2.1. Set-up

The vortex meandering experiment was carried out in a free-surface recirculating water channel with a test section of dimensions 150 cm (length) $\times$ 50 cm (height) $\times$ 38 cm (width), with a set-up similar to the one used by Roy et al. (Reference Roy, Leweke, Thompson and Hourigan2011), see figure 1. The vortex was generated by a rectangular airfoil made of polyvinyl chloride, with a NACA 0012 cross-section profile and a rounded tip with a varying tip diameter equal to the local thickness of the wing. The airfoil was mounted vertically on a U-frame positioned along the side and bottom boundaries of the channel, with the leading edge positioned 20 cm from the test section entry. The wing had a chord $c = {9.8}$ and a 15 cm span, resulting in a vertical position of the tip vortex at approximately one third of the water height in the channel. This was chosen to limit the influence of the free surface, which exhibits a steady deformation at higher flow velocities (see movie 1 in the Supplementary material available at https://doi.org/10.1017/jfm.2026.11698). The airfoil was placed at an angle of attack of ${9}^{\circ }$ .

Figure 1.

Schematic of the water channel experiment. The rectangle around the wing tip shows the area in which the flow in the empty channel was measured. The blue line represents the stationary deformed free surface for the present experimental conditions.

Figure 1. Long description

The schematic illustrates a water channel experiment with a wing tip generating a vortex. The measurement plane is marked in green, intersecting the vortex. A camera is positioned to capture the flow dynamics. The blue line represents the stationary deformed free surface under the experimental conditions. The rectangle around the wing tip indicates the area where the flow in the empty channel was measured. The setup includes a continuous wave laser and a coordinate system with axes labeled x, y, and z. The flow direction is indicated by an arrow labeled U0.

2.2. Free stream properties

The characteristics of the free stream were determined by measuring the velocity field in the empty channel in a rectangular area around the position of the wing tip (figure 1), using particle image velocimetry (PIV). The mean free stream velocity for the current experiment was $U_0 = (79.1\pm 0.3)\,\textrm {cm s}^{-1}$ , resulting in a chord-based Reynolds number $Re= U_0c/\nu \approx 8\times 10^4$ , where $\nu$ is the kinematic viscosity of water. The time-averaged free stream velocity is spatially uniform to within 1 % and the turbulence intensity ( ${\textit{Tu}}$ ), determined by correcting for PIV noise and not including fluctuations related to the sloshing phenomenon described below, is close to 0.6 %.

A well-known phenomenon occurring in free-surface basins and channels is a low-frequency sloshing of the fluid (Ibrahim Reference Ibrahim2005), a gravity wave propagating back and forth between the extremities of the facility, with a wavelength given by twice its length. This small-amplitude oscillation is superposed on the steady recirculating flow in the channel. Figure 2 illustrates the sloshing in our facility by the time history of the streamwise and vertical velocity components, averaged over a horizontal strip of 2 cm width in the area shown in figure 1, at the height of the wing tip. A low-frequency oscillation is clearly present in the streamwise velocity, much less so in the vertical component. This is confirmed by the corresponding power spectra in figures 3( $a$ ) and 3( $b$ ). Movie 2 in the Supplementary material further demonstrates the sloshing motion, by showing the dominant mode of a dynamic mode decomposition of a series of PIV velocity fields.

Figure 2.

Time history of the ( $a$ ) streamwise and ( $b$ ) vertical component of the free stream velocity in the empty channel, measured at a height $y=16.2\,\textrm {cm}$ above the channel floor. The respective time-mean velocities are shown by grey lines.

Figure 2. Long description

Two line graphs illustrate the time history of the streamwise and vertical components of the free stream velocity in an empty channel. The x-axis represents time in seconds, while the y-axis on the left represents the streamwise velocity in centimeters per second, and the y-axis on the right represents the vertical velocity in centimeters per second. The streamwise velocity graph shows fluctuations around a mean value of approximately 79 centimeters per second, with peaks reaching up to 81 centimeters per second and troughs down to 77 centimeters per second. The vertical velocity graph displays smaller fluctuations around zero, ranging from negative 2 to positive 2 centimeters per second. Grey lines indicate the respective time-mean velocities for both components. The streamwise velocity exhibits significant periodic variations, while the vertical velocity remains relatively stable with minor oscillations. All values are approximated.

Figure 3.

( $a$ , $b$ ) Power spectral density (PSD) of the free stream velocities in figure 2. The grey areas represent the power contained in a small frequency band centred at the observed sloshing frequency. ( $c$ ) Amplitude ratio of the velocity oscillations in the vertical and streamwise directions as a function of $h$ . The grey dot represents the data in (a), (b) and figure 2, and the line the theoretical relation (2.2), with a fitted value $L=7.5\,\textrm {m}$ .

Figure 3. Long description

The image contains three graphs related to a fluid dynamics study. The first graph (a) is a power spectral density (PSD) plot showing the free stream velocities with a prominent peak at 0.11 hertz. The second graph (b) is another PSD plot with multiple peaks, including one at 0.11 hertz. The third graph (c) is a scatter plot with a line, showing the amplitude ratio of velocity oscillations in the vertical and streamwise directions as a function of a variable. The grey areas in the PSD plots represent the power contained in a small frequency band centered at the observed sloshing frequency. The grey dot in the scatter plot represents data from the first two graphs and a theoretical relation with a fitted value. The graphs illustrate the dynamics of vortices and waves in fluid flows, focusing on the interaction between vortices and free surfaces.

For a rectangular container of depth $d$ and length $L$ , the sloshing frequency $f_{{s}}$ is given by

(2.1) \begin{equation} f_{{s}} = \frac {1}{2\pi }\sqrt {\frac {\pi g}{L}\tanh \frac {\pi d}{L}}, \end{equation}

where $g$ is the acceleration of gravity. With a water depth of $d = {45}\,\textrm {cm}$ , the observed value of $f_{{s}} = {0.11}\,\textrm {Hz}$ would correspond to a length $L = {9.5}\,\textrm {m}$ , which is of the order of the overall dimension (6.5 m) of the channel (which is in fact not a rectangular container). For finite-depth free-surface gravity waves, the amplitude ratio of the horizontal and vertical velocity oscillations is a function of the vertical coordinate. If $h$ is the distance from the rigid bottom, this ratio is given by

(2.2) \begin{equation} \frac {A_y}{A_z} = \tanh \frac {\pi h}{L}. \end{equation}

Experimentally, it can be estimated by the square root of the ratio of the areas underneath the spectral peaks corresponding to the sloshing motion (highlighted in grey in figures 3 a and 3 $b$ ). The result in figure 3( $c$ ) compares reasonably well with the theoretical prediction, assuming a container dimension $L={7.5}\,\textrm {m}$ , which is again compatible with the channel dimensions. The preceding results represent qualitative and, to a certain degree, quantitative evidence that a sloshing oscillation is present in our experimental facility, which must be taken into account in the further analysis of the tip-vortex dynamics.

2.3. Structure and dynamics of the wing tip vortex

Figure 4.

( $a$ ) Instantaneous vorticity field of the tip vortex at $z/c = 11.2$ (see also movie 3 in the Supplementary material). The origin represents the mean position of the vortex centre, and the black lines the instantaneous centre. The free stream velocity $U_0$ is out of the measurement plane along $z$ , which together with $x$ and $y$ defines the laboratory-fixed reference frame. ( $b$ ) Collection of all instantaneous vortex-centre positions of one recording (grey dots). The principal axes $\boldsymbol{e}_\alpha$ of this distribution define the principal-axes coordinate system $x_\alpha$ with standard deviations $\sigma _\alpha$ in the measurement plane.

Figure 4. Long description

The first graph shows the instantaneous vorticity field of a tip vortex with a color gradient indicating vorticity values. The origin marks the mean position of the vortex center, and black lines indicate the instantaneous center. The free stream velocity is out of the measurement plane along the x-axis, with the y-axis and z-axis defining the laboratory-fixed reference frame. The second graph displays a collection of all instantaneous vortex-centre positions from one recording, represented as grey dots. The principal axes of this distribution define the principal-axes coordinate system with standard deviations in the measurement plane.

Figure 5.

Time history of the (a) horizontal and (b) vertical vortex displacement.

Figure 5. Long description

Two line graphs illustrate the time history of vortex displacement. The first graph (a) shows the horizontal displacement (X/r_c) over time (t in seconds), with values ranging from -0.6 to 0.6. The second graph (b) displays the vertical displacement (Y/r_c) over the same time period, with similar value ranges. Both graphs exhibit fluctuating patterns, indicating variations in vortex displacement over time.

The properties and motion of the wing tip vortex were measured in a plane perpendicular to the free stream located at a downstream distance $z/c = 11.2$ from the trailing edge of the wing. The transverse velocity field was measured using PIV at a frequency of 1 kHz. The flow was seeded with silver-coated hollow glass spheres of diameter 10 μm, and illumination was achieved with a 10 W continuous-wave (CW) solid-state fibre laser (Azurlight Systems). Images were recorded by a Phantom VEO 410L high-speed digital camera and processed by an in-house algorithm (Meunier & Leweke Reference Meunier and Leweke2003; Aguilar-Cabello, Parras & del Pino Reference Aguilar-Cabello, Parras and del Pino2022). A total of $N = 38\,266$ velocity fields was obtained, covering a time interval of 38.3 s.

Figure 4( $a$ ) shows an example of the instantaneous vorticity distribution of the tip vortex in the measurement plane. The vortex centre ( $X$ , $Y$ ), indicated by the black lines, was found using (1.1) for the vorticity centroid. Only vorticity values larger than 50 % of the maximum vorticity were taken into account, in order to reduce the effect of low-level vorticity at large distances from the centre, that are mostly due to noise from the PIV method. A time sequence of the vorticity field illustrating the meandering motion can be seen in movie 3 of the Supplementary material. In figure 4( $b$ ), the instantaneous positions from the entire time series are collected. This figure also shows the vortex-centre principal-axes system $x_1, x_2$ used in the analysis in § 3. An example of the time history of the vortex position is presented in figure 5.

Figure 6.

Radial profiles of the ( $a$ ) mean azimuthal velocity and ( $b$ ) axial vorticity of the wing tip vortex at $z/c=11.2$ . The dashed lines, representing $\overline {u}_\theta$ and $\overline {w}_z$ , were obtained from the average velocity field of all 38 266 measurements. For the solid lines, representing $\mathfrak{u}_\theta$ and $\mathfrak{w}_z$ (see § 3.2), 5000 fields were averaged after recentring the vortex, using the displacement data in figure 5.

Figure 6. Long description

The image contains two line graphs side by side. The first graph on the left shows the radial profile of the mean azimuthal velocity (u theta) in centimeters per second (cm/s) against the radial distance (r) in centimeters (cm). The graph includes two lines: a dashed line representing the velocity profile without recentering and a solid line representing the velocity profile with recentering. The peak velocity occurs at a radial distance of approximately 0.56 centimeters. The second graph on the right shows the radial profile of the axial vorticity (w z) in per second (s inverse) against the radial distance (r) in centimeters (cm). Similar to the first graph, it includes a dashed line for the profile without recentering and a solid line for the profile with recentering. The axial vorticity decreases rapidly with increasing radial distance. The data for the dashed lines are obtained from the average velocity field of all 38,266 measurements, while the data for the solid lines are obtained by averaging 5,000 fields after recentering the vortex using the displacement data in figure 5.

The structure of the tip vortex was determined from the time-averaged velocity field. Velocities were interpolated onto a polar grid around the vortex centre to determine the radial profile of the azimuthally averaged azimuthal velocity and axial vorticity. The results for a time average relative to the mean vortex centre are shown in figure 6 as dashed lines. The core radius $r_{{c}}$ , which is a characteristic length scale of the vortex (used to non-dimensionalise the coordinates in figures 4 and 5), is defined as the radius of the maximum azimuthal velocity. The meandering motion of the vortex leads to an apparent larger core radius and lower maximum vorticity (Baker et al. Reference Baker, Barker, Bofah and Saffman1974; Devenport et al. Reference Devenport, Rife, Liapis and Follin1996). This effect can be eliminated by recenterng each velocity field according to the measured instantaneous vortex displacement (figure 5) before averaging (Heyes, Jones & Smith Reference Heyes, Jones and Smith2004). The corresponding results in figure 6 (solid lines) indeed reveal a larger maximum vorticity and smaller core size. The latter is determined as ${r_{{c}}} = {0.56}\,\textrm {cm}$ for the present experiment. The analysis of Devenport et al. (Reference Devenport, Rife, Liapis and Follin1996) (see also § 3.2) for a Batchelor vortex subject to a two-dimensional Gaussian meandering motion of amplitude (standard deviation) $\sigma$ , predicts an increase of the measured core radius by a factor $\sqrt {(1+2\kappa \sigma ^2/r_c^2)}$ , and a decrease of the vorticity maximum by a factor $(1+2\kappa \sigma ^2/r_c^2)^{-1}$ , with $\kappa =1.25643$ . From the data in figure 5, we find $\sigma \approx {0.11}\,\textrm {cm}$ . The corresponding predicted core radius increase of 5 % and vorticity decrease of 9 % are in good agreement with the measurements in figure 6.

The measured azimuthal velocity profile, corrected for the meandering motion, is extremely close to the theoretical distribution derived by Moore & Saffman (Reference Moore and Saffman1973) for laminar vortices resulting from the vortex sheet roll-up behind a lifting surface,

(2.3) \begin{equation} \mathfrak{u}_\theta = \frac {\beta }{a^n}\varGamma \left (\frac {3-n}{2}\right ) \frac {r}{a} {_1F_1}\left (\frac {1+n}{2},2,-\frac {r^2}{a^2}\right )\!, \end{equation}

where $\varGamma$ is here the gamma function (not to be confused with the vortex circulation $\varGamma$ ) and $_1F_1$ the confluent hypergeometric function of the first kind. The radial length scale $a$ is proportional to the core radius $r_{{c}}$ and the parameters $\beta$ and $n$ represent the vortex strength (circulation) and the outer velocity profile structure ( $\mathfrak{u}_\theta \sim r^{-n}$ , $r/a \gg 1$ ). For the current experiments, we find $a={0.43}\,\textrm {cm}$ , $\beta ={25.6}\, {\textrm {cm}^{(n+1)}\,\textrm{s}^{-1}}$ and $n=0.67$ .

The current measurements can be compared with the turbulence-induced meandering amplitude proposed by van Jaarsveld et al. (Reference van Jaarsveld, Holten, Elsenaar, Trieling and van Heijst2011): $\sigma _t/{r_{{c}}} \approx 4\, Tu\,\sqrt {zU_0/\varGamma }.$ For the experimental parameter values ( ${\textit{Tu}}=6\times 10^{-3}$ , $z={110}\,\textrm {cm}$ , $U_0={79.1}\,\textrm {cm s}^{-1}$ , $\varGamma ={270}\,{\textrm {cm}^{2} \,\textrm{s}^{-1}}$ ), the formula predicts $\sigma _t/{r_{{c}}} \approx 0.14$ , whereas the total meandering amplitude is $\sigma /{r_{{c}}}=0.20$ . It is shown in § 5.1 below that the sloshing wave is responsible for a vortex displacement of the same order as the turbulence-induced meandering. Both contribute to the measured total amplitude.

Overall, the vortex structure and meandering dynamics observed in the present experiments are all consistent with previous results reported in the literature.

3.1. General evolution equation for the vortex centre

In this study, we are concerned with the apparently random lateral motion of the vortex as a whole, referred to as meandering. These vortex kinematics are conveniently described by the leading integrals of the vorticity field in the measurement plane, recalled here from (1.1) for convenience,

(3.1) \begin{equation} \varGamma (z,t) = \int _M{\text{d}}^{2}x\, w_z(\boldsymbol{x},z,t) ,\quad X_\alpha (z,t) = \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, x_\alpha w_z(\boldsymbol{x},z,t) \quad (\alpha = 1,2) . \end{equation}

These integrals give the circulation $\varGamma$ and the vortex centre coordinates $X_\alpha$ , respectively. We then define vortex meandering as the motion of the vortex centre $X_\alpha (z,t)$ in time, at a given position $z$ . Inserting the Reynolds decomposition into (3.1) and respecting the characteristic symmetries of the vorticity field yields the leading-order approximations (Bölle Reference Bölle2023, Reference Bölle2024)

(3.2) \begin{equation} \varGamma = \int _M{\text{d}}^{2}x\, \overline {w}_z(\boldsymbol{x}) ,\quad X_\alpha (z,t) = \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, x_\alpha w_z^{\prime}(\boldsymbol{x},z,t) \quad (\alpha = 1,2) . \end{equation}

The measurement plane $M$ is a non-material area containing the respective $z$ -slice of the unsteady vortex over the entire measurement time. The mean vorticity distribution spreads downstream, as a consequence of the meandering motion (see § 3.2 and Baker et al. (Reference Baker, Barker, Bofah and Saffman1974) and Devenport et al. (Reference Devenport, Rife, Liapis and Follin1996) for details). We suppose that $M$ is chosen large enough to contain the spreading mean vortex and therefore the circulation does not change appreciably downstream.

In the following, we work in the vortex-centre principal-axes system $x_i = (x_\alpha , z) = (x_1, x_2, z) = (\boldsymbol{x}, z)$ spanned by the $\boldsymbol{e}_\alpha$ and $\boldsymbol{e}_z$ (figure 4 $b$ ). In summations and products, Greek indices ( $\alpha , \beta , \ldots$ ) take on values 1 and 2, whereas roman indices run from 1 to 3.

To derive the equation of motion for the vortex-meandering dynamics, we take the temporal derivative of (3.2). Since the integration domain $M$ is non-material, derivation and integration commute (Batchelor Reference Batchelor2000) and we obtain

(3.3) \begin{align} \frac {\partial X_\alpha }{\partial t} &= \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, x_\alpha \frac {\partial w_z^{\prime}}{\partial t} \nonumber \\ &= \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, x_\alpha \left [ \overline {w}_i\frac {\partial v_z^{\prime}}{\partial x_i} + w_i^{\prime}\frac {\partial \overline {u}_z}{\partial x_i} - U_0\frac {\partial w_z^{\prime}}{\partial z} - \overline {u}_i\frac {\partial w_z^{\prime}}{\partial x_i} - v_i^{\prime}\frac {\partial \overline {w}_z}{\partial x_i} \right ] + \xi _\alpha (z,t) . \end{align}

In the second line of (3.3), we have used the linearised fluctuation vorticity equation of an incompressible, inviscid fluid (Saffman Reference Saffman1995; Batchelor Reference Batchelor2000). The forcing $\xi _\alpha (t)$ may represent the net effect of the omitted nonlinear advection terms or any externally applied excitation (Bölle et al. Reference Bölle, Brion, Robinet, Sipp and Jacquin2021).

Vortex eigenmodes with non-zero axial wavenumbers have an exponential decay in the radial coordinate (Ash & Khorrami Reference Ash and Khorrami1995; Fabre & Jacquin Reference Fabre and Jacquin2004). This effectively compact support is also empirically observed for the leading proper orthogonal decomposition (POD) modes of the experimental velocity and vorticity fields (see figures 10 and 12). We therefore assume that the fluctuation fields in (3.3) vanish on the integration boundary $\partial M$ . Further recalling that the velocity and vorticity fields are solenoidal, the equation of motion (3.3) can be simplified through integration by parts, yielding

(3.4) \begin{equation} \frac {\partial X_\alpha }{\partial t} = -U_0\frac {\partial }{\partial z}\frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, x_\alpha w_z^{\prime} + \frac {1}{\varGamma }\int _M{\text{d}}^{2}x \big[\overline {u}_\alpha w_z^{\prime} + \overline {w}_z v_\alpha ^{\prime} - \overline {w}_\alpha v_z^{\prime} - \overline {u}_z w_\alpha ^{\prime} \big]\! + \xi _\alpha (z,t) . \end{equation}

We now assume a Helmholtz decomposition of the fluctuation-velocity fields $v^{\prime}_i = U^{\prime}_i + u^{\prime}_i$ , denoting the potential and rotational flow components by capital and lower-case letters, respectively. The former represent an external forcing and will be related to the free-surface gravity-wave flow present in the water channel experiment (see §§ 2.2 and 5.1). The latter relate to the fluctuation velocity field of the vortex. With this, the equation of motion of the vortex centre reads

(3.5) \begin{align} \frac {\partial X_\alpha }{\partial t} + U_0\frac {\partial X_\alpha }{\partial z} &= \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \overline {u}_\alpha w_z^{\prime} + \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \overline {w}_z u_\alpha ^{\prime} + \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \overline {w}_z U_\alpha ^{\prime} \nonumber \\ &\quad- \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \overline {u}_z w_\alpha ^{\prime} - \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \overline {w}_\alpha u_z^{\prime} - \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \overline {w}_\alpha U_z^{\prime} \nonumber \\ &\quad + \xi _\alpha (z,t) . \end{align}

This shows that vortex meandering $X_\alpha$ , while being advected at the free stream velocity $U_0$ , is principally subject to various feedback or forcing mechanisms appearing on the right-hand side of (3.5). These are organised such that the first line of (3.5) contains all terms that may affect a mean vortex with infinite swirl number (e.g. a Lamb–Oseen vortex (Fabre, Sipp & Jacquin Reference Fabre, Sipp and Jacquin2006)). The second line comprises the additional terms for vortices with axial core flow (finite swirl numbers, e.g. Batchelor or Moore–Saffman vortices (Moore & Saffman Reference Moore and Saffman1973; Batchelor Reference Batchelor2000)). The last forcing $\xi _\alpha$ is assumed to apply in both cases unchanged. To proceed, we have to evaluate the model integrals in (3.5). We do this analytically in the following § 3.2 and check the consistency of this result with our experiment in § 4.2.

3.2. Analytical evaluation of the model integrals

Our further evaluation is based on the fundamental conjecture that due to vortex meandering we have the two representations

(3.6) \begin{equation} w_z(\boldsymbol{x},z,t) = \overline {w}_z(\boldsymbol{x},z) + w_z^{\prime}(\boldsymbol{x},z,t) = \mathfrak{w}_z(\boldsymbol{x}-\boldsymbol{X}(z,t),z) \end{equation}

of the instantaneous flow fields. We here consider in detail only the streamwise vorticity; other vorticity or velocity components can be treated equivalently. The first representation is a Reynolds decomposition of the random flow field, as introduced above. We refer to the second as the vortex-meandering postulate: vortex meandering corresponds to a laminar (turbulent) vortex $\mathfrak{w}_z$ being displaced laterally as a whole (Baker et al. Reference Baker, Barker, Bofah and Saffman1974; Bölle Reference Bölle2021). The instantaneous vortex-centre position is denoted $\boldsymbol{X}(z,t)$ and constitutes a stationary random process with probability density $p(\boldsymbol{X}_0|\boldsymbol{X},z)$ , where $\boldsymbol{X}_0 = \boldsymbol{X}(z=0)$ . The standard deviations associated with the components $X_\alpha$ are denoted $\sigma _\alpha$ and are commonly referred to as meandering amplitudes. With respect to the experimental configuration, we are interested in the dynamics of the vortical flow at a fixed downstream position $z=z_M\neq 0$ . For convenience, we therefore drop the reference to $z$ in the following equations. We impose the compatibility condition (Baker et al. Reference Baker, Barker, Bofah and Saffman1974; Devenport et al. Reference Devenport, Rife, Liapis and Follin1996)

(3.7) \begin{equation} \int _{\mathbb{R}^2}{\text{d}}^{2}X\, p(\boldsymbol{X}) \mathfrak{w}_z(\boldsymbol{x}-\boldsymbol{X}) = \overline {w}_z(\boldsymbol{x}) . \end{equation}

In particular, for a Gaussian vortex $\mathfrak{w}_z(\boldsymbol{x})=2\varOmega _0\exp (-|\boldsymbol{x}|^2/r_0^2)$ , where $\varOmega _0$ is the fluid angular velocity at the vortex centre, $r_0$ the Gaussian core radius and $|\boldsymbol{x}|^2 := x_1^2 + x_2^2$ denotes the Euclidean vector norm. When subject to meandering obeying a Gaussian distribution $p$ with standard deviations $\sigma _\alpha$ along the principal axes $x_\alpha$ , the mean streamwise vorticity becomes

(3.8) \begin{equation} \overline {w}_z(\boldsymbol{x};\boldsymbol{\sigma }) = 2\varOmega _0 r_0^2 \prod _{\alpha =1}^2 \frac {\textrm{e}^{-\tfrac {x_\alpha ^2}{r_0^2 + 2\sigma _\alpha ^2}}}{\sqrt {r_0^2 + 2\sigma _\alpha ^2}}. \end{equation}

There is ample experimental evidence for the Gaussian distribution of vortex meandering in this (figure 13) and previous studies (Bailey & Tavoularis Reference Bailey and Tavoularis2008; Dghim et al. Reference Dghim, Ferchichi and Fellouah2020, Reference Dghim, Ben Miloud, Ferchichi and Fellouah2021; Bölle Reference Bölle2024). For this specific case of a Gaussian vortex subject to meandering obeying a Gaussian probability distribution, series expansion of (3.8) in the standard deviation (meandering amplitude) $\sigma _\alpha$ , and using $\overline {w}_z(\boldsymbol{x};\boldsymbol{\sigma }=0) = \mathfrak{w}_z(\boldsymbol{x})$ , yields

(3.9) \begin{align} \overline {w}_z(\boldsymbol{x};\boldsymbol{\sigma }) &= \mathfrak{w}_z(\boldsymbol{x}) + \frac {1}{2}\frac {\partial ^2 \overline {w}_z}{\partial \sigma _\alpha ^2}\bigg |_{\sigma _\alpha =0}\sigma _\alpha ^2 + O\big(\boldsymbol{\sigma }^4\big) \nonumber \\ &= \mathfrak{w}_z(\boldsymbol{x}) + \mathfrak{w}_z(\boldsymbol{x})\left (\frac {|\boldsymbol{x}|^2}{r_0^2} - 1\right )\bigg |\frac {\boldsymbol{\sigma }}{r_0}\bigg |^2 + O\big(\boldsymbol{\sigma }^4\big) \end{align}

for the leading-order representation of the mean flow as a consequence of meandering.

Figure 7.

Comparison between meandering postulate and experiment. $(a)$ Effect of vortex meandering on mean vorticity distribution. $(b)$ Fluctuation vorticity associated with a typical vortex displacement $X_\alpha /{r_{{c}}} = 0.2$ , compared with the first-order approximation in (3.11) and the leading POD mode $\hat {\phi }^{(\alpha )}$ profile along $x_\alpha$ (see § 4.1).

Figure 7. Long description

The image contains two line graphs side by side. The left graph shows the mean vorticity distribution with different lines representing various approximations and measurements. The x-axis is labeled as xα/r0, and the y-axis is labeled as wz in units of s^-1. The right graph illustrates the fluctuation vorticity associated with a typical vortex displacement, compared with the first-order approximation and the leading POD mode profile. The x-axis is labeled as xα/r0, and the y-axis is labeled as w'z in units of s^-1. The graphs depict how vortex meandering affects vorticity distribution and fluctuations, providing a comparison between theoretical postulates and experimental data.

We tested the validity of these results empirically by a numerical experiment as follows. A Gaussian vortex $\mathfrak{w}_z$ is subjected to a Gaussian stochastic meandering $X_\alpha ^{(n)} \sim \mathcal{N}(0,\sigma _\alpha /r_0), n = 1,2,\ldots , N$ . The values of the parameters $\varOmega _0$ , $r_0$ and $\sigma _\alpha$ were determined from the experiment (see § 2.3, $r_0\approx {r_{{c}}}/1.12$ for a Gaussian vortex). A sample of $N = 10^5$ displaced vortices is generated with an ensemble mean $\overline {w}^N_z = N^{-1}\sum _{n=1}^N \mathfrak{w}_z((x_\alpha -X_\alpha ^{(n)})/r_0)$ . This mean estimator is compared in figure 7( $a$ ) with $\overline {w}_z$ according to (3.8) and with the leading-order approximation (3.9), showing very good agreement.

The meandering postulate (3.6), together with the compatibility condition (3.7), imply that fluctuations associated with the meandering motion have to vanish on average, i.e.

(3.10) \begin{equation} \int _{\mathbb{R}^2}{\text{d}}^{2}X p(\boldsymbol{X}) w_z^{\prime}(\boldsymbol{x},t) = 0 \;\implies \; w_z^{\prime}(\boldsymbol{x},t) = c_\alpha (\boldsymbol{x})X_\alpha (t) + O\big(\boldsymbol{X}^3\big) , \end{equation}

where the $c_\alpha (\boldsymbol{x})$ are stationary scalar fields. We assume that $p(\boldsymbol{X})$ is symmetric around the origin, which rules out any systematic meandering bias, i.e. neither positive nor negative vortex displacements are preferred on average. It follows immediately that the fluctuation vorticity associated with meandering must be an odd function of the instantaneous vortex centre position. All even terms, including a constant, have to vanish, since otherwise (3.10) cannot average to zero.

Using the meandering postulate (3.6) along with expansion (3.9) of the mean flow, we obtain the leading-order series expansion of the fluctuation vorticity

(3.11) \begin{align} w_z^{\prime}(\boldsymbol{x},t) &= \mathfrak{w}_z(\boldsymbol{x}-\boldsymbol{X}(t)) - \overline {w}_z(\boldsymbol{x};\boldsymbol{\sigma }) \nonumber \\ &= -\frac {\partial \mathfrak{w}_z}{\partial x_\alpha }X_\alpha + \frac {1}{2}\frac {\partial ^2 \mathfrak{w}_z}{\partial x_\alpha ^2}X_\alpha ^2 - \frac {1}{2}\frac {\partial ^2 \overline {w}_z}{\partial \sigma _\alpha ^2}\bigg |_{\sigma _\alpha =0}\sigma _\alpha ^2 + O\big(\boldsymbol{X}^3\big) + O\big(\boldsymbol{\sigma }^4\big) . \end{align}

The first term of this result is consistent with the general functional structure of the fluctuation field anticipated in (3.10). However, the third term, contributed from (3.9), is non-random, non-zero and symmetric. It can therefore not vanish upon averaging with $p(\boldsymbol{X})$ . Taking the average of (3.11) with respect to $p(\boldsymbol{X})$ , we have

(3.12) \begin{align} 0 &= \left [\frac {\partial ^2 \mathfrak{w}_z}{\partial x_\alpha ^2} - \frac {\partial ^2 \overline {w}_z}{\partial \sigma _\alpha ^2}\bigg |_{\sigma _\alpha =0}\right ]\frac {\sigma _\alpha ^2}{2} + O\big(\boldsymbol{\sigma }^4\big) \nonumber \\ &= \left[\mathfrak{w}_z\left(2x_\alpha ^2 - 1\right) - \mathfrak{w}_z\left(x_\alpha ^2 - 1\right)\right]\sigma _\alpha ^2 + O\big(\boldsymbol{\sigma }^4\big) = \mathfrak{w}_z x_\alpha ^2 \sigma _\alpha ^2 + O\big(\boldsymbol{\sigma }^4\big) , \end{align}

where we used that $\overline {X_\alpha ^2} = \sigma _\alpha ^2$ and that $\overline {X^{2n+1}} = 0, n \in \mathbb{N}_0$ . The evaluation in the second line of (3.12) is for a Gaussian vortex subject to Gaussian meandering. Apparently, the second-order terms in (3.11) have a very similar, though not identical, functional structure. Consequently, they do not cancel identically and, in order to fulfil (3.10), must be cancelled by the higher-order even terms. We checked these results empirically, again by generating a set of 10 $^5$ vorticity fields using a Gaussian distribution of displacements. The sample mean of $w^{\prime}_z$ indeed vanishes within machine accuracy.

For meandering amplitudes $\sigma _\alpha$ much smaller than the core radius $r_{{c}}$ , we can neglect second-order terms in (3.9) and (3.11). For the present experiment $\sigma _\alpha /{r_{{c}}} \sim 10^{-1}$ (§ 4.3), which is a typical value for meandering amplitudes (Devenport et al. Reference Devenport, Rife, Liapis and Follin1996; Bailey & Tavoularis Reference Bailey and Tavoularis2008; van Jaarsveld et al. Reference van Jaarsveld, Holten, Elsenaar, Trieling and van Heijst2011; Bölle Reference Bölle2021). We therefore assume the leading-order, linear approximation $\overline {w}_z = \mathfrak{w}_z$ and $w^{\prime}_z = -( {\partial \mathfrak{w}_z}/{\partial x_\beta })X_\beta$ (see figure 7) and alike for all other mean and fluctuation fields from now on. Inserting these expressions into (3.5) yields

(3.13) \begin{align} \frac {\partial X_\alpha }{\partial t} + U_0\frac {\partial X_\alpha }{\partial z} = &-\frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \left (\mathfrak{u}_\alpha \frac {\partial \mathfrak{w}_z}{\partial x_\beta }\right ) X_\beta - \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \left (\mathfrak{w}_z \frac {\partial \mathfrak{u}_\alpha }{\partial x_\beta }\right ) X_\beta \nonumber \\ &+ \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \left (\mathfrak{u}_z \frac {\partial \mathfrak{w}_\alpha }{\partial x_\beta }\right ) X_\beta + \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \left (\mathfrak{w}_\alpha \frac {\partial \mathfrak{u}_z}{\partial x_\beta }\right ) X_\beta \nonumber \\ &+ \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \mathfrak{w}_z U_\alpha ^{\prime} - \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \mathfrak{w}_\alpha U_z^{\prime} + \xi _\alpha (z,t) . \end{align}

Integration by parts of the first and second integrals in (3.13) and assuming that the laminar vortex has effectively compact support in $M$ (Saffman Reference Saffman1995), it readily follows that the integrals in the first and second line of (3.13) cancel mutually. The last line of (3.13) contains the external forcing $U^{\prime}_i$ due to linear potential flow (i.e. boundary effects) and otherwise neglected vortical nonlinearities. At this point there is no conceptual difference between these three terms, so that we assemble them into one generic forcing $f_\alpha$ . With this, the final problem of vortex meandering becomes

(3.14) \begin{align} \frac {\partial X_\alpha }{\partial t} = - U_0\frac {\partial X_\alpha }{\partial z} + f_\alpha (z,t)&,\quad 0 \le z \lt \infty ,\quad -\infty \lt t \lt \infty ,\end{align}

(3.15) \begin{align} \text{initial condition,}\quad &X_\alpha (z,t=t_0) = I_\alpha (z) \end{align}

(3.16) \begin{align} \text{boundary (inflow) condition,}\quad &X_\alpha (z=0,t) = X_{0\alpha }(t) . \end{align}

This is a spatiotemporally forced advection problem in which initial perturbations $X_{0\alpha }$ at $z=0$ are transported along the characteristic defined by the free stream velocity $U_0$ . In a frame moving with the free stream, perturbations are continuously amplified and grow without bound on average (Wiener process). This is not realistic and does not reflect the fact that vortices have a response dynamics in the form of eigenmodes propagating along the vortex. For typical experimental configurations, vortices are in a linearly stable regime and all eigenmodes are damped (Edstrand et al. Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016; Qiu et al. Reference Qiu, Cheng, Xu, Xiang and Liu2021; Bölle et al. Reference Bölle, Brion, Couliou and Molton2023; Bölle Reference Bölle2024).

3.3. General solution of the vortex-meandering equation of motion

In this section, we derive the general solution of the above meandering problem under the assumption that the actual response dynamics is given by an infinity of vortex eigenmodes subject to advection by the free stream flow. Vortex meandering as described by (3.14)–(3.16) is a stationary, spatial problem. For its solution, we therefore assume a Laplace transform in space (denoted $\tilde {(\boldsymbol{\cdot })}$ ) and Fourier transform in time (denoted $\hat {(\boldsymbol{\cdot })}$ ). This yields

(3.17) \begin{equation} \hat {\tilde {X}}_\alpha (s,\omega ) = \frac {\hat {\tilde {f}}_\alpha (s,\omega ) +U_0\hat {X}_{0\alpha }(\omega )}{\text{i}(\omega _{{a}} + \omega _{{v}}) + sU_0} = \frac {1}{U_0}\frac {\hat {\tilde {f}}_\alpha (s,\omega ) +U_0\hat {X}_{0\alpha }(\omega )}{\text{i}U_0^{-1}(\omega _{{a}} + \omega _{{v}}) + s} , \end{equation}

where $\omega = \omega _{{a}} + \omega _{{v}}$ denotes the real frequency, having a contribution $\omega _{{a}}$ from the advection at $U_0$ and an intrinsic dynamics $\omega _{{v}}$ in terms of eigenmodes propagating along the vortex core. The complex frequency from the Laplace transform is denoted $s = \varsigma _{{s}} + \text{i} k$ , with $\varsigma _{{s}}$ and $k$ denoting the spatial damping rate and wavenumber, respectively. Taking the inverse Laplace transform of (3.17) yields the general equation of motion for the downstream evolution of the temporal Fourier amplitudes

(3.18) \begin{equation} \hat {X}_\alpha (z,\omega ) - \hat {X}_{0\alpha }(\omega )e^{-\text{i}\tfrac {\omega }{U_0}z}e^{-\varsigma _{{s}} z} = \int _0^z\text{d} z'\, e^{-\varsigma _s (z-z')}e^{-\text{i}\tfrac {\omega }{U_0}(z-z')} \frac {\hat {f}(z',\omega )}{U_0} . \end{equation}

We note that the spectral damping $\varsigma _{{s}}$ and intrinsic frequencies $\omega _{{v}}$ are arbitrary in this solution and are fully determined by the eigenmodes associated with vortex meandering.

For now, let us ignore any forcing by potential flow and assume a vortex embedded in the residual turbulence of the experimental facility. (This being permitted due to the linearity of the problem.) We assume the surrounding turbulence to be of much smaller scale than the vortex response. This allows us to model the forcing exerted by the turbulence on the vortex as a sequence of spatiotemporally uncorrelated Dirac impacts. We therefore assume the ‘Langevin’ condition on the forcing (an asterisk denotes complex conjugation)

(3.19) \begin{equation} \overline {\hat {\xi }}(z,\omega ) = 0 ,\quad \overline {\hat {\xi }(z,\omega )\hat {\xi }^*(z',\omega ')} = 2D\delta (\omega -\omega ')\delta (z-z') , \end{equation}

stating that the forcing balances on average and that a Dirac-localised forcing $\xi$ excites all wavenumbers and frequencies equally with an amplitude parameter $D$ . Averaging (3.18) using this assumption, the mean becomes

(3.20) \begin{equation} \overline {\hat {X}}_\alpha (z,\omega ) = \hat {X}_{0\alpha }(\omega )e^{-\text{i}\tfrac {\omega }{U_0}z}e^{-\varsigma _{{s}} z} , \end{equation}

given that the inflow condition $X_{0\alpha }$ is a sure event. We notice that (3.20) corresponds to the homogeneous solution of (3.18). Introducing the fluctuation around the mean $\hat {X}_\alpha ^{\prime}(z,\omega ) = \hat {X}_\alpha (z,\omega ) - \overline {\hat {X}}_\alpha (z,\omega )$ and comparing with (3.18) shows that the fluctuation can be identified with the particular solution. The variance follows from

(3.21) \begin{align} \overline {|\hat {X}_\alpha ^{\prime}|^2}(z,\omega ) &= \int _0^z\text{d} z'\!\int _0^z\!\text{d} z^{\prime\prime} e^{-\varsigma _{{s}} (z-z')}e^{-\varsigma _{{s}} (z-z^{\prime\prime})}e^{-\text{i}\tfrac {\omega }{U_0}(z-z')}e^{\text{i}\tfrac {\omega }{U_0}(z-z^{\prime\prime})} \frac {\overline {\hat {\xi }(z',\omega )\hat {\xi }^*(z^{\prime\prime},\omega )}}{U_0^2} \nonumber \\ &= 2\frac {D}{U_0^2} e^{-2\varsigma _{{s}} z}\int _0^z\text{d} z'\,\int _0^z\text{d} z^{\prime\prime}\, e^{\varsigma _{{s}} (z'+z^{\prime\prime})} e^{\text{i}\tfrac {\omega }{U_0}(z'-z^{\prime\prime})} \delta (z'-z^{\prime\prime}) \nonumber \\ &= 2\frac {D}{U_0^2} e^{-2\varsigma _{{s}} z}\int _0^z\text{d} z'\, e^{2\varsigma _{{s}} z'} = \frac {D}{U_0^2\varsigma _{{s}}} \left [1 - e^{-2\varsigma _{{s}} z}\right ]\! . \end{align}

In experiments, we measure the temporal variation of $X_\alpha$ in a fixed measurement plane at $z = z_M$ . The streamwise gradient in this measurement plane, appearing on the right-hand side of (3.14), should reflect these internal dynamics due to excitation of vortex eigenmodes. This is reflected in the solution (3.20) and (3.21) by spatially evolving eigenmodes. However, streamwise gradients and streamwise propagating eigenmodes cannot be inferred from our experiment and have to be estimated from theory.

3.4. Estimation from the dispersion relation of the Moore–Saffman vortex

As formalised in (3.2), the instantaneous vortex-centre position is a weighted average of the stochastic fluctuation vorticity $w_z^{\prime}$ in a Reynolds decomposition. Consequently, our derivation of the vortex-meandering equation of motion (3.3) uses a Reynolds decomposition of the velocity and vorticity fields. On the other hand, theoretical studies of the linearised vortex dynamics assume deterministic perturbations of a laminar base-state vortex. It is not a priori obvious that these stochastic fluctuations around the mean can be identified with deterministic perturbations of the base flow (North Reference North1984).

We have shown above that within our leading-order approximation the mean and base flows can be identified. Assuming that the eigenmodes constitute a complete description of the linearised dynamics (Arendt, Fritts & Andreassen Reference Arendt, Fritts and Andreassen1997; Fabre et al. Reference Fabre, Sipp and Jacquin2006; Roy & Subramanian Reference Roy and Subramanian2014), arbitrary fluctuations can be represented as linear combinations of these eigenmodes. The meandering postulate (3.6) implies that the fluctuation vorticity associated with meandering must be dipolar. On the other hand, Bölle (Reference Bölle2023) demonstrated theoretically that the vortex-centre positions $X_\alpha$ are directly related to the leading two POD modes $\hat {\phi }^{(\alpha )}$ of the vorticity field, which exhibit a similar dipolar pattern (see figure 10). Ample experimental evidence for this $X_\alpha$ – $\hat {\phi }^{(\alpha )}$ correspondence can be found in the literature (Edstrand et al. Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016; Karami et al. Reference Karami, Hangan, Carassale and Peerhossaini2019; Bölle et al. Reference Bölle, Brion, Couliou and Molton2023). However, agreement between the leading POD mode patterns and fluctuations predicted by the meandering postulate has not been shown before. Therefore, in order to check the appropriateness of our hypotheses, figure 7( $b$ ) displays the vorticity fluctuation leading to a vortex displacement, computed for a Gaussian vortex. For small displacements, this is well represented by the first-order approximation (3.11) used here. Both are qualitatively and quantitatively similar to the leading POD mode, which we know is associated with meandering.

As already recognised before (Jacquin et al. Reference Jacquin, Fabre, Geffroy and Coustols2001; Edstrand et al. Reference Edstrand, Davis, Schmid, Taira and Cattafesta2016), the fluctuation shown in figures 7( $b$ ) and 10 is well represented by a displacement (D) Kelvin eigenmode of linear vortex dynamics (Fabre et al. Reference Fabre, Sipp and Jacquin2006; Bölle et al. Reference Bölle, Brion, Robinet, Sipp and Jacquin2021). This suggests that vortex meandering is due to surrounding turbulence exciting an infinity of displacement Kelvin waves propagating along the vortex (Melander & Hussain Reference Melander and Hussain1993; Marshall & Beninati Reference Marshall and Beninati2005; Pradeep & Hussain Reference Pradeep and Hussain2010). Displacement-like perturbations are found very generically as the most energetic vortex-response modes (e.g. Gutierrez-Castillo et al. Reference Gutierrez-Castillo, Garrido-Martin, Bölle, García-Ortiz, Aguilar-Cabello and del Pino2022; Bölle et al. Reference Bölle, Brion, Couliou and Molton2023; Garrido-Martin et al. Reference Garrido-Martin, Blanco-Rodríguez, Gutierrez-Castillo, Bölle and del Pino2025). We therefore propose to identify $\omega _{{v}}$ and the corresponding $\varsigma _{{s}}$ in § 3.3 with the theoretical relations of displacement vortex eigenmodes. We determined the dispersion relation of the D mode of the vortex in our experiment, using the procedure given by Fabre & Jacquin (Reference Fabre and Jacquin2004) and the analytical expression (2.3) for the swirl velocity profile. The result for the frequency $\omega _{1,0}$ can with good accuracy be expressed in the form proposed by Fabre (Reference Fabre2002), which has the correct asymptotic behaviour:

(3.22) \begin{equation} \frac {\omega _{1,0}}{\varOmega _0}(k{r_{{c}}}) = -\frac {(k{r_{{c}}})^2}{2 + C_1|k|{r_{{c}}} + C_2(k{r_{{c}}})^2}\left [ \ln \left (\frac {2 + C_3|k|{r_{{c}}}}{|k|{r_{{c}}}}\right ) + \frac {1}{4}-\gamma \right ]\!, \end{equation}

with $C_1 = 2.6742 , C_2 = 0.7613, C_3 = 2.9698$ , Euler’s constant $\gamma = 0.5772\ldots$ and $k$ the axial wavenumber. The ratio between core radius $r_{{c}}$ and the parameter $a$ in (2.3) depends on the value of $n$ and is ${r_{{c}}}/a\approx 1.30$ in the present case.

The D modes are neutrally stable to leading order (in Reynolds number) and regularly perturbed by viscosity. For our experimental vortex, we therefore consider a damping rate given to a good approximation by $\nu k^2$ , in agreement with Fabre (Reference Fabre2002). Together with (3.22), this yields the viscous dispersion relation of the D mode

(3.23) \begin{equation} \frac {\omega _{\texttt {D}}(k{r_{{c}}})}{\varOmega _0} = \frac {\omega _{1,0}(k{r_{{c}}})}{\varOmega _0} + \text{i} \frac {(k{r_{{c}}})^2}{Re_{{c}}}, \end{equation}

where $Re_{{c}} = \varOmega _0{r_{{c}}}^2/\nu$ is the core Reynolds number. According to (3.13), all vortex eigenmodes are subject to advection by the free stream flow, for which the dispersion relation (for a fixed measurement position) reads

(3.24) \begin{equation} \frac {\omega _{{a}}(k{r_{{c}}})}{\varOmega _0} = - \frac {U_0}{{r_{{c}}}\varOmega _0}k{r_{{c}}} . \end{equation}

Adding (3.23) and (3.24) yields the dispersion relation for the advected D modes

(3.25) \begin{equation} D[\omega , k; \nu ] = \frac {\omega _{1,0}(k{r_{{c}}})}{\varOmega _0} - \frac {U_0}{{r_{{c}}}\varOmega _0}k{r_{{c}}} + \text{i} \frac {(k{r_{{c}}})^2}{Re_{{c}}} = 0 . \end{equation}

Figure 8.

Inviscid dispersion relation for the displacement (D) mode of the experimental vortex subject to advection at $U_0$ . The frequency $\omega (k)$ is asymptotically due to advection (3.26), dominating for wavelengths shorter than $r_{{c}}$ .

The real part of (3.25),

(3.26) \begin{equation} \frac {\omega (k{r_{{c}}})}{\varOmega _0} = \frac {\omega _{1,0}(k{r_{{c}}})}{\varOmega _0} - \frac {U_0}{{r_{{c}}}\varOmega _0} k{r_{{c}}} \quad \stackrel {k{r_{{c}}} \gg 1}{\longrightarrow }\quad \frac {\omega (k{r_{{c}}})}{\varOmega _0} = -1 - \frac {U_0}{{r_{{c}}}\varOmega _0} k{r_{{c}}} \end{equation}

corresponding to the inviscid dispersion relation, describes the frequency shift of the D modes due to advection (figure 8). Damping occurs mainly for non-zero wavenumbers of the order of $k{r_{{c}}} \sim 1$ or larger. The frequency $\omega _{1,0}$ of the D mode in (3.22) asymptotes to $\omega _{1,0}/\varOmega _0 \to -1$ as $k{r_{{c}}} \to \infty$ (see also Fabre Reference Fabre2002). This yields the second equality in (3.26), valid for $k{r_{{c}}} \gg 1$ . In this limit of short wavelengths, we can invert (3.26) to obtain $k{r_{{c}}} = (\varOmega _0 - \omega ){r_{{c}}}/U_0$ . Inserting this into the imaginary part of the dispersion relation (3.25) yields the temporal spectral damping

(3.27) \begin{equation} \frac {\varsigma (\omega )}{\varOmega _0} = \frac {1}{Re_{{c}}}\left (1 - \frac {\omega }{\varOmega _0}\right )^2\left (\frac {{r_{{c}}}\varOmega _0}{U_0}\right )^2 \quad (k{r_{{c}}} \gg 1) \end{equation}

as a function of $\omega$ only, which is experimentally measurable. For the particular case of D modes considered here, we then deduce the spatial spectral damping rate

(3.28) \begin{equation} \varsigma _{{s}}(\omega ){r_{{c}}} = \frac {\varsigma (\omega )}{\varOmega _0}\frac {{r_{{c}}}\varOmega _0}{U_0} = \frac {1}{Re_{{c}}}\left (1 - \frac {\omega }{\varOmega _0}\right )^2\left (\frac {{r_{{c}}}\varOmega _0}{U_0}\right )^3 \quad (k{r_{{c}}} \gg 1) . \end{equation}

From (3.26) follows the group velocity $U_0/{r_{{c}}}\varOmega _0$ for $k{r_{{c}}} \gg 0$ , such that the above transformation (3.28) from temporal to spatial damping rates corresponds to Gaster’s transform. Finally, inserting the spatial damping (3.28) for the D mode into (3.21), we obtain

(3.29) \begin{equation} \overline {|\hat {X}_\alpha ^{\prime}|^2}(z,\omega ) = \frac {D U_0}{\nu } \frac {1}{(\varOmega _0 - \omega )^2} \left [1 - e^{-2 \tfrac {\nu (\varOmega _0 - \omega )^2}{U_0^3}z}\right ]\! . \end{equation}

Bölle (Reference Bölle2021, Reference Bölle2023) proposed that meandering essentially corresponds to a Brownian motion of the vortex and can be described adequately by an Ornstein–Uhlenbeck process. The PSD in this case is given by (Yaglom Reference Yaglom1962)

(3.30) \begin{equation} S(z,\omega ) = \frac {\sigma ^2(z)}{\pi } \frac {\lambda }{\omega ^2 + \lambda ^2} , \end{equation}

where $\lambda$ denotes the reciprocal response (damping) time scale of the vortex (Brownian particle). Equation (3.30) was shown to provide a very good fit to experimentally measured spectra of the vortex displacement (Bölle Reference Bölle2024). Whereas an Ornstein–Uhlenbeck process is characterised by a single response time scale $\lambda ^{-1}$ , our model has an infinity of time scales associated with the spectral damping, i.e. the viscous damping of the regular vortex eigenmodes at finite Reynolds numbers.

Matching the asymptotic behaviour of (3.29) and (3.30), one can obtain a relationship between the fluid viscosity $\nu$ appearing in our model and the decay rate $\lambda$ . In particular, we consider the limit as $\omega \to 0$ and $0 \ll \omega \ll \infty$ . We therefore require that

(3.31) \begin{align} \overline {|\hat {X}_\alpha ^{\prime}|^2}(z,0) = \frac {2Dz}{U_0^2} &\stackrel {!}{=} \frac {\sigma ^2}{\pi }\frac {1}{\lambda } = S(z,0) \quad &&(\omega \to 0) ,\end{align}

(3.32) \begin{align} \overline {|\hat {X}_\alpha ^{\prime}|^2}(z,\omega ) = \frac {DU_0}{\nu \omega ^2} &\stackrel {!}{=} \frac {\sigma ^2}{\pi }\frac {\lambda }{\omega ^2} = S(z,\omega ) && (0 \ll \omega \ll \infty ). \end{align}

Taking the limit as $\omega \to \infty$ , we see from (3.32) that the spectral energy vanishes for infinitely large frequencies. Solving (3.32) for $D$ , inserting into (3.31) and solving for $\lambda$ yields

(3.33) \begin{equation} \lambda = \sqrt {\frac {U_0^3}{2\nu z}} . \end{equation}

Evaluating (3.33) for the present experiment (§ 2), viz. $U_0 = {79.1}\,\textrm {cm s}^{-1}$ , $\nu = {1\times {10}^{-2}}\,\textrm {cm s}^{-2}$ and $z = {110}\,\textrm {cm}$ , yields $\lambda = {474}\,\textrm {s}^{-1}$ . Figure 9 displays a comparison between (3.29) and (3.30), using the above scaling, showing that the spectrum predicted from the present model closely resembles the one of an Ornstein–Uhlenbeck process. We show in § 4.3 below that a suitable value for $\lambda$ , from a fit to the experimental data, would rather be $\lambda \approx {2}\,\textrm {s}^{-1}$ , which is two orders of magnitude smaller than the one inferred from the present model based on the viscous decay of vortex Kelvin modes.

Figure 9.

The PSD of vortex meandering motion: comparison between the prediction (3.29) of the present model, using experimental parameter values, and the PSD of an Ornstein–Uhlenbeck process (3.30), where the reciprocal time scale $\lambda$ is set according to (3.33).

Figure 9. Long description

The line graph compares the power spectral density of vortex meandering motion between the prediction of the present model, using experimental parameter values, and the power spectral density of an Ornstein-Uhlenbeck process. The x-axis represents the normalized frequency, while the y-axis represents the power spectral density in arbitrary units. The solid line represents the present model, and the dotted line represents the Ornstein-Uhlenbeck process. The graph shows that both models have similar behavior at low frequencies but diverge at higher frequencies. All values are approximated.

4.1. Proper orthogonal decomposition

A simplification of (3.5) is possible by expansion of the rotational fluctuation fields $w_z^{\prime}$ and $u_\alpha ^{\prime}$ in the leading modes of a POD, as discussed in § 3.4. The expansion of the fluctuation vorticity in POD modes reads (Holmes, Lumley & Berkooz Reference Holmes, Lumley and Berkooz1996)

(4.1) \begin{align} &w_z^{\prime}(\boldsymbol{x},z,t) = \sum _{\mu =1}^\infty a_\mu (z,t)\phi ^{(\mu )}(\boldsymbol{x}) , \end{align}

(4.2) \begin{align} &\text{such that}\quad (\phi ^{(\mu )},\phi ^{(\nu )}) = \sigma _\mu ^2\delta _{\mu \nu } \quad \text{and}\quad \overline {a_\mu a_\nu } = \delta _{\mu \nu } \quad \forall \mu , \nu . \end{align}

Contrary to our general convention, Greek indices in the POD, for now, run through all positive integer values, which we express by writing the sums explicitly. As usual, $(\boldsymbol{\cdot },\boldsymbol{\cdot })$ denotes the $L^2(M)$ -inner product and $||\boldsymbol{\cdot }||$ the induced norm (Holmes et al. Reference Holmes, Lumley and Berkooz1996). The corresponding rotational fluctuation-velocity modes are then obtained from the relevant Biot–Savart integrals (Saffman Reference Saffman1995)

(4.3) \begin{align} \psi _1^{(\mu )}(\boldsymbol{x}) &= -\frac {1}{2\pi }\int _M{\text{d}}^{2}x'\, \phi ^{(\mu )}(\boldsymbol{x}') \frac {x_2-x_2'}{|\boldsymbol{x}-\boldsymbol{x}'|^2} \quad \text{and} \end{align}

(4.4) \begin{align} \psi _2^{(\mu )}(\boldsymbol{x}) &= \frac {1}{2\pi }\int _M{\text{d}}^{2}x'\, \phi ^{(\mu )}(\boldsymbol{x}') \frac {x_1-x_1'}{|\boldsymbol{x}-\boldsymbol{x}'|^2} , \end{align}

where $|\boldsymbol{\cdot }|$ denotes the Euclidean vector norm in the measurement plane (cf. § 3.2).

Figure 10.

Estimates of the $(a)$ first and $(b)$ second vorticity POD modes in the vortex-centre principal-axes system, with positive (negative) values corresponding to red (blue) colour shading and solid (dashed) contours. The axes $\boldsymbol{e}_\alpha$ are scaled by the corresponding meandering amplitudes $\hat {\sigma }_\alpha$ and transformed to the POD system by $\widehat {M}_{\alpha \alpha }^{-1}$ (see § 4.2).

Figure 10. Long description

A heat map displays estimates of the first and second vorticity POD modes in the vortex-centre principal-axes system. Positive values are represented by red color shading and solid contours, while negative values are shown with blue color shading and dashed contours. The axes are scaled by the corresponding meandering amplitudes and transformed to the POD system. The heat map features distinct regions of red and blue, indicating varying vorticity values. The color intensity ranges from deep red to deep blue, highlighting areas of high and low vorticity, respectively. The contours provide additional detail on the distribution and intensity of these vorticity modes.

Figure 10 shows estimators $\hat {\phi }^{(\mu )}$ of the true vorticity POD modes $\phi ^{(\mu )}$ , $\mu = 1,2$ , for the given experiment (Bölle Reference Bölle2024). Using standard notation, statistical estimators are indicated by a circumflex in the following (see § 4.3). Superposition of the (scaled) principal axes illustrates their alignment with the leading POD modes. This alignment, together with the characteristic dipolar spatial pattern, yields the proportionality of the visually manifest meandering motion $X_\mu$ and the leading-order dynamic vorticity fluctuation $a_\mu$ (Bölle Reference Bölle2023, Reference Bölle2024). Figure 11 confirms that the $X_\mu (t)$ and $a_\mu (t)$ time series, when scaled according to our derivation in § 4.2, indeed overlap.

Figure 11.

Comparison of the observed vortex deflection $X_\mu$ (figure 5) with the time series of the amplitudes of the $(a)$ first and $(b)$ second vorticity POD modes. When appropriately scaled by $\widehat {M}_{\mu \mu }$ (see (4.7)) and $\hat {\sigma }_\mu$ , respectively, the time series overlap.

Figure 11. Long description

Two line graphs compare observed vortex deflection with time series of vorticity POD modes. The top graph shows the first vorticity POD mode, while the bottom graph shows the second vorticity POD mode. Both graphs have the same x-axis labeled as t divided by omega sub 0, ranging from 0 to 3000. The y-axes are labeled as X sub 1, a sub 1 for the top graph and X sub 2, a sub 2 for the bottom graph, both ranging from -4 to 4. The black lines represent the observed vortex deflection, and the gray lines represent the time series of the amplitudes of the vorticity POD modes. The time series overlap when appropriately scaled.

Figure 12 shows the rotational fluctuation velocity estimates $\hat {\psi }_\alpha ^{(\mu )}$ along the vortex-centre principal axes, computed by numerically solving (4.3) and (4.4). As assumed in the model derivation (§ 3.1), the leading vortex-response modes shown in figures 10 and 12 have essentially a compact support in $M$ . We note that the Biot–Savart velocities of the leading vorticity POD modes shown in figure 12 are practically identical – qualitatively and quantitatively – to the leading velocity POD modes (not shown).

Figure 12.

Rotational fluctuation velocities associated with the leading vorticity POD modes in the $x_1$ – $x_2$ system, computed from Biot–Savart integrals: $(a)$ $\hat {\psi }_1^{(1)}$ , $(b)$ $\hat {\psi }_2^{(1)}$ , $(c)$ $\hat {\psi }_1^{(2)}$ , $(d)$ $\hat {\psi }_2^{(2)}$ . The vortex-centre principal axes $\boldsymbol{e}_\alpha$ , scaled by the corresponding meandering amplitudes $\hat {\sigma }_\alpha$ and transformed to POD system by $\widehat {M}_{\alpha \alpha }^{-1}$ , are shown as arrows.

Figure 12. Long description

The heat map displays rotational fluctuation velocities associated with the leading vorticity POD modes in the system, computed from BiotSavart integrals. The map consists of four subplots labeled (a), (b), (c), and (d), each showing different distributions of velocities. The x-axis and y-axis are labeled as x1/rc and x2/rc, respectively. The color scale on the right ranges from -7.5 to 7.5 meters per second, with blue indicating lower values and red indicating higher values. Each subplot contains contour lines and arrows representing the vortex-centre principal axes, scaled by the corresponding meandering amplitudes and transformed to the POD system. The arrows point in different directions, indicating the orientation of the principal axes. The contour lines show regions of varying velocity intensities, with red areas indicating higher velocities and blue areas indicating lower velocities. The overall structure of the heat map reveals distinct patterns and gradients in the velocity distributions across the different subplots.

4.2. Numerical evaluation of the model integrals

Injecting the expansions of the fluctuation vorticity and rotational velocity in their respective modes (4.1)–(4.4) into the first line of (3.5) yields

(4.5) \begin{align} \sum _{\mu =1}^\infty \left [\! \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, x_\alpha \phi ^{(\mu )} \!\right ] \frac {\text{d} a_\mu }{\text{d} t} &= \sum _{\mu =1}^\infty \left [\! \frac {1}{\varGamma }\!\int _M\!{\text{d}}^{2}x\, \overline {u}_\alpha \phi ^{(\mu )} \!\right ] a_\mu + \!\sum _{\mu =1}^\infty \!\left [ \!\frac {1}{\varGamma }\int _M\!{\text{d}}^{2}x\, \overline {w}_z \psi _\alpha ^{(\mu )} \!\right ] a_\mu \nonumber \\ &\quad+ \left [ \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \overline {w}_z U_\alpha ^{\prime}(t,\boldsymbol{x}) \right ] + \xi _\alpha (t) \quad (\alpha = 1, 2) . \end{align}

For further simplification, we use the fact that the vortex-meandering motion is essentially determined by the leading two POD modes (Bölle Reference Bölle2023). Hence, we can truncate the POD expansion to $\mu = 1,2$ in the following and express (4.5) in the symbolic matrix-vector form:

(4.6) \begin{equation} {{\unicode{x1D648}}}\left (\frac {\partial \boldsymbol{a}}{\partial t} + U_0\frac {\partial \boldsymbol{a}}{\partial z}\right ) = {{\unicode{x1D646}}}\boldsymbol{a} + \boldsymbol{\eta }(t) + \boldsymbol{\xi }(t) . \end{equation}

The entries of matrices $\unicode{x1D648}$ and $\unicode{x1D646}$ correspond to the left-hand side and right-hand side integrals of the first line of (4.5), respectively. Here $\unicode{x1D646}$ includes contributions from advection by the mean velocity and from stretching by the mean vorticity. The integrals of the second line of (4.5) involving the (given) fluctuating potential flow correspond to an external forcing $\boldsymbol{\eta }(t)$ .

The principal structure and order-of-magnitude values of the matrices $\unicode{x1D648}$ and $\unicode{x1D646}$ can be deduced from the characteristic symmetries and scales of the flow fields. Figures 10 and 12 suggest that $\unicode{x1D4C1} \sim {r_{{c}}} \sim {0.5}\,\textrm {cm}$ is a reasonable length scale estimate over which vorticity and velocity fluctuations associated with vortex meandering change appreciably. The same length scale characterises the mean flow (figure 6). The integration domain’s lateral dimension is $O(10\unicode{x1D4C1})$ and therefore roughly an order of magnitude larger than the characteristic spatial variability. Figure 10 further suggests to take the maximum $\unicode{x1D4CC}' \sim {30}\,\textrm {s}^{-1}$ as the characteristic scale for the vorticity fluctuations. With the Biot–Savart law (4.3)–(4.4), we deduce the corresponding velocity scale $\unicode{x1D4CA}' \sim (2\pi )^{-1} \unicode{x1D4C1}\unicode{x1D4CC}' \sim {3}\,\textrm {cm s}^{-1}$ (cf. also figure 12). Finally, characteristic scales of the mean vorticity and velocity are $\bar {\unicode{x1D4CC}} \sim \varOmega _0 \sim {100}\,\textrm {s}^{-1}$ and $\bar {\unicode{x1D4CA}} \sim 0.6\times {r_{{c}}}\varOmega _0 \sim {30}\,\textrm {cm s}^{-1}$ (figure 6). The matrix structure is deduced from the typical spatial symmetries of the mean fields and POD modes in the principal axes system. For guidance, estimates of the leading vorticity POD modes $\hat {\phi }^{(\mu )}$ and their corresponding velocities $\hat {\psi }_\alpha ^{(\mu )}$ ( $\alpha , \mu = 1,2$ ) computed from the given experiment are shown in figures 10 and 12. Inspection of the characteristic dipolar vorticity pattern of $\hat {\phi }^{(\mu )}$ in figure 10 readily reveals that

(4.7) \begin{equation} M_{\alpha \mu } = \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, x_\alpha \phi ^{(\mu )} \sim \frac {\unicode{x1D4C1} \unicode{x1D4CC}'}{\bar {\unicode{x1D4CC}}} \delta _{\alpha \mu } \sim {0.1}\,\textrm {cm} \; \delta _{\alpha \mu } , \end{equation}

because of the linear weighting with the principal coordinates. The direct proportionality between the experimentally visible vortex meandering $X_\mu$ and the evolution in the phase space spanned by the leading POD modes $a_\mu$ expressed by (4.7) was termed kinematic-dynamic equivalence (Bölle Reference Bölle2023).

Analogously, the contribution to $\unicode{x1D646}$ from advection by the mean velocity is

(4.8) \begin{equation} K_{\alpha \mu }^{\textit{A}} = \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \overline {u}_\alpha \phi ^{(\mu )} \sim \frac {\bar {\unicode{x1D4CA}} \unicode{x1D4CC}'}{\bar {\unicode{x1D4CC}}} \varepsilon _{\alpha \mu } \sim {3}\,\textrm {cm s}^{-1} \; \varepsilon _{\alpha \mu } , \end{equation}

where $\varepsilon _{\alpha \mu }$ denotes the two-dimensional Levi–Civita symbol (symbolically, ${\unicode{x1D640}} = -{\unicode{x1D640}}^{\text{T}}$ ). Equation (4.8) expresses the fact that transport due to advection corresponds to a skew-symmetric matrix and is hence a conservation principle. In the present case of a line vortex, (4.8) corresponds to conserved circular advection in the measurement plane by the vortex mean flow.

From figure 12, we infer that the velocity modes $\psi ^{(\mu )}_\alpha$ in (4.9) should be structurally similar and, moreover, that $\text{sgn}\,\psi ^{(1)}_1 = -\text{sgn}\,\psi ^{(2)}_2$ and $\text{sgn}\,\psi ^{(2)}_1 = -\text{sgn}\,\psi ^{(1)}_2$ , where sgn denotes the sign function. The orientation in the plane is irrelevant for the integrals in (4.9), since the mean streamwise vorticity is essentially rotationally symmetric and, hence, contributes the same weight along all directions. The integrals should therefore yield comparable contributions in magnitude and we thus obtain the estimate

(4.9) \begin{equation} K_{\alpha \mu }^{\textit{S}} = \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \overline {w}_z \psi _\alpha ^{(\mu )} \sim \unicode{x1D4CA}'\varepsilon _{\alpha \mu } \sim {3}\,\textrm {cm s}^{-1} \; \varepsilon _{\alpha \mu } \end{equation}

for the contribution to $\unicode{x1D646}$ from stretching by the mean vorticity.

To check our estimation of the matrices $\unicode{x1D648}$ and $\unicode{x1D646}$ in § 3, we solve (4.7)–(4.9) numerically for the given experiment using the trapezoidal rule,

(4.10) \begin{align} \widehat {{\unicode{x1D648}}} &= \begin{bmatrix} {0.080}\,\textrm {cm} & {-0.001}\,\textrm {cm} \\ {0.004}\,\textrm {cm} & {0.080}\,\textrm {cm} \end{bmatrix} \sim \begin{bmatrix} 10^{-1} & 0 \\ 0 & 10^{-1} \end{bmatrix} \textrm {cm}, \end{align}

(4.11) \begin{align} \widehat {{\unicode{x1D646}}}^{\textit{A}} &= \begin{bmatrix} {0.230}\,\textrm {cm s}^{-1} & {2.830}\,\textrm {cm s}^{-1} \\ {-3.042}\,\textrm {cm s}^{-1} & {0.164}\,\textrm {cm s}^{-1} \end{bmatrix} \sim \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \textrm {cm s}^{-1} ,\end{align}

(4.12) \begin{align} \widehat {{\unicode{x1D646}}}^{\textit{S}} &= \begin{bmatrix} {-0.160}\,\textrm {cm s}^{-1} & {-2.740}\,\textrm {cm s}^{-1} \\ {2.930}\,\textrm {cm s}^{-1} & {-0.120}\,\textrm {cm s}^{-1} \end{bmatrix} \sim \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\,\textrm {cm s}^{-1} , \end{align}

which is consistent with the estimations given in (4.7)–(4.9). The numerically inverted $\widehat {{\unicode{x1D648}}}^{-1}$ is used in figure 11 for scaling. In agreement with our theoretical prediction in § 3.2, $\widehat {{\unicode{x1D646}}}^{\textit{A}}$ and $\widehat {{\unicode{x1D646}}}^{\textit{S}}$ mutually cancel to good approximation and $\widehat {{\unicode{x1D646}}} \approx 0$ .

4.3. Comparison of model characteristics with experiment

In what follows, we consider the principal-component time series $a_\mu (t)$ ( $\mu = 1,2$ ) corresponding to the leading expansion coefficients in a POD development of the streamwise fluctuation vorticity (cf. (4.1), setting $z = z_M = \text{const}$ ). This is a bivariate, stationary, centred random process. However, experimentally available are only finite samples $\{a_\mu (n)\}, n = 1, 2, \ldots , N$ , consisting of autocorrelated, identically distributed realisations probed at constant $\Delta t$ . We emphasise that we can only compute estimators of the actual statistical characteristics for the given finite samples (for a discussion of statistical inference in the context of vortex meandering, see Bölle (Reference Bölle2024)).

A priori we do not know the population probability distribution of $a_\mu$ . Therefore, in order to estimate the uncertainty of the sample statistical characteristics computed from one experimental sample, we use moving-block bootstrapping to account for serial correlation in the data (Wilks Reference Wilks1997; von Storch & Zwiers Reference von Storch and Zwiers2003; Wilks Reference Wilks2006). Our analysis in § 3 suggests that the vortex-response dynamics is characterised by two time scales $\unicode{x1D4C9}_{{v}}$ and $\unicode{x1D4C9}_{{w}}$ . For second-order autoregressive processes the block length $L$ can be estimated from the implicit formula (Wilks Reference Wilks1997, Reference Wilks2006)

(4.13) \begin{equation} L = (N - L + 1)^{\tfrac {2}{3} - \tfrac {1}{3}\sqrt {\tfrac {N'}{N}}e^{-3/2N}} , \end{equation}

where $N'$ denotes the effective sample size (Bölle Reference Bölle2024). Numerically solving (4.13) by bisection yields $L = 1005$ , which is approximately twice the intrinsic response time scale $2\unicode{x1D4C9}_{{v}}/\Delta t$ over which time series are correlated. However, using this block length does not reproduce the correlation structure associated with the low-frequency periodic dynamics, since the periodic time scale $\unicode{x1D4C9}_{{w}}$ induced by the free-surface potential flow is approximately an order of magnitude larger than the intrinsic relaxation scale $\unicode{x1D4C9}_{{r}}$ , viz. $\unicode{x1D4C9}_{{w}}/\unicode{x1D4C9}_{{v}} \sim 10$ . This suggests to take the block length $L$ of the order of $\unicode{x1D4C9}_{{w}}/\Delta t$ . Specifically setting $L = 12752$ , such that the sample of length $N$ is covered by exactly $b = 3$ contiguous blocks, yields plausible results and will be used henceforth. For the bootstrapping, we draw with uniform probability from the original sample with replacement. Repeating this procedure $N_{{b}}$ times, we construct a set of $N_{{b}} = 10^3$ samples that could have been realised instead of the one actually measured. For $\tilde {p} \in (0,1)$ , the $(1\pm \tilde {p})/2$ percentiles then define the $\tilde {p}\times {100}{\%}$ confidence interval (von Storch & Zwiers Reference von Storch and Zwiers2003; Wilks Reference Wilks2006).

Figure 13.

Frequency distribution $\hat {p}(a_{\mu 0}|a_\mu ,z_M)$ histograms of the normalised $(a)$ first and $(b)$ second principal-component time series assuming 60 bins (grey solid line). Grey shading indicates the 95 % confidence interval computed from $N_{{b}} = 10^3$ moving-block bootstrap samples. The solid black lines displays the probability density of a standard normal distribution.

Figure 13. Long description

The image presents a normal distribution curve alongside frequency distribution histograms. The histograms, represented by a grey solid line, show the normalized first and second principal-component time series with 60 bins. Grey shading indicates the 95% confidence interval computed from moving-block bootstrap samples. The solid black line displays the probability density of a standard normal distribution. The x-axis represents normalized values, while the y-axis represents frequency or probability density. The curve is bell-shaped and symmetrical, centered around the mean with standard deviations marked. The shaded areas represent the confidence intervals, highlighting the variability around the mean.

We define the sample variance $\hat {\sigma }_\mu ^2 = ({1}/{N})\sum _{n=1}^Na_\mu ^2(n)$ , which is the maximum likelihood estimator if the $a_\mu$ are normally distributed. For the present experiment, the point estimator and 95 % confidence interval of the standard deviation are $\hat {\sigma }_1/\varOmega _0 = 2.81 \pm 0.02$ , $\hat {\sigma }_2/\varOmega _0 = 2.62 \pm 0.02$ . Normalising $a_\mu$ with these point estimators, figure 13 shows the frequency distribution of the first two principal-component time series $\hat {p}(a_{\mu 0}|a_\mu ,z_M)$ along with the 95 % confidence intervals obtained from moving-block bootstrapping. The frequency distribution and the 95 % confidence intervals coincide reasonably well with a standard normal distribution (figure 13). This is a well-known finding in vortex-meandering experiments (Bailey & Tavoularis Reference Bailey and Tavoularis2008; Dghim et al. Reference Dghim, Ben Miloud, Ferchichi and Fellouah2021; Bölle Reference Bölle2024), although the agreement is somewhat less good than reported previously.

We recall that a perfect Brownian motion-like dynamics would be associated with a normal distribution (Bölle Reference Bölle2023), while a perfectly periodic fluctuation would have a U-shaped probability density with maxima at the oscillation amplitudes (Tennekes & Lumley Reference Tennekes and Lumley1972). This periodicity exerted by the free-surface wave may be visible in the two peaks around $a_\mu \hat {\sigma }_\mu ^{-1} \sim \pm 1$ . The fact that this double-peak structure is apparent also in the 95 % confidence interval suggests that it is a reproducible statistical characteristic.

Neglecting the influence of the free surface, we have shown in § 3.3 that our model power spectrum qualitatively corresponds to that of an Ornstein–Uhlenbeck process describing Brownian motion. From (3.30), we obtain,

(4.14) \begin{equation} \lambda _\mu \frac {\pi S_{\mu \mu }(\omega )}{\sigma ^2} = \frac {\lambda _\mu ^2}{\lambda _\mu ^2 + \omega ^2} \stackrel {\omega \to 0}{\to } 1 \quad (\mu = 1,2\,\text{not summed}) . \end{equation}

Due to linearity, periodic components with a specific frequency $\varOmega$ and amplitude $A_{{w}}$ correspond to Dirac-localised peaks $A_{{w}}^2\delta (\omega -\varOmega )$ superposed to (4.14). It would be expected that any dynamically important periodicity should manifest as a reproducible and statistically significant peak in the sample power spectra. For the normalisation of the sample power spectra $\hat {G}_{\mu \mu }(\omega )$ , we use that $G_{\mu \mu }(\omega =0)\sigma _\mu ^{-2} = 2{\unicode{x1D4C9}}_\mu$ by the Wiener–Khinchin theorem, whereas ${\unicode{x1D4C9}}_\mu = \lambda _\mu ^{-1}$ denotes the integral time scales (Yaglom Reference Yaglom1962; Bölle Reference Bölle2024). Hence,

(4.15) \begin{equation} \lambda _\mu \frac {G_{\mu \mu }(\omega )}{2\sigma _\mu ^2}{ \stackrel {\omega \to 0}{\to }} 1 \quad (\mu = 1,2\,\text{not summed}). \end{equation}

Figure 14 shows a comparison of the normalised sample power spectra (4.15) with the normalised Ornstein–Uhlenbeck process spectra (4.14). The model fits the sample spectra remarkably well and is essentially contained in the 95 % confidence interval computed from the $N_{{b}} = 10^3$ moving-block bootstrap sample. Consequently, for the present experiment, we cannot reject (and hence would accept) the null hypothesis that vortex meandering corresponds to an Ornstein–Uhlenbeck process at the 5 % level of significance (Bölle Reference Bölle2023, Reference Bölle2024). The sample power spectra are normalised by using the respective point estimators in (4.15), where we estimate $\hat {\lambda }_\mu$ from an exponential fit to the sample correlation functions (see below). It is well known that this yields more robust estimators than inferring them from the sample spectra directly (von Storch & Zwiers Reference von Storch and Zwiers2003). It should be stressed that the thus obtained Brownian-motion (reciprocal) response time scale $\hat {\lambda }_\mu \varOmega _0^{-1} \approx 0.02$ (or equivalently ${\unicode{x1D4C9}}_\mu \varOmega _0 \approx 50$ ) is approximately two orders of magnitude smaller than our theoretical estimate of $\lambda _\mu \varOmega _0^{-1} \approx 5.00$ (§ 3.4). This should be contrasted with the overall very good quantitative agreement between (4.14) and (4.15) in figure 14. Finally, we note that there is no clearly discernible peak around the wave period $\varOmega \varOmega _0^{-1} \sim 0.007$ (see § 5.1) in the sample power spectra that would be reproducible and statistically significant (figure 14).

Figure 14.

Non-dimensional sample PSD of the $(a)$ first and $(b)$ second principal-component time series (grey thin line). Grey shading indicates the 95 % confidence interval computed from the $N_{{b}} = 10^3$ moving-block bootstrap sample. Superposed are the normalised power spectra (4.14) of an Ornstein–Uhlenbeck process (black thick line), with an additional peak (indicated by an arrow) at the frequency of the gravity wave in the water channel for the vertical vortex displacement in $(b)$ .

Figure 14. Long description

The image contains two line graphs labeled (a) and (b) side by side. Both graphs plot power spectral density (PSD) on the y-axis against the non-dimensional frequency ratio (omega/Omega_0) on the x-axis. The grey thin lines represent the non-dimensional sample PSD of the first and second principal-component time series, with grey shading indicating the 95% confidence interval computed from the moving-block bootstrap sample. Superimposed on these are black thick lines representing the normalized power spectra of an Ornstein-Uhlenbeck process. In graph (b), an additional peak is indicated by an arrow, representing the frequency of the gravity wave in the water channel for the vertical vortex displacement. The graphs compare empirical data with theoretical models, showing how well the models fit the observed data, particularly noting the presence of a distinct wave frequency in the second graph.

5.1. Forcing from a plane surface gravity wave

In § 2, we provided evidence that in the present experiment the vortex dynamics is subject to potential-flow forcing exerted by gravity waves forming on the free surface of the water channel, with a frequency given by (2.1). We recall from § 2 that the water-channel depth is $d = {45}\,\textrm {cm}$ and that the mean vortex-centre position is at $h_{{v}} \approx {16}\,\textrm {cm}$ above the channel floor. The measured wave frequency is $f_{\textrm{s}} = {0.11}\,\textrm {Hz}$ .

The (two-dimensional) wave velocity field in the channel-fixed coordinate system (figure 4 $a$ ) reads (Landau & Lifshitz Reference Landau and Lifshitz1987)

(5.2) \begin{align} U_z^{\prime}(y,z,t) &= -A \cosh (k(h_{{v}} + y))\sin (kz - \varOmega t)\; =-A_z(y)\sin (kz - \varOmega t), \end{align}

(5.3) \begin{align} U_y^{\prime}(y,z,t) &= A \sinh (k(h_{{v}} + y))\cos (kz - \varOmega t)\; =A_y(y)\sin (kz - \varOmega t) , \end{align}

where $A$ is the amplitude, and $\varOmega =2\pi f_{\textrm{s}}$ and $k=\pi /L$ denote the angular frequency and wavenumber, respectively ( $L$ is the equivalent channel length compatible with the sloshing dynamics, see § 2.2). The amplitudes $A_z$ and $A_y$ of the streamwise and vertical velocity oscillations are related through (2.2). The amplitude at the location of the vortex can be estimated from the time series in figure 2( $a$ ) and the spectral content of the sloshing peak in figure 3( $a$ ) as $A_z(y=0)\approx {1.0}\,\textrm {cm s}^{-1}$ . Since $h_v \ll L$ , relation (2.2) then yields $A_y(y=0) \approx \pi A_z h_v/L \approx {0.07}\,\textrm {cm s}^{-1}$ , in agreement with the observation in figure 2 $(b)$ .

As already emphasised above, for the present experiment, we can evaluate only the forcing due to the vertical component of the wave velocity in the vortex-centre equation of motion (4.5). Hence, with $U^{\prime}_x=0$ for the gravity wave considered here, the forcing exerted by the potential flow on the vortex becomes

(5.4) \begin{align} \eta _1(t) &= \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \overline {w}_z U^{\prime}_x(y,z,t) = 0 , \end{align}

(5.5) \begin{align} \eta _2(t) &= \frac {1}{\varGamma }\int _M{\text{d}}^{2}x\, \overline {w}_z U_y^{\prime}(y,z,t) . \end{align}

Herein, for simplicity, we have used that the $x$ – $y$ and $x_1$ – $x_2$ coordinate systems are approximately aligned (figure 4). Considering further that for the given integration domain $M$ , the vertical coordinate varies in the small range $\Delta y \approx \pm {3}\,\textrm {cm} \approx \pm 5{r_{{c}}}$ , the wave amplitude is taken as constant and equal to $A_y(y=0)$ over the domain to a good approximation. This simplifies (5.5) to

(5.6) \begin{equation} \eta _2(t) \approx A_y(y=0)\cos \varOmega t \end{equation}

for a fixed measurement plane at $z = z_M$ and omitting an arbitrary phase. The corresponding vortex deflection amplitude induced by the free-surface gravity wave is then estimated as $A_y(y=0)/\varOmega$ , and its standard deviation as

(5.7) \begin{equation} \sigma _w = \frac {A_y(y=0)}{\sqrt {2}\varOmega }\approx {0.07}\,\textrm {cm} \approx 0.12 {r_{{c}}}, \end{equation}

which is a significant fraction of the total meandering amplitude measured in the experiment: $\sigma \approx {0.11}\,\textrm {cm} = 0.20 {r_{{c}}}$ . In § 2.3, the amplitude of turbulence-induced meandering was estimated as $\sigma _t\approx 0.14r_c.$ The two estimates for $\sigma _w$ and $\sigma _t$ are of the same order in our configuration, and they add up quadratically quite well to the observed total value.

5.2. Correlation function

In § 4.3, we have shown that empirical power spectra are not ideal to infer significant periodicities in noisy data, since the associated peaks not necessarily separate clearly from the background noise. Rather, the combined periodic–coloured-noise behaviour shows up much better in the autocorrelation functions. Quite generally, correlation functions are expected to be an efficient means to detect ‘hidden’ periodicities in correlated-noise signals if the time scales are sufficiently separated and the energy contained in the periodic component is at least of the same order as the noise. In this case, noise-related correlations are limited to a short time range, while eventually the correlation due to periodicity dominates. The autocorrelation function associated with (5.1) can be computed analytically,

(5.8) \begin{equation} \rho _{\mu \mu }(\tau )=\overline {a_\mu (t+\tau )a_\mu (t)} = \frac {A_{{t}}}{2\lambda _\mu }e^{-\lambda _\mu |\tau |} + \frac {A_{{w}} \delta _{\mu 2}}{\lambda _\mu ^2 + \varOmega ^2}\cos \varOmega \tau \quad (\mu = 1,2\,\text{not summed}) . \end{equation}

To the best of our knowledge, this autocorrelation function of a harmonically forced Langevin equation has not been given before (details are given in Appendix A).

Setting $\tau =0$ in the correlation function (5.8) yields the equilibrium variance

(5.9) \begin{equation} \overline {a_\mu ^2} = \frac {A_{{t}}}{2\lambda _\mu } + \frac {A_{{w}}}{\lambda _\mu ^2 + \varOmega ^2} = 1 , \end{equation}

due to the normalisation, which can be used to estimate $A_{{t}}$ . We recall from § 2 that the wave period is $T_{{w}} = {9}\,\textrm {s}$ and hence $\varOmega = 2\pi /T_{{w}} = {0.70}\,\textrm {rad s}^{-1}$ . The wave amplitude was estimated to be $A_{{w}} = {1}\,\textrm {s}^{-2}$ . Taking $\hat {\lambda }_\mu \approx {2}\,\textrm {s}^{-1}$ from an exponential fit of the sample correlation yields $A_{{t}}$ $\approx {3}\,\textrm {s}^{-1}$ .

Figure 15.

Comparison of the model ( $\rho _{\mu \mu }$ ) and sample ( $\hat {\rho }_{\mu \mu }$ ) autocorrelation functions for the $(a)$ first and $(b)$ second principal-component time series. The solid grey line shows the unbiased sample correlation (5.10) and light-grey shading indicates the 95 % confidence interval computed from $N_{{b}} = 10^3$ samples generated by moving-block bootstrapping. The model autocorrelation functions (5.8) are displayed as solid black lines.

Figure 15. Long description

Two line graphs compare model and sample autocorrelation functions for the first and second principal-component time series. The left graph shows the first principal-component time series with the model autocorrelation function as a solid black line and the sample autocorrelation function as a solid grey line. The right graph shows the second principal-component time series with the model autocorrelation function as a solid black line and the sample autocorrelation function as a solid grey line. The grey shading indicates the 95 percent confidence interval computed from samples generated by moving-block bootstrapping. The x-axis represents the normalized time, and the y-axis represents the autocorrelation values. The model autocorrelation functions are displayed as solid black lines.

We define the sample correlation functions as

(5.10) \begin{equation} \hat {\rho }_{\mu \gamma }(\tau ) = \frac {1}{N-n_\tau }\sum _{n=1}^{N-n_\tau } \frac {a_\mu (n)a_\gamma (n+n_\tau )}{\hat {\sigma }_\mu \hat {\sigma }_\gamma } ,\quad n_\tau \ge 0 , \quad (\mu , \gamma = 1,2\,\text{not summed}), \end{equation}

where $t = (n-1)\Delta t$ , $\tau =n_\tau \Delta t$ and $n = 1, 2, \ldots , N$ , $N = T/\Delta t$ . For a stationary zero-mean process, (5.10) is an unbiased estimator. A typical alternative with less variance but higher bias would be $(N-n_\tau )N^{-1}\hat {\rho }_{\mu \gamma }(\tau )$ . We found that both estimators yielded essentially the same results over the lag range $\tau$ examined (not shown).

Figure 15 shows a comparison of the sample correlations (5.10) with the model correlation functions (5.8) assuming that periodic oscillations due to the free-surface wave are induced only in the vertical coordinate. Our model correctly predicts the combined periodic–coloured-noise behaviour. Grey shading in figure 15 displays the 95 % confidence interval obtained from moving-block bootstrapping with $N_{{b}} = 10^3$ . We notice that the bootstrap sample reproduces the correct correlation structure for time lags $\tau \hat {\lambda }_\mu \lesssim 20$ . In particular, this shows that fluctuations in $\rho _{11}$ for $\tau \hat {\lambda }_\mu \gtrsim 5$ are not statistically reproducible. Our model consistently stays inside the shaded area, indicating that it cannot be rejected (and hence would be accepted) at the 5 % level of significance.