1. Introduction
Tornado-like vortices (TLVs) refer to a class of vortex phenomena that exhibit dynamic features analogous to those of atmospheric tornadoes (Nolan & Farrell Reference Nolan and Farrell1999). The TLVs typically manifest as columnar vortices, with a central core characterised by strong rotational velocity and pronounced vertical vorticity. These vortices are observed in various natural phenomena, including dust devils (Murdoch et al. Reference Murdoch2022), waterspouts (Rodríguez et al. Reference Rodríguez, Bech, Arús, Castán, Figuerola and Rigo2021) and fire whirls (Tohidi et al. Reference Tohidi, Gollner and Xiao2018), as well as in numerous industrial applications, such as cyclone separators (Grimble et al. Reference Grimble, Agarwal and Juniper2017) and hydrocyclones (Zhou et al. Reference Zhou, Kuang, Chu and Yu2010). Consequently, the analysis of TLV motion processes and the understanding of their underlying dynamics have become central research topics in a wide range of fields.
Research has demonstrated that the flow field of a TLV undergoes dynamic evolution as swirl intensity increases, transitioning through stages such as the single-cell vortex, vortex breakdown, dual-cell vortex and multi-cell vortex flow fields (Jischke & Parang Reference Jischke and Parang1974; Zhang et al. Reference Zhang, Liu, Jiang, Jiang and Gao2023a ). Among these, the single-cell TLV has attracted particular attention as the foundational flow field from which subsequent evolutionary stages emerge (Ashrafi et al. Reference Ashrafi, Romanic, Kassab, Hangan and Ezami2021). In this regime, fluid converges centripetally to form a coherent vortex core that does not remain fixed but exhibits time-dependent displacement. This behaviour, characterised by the random motion of the vortex core around the time-averaged centre, is known as vortex wandering (Bailey & Tavoularis Reference Bailey and Tavoularis2008; Bailey et al. Reference Bailey, Pentelow, Ghimire, Estejab, Green and Tavoularis2018). As a key coherent structure within TLVs, vortex wandering interacts with the vortex motion, significantly affecting the dynamic characteristics of the flow field. It is considered a major factor contributing to the complex turbulent behaviour observed in TLVs and has been the subject of considerable research interest (Luo et al. Reference Luo, Tian, Wu, Li, Wang and Zhang2023).
Owing to the inherent complexity of TLV flow fields, recent research has increasingly employed data-driven techniques such as proper orthogonal decomposition (POD) (Ahmed et al. Reference Ahmed, San, Rasheed and Iliescu2021) to extract and decouple key dynamic features. Karami et al. (Reference Karami, Carassale and Hangan2020) utilised particle image velocimetry (PIV) combined with Gaussian vortex simulations to analyse the coherent structures and mechanisms behind TLV velocity and pressure drop using POD. Their findings revealed that vortex wandering dominates velocity fluctuations, occurring at a fixed frequency range of 0.1–4.5 Hz. Similarly, Ishihara & Liu (Reference Ishihara and Liu2014) reproduced TLV flow fields with vortex wandering using large-eddy simulations, demonstrating that the motion is periodic with specific frequency ranges and governs both tangential and radial velocity fluctuations. Ashton et al. (Reference Ashton, Refan, Iungo and Hangan2019) used an MWD-type tornado simulator to show that vortex wandering amplitudes of 5–9 % of the updraft radius led to an enlarged time-averaged vortex core, reduced peak tangential velocity and enhanced turbulence intensity near the core region.
However, the POD method is inherently suited to the analysis of statistically stationary data. To overcome these limitations, dynamic mode decomposition (DMD) was introduced by Schmid (Reference Schmid2010) as an enhancement based on the Koopman operator for dynamic systems. As a data-driven approach, DMD approximates the nonlinear evolution of fluid flows through a linear dynamical system (Schmid Reference Schmid2022). It enables the decomposition of flow-field snapshots into eigenvalues and eigenvectors that represent the dynamic characteristics of the system (Yin et al. Reference Yin, Jia, Yuan and Luo2023). Unlike POD, which tends to encapsulate multi-scale coherent structures within a single mode, DMD offers a key advantage by decomposing each mode with a distinct frequency and growth/decay rate. This characteristic renders DMD particularly effective in isolating and analysing multi-scale coherent structures (Schmid et al. Reference Schmid, Li, Juniper and Pust2011; Le Clainche et al. Reference Le Clainche, Rosti and Brandt2022). Zhang et al. (Reference Zhang, Wang, Xu, Liu and Khoo2023b ) employed a two-dimensional Rankine vortex model to simulate the ideal velocity distribution of TLVs and used DMD to decouple the coherent structures. Their comparison of POD and DMD revealed that both methods effectively decomposed the synthesised flow field of single-cell TLVs with vortex wandering. However, DMD failed to capture the dynamic variations in vortex size, producing only decaying modes. This led to the conclusion that POD is more suitable for analysing vortex size dynamics, and it was subsequently applied in further investigations of velocity fluctuations. Notably, vortex size variations in TLVs are linked to vortex breakdown and expansion due to increased circulation intensity – changes that do not occur in single-cell TLVs without vortex breakdown (Liu & Ishihara Reference Liu and Ishihara2015). Karami et al. (Reference Karami, Hangan, Carassale and Peerhossaini2019) supported this finding, showing that the coherent structures in single-cell TLVs are primarily dominated by vortex motion and vortex wandering, with vortex size changes observed only in vortex breakdown or two-cell flow fields.
Although DMD offers clear advantages in flow structure extraction, it may still yield low-energy or noise-contaminated modes in complex or turbulence-dominated flows, which can obscure key physical features. To overcome this issue, the sparsity-promoting DMD (sp-DMD) method was developed. By introducing an ℓ1-norm regularisation during the amplitude optimisation process, sp-DMD enforces sparsity and retains only the most dynamically relevant modes, thereby improving both the interpretability and robustness of the results (Jovanović et al. Reference Jovanović, Schmid and Nichols2014; Schmid Reference Schmid2022).
Due to limitations in tornado simulators and flow-field measurement systems, most existing studies on TLV mode decomposition rely primarily on data from computational fluid dynamics simulations or idealised vortex models, with experimental investigations remaining relatively limited. To address this gap, this study introduces a novel tornado generator in conjunction with a PIV-based sp-DMD framework. This method applies mode decomposition directly to high-frequency velocity fields captured by PIV, enabling the extraction of large-scale coherent structures and providing spatiotemporal evolution images of the dominant modes associated with vortex wandering. The results provide new insights into the underlying dynamics of TLV flow fields and contribute to a deeper theoretical understanding of swirl-induced instabilities.
2. Experimental configuration and measurement techniques
The experimental platform comprises three primary components: the tornado simulator, the measurement system and the atomised tracer system, as illustrated in figure 1. The tornado simulator is constructed in the form of a hexagonal prism and is equipped with a variable-frequency fan mounted at the top to generate the driving airflow. The design includes a central upward-flow channel and circumferential air inlets with integrated guide vanes at the base. During operation, ambient air is drawn in through the lower inlets, guided by the vanes to impart angular momentum, and subsequently converges and ascends through the central channel, forming a controlled TLV flow. A PIV system was employed to capture instantaneous flow field at a constant sampling frequency of 10 Hz (
$\Delta t=0.1$
s). This temporal resolution was selected based on the characteristic time scales of the vortex structures to ensure sufficient resolution of the dominant unsteady dynamics across all tested swirl ratios. Tracer particles are generated using an ultrasonic atomiser (NS-6T model) with a diluted ethylene glycol solution. Additional details, including design diagrams (figure S1), specific dimensions (table S1) and geometric similarity (table S2) of the tornado simulator, are provided in the supplementary material available at https://doi.org/10.1017/jfm.2025.10862.
Tornado simulator and measurement data. (a) Simulator and measurement set-up. (b) Velocity contours. (c) Pressure distribution. (d) Radial profiles of normalised tangential velocity.
Furthermore, figure 1(b) displays velocity contours obtained via PIV for different swirl ratios. At higher swirl ratios, a distinct dual-cell vortex structure emerges, whereas at lower swirl ratios, the flow maintains a stable single-cell vortex configuration. Additionally, the pressure (figure 1 c) and tangential velocity (figure 1 d) distributions for the single-cell TLV were analysed. The pressure field exhibits a pronounced low-pressure core surrounded by a gradual radial increase, while the tangential velocity rises and then falls with increasing radial distance. These distributions align with theoretical vortex models and observed tornado behaviour (Lee & Wurman Reference Lee and Wurman2005; Wurman & Alexander Reference Wurman and Alexander2005), confirming that the apparatus reliably generates a stable single-cell tornado-like flow field.
3. Flow-field dynamics of single-cell TLV via PIV
Based on PIV data, figure 2(a) presents the instantaneous horizontal resultant velocity, which combines the radial and tangential components, along with the corresponding streamlines. The vortex core centre was respectively identified based on the minimum resultant velocity and the maximum Q-criterion value, and the corresponding coordinates were marked accordingly. As the swirl ratio increases from 0.09 to 0.18, the velocity rises significantly. However, at higher swirl ratios of 0.29 and 0.42, the velocity begins to decrease, and the vortex size expands rapidly. For even higher swirl ratios (
$S=0.60$
and 0.87), the resultant velocity within the high-speed vortex core continues to decrease, accompanied by larger fluctuations and irregular velocity distributions in the high-speed region. This expansion leads to stretching and compression, causing substantial deformation of the vortex structure.
Instantaneous velocity and Q criterion for swirl ratios S of (a) 0.09, (b) 0.18, (c) 0.29, (d) 0.42, (e) 0.60 and (f) 0.87, achieved by setting guide vane angles to 10°, 20°, 30°, 40°, 50° and 60°, respectively.
Snapshots of instantaneous velocity magnitude and Q criterion over the time series at
$S=0.18$
for time instances of
$t=$
(a) 0.1, (b) 0.2, (c) 0.3, (d) 0.4, (e) 0.5, (f) 0.6, (g) 0.7, (h) 0.8 and (i) 0.9 s.
In addition, the Q criterion is applied to analyse the vortex structures (Zhan et al. Reference Yang, Wu, Chen, Spence and Li2019), as shown in figure 2(b). As S increases from 0.09 to 0.18, the vortex intensity strengthens, and the Q criterion becomes more concentrated. However, as S continues to rise, the peak Q-criterion value decreases, and its spatial influence gradually expands. This behaviour suggests the onset of vortex breakdown (Yang et al. Reference Yang, Wu, Chen, Spence and Li2023), although the detailed morphological evolution of the vortex is not explored in this study.
The variation in the vortex core position is the primary focus of our study. Figure 3 illustrates the instantaneous velocity fields across a time series with a time interval of
$\Delta t=0.1$
s. Over time, the vortex core exhibits unstable wandering, commonly referred to as vortex wandering, rather than remaining stationary. Although the flow field consistently maintains a single-cell vortex, its morphology evolves, particularly in the velocity fluctuations observed in the high-speed region of the vortex core. This suggests a strong association between vortex wandering and persistent velocity fluctuations across the flow field. Furthermore, Helmholtz velocity decomposition and Q-criterion calculations were applied to these flow-field snapshots. The vortex wandering motion significantly influences both the extremum values and the spatial distribution of the Q criterion, with these effects being concentrated primarily within the vortex core region. In contrast, turbulent zones farther from the vortex core remain largely unaffected. Furthermore, under varying swirl ratio conditions, the vortex wandering motion exhibits non-stationary and irregular patterns in the instantaneous flow field. Nevertheless, this behaviour contains coherent low-frequency components, which can be effectively captured and characterised using mode analysis.
The Sp-DMD results at
$S=0.09$
. (a) Unit circle of the eigenvalues. (b) Cumulative energy ratio. (c) Distribution of frequency and amplitude. (d) Time evolution of real signals.
4. Mode analysis of single-cell TLV via sp-DMD
The sp-DMD method (Jovanović et al. Reference Jovanović, Schmid and Nichols2014) was employed to analyse 200 time-series snapshots from PIV data in order to extract a sparse set of dynamically significant modes based on modal energy and temporal coherence, and to identify coherent flow structures (the detailed sp-DMD methodology is provided in the supplementary material). To enhance spatial resolution and improve interpretability, all velocity fields were pre-processed using bicubic interpolation before applying sp-DMD. As shown in figure 4(a), the eigenvalues are distributed around the unit circle in the complex plane, indicating that these dynamic modes are expected to persist over long durations (see figure S2 of the supplementary material for other swirl ratios). The red crosses in figure 4(a) denote the dominant modes retained through sp-DMD, which contribute significantly to the overall flow dynamics. Figure 4(b) shows the cumulative energy distribution of the DMD modes. The static mode (zero frequency), which is equivalent to the mean flow, has been excluded from the energy-based ranking to avoid potentially misleading comparisons with dynamic modes. After excluding the static mode, the top three conjugate mode pairs – modes 2 and 3, modes 4 and 5 and modes 6 and 7 – were selected for further analysis, as they account for more than 99.95 % of the total modal energy and enable accurate reconstruction of the dominant unsteady structures. Figure 4(c) shows the relationship between frequency and initial amplitude for each mode, with mode pairs 2 and 3, 4 and 5, and 6 and 7 highlighted to indicate their corresponding frequency positions and amplitude characteristics. Figure 4(d) illustrates the temporal evolution of the real part of the selected modes. Mode pair 2 and 3 displays stable high-frequency oscillations, mode pair 4 and 5 shows lower-frequency periodic behaviour and mode pair 6 and 7 exhibits a modulated waveform, indicative of localised or transient instabilities potentially related to small-scale shear-layer disturbances. To further validate the sp-DMD method, the flow field was reconstructed by superimposing the top n modes, as shown in figure S3 of the supplementary material.
Frequency–energy distribution of each mode for swirl ratios
$S=$
(a) 0.09, (b) 0.18, (c) 0.29, (d) 0.42, (e) 0.60 and (f) 0.87.
Figure 5 shows the frequency and energy distributions of the six most energetic dynamic modes under different swirl intensities. It can be seen that, across all swirl ratios, the second and third modes consistently dominate the spectrum and carry the highest energy, indicating that they represent the most significant periodic components in TLV motion. As the swirl intensity increases, the energy contributions of these modes gradually decrease, and their frequencies also decline, suggesting a reduction in both the periodicity and strength of the associated motion. To further quantify the effect of swirl intensity on vortex dynamics, the Strouhal number was calculated based on the dominant modal frequency, maximum tangential velocity and the time-averaged vortex core radius (detailed calculations and results are provided in figure S7 of the supplementary material). The results indicate that the oscillation frequency decreases with increasing swirl intensity, consistent with the trends observed in modal energy and frequency.
Figure 6 presents the reconstructed results of the first three modes under different swirl ratios. The vortex wandering phenomenon – manifested as random displacements of the vortex core from the geometric centre (marked by a black circle) – can be successfully reproduced by superimposing the second and third modes on the first-order static mode, and this behaviour is observed across all swirl ratios. The temporal evolution of these modes exhibits good continuity without abrupt phase jumps. Based on this, it can be inferred that the second and third modes are possibly not significantly affected by spurious frequencies caused by aliasing and are more likely to reasonably represent the vortex wandering motion in TLVs. It should be emphasised that the PIV experiments were limited by a sampling frequency of 10 Hz, corresponding to a Nyquist frequency of only 5 Hz. Although the obtained modal frequencies are all below this threshold, aliasing cannot be completely ruled out, and thus the relevant conclusions still rely on indirect methods such as modal reconstruction. Nevertheless, this inference cannot replace direct validation based on high-resolution sampling.
Reconstructed flow fields of the first three modes at (a–j) different times under varying swirl ratios.
Since the first mode corresponds to a frequency of zero, it represents a non-periodic single-vortex structure. Its analysis is not the main focus of this study but is presented in figures S4–S6 of the supplementary material as a validation result. The focus here is on analysing the periodic characteristics of the conjugate second and third modes. Since these modes exhibit similar dynamic behaviour, the second-order mode was selected as a representative mode for studying vortex wandering dynamics.
Figure 7 illustrates the temporal evolution of the vortex-wandering-dominated mode, represented by the second-order mode. It is noteworthy that the modes reconstructed by the sp-DMD method inherently exhibit phase differences. To resolve this issue, a continuous-time reconstruction approach was employed, enabling the generation of time-evolving flow-field images (a detailed description of this method is provided in the supplementary material). As shown, the high-velocity regions of this mode consistently remain near the geometric centre throughout the evolution, exhibiting a rotational trend around the centre. A similar rotational behaviour was reported by Edstrand et al. (Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016) during POD analysis of wingtip vortices. The structural features of the vortex-wandering-dominated mode in this study closely resemble those of the DMD-based vortex wandering mode described by Zhang et al. Reference Zhang, Wang, Xu, Liu and Khoo2023b ), both of which reveal significant mainstream motion within the time-averaged vortex core region. However, the data in this study are exclusively derived from PIV measurements, whereas Zhang & Sarkar (Reference Zhang and Sarkar2012) employed a two-dimensional Rankine vortex model to simulate the ideal velocity distribution of the TLV.
Temporal evolution of the vortex-wandering-dominated mode under
$S=$
0.09 for (a) T/8, (b) 2T/8, (c) 3T/8, (d) 4T/8, (e) 5T/8, (f) 6T/8, (g) 7T/8 and (h) T.
Flow structures of the vortex-wandering-dominated mode under
$S=0.09$
at different times. (a,b) Streamlines at
$t=4T/8$
and
$t=6T/8$
; (c,d) Q criterion at
$t=4T/8$
and
$t=6T/8$
; (e, f) Z vorticity at
$t=4T/8$
and
$t=6T/8$
.
Figures 8(a) and 8(b) illustrate the velocity fields and streamlines of the vortex-wandering-dominated mode at two representative time instants within the continuous-time reconstruction cycle. Notably, a pair of vortices with opposite rotation directions is observed. They are symmetrically distributed on either side of the central axis and rotate around the geometric centre, with their intensities varying during the rotation. For instance, in figure 8(a), the upper vortex exhibits counterclockwise-rotating closed streamlines, while the lower vortex rotates clockwise, forming a symmetric vortex structure. To further corroborate this rotational behaviour, figures 8(e) and 8(f) show the corresponding Z vorticity fields, in which the positive and negative peaks respectively align with the upper and lower vortex cores. This confirms the opposite rotational directions of the vortex pair.
Moreover, figures 8(c) and 8(d) illustrate the temporal evolution of the Q criterion corresponding to the vortex-wandering-dominated mode. In contrast to the typical pattern where a single high-Q region appears at the geometric centre – representing the rotational intensity of the time-averaged vortex core – these plots reveal symmetrically arranged zones of high-magnitude positive and negative Q values on either side of the centre, forming a pronounced dipole structure. This symmetric configuration reflects the concurrent presence of rotational and shear effects in the vicinity of the vortex core. The coexistence of opposing Q extremes indicates that the dynamics captured by this mode is not purely rotational but also involves significant shear-induced deformation. The interaction between these two effects drives the vortex core to periodically deviate from its equilibrium position and oscillate around the geometric centre, thereby manifesting as the characteristic vortex wandering motion.
As the swirl ratio increases to a moderate level (
$S=0.29$
), a clear attenuation of the vortex-wandering-dominated mode is observed. As shown in figures 9 and 10, although the mainstream motion remains localised near the geometric centre and exhibits periodic characteristics, the amplitude of the high-velocity regions is substantially reduced compared with the low-swirl-ratio case (
$S=0.09$
). The velocity field becomes more diffuse, and its spatial symmetry deteriorates. Streamline visualisations continue to reveal closed, counter-rotating vortices flanking the mainstream flow. However, their intensity and structural coherence are markedly diminished. Similarly, both the Q criterion and the Z vorticity exhibit reduced absolute values and weaker dipolar structures, indicating a decline in both rotational and shear contributions. Collectively, these trends reflect a loss of coherence, symmetry and strength in the vortex wandering behaviour as the swirl ratio increases, even though the associated dynamic mode persists.
Temporal evolution of the vortex-wandering-dominated mode under
$S=0.29$
for (a) T/8, (b) 2T/8, (c) 3T/8, (d) 4T/8, (e) 5T/8, (f) 6T/8, (g) 7T/8 and (h) T.
Flow structures of the vortex-wandering-dominated mode under
$S=0.29$
at different times. (a,b) Streamlines at
$t=T/8$
and
$t=4T/8$
; (c,d) Q criterion at
$t=T/8$
and
$t=4T/8$
; (e,f) Z vorticity at
$t=T/8$
and
$t=4T/8$
.
At a higher swirl ratio (
$S=0.60$
), the vortex-wandering-dominated mode displays a marked reduction in both intensity and structural coherence. As shown in figures 11 and 12, the velocity associated with the mainstream motion continues to decrease, while the high-velocity regions become increasingly fragmented and spatially irregular. The previously well-defined, symmetric vortex structures observed at lower swirl ratios are no longer evident, indicating that the periodic vortex wandering behaviour has largely degenerated into disorganised fluctuations. In the Q criterion and the Z vorticity, the once-prominent dipole pattern disappears, giving way to scattered, spot-like regions of weak positive and negative values. This signifies a loss of coherent rotational and shear structures near the vortex core and suggests the breakdown of the organised rotational system.
Temporal evolution of the vortex-wandering-dominated mode under
$S=0.60$
for (a) T/8, (b) 2T/8, (c) 3T/8, (d) 4T/8, (e) 5T/8, (f) 6T/8, (g) 7T/8 and (h) T.
Flow structures of the vortex-wandering-dominated mode under
$S=0.60$
at different times. (a,b) Streamlines at
$t=T/8$
and
$t=6T/8$
; (c,d) Q criterion at
$t=T/8$
and
$t=6T/8$
; (e, f) Z vorticity at
$t=T/8$
and
$t=6T/8$
.
In conjunction with the frequency and energy distributions obtained from sp-DMD (figure 5), these observations indicate that, as the swirl ratio increases, the vortex-wandering-dominated mode gradually transitions from a high-energy, high-frequency, coherent and strongly periodic dynamical state at
$S=0.09$
to a low-energy, low-frequency, irregular and structurally disordered state at
$S=0.60$
, reflecting the progressive attenuation of the TLV’s periodic characteristics.
5. Conclusion
As a critical coherent structure in tornado-like flows, vortex wandering plays a significant role in shaping the complexity of flow-field dynamics. In this study, high-resolution spatiotemporal data obtained via PIV experiments, in conjunction with the sp-DMD method, were employed to decouple and analyse the TLV flow structures. This approach effectively extracts the vortex-wandering-dominated modes and characterises their flow features, as well as their evolution across varying swirl ratios. The main conclusions are as follows:
-
(i) With increasing swirl ratio, the vortex structure of the single-cell flow field becomes progressively diffuse, and the vortex intensity decreases significantly. Influenced by intermittent vortex breakdown and descending airflow, the maximum tangential velocity initially increases and then decreases as the swirl ratio rises, while the time-averaged vortex core radius continuously expands. Furthermore, vortex wandering exhibits an approximately Gaussian distribution, with its amplitude significantly affected by the swirl ratio.
-
(ii) A PIV–sp-DMD method was developed to analyse the coherent structures in experimental data from single-cell TLV. The reliability of the sp-DMD algorithm was confirmed through mode-quality evaluation and flow-field reconstruction. Energy growth/decay analysis showed that the first-order mode, along with the second- and third-order conjugate modes, retains high energy and exhibits stable motion, representing the primary coherent structures in TLVs. The velocity and Q-criterion results indicate that the first-order mode is a time-averaged static mode, dominating the flow field and representing the primary vortex motion.
-
(iii) The second- and third-order conjugate modes, identified as vortex-wandering-dominated modes, capture the sustained periodic velocity fluctuations that characterise vortex wandering. These modes exhibit pronounced oscillatory behaviour in the resultant velocity field, with their influence predominantly localised within the vortex core region. Analysis of the velocity field and axial vorticity reveals a dominant mainstream motion accompanied by a pair of vortices that rotate in opposite directions and jointly revolve around the central axis in a symmetric configuration. Corresponding Q-criterion distributions further display a symmetric dipolar structure with coexisting positive and negative values, indicating that these modes are governed by the combined effects of rotational and shear velocity components, primarily acting on the vortex core.
-
(iv) The influence of the swirl ratio on the vortex-wandering-dominated mode was systematically analysed. As the swirl ratio increases, the energy of the vortex-wandering-dominated mode progressively decreases, accompanied by reductions in both velocity and vorticity intensity. The periodicity of velocity fluctuations weakens, and the symmetric vortex distribution, along with the dipole structure, becomes less distinct. Consequently, the motion characteristics shift from high-energy, high-frequency and organised behaviour to low-energy, low-frequency and more disordered dynamics. This trend may further explain the progressive weakening of the vortex wandering motion in TLVs with increasing swirl ratio.
This study performed DMD of TLV motion based purely on PIV experiments. Due to the current limitation of the experimental sampling frequency, the influence of aliasing cannot be completely ruled out. In the future, increasing the sampling frequency could further improve the accuracy.
Supplementary material and movies
Supplementary material and movies are available at https://doi.org/10.1017/jfm.2025.10862.
Funding
The authors are grateful for financial support by the National Natural Science Foundation of China (no. 52400130).
Declaration of interests
The authors report no conflict of interest.
