2.1 Base flow

A simplified model of a two-bladed rotor wake is given by a system of two infinite helical vortex filaments in a cylindrical coordinate system ( $r,\unicode[STIX]{x1D703},z$ ), as shown in figure 1. The defining parameters of the helix geometry are the radius $R$ and the pitch $h^{\prime }$ of the helical centreline of each vortex. The two vortices are separated azimuthally by an angle  $\unicode[STIX]{x03C0}$ , and their unperturbed positions can be written as

(2.1) $$\begin{eqnarray}\boldsymbol{r}_{m}=\left(\begin{array}{@{}c@{}}r_{m}\\ \unicode[STIX]{x1D703}_{m}\\ z_{m}\end{array}\right)=\left(\begin{array}{@{}c@{}}R\\ \unicode[STIX]{x1D703}+m\unicode[STIX]{x03C0}\\ h^{\prime }\unicode[STIX]{x1D703}/(2\unicode[STIX]{x03C0})\end{array}\right),\quad \unicode[STIX]{x1D703}\in \mathbb{R},\end{eqnarray}$$

where $m=1,2$ is the helix index. The distance $h=h^{\prime }/2$ is the separation between neighbouring vortex loops at constant  $\unicode[STIX]{x1D703}$ .

Figure 1.

Schematic of two interlaced infinite helical vortices, including the relevant parameters: pitch $h^{\prime }$ of one helix, separation $h$ of neighbouring helix loops, radius  $R$ , core size $a$ and circulation  $\unicode[STIX]{x1D6E4}$ .

For the present analysis, the internal structure of the vortices is assumed to be that of a Batchelor vortex, characterised by Gaussian distributions of axial vorticity and velocity. In local cylindrical coordinates ( $\unicode[STIX]{x1D70C},\unicode[STIX]{x1D719},\unicode[STIX]{x1D709}$ ) of the vortex, the radial profiles of azimuthal and axial velocities read

(2.2a,b ) $$\begin{eqnarray}v_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D70C})=\frac{\unicode[STIX]{x1D6E4}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}}[1-\exp (-\unicode[STIX]{x1D70C}^{2}/a^{2})]\quad \text{and}\quad v_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70C})=V_{\unicode[STIX]{x1D709}}^{max}\exp (-\unicode[STIX]{x1D70C}^{2}/a^{2}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}$ is the vortex circulation, $a$ is the core radius and $V_{\unicode[STIX]{x1D709}}^{max}$ is the amplitude of the core velocity.

The non-dimensional parameters characterising the flow are the Reynolds number $Re=\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708}$ (where $\unicode[STIX]{x1D708}$ is the kinematic viscosity), the scaled vortex separation $h/R$ and core radius $a/R$ , and the axial flow parameter

(2.3) $$\begin{eqnarray}W=2\unicode[STIX]{x03C0}aV_{\unicode[STIX]{x1D709}}^{max}/\unicode[STIX]{x1D6E4},\end{eqnarray}$$

which represents the ratio between the maximum axial and swirl velocities. As discussed in Quaranta et al. (Reference Quaranta, Bolnot and Leweke2015), the relevant time scale for the development of the long-wave instability studied here is given by $2h^{2}/\unicode[STIX]{x1D6E4}$ , which is used to non-dimensionalise time  $t$ , as well as the instability growth rate $\unicode[STIX]{x1D70E}$ and frequency  $\unicode[STIX]{x1D714}$ , according to $t^{\ast }=t\unicode[STIX]{x1D6E4}/(2h^{2})$ , $\unicode[STIX]{x1D70E}^{\ast }=2\unicode[STIX]{x1D70E}h^{2}/\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D714}^{\ast }=2\unicode[STIX]{x1D714}h^{2}/\unicode[STIX]{x1D6E4}$ (non-dimensional quantities are denoted with an asterisk).

We here also make use of the concept of an equivalent Rankine-type core size. Widnall, Bliss & Zalay (Reference Widnall, Bliss, Zalay, Olsen, Goldberg and Rogers1971), among others, showed that, for long-wave displacement perturbations, a vortex with velocity profiles given by (2.2) exhibits the same self-induced dynamics as an equivalent Rankine vortex (constant core vorticity) without axial flow, having a core radius

(2.4) $$\begin{eqnarray}a_{e}=a\sqrt{2}\exp \left[\frac{1}{4}-\frac{\unicode[STIX]{x1D6FE}}{2}+\frac{W^{2}}{2}\right]\approx 1.36a\exp \left(\frac{W^{2}}{2}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}\approx 0.577$ is Euler’s constant. This relation is used in § 5 for the comparison between experimental and theoretical results.

2.2 Long-wavelength instability

Following the work by Widnall (Reference Widnall1972) and Gupta & Loewy (Reference Gupta and Loewy1974), the stability of the base configuration is analysed by determining the evolution of displacement perturbations of the form

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\boldsymbol{r}_{m}=\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FF}\hat{r}_{m}\\ \unicode[STIX]{x1D6FF}\hat{\unicode[STIX]{x1D703}}_{m}\\ \unicode[STIX]{x1D6FF}\hat{z}_{m}\end{array}\right)\exp [\unicode[STIX]{x1D6FC}t+\text{i}(k\unicode[STIX]{x1D703}+\unicode[STIX]{x1D711}_{m})],\end{eqnarray}$$

where $k$ represents the real azimuthal wavenumber (number of wavelengths per helix turn), and $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D70E}+\text{i}\unicode[STIX]{x1D714}$ is the complex growth rate. Let $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{2}-\unicode[STIX]{x1D711}_{1}$ be the phase difference between the perturbations of the two vortices at the same axial position  $z$ . The pairing instability involves a constant phase shift of the perturbations between neighbouring vortices at the same azimuthal position $\unicode[STIX]{x1D703}$ . Only two values of $\unicode[STIX]{x1D711}$ can produce such a constant phase difference in the two-helix geometry: $\unicode[STIX]{x1D711}=0$ (in-phase perturbations) and $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}$ (out-of-phase perturbations).

Figure 2.

Theoretical growth rate ( $\unicode[STIX]{x1D70E}^{\ast }$ ) and frequency ( $\unicode[STIX]{x1D714}^{\ast }$ ) of the long-wave instability of two helical Rankine vortices with $h/R=0.41$ and $a_{e}/R=0.05$ , as predicted by Gupta & Loewy (Reference Gupta and Loewy1974). (a) Without fixed phase correlation between the two vortices. (b) For perturbations having the same phase at each $z$ ( $\unicode[STIX]{x1D711}=0$ ). Frequencies are shown only for unstable modes with the highest growth rate.

The evolution of the filaments’ shapes is obtained by computing their mutually and self-induced velocities from the Biot–Savart law. For two thin-cored vortices with circulation  $\unicode[STIX]{x1D6E4}$ , the velocity $\boldsymbol{u}$ induced at a point $\boldsymbol{r}_{0}$ in the domain is given by

(2.6) $$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{r}_{0})=-\frac{\unicode[STIX]{x1D6E4}}{4\unicode[STIX]{x03C0}}\mathop{\sum }_{m=1}^{2}\int \frac{[\boldsymbol{r}_{0}-\boldsymbol{r}(l_{m})]\times \text{d}\boldsymbol{l}_{m}}{|\boldsymbol{r}_{0}-\boldsymbol{r}(l_{m})|^{3}},\end{eqnarray}$$

in which $l_{m}$ is the curvilinear coordinate along the $m$ th vortex, whose centreline is described by $\boldsymbol{l}_{m}$ . For points on either one of the centrelines, the corresponding integral diverges. It can be desingularised either through a cutoff method (Crow Reference Crow1970; Widnall Reference Widnall1972), where the singularity is excluded from the integration, or by adding an appropriately chosen finite constant in the denominator (Rosenhead Reference Rosenhead1930; Moore Reference Moore1972; Gupta & Loewy Reference Gupta and Loewy1974). Both the cutoff distance and the constant depend on the core size and the velocity distribution within the core. Details can be found, for example, in Saffman (Reference Saffman1992).

By linearising the evolution equations, based on (2.6), for the two helical filaments with displacement perturbations of the form (2.5), an eigenvalue problem is obtained for the complex growth rate $\unicode[STIX]{x1D6FC}$ and the corresponding eigenvectors $\unicode[STIX]{x1D6FF}\hat{\boldsymbol{r}}_{m}$ , which represent the mode shapes, as a function of the wavenumber  $k$ . Figure 2(a) shows a typical theoretical growth-rate curve $\unicode[STIX]{x1D70E}^{\ast }(k)$ and the corresponding frequencies $\unicode[STIX]{x1D714}^{\ast }(k)$ , calculated with the formalism of Gupta & Loewy (Reference Gupta and Loewy1974) for the flow parameters corresponding to the reference configuration in our experimental study (see § 4). The growth rate has local maxima at integer values of  $k$ , including $k=0$ . The highest values are close to $\unicode[STIX]{x03C0}/2$ . The unstable modes include both in-phase and out-of-phase perturbations (see also figure 7 below). When the phase difference $\unicode[STIX]{x1D711}$ is forced to be zero, the lobes around even values of $k$ disappear from the growth-rate curve, and one obtains the result shown in figure 2(b). This is relevant for the experimental case, when the perturbations are triggered by a modulation of the rotor rotation (see § 3): for a rigid two-bladed rotor, these perturbations are necessarily in phase for the two vortices. The frequencies of the most unstable modes are zero or close to zero, indicating that these perturbations are steady or only slowly oscillating in time.

Figure 3.

Geometrical deformation and local paring predicted by Gupta & Loewy (Reference Gupta and Loewy1974) for the case $h/R=0.45$ . (a) For $k=1$ , perturbations on both vortices are in phase at each  $z$ . (b) For   $k=2$ , perturbations are out of phase. (c) In-phase perturbations with $k=2$ , no pairing occurs. Three-dimensional views on the left and developed plan views on the right. The vertical direction of the plan views is compressed by a factor of 2.

Figure 4.

Theoretical perturbation mode for global pairing ( $k=0$ ); three-dimensional and developed plan views.

The spatial structure of the unstable deformation modes can be seen in figure 3 for the first two maxima of the growth-rate curve with non-zero $k$ . The deformations are in phase for $k=1$ and out of phase for $k=2$ . They result in a local pairing of successive vortex loops at $2k$ locations around the azimuth, as highlighted by the shaded circles in the developed plan views. Figure 3(c) illustrates that an in-phase perturbation with $k=2$ does not lead to local pairing of neighbouring vortex elements, which explains why $\unicode[STIX]{x1D70E}^{\ast }(k=2)$ vanishes in figure 2(b). The same is true for in-phase deformations at all even values of  $k$ .

Figure 5.

Maximum growth rate for global paring ( $k=0$ ), as a function of the relative vortex separation $h/R$ ; comparison of various theoretical results. The analytical studies of Gupta & Loewy (Reference Gupta and Loewy1974) and Okulov & Sørensen (Reference Okulov and Sørensen2010) consider helical Rankine vortices, whereas Selçuk (Reference Selçuk2016) determines the stability of helical Lamb–Oseen vortices numerically. The equivalent core size is $a_{e}/R=0.12$ for all cases. The straight-vortex analogy corresponds to (2.7).

A special kind of unstable perturbation corresponds to the wavenumber $k=0$ . As shown in figure 4, it is out of phase and involves a radial expansion of one helix and a contraction of the other one, resulting in a pairing that is uniform in the azimuthal direction $\unicode[STIX]{x1D703}$ of the flow. In figure 5, various theoretical predictions for the growth rate of this uniform (or global) pairing, as a function of the pitch parameter $h/R$ , are collected and compared. All have in common that they tend to $\unicode[STIX]{x1D70E}^{\ast }=\unicode[STIX]{x03C0}/2$ for $h/R\rightarrow 0$ , and that the growth rate initially increases with $h/R$ . This can partly be understood by considering an analogy with arrays of straight vortices, as depicted in the developed plan views of figure 6. The relevant length scale for the pairing instability in these systems is the shortest distance between the vortices (b in figure 6), which is smaller than  $h$ . From simple geometry one can obtain the corresponding ‘corrected’ expression for the growth rate in the two-helix system as

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D70E}^{\ast }=\frac{\unicode[STIX]{x03C0}}{2}\left[1+\frac{1}{\unicode[STIX]{x03C0}^{2}}\left(\frac{h}{R}\right)^{2}\right],\end{eqnarray}$$

which is plotted as the dash-dotted line in figure 4. This prediction neglects the curvature of the vortices, and it is seen that the results taking into account the full three-dimensional structure of the vortex system rapidly deviate from this simple approach for increasing $h/R$ . In the following section, the straight-vortex-array analogy is further explored also for the local pairing of two helical vortices.

Figure 6.

Schematic of the developed plan view of the double helix, with the definitions of the parameters employed in the analysis.

2.3 Pairing instability

In Quaranta et al. (Reference Quaranta, Bolnot and Leweke2015), a link was established between the (local) pairing instability of a single helical vortex and the pairing of an array of point vortices in two dimensions (Lamb Reference Lamb1932), which represent straight parallel vortices in three dimensions. We here refine this comparison by extending it to the theory of three-dimensional stability of vortex arrays developed by Robinson & Saffman (Reference Robinson and Saffman1982), and apply it to the two-helix system. Following Lamb (Reference Lamb1932), Robinson & Saffman (Reference Robinson and Saffman1982) considered a row of identical Rankine vortices (circulation  $\unicode[STIX]{x1D6E4}$ , core size  $a_{e}$ , separation distance  $l_{RS}$ ), sinusoidal displacement perturbations with a wavenumber $k_{RS}$ along the vortex centrelines, and a phase difference $\unicode[STIX]{x1D6F7}$ between neighbouring vortices. They determine the growth rate of the three-dimensional pairing instability, non-dimensionalised by $\unicode[STIX]{x1D6E4}/(2\unicode[STIX]{x03C0}l_{RS}^{2})$ , as a function of the base flow and perturbation parameters. As for the pairing of straight vortices ( $k_{RS}=0$ ) studied by Lamb (Reference Lamb1932), the growth rate is highest for out-of-phase perturbations ( $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}$ ) between neighbours, but its value decreases with increasing wavenumber  $k_{RS}$ .

We can transpose this result to the case of two interlaced helical vortices by considering the developed plan view in figure 6, where the system appears as a periodic array of inclined straight vortices. From the geometrical relations in this figure, one can find the expressions of the parameters in Robinson & Saffman’s (Reference Robinson and Saffman1982) analysis as a function of the helix parameters:

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle l_{RS}=b=h\frac{2\unicode[STIX]{x03C0}R}{L}=hG^{-1/2}, & \displaystyle\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle k_{RS}=\frac{2\unicode[STIX]{x03C0}k}{L}=\frac{k}{R}G^{-1/2}, & \displaystyle\end{eqnarray}$$

with $G=1+(h/R)^{2}/\unicode[STIX]{x03C0}^{2}$ .

$\unicode[STIX]{x1D6F7}$ is the phase shift between points $\unicode[STIX]{x2460}$ and $\unicode[STIX]{x2463}$ in figure 6. It can be calculated knowing that the phase difference between $\unicode[STIX]{x2460}$ and $\unicode[STIX]{x2461}$ is $\unicode[STIX]{x1D711}$ ( $=0$ or  $\unicode[STIX]{x03C0}$ ), the one between $\unicode[STIX]{x2461}$ and $\unicode[STIX]{x2462}$ is $\unicode[STIX]{x03C0}k$ , and the one between $\unicode[STIX]{x2462}$ and $\unicode[STIX]{x2463}$ is a fraction $-2\unicode[STIX]{x1D6E5}/L$ of the latter, with $\unicode[STIX]{x1D6E5}=2h^{2}/L$ . This leads to

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D711}+\unicode[STIX]{x03C0}kG^{-1}.\end{eqnarray}$$

For a given helix geometry $h/R$ and displacement perturbation with wavenumber $k$ and phase difference  $\unicode[STIX]{x1D711}$ , the parameters  $l_{RS}$ , $k_{RS}$ and $\unicode[STIX]{x1D6F7}$ can be obtained from (2.8)–(2.10). Together with the core size $a_{e}$ , the non-dimensional complex growth rate $\unicode[STIX]{x1D6FC}_{RS}^{\ast }$ for three-dimensional pairing is then found from equations (2.24) of Robinson & Saffman (Reference Robinson and Saffman1982) and transformed into the current scaling via

(2.11) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}^{\ast }=\frac{G}{\unicode[STIX]{x03C0}}\unicode[STIX]{x1D6FC}_{RS}^{\ast }.\end{eqnarray}$$

The (real) growth rates and frequencies obtained from this procedure for in-phase ( $\unicode[STIX]{x1D711}=0$ ) and out-of-phase ( $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}$ ) perturbations of the reference configuration in the experimental study are plotted in figure 7(a). They are compared to the result from the stability analysis by Gupta & Loewy (Reference Gupta and Loewy1974) shown in figure 7(b), which takes the helical geometry of the base flow fully into account. It is quite striking that the growth rates are nearly identical. The more complex variation of the frequencies is also captured qualitatively well by the simplified model.

Figure 7.

Predictions of the growth rate and frequency spectrum. (a,c) Based on the results for three-dimensional pairing of straight vortices by Robinson & Saffman (Reference Robinson and Saffman1982). (b,d) From the full stability analysis of the double-helix system (Gupta & Loewy Reference Gupta and Loewy1974). (a,b) For $h/R=0.41$ (experimental case) and (c,d) for $h/R=1$ . In-phase (red) and out-of-phase (blue) perturbations of the two vortices, $a_{e}/R=0.05$ .

The prediction in (2.11) is based on the assumption that the helical vortex system behaves locally like an array of straight vortices. The results are expected to become invalid for large helix pitch, for which the two-helix system locally tends towards a vortex pair configuration, where other instability mechanisms, such as the Crow instability (Crow Reference Crow1970) are likely to become dominant. Figure 7(c,d) shows the comparison for a larger pitch ( $h/R=1$ , where each helical vortex has pitch  $2R$ ). The range of unstable wavenumbers decreases with increasing pitch. The straight-vortex model still predicts quite accurately the growth rates and frequencies, even if new modes start to appear in the full calculation (e.g. just above $k=2$ in figure 7 d).

The above analysis has shown that the growth rate of the displacement instability of interlaced helical vortices with moderate pitch and small core size can be predicted surprisingly well from the characteristics of three-dimensional pairing of arrays of straight vortices, which demonstrates that pairing is the underlying mechanism for this instability. In the next sections, an experimental study involving a two-bladed rotor is described, which allows the pairing instability to be observed in a real flow, and the theoretical predictions discussed above to be compared to measurements. The nonlinear stages occurring at later times (i.e. far downstream in the rotor wake) are also documented.

4.1 Base flow parameters

In this section, the properties of the undisturbed wake behind the two-bladed rotor, which corresponds to the basic state of the stability study, are discussed. This flow is visualised in figure 12(a). Although the system is unstable, the turbulence intensity in the water channel ( ${<}1\,\%$ ), the vibration levels of the rotor support and the asymmetry in the rotor geometry were sufficiently low, so that no significant perturbation of the regular helix structure could be seen in the field of observation. The latter extended for more than three rotor diameters and was limited in the downstream direction by the support structure of the shaft. The experimental helix system matches well with the theoretical base flow shown in figure 12(b) (see also figure 1).

Figure 13.

(a) Phase-averaged azimuthal vorticity field ( $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}$ ) in the centre plane of the rotor, obtained from 300 PIV measurements. (b) Velocity profile of the vortex inside the circle in (a). The measurements (●) are in close agreement with the theoretical profile provided by Moore & Saffman (Reference Moore and Saffman1973) (solid line, see their equation (3.9) with $n=0.8$ ). The dashed line represents a fit to a Gaussian profile having the same maximum. (c) Log–log plot of the velocity profile, showing that the azimuthal velocity varies as $\unicode[STIX]{x1D70C}^{-0.8}$ outside the core. (d) Vorticity profile of the same vortex. The line is a fit to a Gaussian distribution, using only the core data (first nine measurements).

In figure 13(a), the phase-averaged azimuthal vorticity field in the near-wake centre plane of the rotor is plotted. It clearly shows the tip vortices along with the root vortices of opposite-signed vorticity at $r/R_{0}\approx 0.4$ , and the boundary layer on the shaft. It is notable that the root vortices have a slightly smaller pitch than the tip vortices.

Figure 14.

Characteristic parameters of the double-helix configuration and their downstream evolution. Geometric parameters: (a) helix radius $R$ and (b) separation $h$ between neighbouring helix loops. Vortex parameters: (c) circulation $\unicode[STIX]{x1D6E4}$ and (d) core size  $a$ . The dashed line in (d) is a fit to the viscous expansion of a two-dimensional Gaussian vortex core, $a=\sqrt{4\unicode[STIX]{x1D708}(t-t_{0})}$ .

The radius and pitch of the tip vortex system were measured from the dye visualisations by detecting the centres of the dye loops (corresponding to the vortex cores) at the top and bottom of the helix, and tracing these throughout the wake using image-tracking software (Brown Reference Brown2017). In order to compensate for the scatter in the measurements, due partly to temporal fluctuations in the wake, several vortex ridges were traced and their corresponding trajectories averaged. The resulting helix radius and vortex separation are plotted as a function of the distance from the rotor in figures 14(a) and 14(b), respectively. As expected for a rotor operating in the wind turbine regime, the radius increases with downstream distance from the rotor plane; it reaches its asymptotic value $R_{\infty }$ after two to three radii. The distance between successive helix loops is around 45 % of the blade span, and varies little in the downstream direction. Considering the maxima of the azimuthal vorticity, the parameters $R$ and $h$ could also be obtained from the PIV measurements. As seen in figure 14, the outcomes of the two techniques are in good agreement with each other.

The circulation of the tip vortices was calculated from the velocity field using line integrals on circular paths of diameter  $h$ , as shown in figure 13(a). It increases gradually over a distance of 1.5 rotor radii (figure 14 c), which is a signature of the roll-up process of the initial vortex sheet shed by the blade. The reference circulation ( $\unicode[STIX]{x1D6E4}_{\infty }$ ) for the base flow configuration is the one measured at the downstream end of the PIV field. The root vortex circulation, obtained in a similar way, is only approximately 30 % of the tip vortex circulation. This low value can be understood by considering the boundary layer developing on the rotor shaft. Owing to the zero streamwise velocity on the shaft surface, the boundary layer contains approximately twice as much circulation (per distance  $h$ ) as the tip vortices. Part of this circulation mixes with, and cancels a large fraction of, the root vortex circulation, through the complicated flow behind the rotor hub.

4.2 Vortex core velocities

The swirl velocity profile of the tip vortices was measured from the PIV data using local polar coordinates ( $\unicode[STIX]{x1D70C},\unicode[STIX]{x1D719}$ ) centred on a given vortex. Figure 13(b) shows a profile of the azimuthally averaged azimuthal velocity in these coordinates, which allows a determination of the core size $a_{max}$ . This profile differs from the one of a Gaussian vortex, which underestimates the velocities outside the core, indicating that the outer region also contains circulation. A better fit to the experimental data is provided by the theoretical velocity distribution given by Moore & Saffman (Reference Moore and Saffman1973) for laminar vortices resulting from the vortex sheet roll-up behind a lifting surface. In particular, the azimuthal velocity is found to vary as $\unicode[STIX]{x1D70C}^{-0.8}$ outside the core (figure 13 c).

Since the vortex Reynolds number is here of the order of $10^{4}$ , one might expect some degree of turbulence inside the vortices. Phillips (Reference Phillips1981) has calculated the velocity profiles of turbulent wing-tip vortices, which turn out to be very close to the laminar profiles of Moore & Saffman (Reference Moore and Saffman1973). The identification of the turbulent nature based on the measured swirl velocity or circulation profiles therefore seems hardly possible. In our case, the inner core region (for $\unicode[STIX]{x1D70C}\lesssim a_{max}$ ), which has the main influence on the pairing instability, appears to be laminar, since its size increases on a time scale dictated by the molecular viscosity of the fluid (figure 14 d).

As discussed in more detail in Quaranta et al. (Reference Quaranta, Bolnot and Leweke2015), the core structure influences the self-induced velocity of the vortex through the equivalent Rankine core radius $a_{e}$ (see § 2.1). Theoretical work of Fabre (Reference Fabre2002) suggests that profiles with the same $a_{max}$ have a very similar equivalent core size, so that the development for Gaussian vortices outlined in § 2.1 also can be applied to the experimental vortices. (In addition, the instability growth rate is found to depend only weakly on the vortex core size, in the range of parameters considered in this study.) The downstream evolution of the characteristic (Gaussian) core radius $a\approx a_{max}/1.12$ (derived from (2.2a )) is plotted in figure 14(d); it follows the laminar viscous growth of a Gaussian vortex fairly well. As a representative value to use in the stability calculations, the core radius near $z/R_{0}=2$ is chosen. The core size $a$ can also be obtained from the vorticity profile of the vortex (figure 13 d). For a Gaussian vortex, it is the half-width at a fraction $1/\text{e}$ of the maximum value. The core sizes obtained from the measured vorticity profiles, using this criterion, are roughly the same as the ones obtained from the profiles of the swirl velocity.

The tip vortex generated from the roll-up of a vortex sheet behind a wing or blade also contains an axial velocity component (along the vortex centreline), as derived by Batchelor (Reference Batchelor1964). Inside the core, the pressure is lower compared to the upstream flow, such that fluid elements entering the vortex core may experience a noticeable axial acceleration. As a result, the axial flow inside the core will behave like a jet in the frame of reference of the helix. Taking into account the effects of viscosity and the boundary layer on a wing, Moore & Saffman (Reference Moore and Saffman1973) developed a model which showed that the axial flow inside the core can be either in the same direction as the motion of the wing tip or opposite to it, depending on the load distribution of the wing, the chord and the Reynolds number.

Figure 15.

(a) Visualisation of the axial flow within the vortex cores of the undisturbed helices. The red arrow indicates the location in the rotor plane where dye is injected. (b) Peak velocity along the centreline of the vortices as a function of downstream distance. Symbols represent measurements obtained from visualisations such as in (a); the line is the prediction in (4.1) based on Moore & Saffman (Reference Moore and Saffman1973), using the measured vortex properties. The shaded range represents the values occurring during the growth-rate measurements.

In the present experiments, the axial core flow is visualised using a method similar to that used by Quaranta et al. (Reference Quaranta, Bolnot and Leweke2015), where dyed fluid is injected at a fixed location in the vicinity of the rotor plane ( $z=0$ ). As seen in figure 15(a), the dye follows the motion of the blades. In the reference frame of a blade, this motion corresponds to a velocity deficit within the cores of the tip vortices.

For the purpose of comparing with theoretical results, we assume here that the axial velocity profile may be approximated by a Gaussian profile, with a maximum velocity $V_{\unicode[STIX]{x1D709}}^{max}$ (2.2b ). The value of this maximum velocity, in the direction of the vortex centreline, can be estimated by tracking in time the positions of the tips of the dye pattern in figure 15(a). Several dye tip trajectories were obtained and averaged, and after a straightforward but tedious geometrical transformation and derivation, one obtains the downstream evolution of the maximum core velocity shown in figure 15(b). The figure also shows the prediction that can be derived from the model of Moore & Saffman (Reference Moore and Saffman1973) for our case. For a (straight) viscous trailing vortex with an outer velocity profile given by $v_{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D70C})=\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D70C}^{n}$ , their equation (3.30) predicts the maximum core velocity as

(4.1) $$\begin{eqnarray}V_{\unicode[STIX]{x1D709}}^{max}=\frac{\unicode[STIX]{x1D6FD}^{2}}{U_{s}c_{tip}^{2n}}\left(\frac{s}{c_{tip}}\right)^{-n}Re_{c}^{n}\unicode[STIX]{x1D700}_{n}-0.053\sqrt{\unicode[STIX]{x03C0}}(4-2n)U_{s}\left(\frac{s}{c_{tip}}\right)^{-1/2},\end{eqnarray}$$

where $U_{s}=Lf_{0}$ (see figure 6 for  $L$ ) is the blade-tip velocity with respect to the fluid, $Re_{c}=U_{s}c_{tip}/\unicode[STIX]{x1D708}$ the tip-chord-based Reynolds number, $s=(L/2h)z$ the coordinate along the vortex centreline, and $\unicode[STIX]{x1D700}_{n}$ a constant depending on the exponent  $n$ . From the measured vortex velocity profile in figure 13(c) one finds $n=0.8$ , resulting in $\unicode[STIX]{x1D700}_{0.8}=-0.7$ (figure 2 of Moore & Saffman (Reference Moore and Saffman1973)). The solid line in figure 15(b) shows the resulting prediction for $V_{\unicode[STIX]{x1D709}}^{max}(z/R_{0})$ , where only the parameter $\unicode[STIX]{x1D6FD}$ and the origin for the downstream position were adjusted. The observed evolution (decrease) of the velocity defect inside the tip vortices is well captured by the model. A value representing the mean velocity during the exponential growth phase of the pairing instability is kept for comparison with theory (see table 1). This velocity has approximately the same magnitude as the maximum swirl velocity (see figure 13 b) and corresponds to approximately 7 % of the blade-tip speed.

Table 1.

Experimentally determined flow parameters of the unperturbed vortex system.

The base flow properties and the non-dimensional parameters for the system of two interlaced helical vortices in our experimental rotor wake are summarised in table 1. Uncertainties are based on the scatter of the individual measurements, and on estimates of the accuracy of the measurement procedures.

5.1 Local pairing

When the symmetrically mounted rotor is rotating at a constant rate, the helix structures generated in its wake do not show any significant deformation over the first three diameters behind the rotor (figure 12). However, when slightly modulating the rotation frequency, as outlined in § 3, displacement perturbations are introduced that rapidly amplify and distort the vortex as it advects downstream, demonstrating the instability of the system with respect to such perturbations.

Figure 16.

Rotor wake perturbed with a local pairing mode ( $k=1$ ). (a) Experimental dye visualisation for $d/h_{\infty }=0.05$ ; (b) corresponding theoretical prediction (the perturbation amplitude is kept constant in the grey region).

Figure 16(a) shows a dye visualisation of the perturbation mode $k=1$ , triggered with an initial displacement of amplitude $d/h_{\infty }=0.05$ . This mode corresponds to the first maximum of the in-phase perturbation growth-rate curve in figure 2(b). At the two azimuthal locations where the displacement of the vortex filaments is the largest, local pairing occurs. The phase of the blade rotation modulation was chosen such that this happens at the top and bottom in the view of figure 16(a). Figure 16(b) presents the corresponding theoretical shape from the stability analysis. The temporal growth from the theory is transformed into a spatial growth relevant for the rotor geometry using the vortex convection speed $2hf_{0}$ and an appropriately chosen initial perturbation amplitude and phase. The experimental and theoretical structures agree very well, approximately up to the point where consecutive vortex loops are displaced to the same downstream position (at $z/R_{0}\approx 3$ , near the centre of the figure), which marks the end of the linear regime of the pairing instability (see also § 6).

Figure 17.

Schematic illustrating the displacement of the vortex cores in the upper half of the centre plane due to local pairing. The distances $d_{1}$ and $d_{2}$ are used to estimate the growth rate of the perturbation. (a) Unperturbed and (b) perturbed configuration.

The growth rate of the instability was determined from visualisation sequences, for various perturbation wavenumbers  $k$ . Every sequence contained an initial phase of unperturbed flow (figure 12), which served as the reference configuration, followed by a phase where the flow was perturbed with the rotation modulation (figure 10 b). The positions of the upper and lower sections of the vortex loops for the unperturbed and perturbed flows were measured using tracking software (Brown Reference Brown2017), and the distances between them (marked as $d_{1}$ and $d_{2}$ in figure 17 b) were calculated as a function of time. Multiple individual measurements were made for each case, in order to obtain the average trajectories with precision. Figure 18 shows a typical example of the resulting amplitude evolution, from which the linear growth rate can easily be determined.

Figure 18.

Averaged temporal evolution of the displacement amplitudes $d_{i}$ ( $i=1,2$ ), defined in figure 17(b), for a perturbation with $k=1$ and $d/h_{\infty }=0.03$ . The slope of the thick grey line gives the growth rate  $\unicode[STIX]{x1D70E}$ .

Figure 19.

Growth rate of the long-wave instability due to local pairing, as a function of wavenumber. The symbols correspond to experimental measurements, and the solid line is the theoretical prediction by Gupta & Loewy (Reference Gupta and Loewy1974) for in-phase perturbations on the two vortices. The theoretical results are obtained using the base flow parameters in table 1.

The growth rates for local pairing were measured for various azimuthal wavenumbers in the range $0<k<4$ . In each experiment, it was verified visually that the amplified perturbation was indeed the one corresponding to the imposed wavenumber. In certain cases with off-maximum wavenumbers, the perturbation eventually dominating in the visualisation could have a different wavenumber than intended, with a higher growth rate. Whenever there was a doubt, the measurements were discarded. Figure 19 shows all the valid growth rates that could be obtained in our experiments, as a function of wavenumber. The comparison with the theoretical predictions for in-phase perturbations, using the base flow parameter values corresponding to our experimental configuration, shows good agreement.

5.2 Global pairing

As seen in § 2.2 (figure 7), a system of two interlaced helical vortices is also unstable with respect to displacement perturbations that are out of phase. With the current set-up using a rigid rotor, the only out-of-phase perturbation that could be imposed is the one with $k=0$ , without modulation of the rotor frequency (any such modulation would again result in an in-phase perturbation). As explained in § 3, such a perturbation could be obtained by introducing a small radial offset of the rotor (typically a few per cent of  $R_{0}$ ), resulting in the generation of two helical vortices of slightly different radii undergoing uniform pairing (figure 4). In order to explore the variation of the growth rate for global pairing with the helix geometry ( $h/R$ , figure 5), four different tip speed ratios $\unicode[STIX]{x1D706}$ were used by varying the free-stream velocity in the water channel. For each case, the base flow parameters (helix geometry $R$ and $h$ ; vortex properties $\unicode[STIX]{x1D6E4}$ and  $a$ ) were determined as described in § 4.1.

Figure 20.

Rotor wake perturbed with a global pairing mode ( $k=0$ ). (a) Experimental dye visualisation for the case with tip speed ratio $\unicode[STIX]{x1D706}=5.40$ and radial rotor offset $\unicode[STIX]{x1D6FF}/R_{0}=0.015$ . (b) Corresponding theoretical prediction (the perturbation amplitude is kept constant in the grey region).

Figure 20(a) shows a visualisation of global pairing, obtained with the reference configuration ( $\unicode[STIX]{x1D706}=5.40$ ) and a rotor offset $\unicode[STIX]{x1D6FF}/R_{0}=0.015$ . Pairing occurs simultaneously around the azimuth of the rotor, and the agreement of the spatial structure with the prediction derived from theory is again very good, up to the location of vortex loop swapping at around $z/R_{0}=3$ . Beyond this point, the rotor wake system almost returns to its initial configuration. The late stages of the global pairing instability are further analysed in § 6.

The growth rate of the uniform pairing mode was again determined from video recordings of dye visualisations. Unlike the case with local pairing, these videos had no initial sequence with unperturbed flow, since the rotor offset could only be set in advance. Therefore a different post-processing method was used, similar to the one presented in Bolnot (Reference Bolnot2012). In this method, a group of three consecutive vortex ridges is followed throughout the wake, and their corresponding trajectories determined with the same image tracking software as used before (Brown Reference Brown2017). Figure 21 illustrates this configuration. Assuming a temporal development of the pairing instability that is uniform along the vortex array, and using the fact that for $k=0$ the perturbations of neighbouring vortices are out of phase, it is straightforward to show that $d_{23}^{2}-d_{12}^{2}=8h\unicode[STIX]{x0394}z$ . In the linear regime of the instability, the distances $\unicode[STIX]{x0394}y$ and $\unicode[STIX]{x0394}z$ increase exponentially in time with growth rate  $\unicode[STIX]{x1D70E}$ , just like the distances $d_{1}$ and $d_{2}$ in figure 17. This means that the temporal evolution of the difference $d_{23}^{2}-d_{12}^{2}$ can be used to determine the pairing instability growth rate directly from the perturbed configuration, without knowledge of the unperturbed vortex positions. An example of such a measurement is shown in figure 22.

Figure 21.

Schematic illustrating the displacement of the perturbed vortex cores in the upper half of the centre plane. The distances $d_{12}$ and $d_{23}$ are used to estimate the growth rate of the perturbation. The unperturbed and perturbed vortex positions are marked with black and coloured bullets, respectively.

Figure 22.

Growth of the global pairing instability ( $k=0$ ), as measured in the upper wake cross-section for $\unicode[STIX]{x1D706}=5.40$ . (a) Averaged temporal evolution of $d_{12}$ and $d_{23}$ defined in figure 21. (b) The difference $d_{23}^{2}-d_{12}^{2}$ , whose exponential increase yields the instability growth rate  $\unicode[STIX]{x1D70E}$ . The time $t=0$ marks the instant when the vortex labelled 1 in figure 21 is shed from the blade tip. The upper horizontal axis indicates the corresponding $z$ -position of the ‘second’ vortex loop (labelled 2).

Figure 23.

Growth rate of the long-wave instability due to global pairing, as a function of the distance between neighbouring helix loops. Symbols correspond to experimental measurements, and the solid line is the theoretical prediction by Gupta & Loewy (Reference Gupta and Loewy1974) for $a_{e}/R=0.05$ .

The resulting growth rates for tip speed ratios $\unicode[STIX]{x1D706}=4.47,~5.40,~6.67$ and 8.04 are plotted in figure 23 as a function of the corresponding parameter $h/R$ . They are compared to the theoretical prediction based on the work of Gupta & Loewy (Reference Gupta and Loewy1974) for $k=0$ . The overall agreement is good, even if the uncertainty and scatter in the measurements, and the limited range of accessible values for $h/R$ in our set-up, do not allow for a solid experimental confirmation of the predicted increase of the growth rate with increasing helix pitch.