm \pi}\frac{\left(\,j_1 B_2 - j_2 B_1\right)}{B_1 B_2}. \end{equation}

Figure 3.

Two-dimensional cascade definition.

As, with our normalisation, the relative angular speed is $(M_{t_1}+M_{t_2})$, the time it takes from the interaction between the datum blades to that of the interaction between blade $j_1$ on the front row and blade $j_2$ on the rear row is

(3.2)\begin{equation} \Delta \tau_{j_1,j_2} = \frac{\Delta \phi_{j_1,j_2}}{(M_{t_1}+M_{t_2})} = \frac{2 {\rm \pi}(\,j_1 B_2 - j_2 B_1)}{B_1 B_2 (M_{t_1}+M_{t_2})}. \end{equation}

In that time, the interaction has moved from $\phi _2=0$ to $\phi _2 = 2 {\rm \pi}j_2/B_2$, where $\phi _2=\phi -M_{t_2}\tau$ is the azimuthal angle in the frame of reference of the rear row, so we can deduce that the effective speed with which the interaction has moved in the absolute frame is $M_{j_1,j_2} = M_{t_2} + (2 {\rm \pi}j_2 / B_2 \Delta \tau _{j_1,j_2} )$. Then, after using (3.2), we obtain the effective interaction speed and interaction location as

(3.3a,b)\begin{equation} M_{j_1,j_2} = \frac{j_2 B_1 M_{t_1} + j_1 B_2 M_{t_2}}{j_1 B_2 - j_2 B_1}, \quad \phi_{j_1,j_2} =\frac{2 {\rm \pi}(\,j_2 B_1 M_{t_1} + j_1 B_2 M_{t_2})}{B_1 B_2 (M_{t_1}+M_{t_2})}. \end{equation}

The result for interaction speed in (3.3a,b) is very familiar and, suitably normalised, looks like the standard relationship for circumferential mode-phase speed $\eta /\nu$ in turbomachinery (see e.g. Tyler & Sofrin, Reference Tyler and Sofrin1962) in terms of an angular frequency $\eta =n_1 B_1 M_{t_1}+n_2 B_2 M_{t_2}$ and mode number $\nu =n_2 B_2 - n_1 B_1$. However, the standard relationship applies to the frequency domain and to a range of different combination tones $(n_1,n_2 )$ each of which has its own frequency and azimuthal order so that every one rotates at a different speed. Here it is important to realise that (3.3a,b) is a time-domain result and that $(\,j_1,j_2)$ is used to refer to interactions between specific blades with the fixed values $j_1$ and $j_2$ being determined by the blade numbers. In order to determine which interactions occur in which order, it follows from (3.1) and (3.2) that, after the $0$–$0$ front and rear blades interact, the next interaction occurs a short time $\Delta \tau$ later where

(3.4)\begin{equation} \Delta \tau = \frac{2 {\rm \pi}}{B_1 B_2 (M_{t_1} + M_{t_2})} \times \min_{\begin{array}{c} {\scriptstyle 0 \leq\, j_1 \leq B_1-1} \\ {\scriptstyle 0 \leq\, j_2 \leq B_2-1} \\ {\scriptstyle j_1 B_2 - j_2 B_1 > 0} \end{array}} \{\,j_1 B_2 - j_2 B_1\} , \end{equation}

so that $\Delta \tau$ is the time taken between successive wake interactions (and it is important to note that those interactions do not necessarily occur between adjacent blades). The values $(\,j_1,j_2)$ that are obtained as the solution to (3.4) are related to the blade numbers $B_1$ and $B_2$ and can be treated as fixed for that blade number combination.

By considering the blade row speeds and blade spacings, and using a little simple arithmetic, it is easy to show that the next interaction will occur between blades $(2j_1,2j_2)$ at azimuthal location $\phi _{2j_1,2j_2} =4 {\rm \pi}(\,j_2 B_1 M_{t_1} + j_1 B_2 M_{t_2})/[B_1 B_2 (M_{t_1}+M_{t_2})]$, and so on, so that the effective speed at which the interaction location moves – or precesses – is continuous around the annulus. If we suppose that $l$ and $m$ are the values of $j_1$ and $j_2$ that are required in (3.4), then the speed (or tip Mach number) at which the interaction events precess around the annulus is

(3.5)\begin{equation} M_p = \frac{m B_1 M_{t_1} + l B_2 M_{t_2}}{l B_2 - m B_1}. \end{equation}

In practice, the interactions are discrete but the idea that the interactions are moving continuously becomes more acceptable for higher blade numbers. Indeed, the assumption of a continuum of interactions is completely consistent with the many-bladed asymptotic approach, used in the frequency-domain analyses of Kingan & Parry (Reference Kingan and Parry2019, Reference Kingan and Parry2020a) and Parry & Kingan (Reference Parry and Kingan2019), in which the blade numbers $B_1 \rightarrow \infty$, $B_2 \rightarrow \infty$ because then the time interval between interactions $\Delta \tau \rightarrow 0$.

3.2 Examples

We will consider a few simple examples to illustrate the values of $l$ and $m$. Note first, though, that the integers $l$ and $m$ can be negative, where negative values are obtained if the blades are counted in the opposite (or negative $\phi$) direction, provided that we still maintain $lB_2-mB_1>0$ to ensure that we are moving forwards in time. Of course, the interactions occur in the same ‘firing order’ in both forward and reverse cases; the point is that the interaction disturbances can merely be viewed as moving in either the positive or negative direction.

First, we examine the contra-rotating propeller studied by Kingan & Parry (Reference Kingan and Parry2020b) in their time-domain analysis. The propeller had 10 blades on the front row and a single blade on the rear row, and tip rotational Mach numbers $M_{t_1}=M_{t_2}=0.7$. That case is straightforward, as all the interactions (events) must occur at the leading edge of the same rear blade so that $m=0$; the interactions must also be caused by the wakes from successive blades on the front row so that $l=1$ and $M_p=M_{t_2}$. We emphasise that the interactions are not continuous; from (3.4) they occur discretely at source time intervals of $\Delta \tau = 2{\rm \pi} /B_1(M_{t_1} +M_{t_2})={\rm \pi} /7$, given as the period $T$ by Kingan & Parry (Reference Kingan and Parry2020b). In order to demonstrate the effect of the source we show, in figure 4, a full numerical calculation for this contra-rotating propeller, in the time domain, with the pressure per unit span, $P$, radiated from the mid-span ($r=0.7$) of the blade. Figure 4 has been recalculated and is plotted here in terms of (leading-edge) source time $\tau$ rather than reception time $t$ (as it was in figure 5(a) of Kingan & Parry (Reference Kingan and Parry2020b)). The source period is normalised on $T$ so that pulses occur at multiples of $\tau /T=1$, precisely as expected. The calculation also serves to demonstrate the sharpness of the interaction impulses in a practical wake–blade interaction case. For this configuration, the interactions can be considered to be rotating with the rear blade leading edge and at the speed of the rear blade row.

Figure 4.

The pressure field radiated from $r=0.7$ for a straight propeller, plotted in terms of leading-edge source time $\tau\!$.

Second, we consider the examples used by Kingan & Parry (Reference Kingan and Parry2019) in demonstrating the accuracy of their asymptotic, frequency-domain analysis of contra-rotating propellers. Whilst their example architectures had different degrees of sweep, they all had 12 blades on the front row and 9 blades on the rear row. In a fixed reference frame, the front blades are thus $30^{\circ }$ apart and the rear blades are $40^{\circ }$ apart. Assuming that the zeroth front and rear blades are aligned at $\tau =0$ then, taking into account the direction of motion of the two blade rows, a little arithmetic can be used to show that the next interaction will occur between the third blade of the front row and the second blade of the rear row. (Of course, since for this case there is a common factor of 3 in the blade numbers, the interactions repeat spatially every $120^{\circ }$.) The relevant values are thus $l=3$ and $m=2$, and we obtain the precession speed of $M_p = 8 M_{t_1}+9M_{t_2}$ (which is much faster than that of the case with a single-bladed rear row) and the time interval of $\Delta \tau = 2 {\rm \pi}/36(M_{t_1}+M_{t_2})$ between interaction events. By counting blades in the opposite direction, an alternative solution is also obtained with $l=-1$ and $m=-1$ for which the interactions precess in the reverse direction at $M_p = -(4 M_{t_1}+3M_{t_2})$, which is less than half that of the precession speed in the forward (or positive $\phi$) direction. The interval between interaction events is the same as that of the forward propagating sequence of interactions.

Purely for completeness, we will also consider a third case relating to a rotor–stator interaction. For such architectures, the stator numbers are usually selected to be more than twice the number of rotors in order to ensure the first interaction tone cannot propagate. We thus pick the numbers $B_1=18$ and $B_2=40$. In a fixed reference frame, the rotors are thus $20^{\circ }$ apart and the stators are $9^{\circ }$ apart. Once again, we assume that the zeroth front and rear blades are aligned at $\tau =0$. The next interaction is then easily shown to occur between the fifth blade of the front row and the eleventh blade of the rear row. (Of course, since for this case there is a common factor of 2 in the blade numbers, the interactions repeat spatially every $180^{\circ }$.) The relevant values are thus $l=5$ and $m=11$, and we obtain the precession speed of $M_p = 220M_{t_1}$ and the time interval of $\Delta \tau = 2 {\rm \pi}/360 M_{t_1}$ between interaction events. Alternatively, we can use $l=-4$ and $m=-9$ and obtain $M_p = -81M_{t_1}$ and the same value of $\Delta \tau\!$.

3.3 Interaction location as a function of radius

The analysis of Kingan & Parry (Reference Kingan and Parry2020b) showed that the front and rear reference blades, on a contra-rotating propeller, interact at source times that vary across the blade span due to the effect of blade sweep. Their derivation (given as (2.11) in Kingan & Parry (Reference Kingan and Parry2020b)) is straightforward, as it is based solely on blade geometry and speed. With our normalised parameters and definition of $\tau =0$, the expression simplifies and the reference blade interaction source times are $\tau _0(r)=S(r)$, where both $\tau _0$ and $S$ are zero at the hub: $r=r_h$.

The precession speed is given by (3.5), so that the azimuthal location at which the interactions take place can be expressed as $\phi _{ev}=M_{p}(\tau - \tau _0 )$, where we have used the subscript ‘$ev$’ to reflect the fact that the wake–blade interactions are referred to by Tyler & Sofrin (Reference Tyler and Sofrin1962) as ‘events’. We will require the radial derivative of $\phi _{ev}$ in the following section and we get

(3.6)\begin{equation} \frac{\textrm{d}\phi_{ev}}{\textrm{d}r} = -M_p S'(r). \end{equation}