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# Experiments and simulations on short-wave instabilities in helical vortices

Zhihan Xu, Mark C. Thompson, Thomas Leweke

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Zhihan Xu, Mark C. Thompson, Thomas Leweke. Experiments and simulations on short-wave instabilities in helical vortices. Journal of Physics: Conference Series, 2025, 3016, pp.012028. 10.1088/1742-6596/3016/1/012028. [hal-05234916](https://hal.science/hal-05234916v1)

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# Experiments and simulations on short-wave instabilities in helical vortices

Zhihan Xu¹,², Mark C. Thompson³ and Thomas Leweke²

¹School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China

²IRPHE, Aix-Marseille University, CNRS, Centrale Méditerranée, Marseille, France

³Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Clayton, 3800, Australia

E-mail: xzh11272001@outlook.com

**Abstract.** Short-wave instabilities of helical vortices are investigated experimentally using a one-bladed rotor and numerically with a spectral/spectral-element code. Certain rotor flow configurations, defined by the Reynolds number (Re) and tip speed ratio (λ), exhibit short wave deformations of the vortex core. Increased λ or decreased Re lead to earlier and more pronounced perturbations, possibly due to enhanced helix loop interaction or increased turbulence from flow separation. The perturbations display wavelengths longer than expected for elliptic instability, possibly due to the specific velocity profiles of the vortices in this study. Numerical simulations based on measured velocity profiles align well with experiments. The exact origin of these instabilities remains uncertain, with possible mechanisms including elliptic, curvature/torsion, or swirling jet instabilities.

## 1 Introduction

The near wake behind a rotor comprises a system of interlaced helical vortices, formed at the tips of the rotor blades through the roll-up of the vorticity layer shed from the blade trailing edge. This initially well-organized vortex system is advected downstream and eventually becomes unstable after a distance of several rotor diameters, leading to vortex breakdown and a turbulent far wake. In a wind farm setting, the tip vortices generated by an upstream wind turbine can have detrimental effects on downstream turbines, such as amplified fatigue loading and loss of efficiency due to a delayed wake recovery. Various instabilities of the helical vortices can accelerate vortex breakdown and therefore mitigate these harmful effects. These instabilities can be categorized according to the wavelength of the corresponding perturbations: long-wave displacements of the entire vortex core, and short-wave deformation within the core. [1]

Long-wave instabilities lead to local pairing and leapfrogging of adjacent vortex loops for a single helix. Local pairing also exists for systems of multiple helices, which can in addition exhibit a global pairing with a change in the relative axial positions of the different helices. The linear stability of a single helical vortex filament was first analyzed by Widnall [2], who determined the growth rates of displacement perturbations of different wavenumbers. Gupta and Loewy [3] and Okulov [4] later extended this theory to configurations with multiple interdigitated helices. The phenomenon of long-wave instability has been observed experimentally since Alfredsson and Dahlberg [5], though quantitative comparisons with theoretical predictions were initially scarce. Quaranta et al. [6] performed detailed experiments on a single helix generated by a one-bladed rotor. By imposing controlled perturbations with various wavenumbers, they found growth rates in good agreement with theoretical predictions. This work was then extended to two interlaced helical vortices [7], where both local and global pairing were explored.

<figure><img src="image_3.png" /></figure>
Short-wave instabilities are characterized by the amplification of perturbations scaling on the vortex core size. They may originate from an external strain field, induced, e.g., by a neighboring vortex, or from the curvature and torsion of the vortex itself. Such effects can produce either an elliptic instability, or a curvature instability. Both instabilities can be understood as the resonance between two Kelvin modes of the underlying vortex and a vortex perturbation by the strain or curvature. First described for single columnar vortices subjected to an external strain field [8], elliptic instability has been extensively examined in straight vortex pairs [9] or vortex rings. [10] It was also predicted to appear in helical vortices [11], and has been captured numerically in a double-helix system. [12] In contrast, the curvature instability remains less explored; it was studied theoretically for curved Rankine vortices (uniform-vorticity core) [13, 14] and Batchelor vortices (Gaussian axial velocity and axial vorticity) [15]. Both theory and numerical simulation have revealed the significant role of the axial velocity component on these two instabilities. Previous experiments have provided some evidence of short-wave instability in rotor wakes [1], without a clear identification of the stability modes. [16, 13, 15] In a recent numerical study of the stability of a single helix with axial core flow [17], both elliptic and curvature instabilities were found and the interacting Kelvin modes were identified.

In this work, we perform precise experiments concerning short-wave instabilities in helical vortices, using a small-scale rotor in a water channel. The observations and measurements are complemented by results from direct numerical simulations.

## 2 Experiments

### 2.1 Set up

Experiments were conducted in a recirculating free-surface water channel with a test section of 38 cm (width) × 50 cm (height) × 150 cm (length). A helical vortex was generated by a one-bladed rotor, mounted on an ogive-tipped shaft of 1.5 cm diameter, and driven externally by a computer-controlled stepper motor via a gear box (figure 1a). The blade geometry, presented in figure 1(b), is based on the low-Reynolds number airfoil A18 by Selig et al. [18]; it was designed to operate in the wind turbine regime and produce a constant radial circulation distribution in order to generate a highly concentrated tip vortex. The rotor has a radius $R_0 = 8$ cm and a tip chord $c = 1$ cm. More details on the design methodology and experimental protocols can be found in Bolnot [19] and Quaranta et al. [6]. In the baseline configuration of the present experiments, the blade was rotated at a frequency of $f_0 = 6.25$ Hz, and the flow in the channel had a uniform free-stream velocity of $U_0 = 37.5$ cm/s, yielding a tip-chord-based Reynolds number $Re = 2\pi R_0 f_0 c/\nu \approx 30,000$, where $\nu$ is the kinematic viscosity, and a tip speed ratio $\lambda = 2\pi R_0 f_0/U_0 = 8.4$. These parameters were then varied in the ranges $10,000 \le Re \le 40,000$ and $5.6 \le \lambda \le 14$, in order to trigger short-wave instabilities in off-design conditions.

### 2.2 Measurement methods

The experimental methods used to characterize the structure and evolution of the helical vortices include dye visualization and velocity measurements. The tip vortices generated by the 1-bladed rotor can be

<figure><img src="image_4.png" /><figcaption>**Figure 1:** (a) Experimental set-up (side view of the water channel test section). (b) Schematic of the rotor blade geometry.</figcaption></figure>
clearly visualized using fluorescent dye (fluorescein) painted on the blade tip and illuminated by LED panels emitting at a wavelength of 480 nm. Video sequences of these visualizations were recorded with a standard digital camera operating at 25 images per second. Visualizations were used to obtain information about the helix geometry, including its radius (R) and pitch (h), as function of downstream distance.

Velocity fields were measured for quantitative analyses by phase-averaged high-resolution stereoscopic Particle Image Velocimetry (PIV). A Nd-YAG double-pulsed laser was positioned underneath the test section to generate a light sheet illuminating the mid-plane of the rotor wake. The flow was seeded with tracer particles consisting of 10 µm silver-coated hollow glass spheres. Particle image pairs were recorded by two FlowSense EO cameras, each with a resolution of 3388 × 2712 pixels. They were adjusted precisely using Scheimpflug mounts to ensure full-field focus on the measurement plane. Phase-averaged distributions were obtained from 1500 instantaneous PIV fields. They were used to determine the radial velocity profiles of the tip vortex, including swirl motion and axial flow, and vortex parameters, such as circulation and core size.

More detailed observations of short-wave perturbations were obtained through high-speed dye visualizations at 600 fps. The measurement region was fixed in the middle section of the helix, at the height of the shaft, and at an appropriate downstream distance to observe the perturbation with a sufficient amplitude, but before the breakup of the vortex. In this way, the short-wave structures could be captured clearly, allowing the determination of the perturbation wavelength and corresponding wavenumber.

## 3 Numerical method and setup

### 3.1 Three-dimensional simulations

The flow is simulated using an in-house spectral/spectral-element code based on the approach given in [20, 21]. In particular, in axial planes (across the helical vortex) the flow is captured using a spectral-element discretization, while the variation in the axial direction is represented by a Fourier expansion. For the helical vortices examined in this paper, considerable care needs to be taken to resolve the very compact helical vortical core, which typically has a size of less than 0.02 blade radii. To accomplish this, a polar grid is used to capture helical vortex variation with the grid highly concentrated in the radial direction in the vicinity of the vortex. The grid spacing is ≈ 0.002*R₀* in both the radial and axial directions near the helix. In the azimuthal direction, the spatial resolution is less critical since the helical vortex passes through axial planes at an acute angle in the cases we are most interested in, i.e., for *h/R₀* ~ 0.5 or less.

Typically for single-bladed rotor cases, simulations are run with 5120 5×5 (4th order) spectral elements to capture the variation across two-dimensional axial slices and 256 Fourier planes in the axial direction. This equates to approximately 21 million mesh points. The three-bladed rotor cases discussed briefly at the end of the paper employ 512 Fourier planes in the axial direction. The grid extends out radially to 4*R₀*. The flow is assumed periodic in the axial direction across each helix loop, consistent with the Fourier approximation, hence time integration allows the flow to evolve temporally, rather than spatially as captured in the experiments.

The equations are integrated forward in time using a three-step fractional step method accounting for the effects of the convection, pressure plus continuity, and diffusion terms. The convection term is treated explicitly using the third-order Adams-Bashforth method, and the linear diffusion step is treated using the theta-corrected Crank-Nicolson method in Fourier space. This involves a matrix inversion (LU decomposition, in practice) for each axial Fourier mode as a pre-processing step and subsequently a sparse matrix multiplication for each timestep. The same approach is also taken with the pressure step, where continuity is also satisfied. Full details of the numerical approach can be found in [20, 22].

### 3.2 Setting up the helical vortex

The numerical simulations are initialized using the actual core velocity distributions obtained from the highly resolved stereo-PIV measurements in the experiments. In terms of velocity components based on the local polar coordinate system centered on the vortex core, the experiments provide azimuthal and axial velocities. The azimuthal velocity component, in particular, falls off slowly with distance from the core, well beyond where the axial vorticity drops to zero. Given this, it is not very convenient to use the velocity components directly to set up the initial flow state. Instead, a satisfactory initial velocity field can be generated from the (more compact) derived azimuthal and axial vorticity fields by first determining the vector stream function $\vec{\Psi}$, where $\nabla^2\vec{\Psi} = -\vec{\omega}$, with $\vec{\omega}$ the vorticity vector. In practice, for each grid point in the numerical domain, the shortest distance to the helical core is determined, which allows the vorticity components at that point to be set based on the analytical fits for the vorticity component distributions. The normal derivative of the vector stream function is set to zero at the outer
radial boundary of the domain, providing the outer boundary condition for each component's Poisson equation. In the axial direction, periodicity is assumed, consistent with the Fourier representation in that direction. Once the vector stream function is determined using the same spectral/spectral-element grid used for the subsequent time integrations, the velocity field is determined from $\vec{u} = \nabla \times \vec{\Psi}$. The ability of the mesh to resolve the axial and radial vorticity distributions can be checked by redetermining the vorticity from the initial velocity field. The match with the experimental functional fits is excellent – typically better than 1%. A low-level white-noise perturbation is added to the velocity field at the start of each simulation to initiate the development of instabilities.

It is convenient to set the circulation ($\Gamma$) and the helix radius ($R$) to be unity for the simulations, allowing the circulation-based Reynolds number to be set through the kinematic viscosity. This provides a non-dimensionalization based on length and velocity scales of $R$ and $\Gamma/R$. Non-dimensional time is given by $t^* = t\Gamma/R^2$.

The following procedure is used to determine Fourier representation of the perturbation field along the helical vortex cores. The simulation employs a Fourier expansion in the global helix axis direction, so in real space, this just gives a set of velocity fields corresponding to each axial slice. Because of the inherent symmetry of the unperturbed base state, these slices can be rotated around the helix axis so that the helical vortices line up from one slice to the next. At this stage, this three-dimensional velocity field can be Fourier transformed in the axial direction to derive the helical vortex modes. Finally, we can select the mode(s) of interest and back-transform to real space and unrotate the slices to give back the desired perturbation fields for any modes of interest.

# 4 Results and discussion

## 4.1 *Baseline wake characteristics*

The characteristics of the baseline wake behind the one-bladed rotor are discussed in this section. The rotor geometry was designed for optimal operation at $Re = 30,000$ and $\lambda = 8.4$. This configuration is the reference case denoted by 'Base'. The regular structure of the corresponding helical vortex is visualized in figure 2(a), persisting over three rotor diameters downstream. Such visualizations were used to evaluate the geometrical parameters of the wake vortex. The helix radius and pitch were measured directly from the visualizations, by tracking dye patterns at the top and bottom positions of each vortex loop. 7–9 individual images were processed for this measurement to ensure accuracy. After a brief transient adjustment over 1–2 rotor diameters downstream of the rotor, the radius and pitch reach stable asymptotic values $R$ and $h$, respectively. The measured properties are summarized in table 1.

The velocity field of the tip vortex was measured for a rotor phase angle of 180°, i.e. half a rotation period after the blade passed through the measurement plane, by stereo-PIV, averaging over 1500 instantaneous fields. Local polar coordinates $(r, \theta)$ were placed around the vortex center, which was manually determined from the phase-averaged azimuthal vorticity. Figure 2(b) shows the azimuthally averaged velocity profiles in this coordinate system for the Base case, including the swirl velocity $v_\theta$ and axial velocity $v_z$. The axial velocity shows a large deficit in the vicinity of the vortex axis, the maximum axial

<figure><img src="image_5.png" /><figcaption>**Figure 2:** Wake characteristics for the Base case with $Re = 30,000$ and $\lambda = 8.4$. (a) Dye visualization of the baseline helix. The rotor is on the left, flow is from left to right. (b) Azimuthally averaged velocity profiles of the tip vortex including the swirl velocity $v_\theta$ and the axial velocity deficit $v_z$. $a_{max}$ is the core radius.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Figure 3:** Dye visualization survey of wake structures for various Reynolds numbers (Re) and tip speed ratios (λ).</figcaption></figure>

velocity $v_z^{max}$ is similar to the maximum swirl velocity $v_θ^{max}$, i.e. the corresponding swirl number, given by the ratio of these two velocities ($S = v_θ^{max}/v_z^{max}$), is of order 1. From the azimuthal velocity profile, the core size $a_{max}$ can be determined as the radial position of peak swirl velocity.

## 4.2 Dye visualization survey

The wake structure of the 1-bladed rotor was visualized for a range of different Reynolds numbers and tip speed ratios, in order to detect the conditions in which short-wave instabilities appear. A visual summary of this survey can be seen in figure 3. Short-wave perturbations are found to develop earlier and in a more pronounced way, with respect to the Base case, when either the tip speed ratio is increased or the Reynolds number is decreased.

When varying the tip speed ratio, the helix pitch *h* varies in an approximately inversely proportional way, while the asymptotic radius *R* remains comparable. Smaller *h* results in a stronger strain field felt by a given portion of the vortex, induced by neighboring loops, which may enhance the growth of short-wave core perturbations whose growth rates scale on this strain. Short-wave perturbations are clearly seen in all cases with higher λ in figure 3. For these configurations, the helical wake structure persists over a shorter distance compared to Base case. At about 1.5 rotor diameters downstream of the rotor, strong short-wave instabilities interact with long-wave deformation modes, culminating in a rapid and violent breakdown of the regular vortex system. When varying Re, both the helix radius and pitch approximately maintain their values. For lower Re, the initial wake appears more disturbed than for the Base case. This could be due to flow separation on the blade, which may be expected at this low Re, resulting in unsteadiness and turbulence in the flow around the vortex cores, i.e. in an enhanced triggering of the short-wave perturbations observed further downstream in the visualizations.

Based on the above survey, three representative cases were chosen for further analysis: the design case ('Base', Re = 30,000, λ = 8.4), a case with lower Reynolds number ('Low-Re', Re = 10,000, λ = 8.4), and a case with a higher tip speed ratio ('High-λ', Re = 30,000, λ = 14). The vortex velocity profiles for the Low-Re and High-λ cases are shown in figure 4; they can be compared to the ones for the Base case in figure 2. The Low-Re profiles are qualitatively similar to those of the Base case, with a similar swirl number of order 1, whereas this parameter is almost twice as big for the High-λ case. The circulation Γ of the tip vortex was calculated from the PIV velocity data by integrating the swirl velocity over a circle
<figure><img src="image_7.png" /><figcaption>**Figure 4:** Azimuthally averaged velocity profiles of the tip vortex for (a) the Low-Re and (b) the High-λ cases. Note the different scales of the velocity axes.</figcaption></figure>

<table><caption>**Table 1:** Properties of the helical vortex wake.</caption><thead><tr><th>Case</th><th>Re</th><th>λ</th><th>Γ (cm²/s)</th><th>Re<sub>Γ</sub></th><th>R/R<sub>0</sub></th><th>h/R<sub>0</sub></th></tr></thead><tbody><tr><td>Base</td><td>30,000</td><td>8.4</td><td>107</td><td>10700</td><td>1.14</td><td>0.57</td></tr><tr><td>Low-Re</td><td>10,000</td><td>8.4</td><td>29.0</td><td>2900</td><td>1.11</td><td>0.58</td></tr><tr><td>High-λ</td><td>30,000</td><td>14</td><td>50.9</td><td>5090</td><td>1.21</td><td>0.34</td></tr></tbody><thead><tr><th></th><th>a<sub>max</sub> (cm)</th><th>v<sub>θ</sub><sup>max</sup> (cm/s)</th><th>v<sub>z</sub><sup>max</sup> (cm/s)</th><th>S</th><th>Λ (cm)</th><th>ka<sub>max</sub></th><th>m</th></tr></thead><tbody><tr><td>Base</td><td>0.138</td><td>39.1</td><td>38.7</td><td>1.01</td><td>1.63</td><td>0.53</td><td>35</td></tr><tr><td>Low-Re</td><td>0.156</td><td>11.1</td><td>9.9</td><td>1.13</td><td>2.30</td><td>0.43</td><td>24</td></tr><tr><td>High-λ</td><td>0.108</td><td>27.9</td><td>14.4</td><td>1.94</td><td>1.72</td><td>0.39</td><td>35</td></tr></tbody></table>

of diameter *h* around the vortex center. With the decrease of Re from the Base to the Low-Re case, the vortex Reynolds number $Re_Γ = Γ/ν$ reduces in equal proportion. For the High-λ case, the tip, speed ratio is non-optimal, resulting in a lower effective angle of attack of the blade, and therefore in a lower lift and $Re_Γ$.

The parameters of the wake structure and the properties of the tip vortex, obtained from the visualizations and PIV measurements, are compiled in table 1.

### 4.3 Short-wave instability characteristics

Details of the short-wave perturbation structure were obtained from high-speed close-up recordings of dye visualizations. Example images for each of the three cases are shown in figure 5. All exhibit fairly regular waves, although considerable variations were observed. They appear to be mainly of a sinusoidal nature, although a helical structure cannot be ruled out completely from the visualizations. The instability wavelength $Λ$ was determined from these recordings. After correcting for the vortex curvature, 9–13 individual images yielded the average wavelength values given in table 1. The corresponding dimensionless wavenumbers $ka_{max}$, where $k = 2π/Λ$, are all well below 1. Short-wave elliptic instability has been predicted theoretically for Batchelor vortices with axial flow, involving wavenumbers $ka_{max} > 1$ [16], which appears incompatible with the present observations. It may be that the particular vortex velocity profiles in these experiments, which are quite different from those of a Batchelor vortex, lead to unstable modes with larger wavelengths. Since the swirl number for the Base and Low-Re cases is close to the limit $S ≈ 1$ for the swirling jet instability (see [23]), this mechanism may also be responsible for the short-wave perturbations.
<figure><img src="image_8.png" /><figcaption>**Figure 5:** Short-wave perturbations for (a) Base, (b) Low-Re and (c) High-$\lambda$ cases captured by high-speed dye visualizations. Flow is from left to right.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 6:** Short-wave instability for the Low-Re case. (a) **Left:** Experimental dye visualization, showing the last four loops in figure 3. **Right:** Numerical simulation with the helix visualized using the Q-criterion at $t^* = 2$. (b) Modal multiplication factors between $t^* = 1$ and 2.</figcaption></figure>

Numerical simulations were conducted for the Low-Re case, for which the short-wave perturbations appear more clearly than for the Base case. In figure 6(a), a flow simulation snapshot is compared to the experimental visualization at a similar non-dimensional time, and a similar range of short-wave instabilities can be seen to have developed, demonstrating a reasonable match. The amplification factors of different wavelength modes between times $t^* = 1$ and $t^* = 2$ is provided in figure 6(b), revealing that multiple modes contribute to the perturbation of the helical vortex. While figure 6(b) suggests that virtually all modes are amplified, we think that this reflects non-modal transient growth from an initial random perturbation field. We don't expect that this spectrum corresponds to the modal growth of generic elliptic or curvature instability modes, which should show the more characteristic smooth amplification peaks identified by other authors. This aspect is under further investigation. Clear distortions of the helix are seen in the dye visualizations at approximately 7-8 helix loops downstream of the rotor, which corresponds to a non-dimensional time of $t^* = 1.5-1.7$. The mode number, $m$, represents the number of wavelengths contained in a single loop of the helix. Mode $m = 20$ exhibits the peak growth multiplier, identifying it as the most unstable mode (at the time that helix distortion first appears), while modes $m = 13, 16$ and 17 also make substantial contributions. In the experiments, the average perturbation wavelength for the Low-Re case corresponds to $m = 24$.

Based on the growth multiplier in figure 6, the first two dominant unstable modes, with $m = 20$ and 17, were extracted. Their spatial structure is shown in figure 7. Both modes are concentrated within the vortex core, and their local azimuthal dependence has a wavenumber of 1, indicating a displacement of
<figure><img src="image_10.png" /><figcaption>**Figure 7:** Perturbation vorticity (component along the vortex axis) of the dominant amplified modes for the Low-Re case. (a) $m = 20$; (b) $m = 17$. Three-dimensional views on the left, and distributions in a plane perpendicular to the vortex axis on the right. The black circle has a radius $a_{max}$.</figcaption></figure>

the vortex center. Whereas the mode $m = 20$ corresponds to a sinusoidal perturbation, which appears to lie in the plane of maximum strain experienced by each vortex (inclined by 45° to the horizontal), mode $m = 17$ has the shape of a single helix, which in addition seems to rotate. Sinusoidal core perturbations are found for elliptic short-wave instabilities (although generally at much shorter wavelengths), whereas helical perturbations can appear for both elliptic and curvature instabilities, as well as for the swirling jet instability.

Further preliminary work on short-wave instabilities in helical vortices, involving a three-bladed rotor, has led to additional interesting observations. The experimental visualizations in figure 8(a) show a global view of the wake and close-ups of what appear to be single- and double-helix short-wave perturba-

<figure><img src="image_11.png" /><figcaption>**Figure 8:** Short-wave instability in helical vortices behind a three-bladed rotor. (a) Experimental dye visualization. **Left:** Global view: the rotor is on the left, flow is from left to right. **Right:** Close-up views of single- and double-helix perturbations. (b) Perturbation vorticity of the most amplified mode found by DNS for the same parameters.</figcaption></figure>
tions. The most unstable mode in the corresponding numerical simulation also shows a clear single-helix structure, with a wavelength close to the experimental one. The observed perturbation structures are compatible with both elliptic and curvature instabilities. Experimental measurements of the vortex profiles, and theoretical stability analyses of the measured vortex structures, are currently underway.

The precise identification of the nature of the short-wave perturbations found in the present experiments and simulations will be addressed in future theoretical work.

## 5 Conclusion

Short-wave instabilities of helical vortices were studied experimentally, using a one-bladed rotor, and numerically with a spectral element code. Certain off-design configurations of the rotor flows, with respect to tip-chord-based Reynolds number $Re$ and tip speed ratio $\lambda$, were found to exhibit short-wave deformations of the helical vortex core. In particular, an earlier appearance and more pronounced amplitude were observed in dye visualizations when $\lambda$ was increased or $Re$ decreased. The former may be due to the enhanced interaction between helix loops at higher tip speed ratio (lower helix pitch), whereas the latter could be caused by an increased turbulence level caused by flow separation on the rotor blade at low $Re$. The observed short-wave perturbations involve sinusoidal or helical deformations of the vortex cores, with wavelengths significantly longer than expected, e.g., for elliptic instability of a Batchelor vortex. The qualitatively different velocity profiles measured in the experiments may be responsible for this difference. Numerical simulations using the measured velocity profiles as initial conditions, showed good agreement with the experiments, concerning perturbation mode structures and wavelengths. The precise origin of the short-wave instabilities found in this work is presently not clear. In addition to the elliptic instability of the particular vortices found in the experiments; possible mechanisms include the curvature or torsion instability, or a swirling jet instability, which occurs at low swirl numbers. Further experimental, numerical and theoretical work is underway, in order to understand the current observations.

## Acknowledgements

Z. Xu acknowledges the financial support from the China Scholarship Council (CSC202406230291).

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