3. Experimental results

Figure 2.

( $a$ ) Dye visualisation of the helical blade tip vortex for constant (unperturbed) rotor frequency. Short-wave perturbations appear towards the end of the observation interval. The dashed rectangle represents the area visualised in figure 3. Here, $\zeta$ is the distance from the rotor and $t^*$ the non-dimensional time elapsed since the vortex passed the PIV measurement location at $\zeta /R_0=0.3$ . ( $b$ ) Sequence of close-up views of one helix loop.

Figure 2( $a$ ) shows an overall dye visualisation of the rotor wake generated in the water channel. The tip vortex presents a regular helical structure over a downstream distance ( $\zeta$ ) of approximately five rotor radii. Such visualisations were used to determine the geometric parameters of the helix. The radius and pitch were measured by tracking dye patterns at the top and bottom of each loop. After a short transient, they approach the asymptotic values $R$ and $h$ shown in table 1. The long-wave pairing instability is not visible, its growth is too slow to appear within the observation interval. Short-wave helix distortions become visible around $\zeta /R_0 = 4$ . A closer look is presented in figure 2( $b$ ) as a sequence of images of the same helix loop at several locations/times. The loop exhibits irregular waves, without a clearly marked single wavelength, suggesting that the distortion results from a combination of multiple unstable modes with different wavelengths. An average wavelength of $\lambda =1.4$ cm is determined from these recordings, after correcting for vortex curvature in this view. Based on the vortex properties at $t^*=3$ shown in table 1, this wavelength is approximately 5 times the vortex core radius $a$ , corresponding to a non-dimensional wavenumber $ka = 2\pi a/\lambda = 1.2$ . Judging from the dye pattern, the perturbations appear to involve a displacement of the core as a whole. As the helical vortex loops propagate downstream, the perturbation amplitude grows, leading to a more irregular pattern.

Figure 3.

Perturbation structure for a forced wavenumber $M=16$ . ( $a$ ) Close-up dye visualisation in the region outlined in figure 2( $a$ ). ( $b$ ) Three-dimensional reconstruction from stereoscopic images. The red line represents the axis of the unperturbed vortex. ( $c$ ) Axial view from below along the digitally straightened vortices of ( $b$ ). The mutually induced strain field is shown schematically in grey.

To isolate a single instability mode from these irregular waves, an additional experiment was conducted. The helical vortex was forced by periodically modulating the rotor rotation frequency $f$ with period $(Mf)^{-1}$ . Here, the number of waves per helix loop $M=L/\lambda$ is a measure for the instability scale, where $L=\sqrt {4\pi ^2R^2+h^2}$ is the length of one loop. The value of $M$ is determined by the choice of the target wavelength. In light of the results from the numerical stability analysis presented in § 4, two elliptic modes with wavenumbers $ka=0.47$ ( $M=16$ ) and $ka=1.61$ ( $M=55$ ) are candidates. Due to a limitation of the maximum motor bandwidth, perturbations with small $\lambda$ (large $k$ ) are not experimentally accessible in our set-up. Therefore, the mode with $M=16$ is forced and becomes apparent closer to the rotor as a core displacement, with a well-defined wavelength matching the forcing. A close-up dye visualisation can be seen in figure 3( $a$ ). Stereoscopic imaging from two viewing directions was used, in order to determine the three-dimensional structure of the vortex deformation, following Verhille & Bartoli (Reference Verhille and Bartoli2016). The reconstructed vortex lines are shown in figure 3( $b$ ) for the unperturbed and perturbed cases. After digital straightening of the vortex axes, shown in figure 3( $c$ ), the deformations of the vortex core are clearly visible. They are approximately planar, in a plane between the horizontal and the stretching direction of the strain field that the helix loops induce on each other. Classical elliptic instability is typically observed at much smaller wavelengths and involves more complex core perturbations than the apparent planar displacement of the vortex centre.

The properties of the tip vortex were determined from velocity measurements using stereoscopic PIV. Figures 4( $a$ ) and 4( $b$ ) present the radial profiles of swirl and axial velocity, respectively, at $\zeta /R_0=0.3$ . The measured profiles are close to the ones of the model proposed by Moore & Saffman (Reference Moore and Saffman1973) for trailing wing tip vortices. In particular, the swirl velocity approximately follows $r^{-0.75}$ for large distances $r$ , which implies that there is still significant axial vorticity in this region outside the viscous core, as illustrated by figure 5( $a$ ). As indicated above, the core radius $a$ is here defined as the radial position of maximum swirl velocity $V_m$ . For the same configuration, Quaranta et al. (Reference Quaranta, Bolnot and Leweke2015) have reported a substantially larger value of $a$ , which was likely due to insufficient spatial resolution. The characteristics of the experimental helical vortex determined from visualisations and PIV are summarised in table 1. They also include the total circulation $\varGamma$ , determined using a circular contour of radius $h/2$ around the vortex centre, and the swirl number $S$ , defined as the ratio of maximum swirl to axial velocities.

As mentioned above, the wavelength of the unstable mode in figure 3 is significantly larger than that found for short-wave instabilities in other flows, which would typically have $M\gtrsim 50$ . It is also much smaller than the wavelengths of the pairing instability, which have $M=({1}/{2}), ({3}/{2}), ({5}/{2}), \ldots$ for the case of a single helix. In the next two sections, numerical simulation and stability analysis, as well as a theoretical considerations, will be used to further investigate this unexpected low-wavenumber short-wave displacement instability of a helical vortex.

Figure 4.

Profiles of the mean azimuthally averaged ( $a$ ) swirl ( $V$ ) and ( $b$ ) axial ( $W$ ) velocity components of the tip vortex. Symbols represent the PIV measurements at $\zeta /R_0=0.3$ ( $t^*=0$ ), and the thick line the data at $t^*=3$ obtained from the numerical simulation. ( $c$ ) Temporal evolution of core size $a$ , from numerical simulation. The solid line gives the theoretical viscous evolution.

4. Numerical results

A direct numerical simulation (DNS) of the helical vortex was performed, initialised using experimental data, as explained in § 2.2. The truncated initial vorticity profile is shown qualitatively in figure 5( $a$ ). Due to the small scale and moderate Reynolds number, the core radius increases significantly over the observation interval, as illustrated in figure 4( $c$ ). When low-level white noise is initially added to trigger instability growth, short-wave perturbations appear in the helix (figure 5 $b$ ), with wavelengths and spiral structure resembling those observed in the (unforced) experiments (figure 2). Note that, while only a single helical loop is simulated, the image in figure 5 $(b)$ (and those in figure 7 below) show multiple translated loops for clarity of interpretation.

Figure 5.

( $a$ ) Axial vorticity ( $\eta$ ) profile used for the numerical and theoretical analyses. The dashed line represents a Gaussian distribution. ( $b$ ) Visualisation from DNS ( $Q$ -criterion, see Dubief & Delcayre Reference Dubief and Delcayre2000) of the short-wave instability at $t^*=6$ .

The frozen flow at $t^*=3$ , with the vortex velocity profiles shown as thick lines in figure 4( $a$ , $b$ ), was used for linear stability analysis. The results for the most unstable perturbations are given in figure 6. The growth rate ( $\sigma$ ) versus wavenumber $k$ (or equivalently $M$ ) in figure 6( $a$ ) presents four wide branches, illustrating a large range of unstable wavelengths. Representative mode structures are shown by cross-sectional perturbation vorticity, with three-dimensional views given in figure 7. At $ka=0.47$ , the perturbations consist of plane displacement waves of the vortex, whose orientation is similar to the experimental one for $M=16$ (figure 3). At larger $ka=1.61$ , the unstable modes take the form of higher-order helical bending modes, with a more complex core structure.

Figure 6.

Numerical linear stability analysis of the helical vortex at $t^*=3$ . ( $a$ ) Growth rate of most unstable modes as a function of wavenumber, with corresponding perturbation vorticity in the vortex cross-section (the black circles have radius $a$ ). Here, $\varOmega _0$ is the fluid angular velocity at the vortex centre. ( $b$ ) Frequencies of the modes shown in ( $a$ ), using the same symbols.

Figure 7.

Structure of the unstable modes highlighted in figure 6( $a$ ); ( $a,b$ ) $M=16, ka=0.47$ , ( $c,d$ ) $M=55, ka=1.61$ . Three-dimensional representation (and close-up) of ( $a,c$ ) azimuthal perturbation vorticity and ( $b,d$ ) total vorticity.

In order to further explore the origin of this instability, particularly the effect of vortex curvature, a second stability analysis was performed in the same way, for an array of straight parallel vortices with the same properties as in the helical case and spaced by the helix pitch $h$ . The result in figure 8 again shows a large interval of unstable wavenumbers, with basically the same unstable modes at small and large $k$ , and growth rates of the same order of magnitude. This similarity between the instabilities of the helical vortex and the straight-vortex array strongly suggests that vortex curvature is not at the origin of this phenomenon. The non-dimensional curvature, of order $a/R$ , is indeed quite small in our flow.

Figure 8.

Numerical linear stability analysis of an array of straight vortices with the same velocity profiles and spacing as for the helical vortex. ( $a$ ) Growth rate as a function of wavenumber, with cross-sectional mode structures for $ka=0.45$ and $ka=1.60$ (large insets). The dashed lines correspond to the only unstable perturbations (small insets) arising from Kelvin mode resonances (branch crossings $C_1$ and $C_2$ in figure 9 $b$ ). ( $b$ , $c$ ) Enlargements of ( $a$ ). ( $d$ ) Mode frequencies.

In the following section, we use elements of short-wave instability theory to shed more light on the observations made from experiments and numerical simulation.

5. Theoretical results

Short-wavelength instabilities are usually associated with the resonance of modes of the underlying axisymmetric vortex with either the background strain field (elliptic instability) or corrections induced by vortex curvature (curvature instability). Since curvature can be ruled out here, the resonance would therefore be mediated by a strain field, implying a coupling between modes with azimuthal wavenumbers $m$ and $m+2$ . Both experimental and numerical results demonstrate that the instability involves modes with $|m|=1$ , which motivated us to focus on the vortex modes with azimuthal wavenumbers $m=1$ and $m=-1$ .

We use the underlying axisymmetric vortex profiles obtained from the numerical simulations at $t^*=3$ as the base flow. To isolate a single vortex, the axial vorticity decays smoothly to zero through a spline function for $r\gt h/2$ (figure 5 $a$ ), and the axial velocity is held constant beyond this radius. As explained in § 2.3, the vortex modes are obtained by solving the viscous linear stability problem. We compute Kelvin modes over a range of $k$ for azimuthal wavenumbers $m=1$ and $m=-1$ . The least-damped modes are selected in the calculation, because the instability is not expected when the modes have strong damping rates.

Figure 9.

( $a$ ) Dispersion relations for Kelvin modes of azimuthal wavenumbers $m=1$ (dashed lines) and $m=-1$ (solid lines) for a straight vortex with the same profile as the helical vortex at $t^*=3$ . The grey level indicates the damping rate. ( $b$ ) Close-up view around $\omega =0$ . The symbols marked $S_i$ refer to the mode structures shown in figure 10, and the $C_i$ refer to branch crossings.

Figure 10.

Structure of selected Kelvin modes identified in figure 9( $b$ ). Top row: core region for $r/a\lt 2.5$ ; bottom row: global view for $r/a\lt 10$ . The black circles have radius $a$ . Shown are: $S_1$ ( $m=1$ , $n=1$ ’, $ka=0.45$ ); $S_2$ (−1, 1’, 0.45); $S_3$ (1, 1’, 1.60); $S_4$ (−1, 1’, 1.60); $S_5$ (1, 0, 0.14); $S_6$ (−1, 1, 1.42).

Figure 9( $a$ ) presents the dispersion relation $\omega (k)$ , with the damping rate given by the grey level. Several branches labelled by $n$ are displayed for each $m$ . This label is related to the number of zeros of the radial velocity eigenfunction within the core (Sipp & Jacquin Reference Sipp and Jacquin2003) and provides a measure of the mode complexity. For $m=-1$ , the modes become increasingly damped as $n$ increases. For $m=1$ , two branches are obtained, for $n=0$ whose eigenmode has the same structure as that obtained for the $m=-1$ , $n=0$ branch, and for $n=1$ whose modes are much more damped. The spatial structures of the modes for typical branches $n=0$ and $n=1$ are illustrated in figure 10, $S_5$ and $S_6$ , respectively. The modes for the $n=0$ branches correspond to displacement modes.

Another family of eigenmodes is observed in the frequency range $-0.025\lt \omega \lt 0.025$ shown in figure 9( $b$ ), where parts of the least-damped branches are displayed with special labels $1',2',3'$ . As with the classical branches, the modes become more strongly damped as $n$ increases. The spatial structures of these modes for $n=1'$ are shown in figure 10 ( $S_1$ – $S_4$ ) for the two axial wavenumbers corresponding to highlighted modes in the stability analysis. At small $k$ , the modes for $m=1$ and $m=-1$ are very similar, exhibiting the structure of the displacement mode within the core and a weak structure localised near the edge of the vortex at $r=h/2$ . As $k$ increases, the mode for $m=-1$ transitions within the vortex core to an $n=1$ mode, while for $m=1$ it remains a displacement mode, but with a stronger structure at the edge.

This family of eigenmodes is newly discovered here, it does not appear in standard vortex models. We computed the dispersion relation of an equivalent Batchelor vortex with identical core profiles (see below, figure 11) and only the expected classical branches were found. The modes of the new family seem to be associated with the weak vorticity present outside the vortex core (figure 5 $a$ ). The structure they exhibit at the edge is not associated with a critical layer, which lies much further away, in the region where the vortex is potential. Therefore, they are probably not critical-layer modes, such as the L2 family reported by Fabre et al. (Reference Fabre, Sipp and Jacquin2006) for the Lamb–Oseen vortex.

The $m=1$ and $m=-1$ branches intersect at several points (figure 9 $b$ ), where the corresponding modes can be resonantly coupled via the background strain field, making an elliptic instability possible, as explained in § 2.3. We computed the growth rates via the external strain rate $S_e$ for the array of straight vortices simulated above. The value of $S_e$ acting on a given vortex is obtained by superposing the strain field induced by all other vortices in the array. With one nearby vortex of circulation $\varGamma$ located at a distance $h$ , the induced strain rate is $\varGamma /(2\pi h^2)$ . The total contribution from the vortices on both sides of an infinite array is therefore

(5.1) \begin{equation} S_e = 2\boldsymbol{\cdot }\frac {\varGamma }{2\pi } \left ( \frac {1}{h^2} + \frac {1}{(2h)^2} + \frac {1}{(3h)^2} + {\cdots} \right ) = \frac {\pi \varGamma }{6h^2}. \end{equation}

The dimensionless strain rate for our configuration is $S_e/\varOmega _0=0.016$ .

Figure 11.

( $a$ ) Dispersion relations for Kelvin modes of azimuthal wavenumbers $m=1$ (dashed lines) and $m=-1$ (solid lines) for an equivalent Batchelor vortex, with the same core properties as the helical vortex at $t^*=3$ . The grey level indicates the damping rate. ( $b$ ) Close-up view around $\omega =0$ .

Previous work on the Lamb–Oseen vortex (Eloy & Le Dizès Reference Eloy and Le Dizès1999) and on a strained Batchelor vortex (Lacaze et al. Reference Lacaze, Ryan and Le Dizès2007; Roy et al. Reference Roy, Schaeffer, Le Dizès and Thompson2008) has demonstrated that resonances between modes with the same radial structure, i.e. with the same label $n$ for corresponding branches, are usually the most unstable. They are called ‘principal modes’. Among the resonant points, the sole principal mode for which the corresponding Kelvin modes share the same label $n=1$ was stable, because the mode for $m=1,n=1$ is too strongly damped. Only two resonances, denoted by $C_1$ and $C_2$ in figure 9( $b$ ), were found to be unstable. However, their coupling coefficients are small due to the mismatch in radial structure. As a result, although $C_1$ and $C_2$ occur near the wavenumbers associated with the maximum amplification in the numerical stability analysis (figure 8), the resulting theoretical growth rates are approximately one order of magnitude lower than those predicted numerically. The mode resonance also cannot explain the large instability bands.

In order to further assess the role of classical resonances in our configuration, a stability analysis was performed with the same procedure for the equivalent Batchelor vortex, whose vorticity profile is shown in figure 5( $a$ ) as a dotted line, and considering the same external strain ( $S_e/\varOmega _0=0.016)$ . The growth rates of the resonances (branch crossings) in figure 11( $b$ ) were computed exhaustively, but no unstable modes were found among them. This result is consistent with the one reported in figure 12(a) of Lacaze et al. (Reference Lacaze, Ryan and Le Dizès2007), where an instability map is presented for a similar strain rate $S_e/\varOmega _0=0.01$ . Our configuration falls outside the unstable regime. Therefore, the new instability cannot be explained by resonances between Kelvin modes from the classical branches.

We suspect that the instability arises from interaction between the new $n=1'$ branches for $m=1$ and $m=-1$ . Although these branches do not intersect, their frequencies remain close over a wide range of $k$ . A sufficiently strong strain field could then still couple these modes and induce growth that overcomes the damping associated with the weak frequency detuning between them. This mechanism would explain the structure of the unstable mode at $k=0.45$ , which corresponds to a superposition of the $m=1$ and $m=-1$ displacement modes with comparable amplitude, leading to a planar sinuous deformation of the vortex, as observed in figures 3 and 7( $a,b$ ). At $k=1.60$ , the two $n=1'$ modes differ within the core, which would explain that their superposition forms a helical structure, possibly dominated by the structure of the $m=-1$ mode, in agreement with figure 7( $c,d$ ).

A quantitative proof of this hypothesis is not provided here, because a numerically robust prediction of the growth rate between modes with special labels is beyond the present theoretical framework. In contrast to the classical resonances analysed above, this coupling has large contributions from regions far from the vortex core, where the elliptic correction is no longer a small perturbation. A new theoretical framework, valid up to the hyperbolic point, needs to be developed to correctly estimate this contribution.