2.1. Helical vortex configuration

We consider an infinite helical vortex filament in a cylindrical reference frame ( $r$ ,  ${\it\theta}$ ,  $z$ ), as depicted in figure 1, which represents a simplified model of the wake behind the single-bladed rotor used in the experiments described in §§ 3–6. The helix geometry is defined by the radius $R$ and the pitch $h$ of the vortex centre line, whose coordinates $\boldsymbol{r}_{\boldsymbol{h}}$ are

$R$ and $h$ are both considered as constant. Additional parameters are the vortex circulation ${\it\Gamma}$ and the core radius $a$ .

Figure 1.

Schematic of an infinite helical vortex including the relevant parameters: pitch $h$ , radius $R$ , core size $a$ and circulation ${\it\Gamma}$ .

Figure 2.

Schematic of the azimuthal ( ${\it\phi}$ ) and axial ( ${\it\xi}$ ) velocity profiles of the vortex.

The structure of the filament is further characterised by the distributions of azimuthal and axial velocities in the local cylindrical reference frame ( ${\it\rho}$ ,  ${\it\phi}$ ,  ${\it\xi}$ ) of the vortex (see figure 2), which are axisymmetric at leading order. Previous theoretical studies have mostly considered Rankine-type vortices, with constant vorticity inside the core, zero vorticity outside and without axial flow. In the present analysis, we allow for a more realistic smoother vorticity (and swirl velocity) profile, as well as an axial core flow, as shown in figure 2. One model which includes these features is the Batchelor vortex with a Gaussian vorticity distribution, whose azimuthal and axial velocities are respectively given by

The core radius $a$ in this model is related to the radius $a_{max}$ of maximum swirl velocity (see figure 2), which can easily be determined from experimental measurements, by the relation

The non-dimensional parameters describing a helical vortex with a structure given by (2.2) are the Reynolds number $Re={\it\Gamma}/{\it\nu}$ ( ${\it\nu}$ : kinematic viscosity), the scaled pitch $h/R$ and core radius $a/R$ and an axial flow parameter

representing the ratio between maximum axial velocity and maximum swirl velocity. It is equivalent to an inverse swirl number. Typical parameter values for the present study are: $Re=10^{4}$ ; $h/R=0.5$ ; $a/R=0.05$ ; and $W=0.5$ . For an infinite helical vortex, the fluid velocities in the direction $z$ of the axis of symmetry differ by an amount ${\it\Gamma}/h$ between the inside and the outside of the cylinder containing the helix. In a reference frame where the $z$ -velocity vanishes at large $r$ , the helix is convected with a velocity of approximately ${\it\Gamma}/2h$ (a small difference comes from the self-induced motion due to the vortex curvature). In the following, the pitch $h$ and the velocity ${\it\Gamma}/2h$ are used for normalisation. In particular, the time $2h^{2}/{\it\Gamma}$ is used to obtain non-dimensional instability growth rates. It corresponds roughly to the time it takes the helix to move one pitch in the axial direction under its own influence.

In order to be able to use previous theoretical results for Rankine vortices, we recall the concept of an equivalent core size first introduced by Widnall, Bliss & Zalay (Reference Widnall, Bliss, Zalay, Olsen, Goldberg and Rogers1971). The distributions of azimuthal and axial velocity have an effect on the self-induced motion of the vortex. It can be shown that, for long-wave displacement perturbations such as the ones considered in the present work, the self-induced dynamics of a vortex with velocity profiles $v_{{\it\phi}}({\it\rho})$ and $v_{{\it\xi}}({\it\rho})$ , evolving on a characteristic radial scale $a$ , exhibit the same self-induced dynamics as an equivalent Rankine vortex without axial flow, with the same circulation and a core radius given by

For the Batchelor vortex (2.2) one finds

${\it\gamma}\approx 0.577$ is Euler’s constant. Equation (2.7), together with (2.3), is used in § 3 for the comparison between experimental and theoretical results (the fact that the experimental vortex is not exactly Gaussian is discussed in § 4).

2.2. Displacement instability

In its unperturbed base state, the helical vortex rotates around its axis and translates in the direction of this axis due to its self-induced velocity, without changing shape. The stability of this configuration is analysed by considering the evolution of displacement perturbations of the base geometry in (2.1). Following Widnall (Reference Widnall1972) and Gupta & Loewy (Reference Gupta and Loewy1974), perturbations of the form

with azimuthal wavenumber $k$ and (complex) growth rate ${\it\alpha}$ , are considered. The evolution of the filament shape is obtained by computing its self-induced velocity from the Biot–Savart law. For a vortex of small core size, the velocity $\boldsymbol{u}$ induced at a point $\boldsymbol{r}_{\mathbf{0}}$ in the fluid is given by

where $l$ is the curvilinear coordinate along the vortex centre line. For a point on this line, i.e. $\boldsymbol{r}_{\mathbf{0}}=\boldsymbol{r}(l_{0})$ , the integral in (2.9) diverges, and the detailed core structure has to be taken into account in order to determine the correct self-induced velocity. Widnall (Reference Widnall1972) employed the cutoff method, in which the interval $l_{0}-{\it\varepsilon}<l<l_{0}+{\it\varepsilon}$ containing the singularity is excluded from the integration. For a Rankine vortex with core radius $a_{e}$ and without axial flow, the correct cutoff distance is given by ${\it\varepsilon}=(a_{e}/2)\exp (1/4)$ . Gupta & Loewy (Reference Gupta and Loewy1974) used a different approach, initially proposed by Rosenhead (Reference Rosenhead1930), to desingularize the integrand in (2.9). The denominator is replaced by $|[\boldsymbol{r}_{\mathbf{0}}-\boldsymbol{r}(l)]^{2}+{\it\mu}^{2}|^{3/2}$ , where ${\it\mu}$ is a length proportional to the core size. If ${\it\mu}=(2/e){\it\varepsilon}=a_{e}\exp (-3/4)\approx 0.47a_{e}$ , this procedure is equivalent to the cutoff method (Saffman Reference Saffman1992). Gupta & Loewy (Reference Gupta and Loewy1974), however, used ${\it\mu}=a_{e}$ , which explains the discrepancies with Widnall’s (Reference Widnall1972) results for large core size.

Linearisation of the evolution equations for the perturbed helix shape, using one or the other adapted versions of (2.9), then leads to an eigenvalue problem for the growth rate ${\it\alpha}$ , whose real part ${\it\sigma}$ characterises the growth or decay of the perturbation amplitude. The corresponding eigenvectors $({\it\delta}r_{h},{\it\delta}{\it\theta}_{h},{\it\delta}z_{h})$ define the mode shape. Details of the calculations can be found in Widnall (Reference Widnall1972) and Gupta & Loewy (Reference Gupta and Loewy1974). Figure 3 shows an example of the non-dimensional growth rate ${\it\sigma}^{\ast }={\it\sigma}(2h^{2}/{\it\Gamma})$ as a function of the perturbation wavenumber $k$ . It corresponds to one of the cases discussed by Widnall (Reference Widnall1972), with values of the helix pitch and core size close to the ones considered in the present study. The difference between Widnall’s analytical result and the corresponding figure in her paper is believed to be due to the difficulties (at the time) in numerically evaluating the rather complex functions and integrals appearing in the expression of the growth rate, or simply to a plotting error. The growth rate has local maxima for wavenumbers which are odd multiples of $1/2$ . The highest growth rate is obtained for $k=1/2$ , its value is close to ${\it\sigma}^{\ast }={\rm\pi}/2$ .

Figure 3.

Growth rate as function of wavenumber for a helical Rankine vortex with $h/R={\rm\pi}/5$ . Black (thin) lines: figure 5(d) from Widnall (Reference Widnall1972), the numbers indicate $a_{e}/R$ (reproduced with permission); red (thick) line: analytical result from Widnall (Reference Widnall1972) for $a_{e}/R=0.1$ . The long-wave assumption used in the theory is not valid in the shaded area.

The theoretical development by Widnall (Reference Widnall1972) assumes that the wavelength of the displacement perturbations is large compared to the core size, which corresponds to an upper limit of validity for the (dimensional) wavenumber $k_{l}$ along the vortex centre line. Leweke & Williamson (Reference Leweke and Williamson1998) have estimated this limit for a straight Rankine vortex of radius $a_{e}$ , based on the exact dispersion relation for perturbation waves of such a vortex (Kelvin waves), to be $k_{l}a_{e}\approx 0.4$ . This result can be used to estimate the range of validity of the theory for the helical geometry. From the definition of the non-dimensional wavenumber $k$ ( $=$  number of perturbation wavelengths in one helix turn of length $L=[h^{2}+(2{\rm\pi}R)^{2}]^{1/2}$ ), it is straightforward to show that the long-wave condition corresponds to

For the case shown in figure 3 ( $a_{e}/R=0.1$ ), one obtains $k\lesssim 4.0$ (for $a_{e}/R=0.33$ , the range of validity is even reduced to $k\lesssim 1.2$ , but this configuration is unrealistic since successive helix turns have overlapping cores). In the experimental study presented in § 3, we observe very thin-cored vortices ( $a_{e}/R\approx 0.05$ ), for which the limit of validity of the long-wave theory is not a concern.

The effect of viscosity was neglected in the above analysis. The correction of the growth rate due to viscous damping can be estimated to be of the order of $-{\it\nu}k_{l}^{2}$ , which in non-dimensional form equals $-(8{\rm\pi}^{2}/Re)k^{2}(h/L)^{2}$ . For the parameter ranges considered in this study ( $h/R<1$ , $k<4$ , $Re=O(10^{4})$ ), the non-dimensional viscous correction is of $O(10^{-2})$ and will therefore not be considered further here (for full-scale applications such as wind turbines, where the Reynolds number is several orders of magnitudes larger, viscous effects on the pairing instability are even less relevant).

The instability growth rates obtained with the formalism of Gupta & Loewy (Reference Gupta and Loewy1974), using the correct parameter ${\it\mu}$ , are identical to the ones from Widnall’s (Reference Widnall1972) analysis, within the range of validity of the theory. For the comparison with our experimental measurements, we use the former procedure, which is appreciably easier to implement.

Figure 4.

Geometrical deformation and local paring predicted by Widnall (Reference Widnall1972) for the case $h/R=0.5$ . (a,b) $k=3/2$ ; (c,d) $k=4/3$ . Three-dimensional view (a,c) and developed plan view (b,d). The vertical scale of the plan views is compressed by a factor 2.

Figure 4 presents two examples of deformations of the initially helical vortex due to the displacement instability by plotting the perturbed helix geometry $\boldsymbol{r}_{\boldsymbol{h}}+{\it\delta}\boldsymbol{r}_{\boldsymbol{h}}$ ((2.1) and (2.8)). The perturbation ${\it\delta}\boldsymbol{r}_{\boldsymbol{h}}$ is the unstable eigenmode corresponding to the given wavenumber $k$ . Figure 4(a) shows one of the most unstable perturbations, for $k=3/2$ . For this value of the wavenumber, as for all odd multiples of $1/2$ , the displacement perturbations of successive helix loops are exactly out of phase at a given azimuthal position ${\it\theta}$ . The developed plan view reveals that these perturbations correspond to a local pairing of neighbouring helix turns, with the pattern repeating after two turns. For wavenumbers away from the local maxima of the growth rate curve, e.g. for $k=4/3$ in figure 4(b), the perturbation repeats at constant ${\it\theta}$ after a longer period, but a local grouping of two successive helix loops is still found at various azimuthal positions. One can easily work out that a perturbation with a wavenumber of the form $k=m/n$ repeats after $n$ cycles and exhibits $m$ azimuthal locations where local grouping occurs. Further examples of unstable perturbation modes can be seen in figures 17 and 18.

2.3. Pairing of vortex arrays

As seen in the previous section, the long-wave displacement instability of a helical vortex is characterised by a periodic local pairing of successive turns. This is further illustrated by considering a longitudinal plane containing the helix axis. In this plane, the helical vortex appears in the form of two periodic rows of concentrated vortices (see, e.g. figure 21). These rows bear a resemblance to arrays of point vortices in two dimensions, which are also known to be unstable with respect to the pairing mechanism. In addition to this qualitative similarity, we now establish a quantitative link between the helix instability and the classical pairing of a point vortex array.

Figure 5.

Schematic of the developed plan view of the helix, showing the parameters used in the analysis.

The stability of an infinite row of identical point vortices was treated by Lamb (Reference Lamb1932) based on earlier work by von Kármán & Rubach (Reference von Kármán and Rubach1912). The vortices of circulation ${\it\Gamma}$ are separated by a constant distance $b$ from their next neighbours, and this base configuration is stationary. The position of each vortex is then perturbed periodically along the array, with a phase difference ${\it\Phi}$ for the displacements of two adjacent vortices. It is found that the array is unstable for all perturbations with non-zero ${\it\Phi}$ ( ${\it\Phi}=0$ corresponds to a neutral translation of the entire row), and that the instability growth rate ${\it\sigma}_{p}$ is given by:

The maximum is reached for ${\it\Phi}={\rm\pi}$ , i.e. for out-of-phase displacements of neighbouring vortices, which corresponds to the pairing shown schematically in figure 21(b). The growth rate for this case is ${\it\sigma}_{p}={\rm\pi}{\it\Gamma}/4b^{2}$ or, in non-dimensional form, ${\it\sigma}_{p}(2b^{2}/{\it\Gamma})={\rm\pi}/2$ .

We now transpose this result to the case of a helical vortex by considering the developed plan view of this configuration in figure 5. In this representation, the helix appears as a periodic array of inclined straight vortices that can be associated with a row of point vortices in a plane going through $\unicode[STIX]{x2460}$ and $\unicode[STIX]{x2461}$ . From the form of the perturbations of the helix instability in (2.8), the phase difference between $\unicode[STIX]{x2460}$ and $\unicode[STIX]{x2462}$ , separated by a complete turn, equals $2{\rm\pi}k$ . The phase difference ${\it\Phi}$ between $\unicode[STIX]{x2460}$ and $\unicode[STIX]{x2461}$ is reduced from this value by a fraction ${\it\Delta}/L$ , where $L$ is again the length of one helix turn. The geometrical relations in figure 5 lead to ${\it\Delta}=h^{2}/L$ , and therefore:

Combining (2.11) and (2.12), normalising by $2h^{2}/{\it\Gamma}$ and using $b^{2}=h^{2}-{\it\Delta}^{2}$ , one obtains an expression for the dimensionless growth rate of the pairing instability in the helical geometry:

This result is plotted in figure 6(a) for the first lobe of the growth rate curve ( $0\leqslant k\leqslant 1$ ) (see figure 3), for three values of the non-dimensional pitch $h/R$ . For $h\rightarrow 0$ , the maximum growth rate is ${\it\sigma}_{p}^{\ast }={\rm\pi}/2$ for $k=1/2$ . For increasing pitch, the maximum growth rate increases, and it is reached at slightly higher wavenumbers. This is in very close agreement with the results from the full stability analysis of the helical geometry, an example of which is shown in figure 6(b).

Figure 6.

(a) Prediction of the growth rate for various $h/R$ , based on the result (2.13) for two-dimensional pairing (Lamb Reference Lamb1932). (b) Growth rate from the full stability analysis of a helical vortex with $a_{e}/R=0.1$ (Gupta & Loewy Reference Gupta and Loewy1974).

The prediction of the growth rate in (2.13) is based on the assumption that the helical geometry can locally (in the azimuthal direction ${\it\theta}$ ) be approximated by an array of straight vortices. The curvature of the vortex loops is neglected in this approach, as is the self-induced velocity due to this curvature. The core size $a_{e}$ , which influences the curvature-induced velocity, does not appear in the analysis, which is based on a point vortex system. The dependence on this parameter in the full analysis is weak for thin-cored vortices ( $a_{e}/R\lesssim 0.1$ ). Levy & Forsdyke (Reference Levy and Forsdyke1927) have studied the stability of an infinite series of vortex rings, a configuration which, for small separation distances between the rings, resembles a helical vortex of small pitch. They also find a weak dependence of the growth rate on the curvature and core size in this parameter range, and as $h/R$ (the normalised curvature) tends to zero, the non-dimensional maximum rate converges to the value ${\rm\pi}/2$ for two-dimensional pairing.

The result in (2.13) is expected to become invalid for large helix pitch, for which the mutual interaction between neighbouring loops becomes much weaker and the dynamics are dominated by the local induction from the vortex curvature and torsion.

The present analysis can in principle be extended to wavenumbers $k$ larger than 1. Since the phase difference ${\it\Phi}$ in the growth rate (2.11) for point vortex arrays is to be taken modulo $2{\rm\pi}$ , the lobe of the growth rate curve in figure 6(a) repeats itself in the same way for higher $k$ . However, as $k$ increases, i.e. as the wavelength of the displacement perturbation decreases, the assumption of arrays of locally straight vortices becomes increasingly less valid. Robinson & Saffman (Reference Robinson and Saffman1982) have investigated the pairing instability for arrays of sinusoidally perturbed vortices. They found that the maximum growth rate decreases with increasing perturbation wavenumber, which is consistent with the decrease of the local maxima for the higher wavenumbers observed in figures 3 and 23.

It can be concluded that the growth rate of the long-wave displacement instability of a helical vortex with moderate pitch ( $h\lesssim R$ ) and small core size ( $a_{e}/R\lesssim 0.1$ ) is predicted remarkably well using the results from the classical pairing (in two or three dimensions) of infinite vortex arrays, showing that vortex pairing is the dominant mechanism involved in this phenomenon.

Figure 7.

(a) Experimental set-up (side view of the water channel test section) and (b) schematic of the rotor blade geometry.

In the following sections, we describe an experimental study aimed at clearly identifying the pairing instability of a helical vortex in a real flow (wake of a rotor), and to compare measurements of its growth rate with theoretical predictions. Observations of the later stages of this phenomenon, not accessible by linear theory, are also shown.

3.1. Set-up

Experiments were carried out in a recirculating free surface water channel with a test section of dimensions 38 cm (height) $\times$ 50 cm (width) $\times$ 150 cm (length). A helical vortex was generated near the test section entry by a single-bladed rotor mounted on an ogive-tipped shaft (diameter 15 mm) and driven by a computer-controlled stepper motor outside the channel using a belt (figure 7 a). The rotor blade geometry, shown schematically in figure 7(b), is based on the low Reynolds number airfoil A18 by Selig et al. (Reference Selig, Guglielmo, Broeren and Giguere1995). It is designed to operate in the wind turbine regime and produce a constant radial circulation distribution (Joukowsky rotor) over the outer $75\,\%$ of the span, in order to generate a highly concentrated tip vortex. The chord and twist distributions of the blade are given in figure 8. The rotor has a radius $R_{0}=80~\text{mm}$ and a tip chord $c_{tip}=10~\text{mm}$ . Details of the design procedure can be found in Bolnot (Reference Bolnot2012). For the present experiments, the blade is rotated at a frequency $f_{0}=6~\text{Hz}$ , and placed in a uniform incoming flow of velocity $U=36~\text{cm}\,\text{s}^{-1}$ . This results in a tip chord-based Reynolds number of approximately 30 000 and a tip speed ratio $2{\rm\pi}f_{0}R_{0}/U=8.4$ , representing the design conditions for the blade geometry. Experiments were also carried out for two higher tip speed ratios (9.6 and 12.3), by varying the free stream velocity $U$ , leading to lower values of the (non-dimensional) pitch. The results are qualitatively similar to the ones of the design case presented in the following sections, leading to the same overall conclusions.

Figure 8.

Radial distributions of (a) chord $c$ and (b) twist angle ${\it\zeta}$ of the blade.

Figure 9.

Schematic of the experimental helix perturbation consisting of an axial displacement ( $k=1/2$ , $d/h=0.2$ ). The dashed line represents the unperturbed geometry. The amplitude is chosen here to be much higher than in the experiments for improved visual clarity.

3.2. Perturbation method

Although the helical vortex generated with this set-up is known to be unstable from theory, the turbulence intensity in the channel ( ${<}1\,\%$ ) and the level of vibrations of the rotor support were low enough so that no significant perturbation of the helix shape was visible in the field of observation, which covered approximately 10 turns (see figure 11). This made it possible to impose controlled displacement perturbations of the helix, by introducing suitable periodic modulations of the initially constant blade rotation frequency. Varying the rotation speed results in a shift of the tip vortex in the axial direction with respect to its unperturbed position. Based on the form of the unstable modes of the instability given by (2.8), we aim for a helix with a perturbed $z$ -coordinate given by

where $d$ is the amplitude of the displacement. An example with an out-of-phase displacement of successive helix turns ( $k=1/2$ ) is shown schematically in figure 9. Given that, in the frame of reference of the helical vortex, the rotor moves with a velocity $hf_{0}$ in the axial direction, we may write $z=hf_{0}t$ . Derivation with respect to time and the relation $\dot{{\it\theta}}=2{\rm\pi}f$ then lead to

which gives the rotation rate of the rotor as function of its angular position. Numerical integration of (3.2) then provides the rotation rate as function of time, which is programmed into the step motor controller. Figure 10 contains an example of such a modulation, for the case shown in figure 9, with a typical value of the amplitude used in the experiments. The modulation has a period of 2 ( $=1/k$ ) rotor cycles. The rotation is faster than the mean in the first cycle and slower in the second. Whereas the rate varies almost sinusoidally with the azimuthal position (for small $d/h$ ), the dependence on time is more distorted. The motor (Parker CompuMotor AMP 5034-232) and controller (Parker Gemini GT6K) have a maximum resolution of 128 000 microsteps per revolution, which enables them to execute the prescribed modulation precisely. Amplitudes of only a few per cent of the helix pitch were sufficient to trigger the desired instability mode.

Figure 10.

Modulation of the rotor rotation rate for a perturbation with $k=1/2$ and $d/h=0.03$ : (a) as function of the angular position ${\it\theta}$ , and (b) as function of time $t$ .

Figure 11.

Dye visualisation of the unperturbed helix. The rotor is on the left, flow is from left to right. The vertical shaft support is located on the right, just outside the field of view.

The amplitude of the frequency modulation in figure 10 corresponds to about 10 % of its mean value. Surprisingly, this rather large variation does not induce a modulation of the vortex parameters of the same order, as might have been expected. It was found from particle image velocimetry (PIV) measurements that the circulation and core sizes vary by only a few per cent during a typical perturbation cycle, as long as the deformation amplitude $d/h$ remains below 5 %. This variation lies within the uncertainty intervals for these parameters in the measurements of the unperturbed flow presented in § 4 below. We therefore conclude that the perturbation method does not significantly modify the helical base flow properties.

3.3. Visualisations and measurements

The helical vortex structure was visualised using fluorescent dye (Leweke Reference Leweke2012). The rotor set-up could be tilted to be able to apply the fluorescent paint (thickened fluorescein solution) to the blade tip outside the water. When the spinning rotor was lowered into channel test section with the water flowing, the dye was washed off and clearly marked the tip vortex. Volume illumination was achieved with light from an argon ion laser (model Stabilite 2017 by Spectra Physics), coupled to an optical fibre equipped with a spherical lens. The helical wake established itself after a short transient, and observation was possible for about 30 s (180 blade rotations) before the dye disappeared. Images and video sequences were recorded with a standard digital camera (Canon IXUS 220 HS) in high-speed mode at 120 images per second. These visualisations were also used to obtain quantitative information about the vortex geometry (radius, pitch) and to measure the instability growth rates, as explained in § 5.2. Additional visualisations were obtained by injecting a dye solution at a fixed location in the rotor plane in order to observe and measure the flow along the vortex centre (see figure 16 below).

Velocity fields and vorticity distributions were obtained from PIV. A Nd-YAG pulsed laser (Quantel Twins Ultra 190) was positioned underneath the test section to generate a 3 mm vertical light sheet illuminating the centre plane of the rotor flow (only half of the plane was illuminated behind the rotor since the shaft blocked the light). The flow was seeded with $30~{\rm\mu}\text{m}$ plastic particles, and images were recorded with a resolution of 2000 $\times$ 1336 pixels using a Roper Redlake digital camera. The images were processed using an in-house code based on an algorithm developed by Meunier & Leweke (Reference Meunier and Leweke2003), which has been successfully employed in previous experimental studies of vortex flows (see, e.g. Meunier & Leweke Reference Meunier and Leweke2005; Roy et al. Reference Roy, Leweke, Thompson and Hourigan2011). PIV measurements were used to determine the base flow properties of the helical rotor wake (circulation, core size, etc.).

Further details about the experimental set-up and procedures are given in Bolnot (Reference Bolnot2012).

5.1. Unstable modes

Figure 17.

Most unstable perturbation modes. (a,c,e,g) Experimental dye visualisations; (b,d,f,h): sketches of the theoretical mode shapes. (a,b) $k=1/2$ ; (c,d) $k=3/2$ ; (e,f) $k=5/2$ ; (g,h) $k=7/2$ .

Figure 18.

Subharmonic instability modes. (a,c,e) Experiment; (b,d,f) theory. (a,b) $k=1/3$ ; (c,d) $k=1/4$ ; (e,f) $k=2/5$ . The deformation pattern repeats after 3, 4 and 5 helix/rotor turns, respectively.

As seen in figure 11, the helical vortex generated by the blade spinning at a constant rate does not show any significant deformation over the first 3–4 diameters behind the rotor, even if some ‘scatter’ occurs towards the end of this interval (figure 20 a), likely caused by some interference with the rotor support structure located just outside the field of view. As soon as the flow is perturbed through a modulation of the blade rotation, as explained in § 3.2, deformations of the initially regular helix become visible a short distance behind the rotor, and they are rapidly amplified as the vortex moves downstream, showing that the vortex is indeed unstable with respect to these perturbations. Figures 17 and 18 contain dye visualisations of a number of different perturbation modes, for various azimuthal ( ${\it\theta}$ ) wave numbers $k$ , along with the theoretical counterparts from the stability analysis.

The modes in figure 17 correspond to the first four maxima of the growth rate curve (figures 3 and 23), for which $k$ is an odd multiple of $1/2$ . For these cases, the perturbation repeats itself after two blade revolutions, and the displacements of neighbouring helix loops are exactly out of phase. Local pairing occurs at azimuthal locations where the displacement is maximal. The phase of the blade rotation modulation was chosen so that pairing always occurs at the top. For $k=1/2$ , this is the only location of pairing; for the higher wavenumbers, additional locations around the azimuth of the helix can be identified. The pattern for $k=3/2$ in figure 17(b) is the same as found by Walther et al. (Reference Walther, Guénot, Machefaux, Rasmussen, Chatelain, Okulov, Sørensen, Bergdorf and Koumoutsakos2007) in their figure 9. The theory in § 2.2 predicts a spatially uniform temporal growth in an infinite helical vortex. In the experimental configuration, the vortex is generated continuously by the rotor blade, and perturbations grow (in time) as they are convected downstream. In the frame of reference of the rotor, this results in a spatial growth of the perturbation amplitude. The amplitude at a given downstream position depends on its initial value, as well as on the growth rate of the mode. For the examples shown in figures 17 and 18, forcing amplitudes $d/h_{\infty }$ (see (3.2)) between 3 % and 5 % were used. The dependence of the wake evolution on this parameter is further discussed in § 6.

Perturbation amplitudes of the same order as for the theoretical shape on the right-hand side of figure 17 are found near the centre of the experimental visualisation on the left. The observed deformations are extremely close to the theoretical predictions for an infinite helix. This demonstrates that we are indeed observing a convective long-wave (pairing) instability in our flow, and it also shows that the the pairing mechanism depends mainly on the immediate vicinity of a given vortex group.

Figure 18 presents further instability modes, with wavenumbers below the first maximum of the growth rate curve. These perturbations are more complex than the ones of figure 17, their shapes repeat after a larger number of cycles. Their amplitudes nevertheless grow downstream, showing their unstable nature. Again, a close similarity between experimentally observed and theoretically predicted deformation shapes is found.

Finally, figure 19 shows a case where the tip vortex is perturbed with a wavenumber $k=1$ . For integer values of $k$ , theory predicts a growth rate close to zero. Indeed, the perturbed flow in figure 19 is practically indistinguishable from the base flow in figure 11: the imposed perturbations are only very weakly amplified.

Overall, our experiments show that the rotor wake is highly receptive to a large variety of deformation perturbations corresponding to the modes of the theoretically predicted long-wave instability of a helical vortex.

Figure 19.

Helical vortex perturbed with $k=1$ and $d/h_{\infty }=0.03$ . The displacement is in phase between successive helix turns. No amplification of the perturbation can be detected.

Figure 20.

Superposition of 10 instantaneous dye images recorded at the same phase of the blade motion. (a) Unperturbed flow; (b) perturbed flow with $k=1/2$ and $d/h_{\infty }=0.03$ . The white rectangles show the regions represented in figure 21.

Figure 21.

Schematic illustrating the determination of the base flow and perturbation properties from dye visualization. (a) Unperturbed and (b) perturbed helix. The dots represent the vortex centre positions in the upper part of the centre plane.

5.2. Growth rates

It was possible to determine the growth rates of the various deformation modes seen in the previous section from video sequences of the helical vortex visualised with dye. For each case, the sequence started with a phase of unperturbed flow, serving as the reference configuration, before the rotation modulation was switched on. In figure 20, several images with the same rotor position are superposed for each of the two phases of one such recording. The mean positions of the upper sections of the vortex loops were then detected for the unperturbed and perturbed flows, and the distance between them ( $d_{1}$ and $d_{2}$ in figure 21) calculated as function of the downstream position. The evolution of this distance with time can be directly deduced from the rotor mean frequency $f_{0}$ and the frame rate of the camera. Measurements of the amplitude were made up to the point where two successive helix loops first swap their axial position $z$ (near the centre of the images in figure 20). Up to this point, the patterns are very repeatable, so that five images were sufficient to obtain the average positions with precision for each case.

Figure 22 shows a typical growth of the perturbation amplitude as function of time. At $t=0$ the tip vortex leaves the rotor plane. After an initial transient phase, in which the deformation adjusts from the imposed axial displacement to the proper eigenmode of the instability, the amplitude increases exponentially for about 0.7 s, which corresponds to a distance of about $4h_{\infty }$ or $2R_{0}$ . The end of this exponential growth corresponds roughly to the time/location where the loop swapping occurs, with a displacement amplitude close to $0.5h_{\infty }$ . The growth rate is found by fitting an exponential function to the measured data.

The above procedure was repeated for 11 different wavenumbers, including all examples shown in the previous section. The resulting growth rates are plotted in figure 23, non-dimensionalised by the measured helix properties. Figure 23 also contains the theoretical prediction for the pairing instability of an infinite helical vortex, using the parameter values listed in table 1. The agreement between experiment and theory is very good. In the theoretical analysis of § 2.2, the presence of an inner blade root vortex is not taken into account. Adding a straight vortex of the same circulation as the tip vortex (but of opposite sign) on the helix centre line in the calculations, results in a variation of the instability growth rate of only a few per cent. The root vortex in our experiments (figures 12 a and 13) is much weaker than the tip vortex, probably due to the presence of the shaft. Its effect on the growth rate therefore lies well within the uncertainty interval of the measurements and was therefore not considered in the theory.

Figure 22.

Growth of the displacement amplitudes $d_{i}$ for a perturbation with $k=3/2$ and $d/h_{\infty }=0.03$ . The slope of the linear part represents the growth rate ${\it\sigma}$ .

Figure 23.

Growth rates of the long-wave instability. The symbols represent experimental measurements, and the line the corresponding theoretical prediction from Gupta & Loewy (Reference Gupta and Loewy1974) using the values in table 1.