# Wingtip vortex dynamics at low Reynolds numbers under the influence of turbulence

Maël Benlarbi, Marie Couliou, Geoffrey Tanguy, Vincent Brion, Jean-Philippe Laval

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Maël Benlarbi, Marie Couliou, Geoffrey Tanguy, Vincent Brion, Jean-Philippe Laval. Wingtip vortex dynamics at low Reynolds numbers under the influence of turbulence. Experiments in Fluids, 2026, 67 (3), pp.34. 10.1007/s00348-026-04183-5. hal-05517619

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# Wingtip vortex dynamics at low Reynolds numbers under the influence of turbulence

Maël Benlarbi¹*, Marie Couliou², Geoffrey Tanguy¹, Vincent Brion², Jean-Philippe Laval¹

¹ Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale Lille Institut, UMR 9014 - LMFL - Laboratoire de Mécanique des Fluides de Lille - Kampé de Fériet, F-59000 Lille, France.

²DAAA, ONERA, Institut Polytechnique de Paris, 92190, Meudon, France.

*Corresponding author(s). E-mail(s): mael.benlarbi.loyau@gmail.com;

## Abstract

The influence of turbulence on the aerodynamics of a NACA0012 wing and the behavior of the associated wingtip vortex is investigated at a chord-based Reynolds number of 5000. The study is conducted through water tunnel experiments using a half-wing setup with an aspect ratio of 6 and an angle of attack of 10 degrees. At this low Reynolds number, relevant for aerodynamics at low-speed, the flow over a stationary wing exhibits laminar separation from the upper surface. Longitudinal Particle Image Velocimetry (PIV) measurements show a laminar flow separation over the wing upper surface and its reduction with increased incoming turbulence. Four turbulence conditions are tested: three different passive grids and one baseline (no-grid) configuration, yielding turbulence intensities ranging from 1.4% to 8.2%. Transverse PIV measurements are conducted over three cross-sections up to 24 chord lengths downstream of the wing trailing edge, to characterize the vortex dynamics. At the wingtip, the flow rolls up into a single coherent tip vortex, which exhibits significant unsteadiness in the form of a dominant displacement mode, as Proper Orthogonal Decomposition (POD) shows. In the no-grid case, the vortex exhibits significantly lower-frequency and higher-amplitude motion than in the grid-generated turbulence cases. This indicates that, at low Reynolds numbers, vortex meandering can arise either from unsteady perturbations linked to large-scale flow separation on the wing under low turbulence intensity, or from the direct receptivity of the vortex to freestream turbulence at higher turbulence levels. These results highlight the coexistence and relative importance of two distinct sources of disturbance - wing-separated flow and inflow turbulence - in governing wingtip vortex dynamics.

## 1 Introduction

Aerodynamics at low Reynolds numbers is of particular interest from a fundamental perspective, since viscous effects have a strong influence on flow separation, transition, and vortex formation. The transition between low and high Reynolds number aerodynamics typically occurs around a

chord based Reynolds number of $Re_c \approx 10^5$. However, this threshold can also depend on the wing geometry and surface roughness (Lissaman, 1983). Aerodynamics in the low Reynolds number regime differs significantly from that at higher Reynolds numbers, primarily because flow separation can occur on the wing surface (Winslow et al, 2018),
driven by the stronger viscous effects. This often
leads to the formation of a laminar separation
bubble, which can significantly reduce the flight
efficiency of the vehicle.

In this study, we focus on the aerodynamics of
fixed wings at a chord-based Reynolds number of
5 000. Previous experimental (Mueller et al, 1983;
Kim et al, 2011) and numerical studies (Kojima
et al, 2013; Chiu et al, 2023; Taira and Colonius,
2009; Winslow et al, 2018) have been conducted
to improve the understanding of the behavior of
flows at low Reynolds numbers. The tip vortex
significantly influences the wing performance com-
pared with an idealized two-dimensional airfoil
case, as it reduces the effective angle of attack
and helps prevent flow separation. In turn, as the
tip vortex rolls-up the flow past the wing, any
detail of the wing aerodynamics is expected to
modify the tip vortex dynamics. Subtle effects
can also arise. For instance the Reynolds number
can influence the tip vortex formation mechanism.
At higher Reynolds numbers (on the order of
$10^5$), multiple vortices may form at the wingtip,
whereas typically only one single vortex appears
at $Re \approx 10^4$ (Smith and Ventikos, 2021). Given
the peculiar aerodynamics at low Reynolds num-
bers, it is worth exploring how these conditions
affect the dynamics of the tip vortex, and whether
it changes from that occurring at higher Reynolds
numbers. In particular, experiments conducted at
higher Reynolds number show that tip vortices are
prone to "meandering", a phenomenon described
by an erratic lateral motion of the vortex core.
An empirical model has been developed previ-
ously (Van Jaarsveld et al, 2011), which suggests
that the amplitude of vortex meandering depends
on flow turbulence and vortex strength. However,
there is a lack of literature reporting on tip vortex
dynamics at lower Reynolds number flows and it
is worthwhile to know whether the high-Reynolds
number law can be used to extrapolate to the
lower Reynolds regime, given the specific wing
aerodynamics in this case.

A possible difference comes from the fact
that flow separation at low Reynolds numbers
inherently leads to unsteady flow patterns. It
is questioned then whether such unsteadiness
could influence the vortex dynamics in terms of
amplitude and structure. Huang and Lin (1995)
classified various vortex shedding regimes in the

wake of a NACA0012 airfoil, based on chord-
based Reynolds number and angle of attack α.
Using hot-wire anemometry, they identified spe-
cific Strouhal number (Stc) values - based on
chord length and freestream velocity - in the range
[0.5, 3]. These frequencies fall within the same
range as those associated with vortex meandering
(Bailey et al, 2018), suggesting that the tip vortex
may be subject to forcing.

Aerodynamics at low Reynolds number being
related to low altitude flying objects, it also faces
the influence of turbulent environments peculiar
to the atmospheric boundary layer flow. Turbu-
lence intensity $T_u = u_{rms}/\bar{u}$ (with the mean
velocity $\bar{u}$ and standard deviation $u_{rms}$ of the
flow) can reach 10% to 20% with length scales
up to ten times the chord length of the aircraft
(Watkins et al, 2006). Since the flow separa-
tion can be impacted by upstream disturbances,
freestream turbulence can significantly alter the
aerodynamic characteristics of the wing. Wang
et al (2014) demonstrated this by varying the tur-
bulence intensity upstream of a NACA0012 airfoil
over a Reynolds number range from $5.3 \times 10^3$
to $2.0 \times 10^4$. Their study shows that increasing
turbulence delays boundary layer separation.

The objective of this study is to investigate
the effect of flow turbulence on the tip vor-
tex meandering motion at low Reynolds regimes.
An experimental study was conducted in a low-
speed water tunnel using a finite wing mounted
on the side wall of the test section to generate
a tip vortex. The paper is organized as follows.
Section 2 describes the experimental setup and
measurement techniques. Section 3 examines the
flow dynamics over the wing and the influence
of freestream turbulence. Section 4 is devoted to
the analysis of the tip vortex. The conclusions are
presented in the final section.

2 Experimental setup

## 2.1 Facility and test model

The experimental work is conducted in a small
scale water tunnel located at the ONERA Meudon
research center. The facility operates under
gravity-driven flow, with an upper reservoir posi-
tioned 2.88 meters above the horizontal test
section centerline. Water flows from this reser-
voir through a diffuser, a settling chamber and a
<figure><img src="image_2.png" /><figcaption>**Fig. 1:** Schematic of the experimental setup (a) Top view. Red crosses represent the LDV measurement locations, and dotted lines indicate the PIV measurement planes (longitudinal PIV in magenta and transverse PIV in blue). (b) Schematic of the grid geometry.</figcaption></figure>

converging nozzle towards the test section. The rectangular test section measures 1.6 m in length, 0.3 m in width, and 0.15 m in height. It is mostly enclosed with glass walls, providing wide optical access for flow visualization and optic based measurements. Downstream of the test section, the water is collected in a lower reservoir and recirculated to the upper tank via a pump, enabling continuous operation of the facility. The maximum flow velocity achievable in the tunnel is 0.15 m/s. The tunnel velocity is controlled with a valve located just upstream of the lower reservoir. The volumetric flow rate $q$ is obtained with an electromagnetic flowmeter (Krohne Optiflux 2000) installed in the return line of the circuit. Water temperature is monitored with two PT100 resistance temperature sensors, offering an accuracy of ±0.5 °C.

The test model consists of a half-wing with a rectangular planform, sharp tips and a NACA0012 airfoil profile, attached to the side wall of the test section. The model is 3D printed and made of ABS plastic, with a chord length $c = 5$ cm and a semi-span $b = 15$ cm. It is mounted adjacent to a vertical symmetry wall, which effectively mirrors the geometry and flow, creating the equivalent of a full-span wing of span $2b$. The resulting aspect ratio is $AR = 2b/c = 6$. The angle of attack $\alpha$ is set to 10°. The wing leading edge is positioned three chords downstream from the entrance of the test section. A schematic of the setup and the Cartesian coordinate system $[x, y, z]$ is shown in Figure 1.(a). The $x$-axis is aligned with the freestream direction, while $y$ and $z$ represent the horizontal and vertical directions, respectively. The origin $0$ is located at the tip of the wing trailing edge when $\alpha = 0$. The velocity field is denoted

by $\mathbf{u} = [u, v, w]$. Time averaged of a given variable $a$ is denoted by $\bar{a}$ and the fluctuation by $a'$.

The chord $c$ and inflow velocity $U_0$ are used as characteristic dimensions of the flow, where $U_0$ is the axial flow velocity measured at the location $(x, y, z) = (0, 0, 0)$ in the absence of the wing model in the test section. This yields the Reynolds $Re = U_0c/\nu$, which is set to 5000 ($\nu = \mu/\rho$ is the kinematic viscosity of water, $\mu$ dynamics viscosity and $\rho$ water specific mass). Given the strong sensitivity of $\mu$ upon temperature, the flow rate $q$ is adjusted to account for temperature variations during the operation of the facility (water heats up, mainly through the pump). The test section is straight, and the natural growth of the boundary layers along the walls leads to a slight increase in flow velocity. Laser Doppler Velocimetry measurements in the empty test section indicated an increase of approximately 5% in centerline velocity from the beginning of the test section up to the location $x/c = 24$ (Figure 2). This increase in velocity corresponds to a favorable pressure gradient, with estimated values of $dC_p/dx \times 10^3 = -1, -1.5, -1.2$, and $-1.5$ in units of $m^{-1}$ for the reference case and for grids A, B, and C, respectively, where $C_p$ is defined using the local static pressure relative to the free-stream reference pressure at $x = 0$. Overall, the estimated streamwise pressure gradients in the test section are in the range [-4.4 : -7.9] Pa.m$^{-1}$, which is one to two orders of magnitude smaller than the radial pressure gradients induced by the vortex ([50 : 500] Pa.m$^{-1}$). This indicates that the axial pressure variation is not expected to significantly affect the vortex dynamics, although this remains an inherent limitation of the present experimental configuration.
<figure><img src="image_3.png" /><figcaption>Fig. 2: Streamwise evolution of the center-line mean velocity $\bar{u}$ in the empty test section, obtained by LDV measurements.</figcaption></figure>

## 2.2 Measurements methods

### 2.2.1 Laser Doppler Velocimetry measurements

Laser Doppler Velocimetry (LDV) is used to characterize the flow in the water tunnel without the wing model installed. The longitudinal velocity component is measured at discrete locations along the tunnel centerline using a one-component LDV system (see Figure 1). The setup consists of a Dantec LDV system. The velocity signal is processed with a BSA processor operating at 40 MHz. Polyamide tracer particles with a mean diameter of $d_p = 5 \ \mu m$ and a density of $\rho_p = 1.03 \ g/cm^3$ are used to seed the water. At each measurement point, 20 000 samples are recorded to ensure statistical convergence. The Stokes number based on the particle characteristics is defined as

$$S = \frac{(\rho_p - \rho) d_p^2 U_0}{18 \mu L},$$

where $\rho_p$ is the particle densities and $L$ is the characteristic length scale associated with the smallest spatial variations of the flow (estimated here from the Kolmogorov length scale $\eta$ discussed in section 2.3). Under the present experimental conditions, the Stokes number is of the order of $10^{-3}$, and therefore much smaller than unity, which ensures that the seeding particles have negligible inertia and faithfully follow the fluid motion (Greenwell, 2002).

### 2.2.2 Longitudinal PIV measurements

A planar Particle Image Velocimetry (PIV) system is implemented to perform flow velocity measurements in a longitudinal plane ($0xz$) both with and without the wing model. The objective is to characterize the freestream conditions induced by the different turbulence-generating grids and to describe the flow past the wing. The measurement domain corresponds to the sectional plane $y = 0$ i.e located half-way between the wingtip and the lateral wall, as illustrated in Figure 1. Measurements are conducted with the same seeding particles as used for the LDV. Illumination is provided by a Litron dual-cavity Nd:YAG laser of 22 mJ pulse. The laser sheet is approximately 1 mm thick. Reflections from the wing upper surface obliterated the PIV measurements at distance shorter than approximately 0.5 mm. Particle images are captured using a Phantom V710 high-speed camera with a sensor resolution of $1280 \times 800 \ px^2$. The spatial resolution is $8.5 \ px.mm^{-1}$. Measurements are recorded with the laser pulsing at $f_{aq} = 200 \ Hz$, synchronized with the camera operating in single-frame mode. For each configuration, a total of 11 000 images are recorded, corresponding to an acquisition time of $t_{acq} = 55 \ s$. Velocity fields are reconstructed using the ONERA - developed PIV software FOLKI (Champagnat et al, 2011), employing interrogation windows of $15 \times 15 \ px^2$ (corresponding to $1.8 \times 1.8 \ mm^2$). The FOLKI algorithm operates with maximal overlap, without a fixed percentage. A vector is retained only if its interrogation window overlaps the valid image region by at least 20%. The displacement field is refined using a multi-resolution Gauss-Newton optimization, typically converging within three to five iterations (seven were used here to ensure full convergence).

### 2.2.3 Transverse PIV measurements

Stereoscopic PIV (SPIV) measurements are conducted in transverse planes ($0yz$) at several streamwise locations : $x/c = 3, 12$ and $24$. This serves to monitor the tip vortex in the near wake. For this setup, two Phantom V710 high-speed cameras are positioned on opposite sides of the test section. The spatial resolution is $15 \ px.mm^{-1}$. The laser is operated at repetition rates between 160 and 200 Hz and synchronized with the cameras in single-frame mode. A total of ten sets of
11000 images are recorded per configuration to ensure convergence of first-order statistics. The time interval between image pairs was adapted at each measurement location to account for the decreasing azimuthal velocity of the vortex with downstream distance. Consequently, the total acquisition time ranged from 55 s to 68 s. The stereoscopic calibration is performed using a LaVision 3D calibration target and the mapping between the images and the measurement plane was made with an in-house software called Afix. The velocity field are reconstructed in the same way as for longitudinal PIV measurements.

## 2.3 Grid generated turbulence

The turbulent characteristics of the incoming flow upon the test model are modified using a set of three turbulent grids labeled A, B and C, the no-grid case serving as reference. These grids are installed at the inlet of the test section, three chords upstream of the wing leading edge. They feature square openings of size M separated by rectangular rods of thickness e and depth d as sketched in Figure 1(b). The grid porosity β is given by $β = M^2/(M + e)^2$ and is set to 60% for all the grids. The grid parameters are chosen according to the empirical decay law of the turbulent intensity $T_u$, proposed by Roach (1987) and validated by Mohamed and Larue (1990), that writes

$$T_u = F \left( \frac{x_{grid}}{M} \right)^{-n} \quad (1)$$

where $x_{grid}$ is the distance downstream from the grid, n is the decay exponent, and F is an empirical constant that depends on e and M. The grids are 3D printed from ABS plastic.

The turbulent flow generated by these grids is evaluated using LDV and longitudinal PIV measurements. The LDV is used for the statistics, as it is not homogeneously sampled, while fast PIV measurements are used to evaluate the velocity spectra of turbulence.

Table 1 reports the grid main characteristics and $T_u$ at $x/c = -1$, showing that turbulent rates of 1.4, 4.5, 4.9, and 8.2% are obtained for the reference case and A, B and C grids, respectively. The uncertainty on $T_u$ reported in the table is estimated based on the convergence uncertainty of the velocity data obtained from LDV measurements. The uncertainty on $T_u$ reported in the

table is estimated based on the convergence uncertainty of the velocity data obtained from LDV measurements. Figure 3(a) shows the evolution of the turbulence intensity $T_u$ as a function of the downstream distance $x/c$. In the absence of a grid, the turbulence intensity exhibits no significant streamwise evolution and remains approximately constant at $T_u \approx 1.4\%$ over the entire test section. For the cases with a grid, $T_u$ decreases following a power-law decay, in accordance with the correlation established by Roach (1987). Figure 3(b) shows the evolution of $T_u$ in a log-log scale as a function of the non dimensional downstream distance $x_{grid}/M$. For grid C, a change in slope is observed around $x_{grid}/M \approx 20$. This departure from the power-law decay of Roach (1987) is consistent with near-field development effects, as the corresponding downstream locations remain within the turbulence development region of the grid, according to the criteria proposed by Gomes-Fernandes et al (2012). For each inflow case, a fit has been performed using relation (1). For grid C, the fit was restricted to the region where $x_{grid}/M > 20$ so as to exclude the developing near-field region. The fitted curves are shown in solid lines in Figure 3, yielding values of F of 0.17, 0.14 and 0.17 and of n of 0.37, 0.48 and 0.48 for grid A, B and C respectively. The quality of the fits is good, with coefficients of determination $R^2$ equal to 0.963, 0.997, and 0.999 for these grids. It worth noting that, for all these grids, the values of n remain significantly lower than those predicted by the empirical relation of Roach (1987) (n = 5/7) or the measurements reported by Mohamed and Larue (1990), which typically fall within the range [1 : 1.45]. One possible explanation for this discrepancy lies in the relatively low rod-based Reynolds numbers used in the present experiments compared to those investigated in earlier studies. Roach (1987) found that the decay law holds for rod-based Reynolds numbers defined as $Re_e = U_0e/\nu$ in the range [10² : 10⁴]. In our experiments, grid C exhibits a Reynolds number $Re_e = 351$, which lies within this valid range. However, grid B has $Re_e = 220$, close to the lower threshold, and grid A has $Re_e = 88$, which falls below it. This likely accounts for the observed deviation from the canonical decay behavior, particularly for the smaller grids.
<table><caption>**Table 1:** Grid geometry characteristics with *M* the mesh size (distance between two grid rods), *d* the rod thickness, and *e/d* the rod aspect ratio. The Reynolds number based on the rod thickness is defined as *Re*<sub>e</sub> = *U*<sub>0</sub>*e*/*ν*, where *U*<sub>0</sub> is the mean inflow velocity and *ν* the kinematic viscosity. The turbulence characteristics are measured at 4*x*/c downstream of the grid: *T*<sub>u</sub> is the turbulence intensity, *L*<sub>x</sub> the integral length scale, and *f*<sub>η</sub> the frequency associated with the Kolmogorov dissipation scale.</caption><thead><tr><th>CASE</th><th><i>M</i> (mm)</th><th><i>d</i> (mm)</th><th><i>e</i>/<i>d</i></th><th><i>Re</i><sub>e</sub></th><th><i>T</i><sub>u</sub> (%)</th><th><i>L</i><sub>x</sub> (mm)</th><th><i>f</i><sub>η</sub> (Hz)</th></tr></thead><tbody><tr><td>no-grid</td><td>-</td><td>-</td><td>-</td><td>-</td><td>1.4±0.1</td><td>0.5±1</td><td>2±2</td></tr><tr><td>A</td><td>3.1</td><td>5.0</td><td>0.2</td><td>88</td><td>4.5±0.2</td><td>3.5±0.7</td><td>6.9±0.5</td></tr><tr><td>B</td><td>7.4</td><td>5.0</td><td>0.5</td><td>220</td><td>4.9±0.2</td><td>4.5±1.4</td><td>5.6±0.8</td></tr><tr><td>C</td><td>13.0</td><td>5.0</td><td>0.8</td><td>351</td><td>8.2±0.4</td><td>5.5±1.4</td><td>9.4±1.3</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 3:** Evolution of the longitudinal turbulence intensity *T*<sub>u</sub> with downstream distance, obtained from LDV measurements (a) plotted in linear scale as a function of *x*/c (b) plotted in log-log scale as a function of *x*<sub>grid</sub>/M. The markers represent the experimental data, while the solid lines correspond to the fitted curves. The dashed magenta line indicates the -5/7 decay slope reported by Roach (1987). The empty circular symbols indicate the locations of interest for each grid, corresponding to *x*/c = 3, 12, and 24.</figcaption></figure>

Table 1 also reports on the integral length scale $L_x$, which is obtained from $L_x = U_0 \int_0^\infty R_u(\tau) d\tau$ (where $R_u(\tau)$ is the temporal autocorrelation). The uncertainty on the value of $L_x$ is estimated by propagating the measurement errors of the instantaneous PIV velocities and the uncertainty on the zero-crossing time of the autocorrelation function. The integral scale $L_x$ is a fraction of the mesh size *M*, with values in agreement with those established by Roach (1987), despite the difference in rod shape (rectangular versus square). It is useful to evaluate the PIV temporal resolution with regard to the Kolmogorov dissipation scale. The latter is estimated through $\eta = (\nu^3/\epsilon)^{1/4}$ with the time scale given by $\tau_\eta = (\nu/\epsilon)^{1/2}$, where $\epsilon$ is the turbulent dissipation rate, which can be estimated from $\epsilon \sim u_{rms}^3/L_x$ using the LDV-derived

$u_{rms}$ and PIV-derived $L_x$ given in the Table 1. The uncertainty on $\epsilon$ and on the derived quantities, $\eta$, $\tau_\eta$ is evaluated from the propagation of the uncertainties on $L_x$ and $u_{rms}$. The values of $\eta$ for the cases with a grid are in the range [0.3 : 0.5] ± 0.2 mm, while $\tau_\eta$ varies between 0.09 ± 0.01 - 0.19 ± 0.02 s. Consequently, the PIV system with a sampling time of $\tau_{aq} = 0.005$ s is able to capture the temporal evolution of the turbulent features of the flow. However, due to the interrogation window of size 1.8 mm, PIV introduces spatial filtering on the sub-millimeter turbulent structures.

The power spectral density of the time-series $u'(t)$ is calculated using the 2D PIV velocity fields in the empty tunnel, at the location where the wing leading edge would have been if it were
present. The spectra are displayed as a function of frequency and chord based Strouhal number $St_c = f \times c/U_0$ in Figure 4 showing that all the grids feature a similar turbulent pattern with a clear inertial range identified by the -5/3 slope. The spectra appear as staggered, displaying increased amplitude going from no-grid to grids A, B and C. The no-grid case features fluctuation energy essentially in the low frequency range $St_c < 1$ while the grid cases feature elevated fluctuation levels up to $St_c \simeq 1$ before the turbulent transfer to lower scales begins to operate. Peaks appear for all grid configurations in the high frequency range of the spectra, i.e. 8Hz and 83Hz ($St_c$ = 4 and 42, respectively), the origin of which remains unclear. As these frequencies are significantly higher than those relevant to the phenomena investigated here, they are not discussed further in the following analysis.

<figure><img src="image_5.png" /><figcaption>**Fig. 4:** Spectra of the longitudinal velocity fluctuations at $x/c = 0$ given in terms of frequency $f$ (top axis) and Strouhal number $St_c = fc/U_0$ (bottom axis).</figcaption></figure>

## 3 Flow around the wing

As a first illustration of the flow around the wing, the normalized mean axial velocity field $\bar{u}/U_0$ is shown in Figure 5 based on the longitudinal PIV measurements. The flow is mainly separated over the wing at the chosen angle of attack. Although not shown here, a similar flow separation is also

observed at lower angles of attack. Flow separation at these angles of attack is the mark of the low Reynolds number dynamics. The size of the separation region can be identified using the iso-line of zero axial velocity. When the turbulence intensity is low (no-grid case), the flow separates at approximately 25% of the chord, leading to a large region of reverse flow extending down to the trailing edge. In this case, the maximum height of the separation region is about 40% of the chord.

The extent of the separated region can be roughly delineated by the iso-line of zero axial velocity. For the lowest turbulence level (no-grid case, $T_u$ = 1.4%), a large recirculation zone develops over a significant portion of the chord, extending downstream toward the trailing edge. Increasing the inflow turbulence intensity through the use of grids markedly affects this structure. For grids A and B, corresponding to $T_u$ = 4.5% and 4.9%, respectively, the separated region becomes noticeably thinner and closer to the wall, suggesting an earlier transition to a more attached flow regime. In the case of grid C ($T_u$ = 8.2%), the separated region appears even more confined and remains limited to a small near-wall bubble. However, due to the overexposed region in the PIV images near the wall, the exact location of the separation and possible reattachment points cannot be determined with certainty. It should also be noted that the present analysis relies on planar PIV measurements acquired in a single two-dimensional plane. As a result, potential three-dimensional features of the separation and reattachment structure cannot be fully resolved. The results presented here should therefore be interpreted in the context of these planar measurements.

Overall the sensitivity of the separated flow to upstream turbulence and the effect of turbulence to reduce flow separation are coherent with known results (see, e.g., Wang et al (2014)). Flow separation generates flow unsteadiness in the wake of the wing, promoted in the shear layers bounding the concerned area. This can be identified from the instantaneous flow fields shown in Figure 6 for the different inflow turbulent conditions, and using spanwise vorticity as field variable. At first, it can be seen that all cases feature a shear layer close to the wing leading edge that rapidly destabilizes and sheds vortices downstream. A second shear layer forms at the trailing edge, of opposite
<figure><img src="image_6.png" /><figcaption>**Fig. 5:** Iso-contours of the normalized mean axial velocity $\bar{u}/U_0$. The airfoil contour is drawn with a solid line. dotted lines correspond to $\bar{u} = 0$. The gray shaded area corresponds to zone of the laser lighted area hidden by the wing, where no velocity measurements are available.</figcaption></figure>

sign to that of the leading-edge shear layer. This second shear layer also deteriorates into vortices, as can be well seen in the no-grid case. Similar observations are made by Huang and Lin (1995). Despite the vortex shedding processes above and below the separated flow region, the downstream flow exhibits no clear coherent structures, suggesting a rapid onset of three-dimensionality in the wake.

This downstream flow may act as a source of unsteady perturbations influencing the tip vortex dynamics. Its frequency content is therefore of interest, and the time-dependent velocity signal downstream of the wing is analyzed. The signal is extracted within the wake at $x/c = 1$ and $z = 0$ (see × markers in Figure 7) and a Fourier analysis is performed (Figure 6). It is found that shedding occurs primarily at 2 Hz ($St_c \approx 1$) at the chosen position, as indicated by the peak in the spectra for each grid configuration, see Figure 7. This peak is rather large indicating that multiple frequencies are at play. For grids A, B, and C, the spectra exhibit a broad energy distribution over a wide range of frequencies, without any pronounced peak. In contrast, the no-grid case shows an additional peak at $St_c = 1.5$. Canonical vortex shedding with a well-defined frequency is not observed, in agreement with the previous report of a quite unstructured wake pattern. Instead, a broad-band signal prevails. The frequency around $St_c = 1$ is fairly close to the shedding frequency for a NACA0012 airfoil, determined experimentally by Huang and Lin (1995) at $St_c = 1.1$ in similar conditions (i.e. $Re = 11 \times 10^3$ and $\alpha = 10^\circ$). Gupta et al (2023) also report $St_c = 0.7$ from numerical analysis at $Re = 5 \times 10^3$ and $\alpha = 10^\circ$.

It should be noted that, in the present Reynolds-number range, Gupta et al (2023) reported a strong sensitivity of the shedding frequency to both Reynolds number and angle of attack. The latter sensitivity could explain that the present pattern is not clear when a finite wing is used, since the presence of the tip vortex induces a modification to the effective angle of attack that depends on the spanwise location. Hence vortex shedding frequency can change along the span of the wing.

# 4 Wingtip vortex dynamics

## 4.1 Time-averaged vortex

Figure 8 shows the dimensionless time-averaged axial vorticity field $\bar{\omega}_{xc}/U_0$ for the different grid configurations and highlights the presence of the tip vortex. At the station $x/c = 3$, the vorticity sheet issued from the trailing edge of the wing is also visible. Measurements for the no-grid and grid C configurations demonstrate a spatial fluctuation of the vorticity within the vortex sheet which is less prominent compared with configurations A and B. For the downstream positions located at $x/c = 12$ and 24, the vortex sheet cannot be identified suggesting that the roll up mechanism of the wingtip vortex is completed at those positions. The vortex has further reached a state close to full azimuthal symmetry.

The various test cases exhibit significant differences in vortex strength at the first measurement station $x/c = 3$, highlighting the influence of wing aerodynamics and inflow turbulence. Based on the time-averaged PIV measurements, the maximum axial vorticity levels located at the core of
<figure><img src="image_7.png" /><figcaption>**Fig. 6:** Time sequence of the instantaneous spanwise vorticity in normalized units $\omega_y c/U_0$ for the reference no-grid and grid A, B and C configurations (from top to bottom row). The × markers represent the positions of the probing point used in the analysis of the wake unsteadiness as shown by the power density spectra in Figure 7.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 7:** Power spectral density of the vertical velocity component taken in the wing wake at the location marked by × in Figure 6</figcaption></figure>

the vortex are strongly impacted by the use of a grid. When the flow is altered with a grid, the increase in $T_u$ from 1.4% to 8.2% tends to decrease the maximum value of dimensionless axial vorticity, $\bar{\omega}_x c/U_0$, reached at the vortex core. At $x/c = 3$, the maximum value of $\bar{\omega}_x c/U_0$ is 12.1 for $T_u = 4.5\%$, whereas it decreases to 6.4 for $T_u = 8.2\%$. The axial evolution is marked by a

strong decrease of the dimensionless axial vorticity levels as well as a thickening of the vortical area. As highlighted in the introduction, previous studies have shown that tip vortices in the near field are subjected to a meandering phenomenon and that an increase in the incoming turbulence intensity leads to an increase in the meandering amplitude (Bailey and Tavoularis, 2008; Pentelow, 2014; Ben Miloud et al, 2020). Part of the diffusion of the vortex properties that are observed are thus likely due to the averaging effect implied by the meandering vortex motion.

## 4.2 Conditional-averaged vortex characteristics

To visualize and study the changes caused by the turbulence upon the wingtip vortex, each instantaneous velocity and vorticity field is first recentered on the mean vortex position before being temporally averaged, providing a more meaningful quantity for analysis rather than simple time-averaged fields. This is thereafter referred to as conditional (time-) averaging. The fields obtained using this procedure are referred to as recentered
<figure><img src="image_9.png" /><figcaption>Fig. 8: Dimensionless time-averaged axial vorticity field $\overline{\omega}_x c/U_0$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9: Instantaneous vortex center positions at $x/c = 12$ - (a) shows the no-grid case, while (b), (c), and (d) correspond to grid A, grid B, and grid C, respectively. Covariance ellipses and their principal axes are plotted as solid lines.</figcaption></figure>

mean fields and are denoted by $\hat{a}$, $a$ being here a dummy variable.

The vortex center $\mathbf{x}_c(t)$, defined as the barycenter of the instantaneous vorticity field

$$ \mathbf{x}_c(t) = \frac{\iint_S \mathbf{x} \omega_x(\mathbf{x}, t) \mathbf{1}_{\omega_x > \omega_{th}} d\mathbf{x}}{\iint_S \omega_x(\mathbf{x}, t) \mathbf{1}_{\omega_x > \omega_{th}} d\mathbf{x}}, $$

is first identified in every PIV snapshots. In this center localization process, $S$ is taken as an area of maximum radius $0.2c$ centered about the location of maximum axial vorticity $\omega_x$ and a threshold, defined as $|\omega_x| > |\omega_{th}| = 0.1 \max(\overline{\omega}_x(y, z))$, is used to prevent taking into account the vorticity outside of the vortex. The sensitivity of the vortex-center detection to the choice of parameters has been assessed and validated. If the integration radius is too large, spurious vorticity unrelated to the vortex core may be included, while a radius that is too small compared to the core size makes the barycenter calculation equivalent to simply locating the vorticity maximum. Our sensitivity analysis shows a weak sensitivity to the integration radius within the range 0.15c-0.3c, with variations in the detected vortex position remaining below 0.01c. The robustness of the barycenter method was further validated
by comparison with the Γ₁ criterion (Graftieaux et al, 2001), showing that the vortex-center positions obtained with both methods are very close, with average differences below 0.01c across the entire dataset.

The scatter plot of the vortex core position at $x/c = 12$ is shown in Figure 9. The meandering motion is apparent, with an amplitude which is higher in the no-grid and grid C cases, minimum for grid A and intermediate for grid B. The meandering motion and amplitude is detailed later in section 4.3.

Figure 10 shows the recentered mean axial vorticity fields $\hat{\omega}_x$. The axial evolution remains similar to that observed in the non-recentered fields, with a strong decrease in the levels of non-dimensional vorticity and a gradual thickening of the vortex region. In contrast, compared to the raw mean fields (see Figure 8), the distribution appears more compact and the maximum vorticity at the vortex center is enhanced. For instance, in the case without a grid, an increase of 45% in the maximum vorticity of the recentered field is observed compared to the non-recentered field. As noted by Devenport et al (1996), the displacement of the vortex over time induces, in the raw mean fields, an artificial diffusion of vorticity. The recentering procedure corrects this effect, thus allowing a more accurate characterization of the tip vortex.

The azimuthal velocity is determined by using the velocity components *v* and *w* in the Cartesian coordinate system and then converted into polar coordinates: $u_{\theta} = w \cos(\theta) - v \sin(\theta)$. The conditional and azimuthal average of the vortex azimuthal velocity $\langle \hat{u}_{\theta} \rangle$ is evaluated as a function of the radial distance *r* about the barycenter of vorticity of the conditionally averaged vorticity field, given that $\langle \cdots \rangle$ represents azimuthal averaging. Figure 11 shows the evolution of the $\langle \hat{u}_{\theta} \rangle$ profile as a function of the radial distance for the different inflow turbulence levels.

The vortex radius $r^*$ is taken as the radius where the maximum of azimuthal velocity $\langle \hat{u}_{\theta} \rangle$ occurs. For the no-grid case, we observe in Figure 11.(a) that the vortex radius increases downstream from $r^*/c = 0.09$ at $x/c = 3$ to 0.16 at $x/c = 24$. At $x/c = 12$, the core radius is not impacted by the $T_u$ level and remains the same for grid A, B and C with a value around 0.13. This is consistent with the findings of Dghim et al

(2021), who reported that increasing freestream turbulence has little effect on the conditionally averaged vortex radius.

Figure 11.(b) reveals, taking example the flow at $x/c = 12$, that $\langle \hat{u}_{\theta} \rangle$ is maximum for the grid cases and minimum for the no-grid case. The ranking of Figure 11 between grid A, B and C follows that of the turbulence level. The markedly reduced azimuthal velocity observed in the no-grid case is linked to the large separation region over the wing, which diminishes lift and, consequently, weakens the tip vortex. A comparison of the vortex circulation, estimated as $\Gamma^* = 2\pi r^* \langle \hat{u}_{\theta}(r^*) \rangle$, between the no-grid case and cases A to C at $x/c = 12$ yields values of 7.7, 9.0, 8.2, and 6.6, respectively, in units of $10^{-4} \text{ m}^2 \text{ s}^{-1}$. Since the radius $r^*$ is unchanged between cases, this matches the ranking of maximum $\langle \hat{u}_{\theta} \rangle$. Eventually, it is interesting to note that the strongest vortex, in grid A, is also the one that moves the least.

## 4.3 Vortex meandering

The meandering effect is assessed by analyzing the tip vortex motion. The scatter plot of the vortex core position at $x/c = 12$ is shown in Figure 9. The meandering motion is apparent, with an amplitude which is higher in the no-grid and grid C cases, minimum for grid A and intermediate for grid B. Within Figure 9, the standard deviation of the lateral $\sigma_y$ and vertical $\sigma_z$ positions of the vortex core is represented with an ellipse centered about the mean vortex center. The circular shape of the curve suggests an isotropic displacement of the vortex core for each configuration.

The parametric analysis of the amplitude of the vortex motion is presented in Figure 12 with the total standard deviation $\sigma = (\sigma_y^2 + \sigma_z^2)^{1/2}$ given as a function of axial distance downstream of the wing. Unsurprisingly, and consistent with the behavior of the vortex-center position, the value of $\sigma$ shows only a weak sensitivity to the detection method, the vorticity threshold, or the chosen integration radius. For all configurations, the meandering increases with streamwise distance. Surprisingly, the meandering amplitude of the no-grid case ($T_u = 1.4\%$) is as large as in grid C case ($T_u = 8.2\%$). Furthermore for configurations A and B, the meandering amplitude almost linearly increases with distance from the wing
<figure><img src="image_11.png" /><figcaption>**Fig. 10:** Dimensionless conditional-averaged axial vorticity field $\hat{\omega}_x c/U_0$. For the position $x/c = 3$, the region corresponding to the roll-up of the vorticity sheet has been masked, as its interpretation is not meaningful in the context of an averaging procedure centered on the instantaneous vortices.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 11:** Examples of normalized conditional and azimuthal averaged profiles of azimuthal velocity $\langle \hat{u}_{\theta} \rangle / U_0$ (a) in the reference case (no-grid) and (b) at $x/c = 12$ for all grid configurations. The error bars represents the standard deviation of $\langle \hat{u}_{\theta} \rangle / U_0$ at the location of the maximal azimuthal velocity</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 12:** Intensity of vortex movement directions versus streamwise distance for all configurations. Each point is averaged over 10 realizations, with error bars showing the corresponding standard deviation.</figcaption></figure>

while, for the no-grid and grid C cases, the mean-dering amplitude increases first and then plateaus beyond $x/c = 12$.

Further scrutiny is achieved by looking at the frequencies of the vortex motion. Figure 13 displays the pre-multiplied spectra of the vortex motion obtained from the vortex trajectories

measured using the transverse PIV, in the horizontal direction, both as a function of the chord based Strouhal number $St_c$ and the vortex radius based normalization ($St_{r^*} = fr^*/U_0$) (Bailey and Tavoularis, 2008). It is worth noting that the cross-spectral method proposed by Bailey and
<figure><img src="image_14.png" /><figcaption>**Fig. 13:** Premultiplied frequency spectra of $y_c$ plotted as a function of the Strouhal number based on the chord and the free-stream velocity. The highlighted frequency band containing most of the fluctuation energy is shown in light gray for the reference case (no grid), and in light yellow for the cases with a grid.</figcaption></figure>

Tavoularis (2008) could alternatively be used to estimate the meandering frequency, as it offers enhanced temporal resolution through hot-wire anemometry measurements. In our case, however, the high-speed PIV operating between 160 and 200 Hz provides sufficient temporal resolution to capture the dynamics of the vortex motion, as evidenced by the spectra showing a maximum frequency around 10 Hz ($St_c = 5$) in Figure 13(c). Due to the broadband nature of the meandering spectra, no single well-defined peak can be attributed to the meandering motion. Nevertheless, the frequency bands containing most of the fluctuation energy can still be analyzed to characterize the dominant dynamics of the motion. Owing to the noise level in the signals, the shaded frequency bands are introduced solely to provide a qualitative and approximate indication of the dominant frequency ranges, rather than to establish precise limits. At the streamwise positions $x/c = 3$ and $x/c = 12$ (see Figure 13(a) and (b)) the spectra show that vortex motion occurs over a different frequency range in the no-grid and grid cases. While the tip vortex in the no-grid case exhibits a motion centered around $St_c \approx 10^{-1}$ (frequency band highlighted in light gray), the dominant frequency in the grid cases is higher, around $St_c \approx 0.6$ (frequency band highlighted in light yellow). At $x/c = 24$ (see Figure 13(c)) the no-grid energy distribution becomes broader and spans a frequency range comparable to that of the grid cases. The only low frequency unsteadiness that is present in the no-grid flow comes from the inflow (see Figure 4), suggesting receptivity of the

tip vortex to the inflow dominant frequency. A similar comment is made for the grid cases: the inflow reveals most of its fluctuation energy close to $St_c \approx 1$ which also suggests the receptivity of the tip vortex to the inflow perturbations.

The evolution in the axial direction of the vortex frequency is also notable. For the no-grid case, one observes a strong increase of the vortex amplitude at the low frequency range from the station $x/c = 3$ to $x/c = 12$ and a subsequent decrease, while a higher frequency motion emerges in the same range as that of the grid cases. For the cases with grid, one also observes an increasing amplitude, particularly for grid C, yet the frequency range of the motion is maintained. These trends show some similarity to the observations of (Bailey and Tavoularis, 2008) in the no-grid case, but differ significantly for the turbulent cases. Indeed Bailey and Tavoularis (2008) report that high-frequency (short-wavelength) components decay downstream, while low-frequency (long-wavelength) motions progressively amplify and eventually dominate the vortex meandering motion.

A matter of interrogation remains concerning the high amplitude of vortex motion in the no-grid case. Figures 9 and 12 indicate that the amplitude is almost as large in the no-grid case as in grid C case, which seems to contradict the significant difference in turbulence levels. A possible solution to this conflict is provided by Figure 14 which directly compares the turbulence level, amplitude of disturbance induced by the
separated flow about the wing and the amplitude of vortex motion. From a qualitative point of view, it appears that the only source of strong fluctuations in the no-grid case is the separated flow, which is maximum among the test cases. It stems from this observation that the most likely cause of large amplitude of vortex displacement in the no-grid case is the larger amplitude of separated flow in this case, which promotes larger fluctuations close to the vortex. As these wake fluctuations decrease with the use of turbulent grids, the inflow turbulence takes over and drives the vortex motion in the grid cases.

A frequency analysis performed on the PIV data along the wing, within the shear-layer regions, revealed no spectral peak matching the low meandering frequency of the no-grid case ($St_c \approx 0.1$). It is worth noting that properly resolving such low frequencies over the wing would require acquisition durations about ten times longer than those available in the present longitudinal PIV dataset, which would be necessary to verify whether a very low-frequency component at $St_c \approx 0.1$ is indeed present in the reference (no-grid) case. A possible explanation for this behavior, consistent with the lifting-line framework, can be proposed: in this approach, variations in the local circulation directly influence the position of the trailing vortex (Saffman, 1993). In the reference case (no-grid), the unsteady separation along the wing induces temporal fluctuations of the local lift distribution, which in turn generate instantaneous variations of circulation along the span. The tip-vortex trajectory then adjusts quasi-instantaneously to these changes, providing a plausible mechanism for the coupling between wing-flow unsteadiness and the amplitude of the vortex displacement.

In conclusion, regarding the vortex motion and its associated frequencies, the present observations indicate that the vortex responds to the frequency content of the inflow. We surmise that this also implies that the frequency content of the inflow lies within the receptivity band of the tip vortex; otherwise, such inflow frequencies would have been filtered out. A further conclusion is that flow separation on the wing likely plays a decisive role in setting the amplitude of the vortex motion in the no-grid case.

It is worth noting that, although the turbulence characteristics of grids A and B remain

<figure><img src="image_15.png" /><figcaption>Fig. 14: Perturbations in the freestream and in the wake measured at $x/c = 3$. The parameter $k$ denotes the mean turbulent kinetic energy measured in the wake, within the transverse PIV field of view, excluding the vortex region.</figcaption></figure>

relatively close - $T_u$ is $4.5 \pm 0.15\%$ for grid A and $4.9 \pm 0.2\%$ for grid B and the integral length scale $L_x$ is $3.5 \pm 0.7$ for grid A and $4.5 \pm 1.4$ for grid B - the resulting vortex behaviors differ markedly. The meandering amplitude is higher for grid B, increasing by about 60% and 24% at $x/c = 12$ and $x/c = 24$, respectively. Moreover, the spectral content differs at the upstream station ($x/c = 3$), where grid A exhibits a narrower energy band, indicating less broadband motion. No clear explanation could be provided so far for these differences. This however reflects that tip-vortex dynamics is highly sensitive to small variations in the incoming turbulence parameters. Beyond the slight differences in $T_u$ and $L_x$, another noticeable distinction between grids A and B lies in their turbulence spectra, as shown in Figure 4 the inflow spectrum of grid A exhibits a marked peak around $St_c \approx 0.7$, a feature absent from the spectrum of grid B, which could contribute to the differences observed in the meandering behavior between the two grids.

Based on available data in the literature Van Jaarsveld et al (2011) proposed a law in order to describe the vortex dynamic within turbulent flows and Bölle (2024) recently proposed a model to demonstrate this scaling. The latter formulates equations based on the premise that the transverse motion of the vortex arises from a dispersion process, resulting from a balance between forcing by external disturbances and a restoring force associated with the vortex strength $\Gamma^*$. The law
<figure><img src="image_16.png" /><figcaption>**Fig. 15:** Scaling of the vortex motion intensity for all configurations. The solid line represents the linear fit to the data made Van Jaarsveld et al. (2011), while the dashed lines indicate the standard error of the intercept. The uncertainties associated with $r^*$ and $\Gamma^*$ are estimated by calculating the typical standard deviation over the time series.</figcaption></figure>

proposed by Van Jaarsveld et al. (2011) is written as $\sigma/r^*$ as a function of $u_{rms}/u_{vortex}$ where $u_{vortex} = \sqrt{\Gamma^*/t}$ is an estimate of the azimuthal velocity within the vortex. Figure 15 compares the present results with the proposed scaling. Note that axial distance has been transformed into time assuming that the vortex disturbances are convected axially with the inflow velocity in the test section, such that $x = U_0t$. The three measurement stations thus provide three different evaluation times of the vortex displacement phenomenon. In the absence of grid, the amplitude of vortex motion slightly exceeds the prediction given by Van Jaarsveld’s scaling. In contrast, when turbulence is introduced, the amplitude of vortex motion falls slightly below the scaling prediction. These observations suggest an acceptable agreement with Van Jaarsveld’s scaling but leaves room for other effects than those incorporated in the defining law. In particular for the no-grid case, the likely influence of the separated flow could cause the slightly higher amplitude of vortex motion compared to the scaling.

## 4.4 Proper orthogonal decomposition analysis

In this work, Proper Orthogonal Decomposition (POD) is used as a standard diagnostic tool to more clearly identify and quantify the dominant

unsteady structures. The analysis (Taira et al, 2017) was performed on the instantaneous vorticity fields in order to investigate the impact of turbulence conditions on the tip vortex dynamical features. Prior to the decomposition, the temporal mean field was removed from each time series so that the POD captures only the fluctuating component of the flow. Importantly, this analysis was performed on the original (non-meandering corrected) fields, since recentering the vortices would artificially suppress the meandering motion. Snapshot-POD (Sirovich, 1987) was applied to the time-series of the streamwise vorticity field using the parallelized Python library developed by Brunton et al (2014). The fluctuating streamwise vorticity $\omega_x'$ (where $\omega_x = \bar{\omega}_x + \omega_x'$) was decomposed as follows: $\omega_x'(y, z, t) = \sum_n \lambda_n \Theta_n(t) \Psi_n(y, z)$, where $\Psi_n(y, z)$ are the eigenfunctions (spatial POD modes) that represent the spatial organization of fluctuations. The eigenvalues $\lambda_n$ denote the energy of each mode, and $\Theta_n(t)$ are the temporal coefficients, with $n$ representing the mode number. The modes are ranked in descending order based on their share of enstrophy of the total disturbance flow, as shown in Figure 17.

The dynamics of the wingtip vortex appears dominated by the 5 first modes in the general case, and mostly the 3 first modes in the grid A case. Other modes are of low enstrophy but with a level that does not decrease much when scrolling further in the hierarchy. In the following the analysis concentrates on the dominant first 5 modes. Figure 16 shows the vorticity fields associated with the first 4 modes, with the left and right sides featuring $x/c = 12$ and 24, respectively. Mode 5 is not shown as it is similar to mode 4.

A first observation is that the mode are similar regardless of the test case. The first two POD modes exhibit a clear dipole structure, indicating that they correspond to a global displacement of the vortex core rather than to an internal deformation of the vortex. This dipole organization is the typical signature of a helical displacement mode with azimuthal wavenumber $|m| = 1$, where $m$ denotes the azimuthal wavenumber of the perturbation applied to the vortex. In such a mode, the vortex core oscillates laterally around its mean position while being convected downstream, producing the characteristic meandering motion of the vortex. The two spatial modes obtained from
<figure><img src="image_17.png" /></figure>

$x/c = 12$

$x/c = 24$

<figure><img src="image_18.png" /><figcaption>Fig. 17: Hierarchy of POD mode contributions to the perturbation entrophy at $x/c = 12$. Error bars indicate the standard deviation computed over 10 realizations.</figcaption></figure>

the POD analysis are almost identical in shape but rotated by about 90° with respect to each other. This quadrature relationship means that, when combined, they describe a helical displacement of the vortex core. Such a pair of orthogonal displacement modes is a well-known feature of the $|m| = 1$ helical wave (Fabre et al, 2006; Lacaze et al, 2007), consistent with the meandering-like behavior reported in previous experimental and numerical studies (Edstrand et al, 2016; Dghim et al, 2021). Mode 3 (see Figure 16.(i-l)) shows a central $m = 0$ vorticity pattern indicating axisymmetric vortex deformation, encircled by

opposite sign vorticity of close to axisymmetric shape. A difference is observed in the no-grid case, where mode 3 (Figure 16.(i)) exhibits a non-axisymmetric $m = 1$ disturbance. Its structure suggests a contraction of the vortex along the specific direction connecting the centroids of the various perturbation vorticity patches. It should be noted that this mode sometimes appears with an opposite sign in the POD decomposition, but this has no physical implication. The sign inversion is a mathematical artefact of the decomposition and represents the same physical structure rather than a distinct mode. Mode 4 (see Figure 16.(m-p)) is a quadrupole ($m = 2$) indicating, following Fabre et al (2006), a so-called flattening wave of the vortex, as each patch of the quadrupole modifies the vortex into an elliptical shape. With an axial wavelength, this elliptic deformation induce a helical pattern. Similar mode structures have been observed experimentally, as reported by Del Pino et al (2011) and Dghim et al (2021) at higher Reynolds number.

Figure 17 shows the contribution of each mode to the total entrophy for all configurations at $x/c = 12$. The displacement modes (modes 1 and 2) represent 15.5, 9.3, 12.4, and 9.6% of the total entrophy for the reference case and
for grids A, B, and C, respectively. These modes account for approximately 70% of the total energy contained within the first five modes across all inflow conditions. This is coherent with the observation of a clear meandering pattern of the tip vortex when previously analyzing its motion. The axisymmetric mode $m = 0$ (mode 3), by contrast, contributes a smaller fraction of the total enstrophy, namely 4.5, 1.9, 2.9, and 3.2% for the reference case and for grids A, B, and C, respectively. Although dominant, the displacement modes represent a smaller fraction of the total enstrophy compared to the results of Couliou et al. (2026). The results of Couliou et al. (2026) are obtained at the same streamwise position ($x/c = 12$) but at lower turbulence intensities ($T_u$) and for a higher Reynolds number ($Re_c = 10^5$). In their work the displacement modes account for 39% and 50% of the enstrophy budget ($T_u$ = 0.3% and 1.83%, respectively).

An important finding of the present study, given from the evolution of the dominant POD modes between $x/c = 12$ and 24 in Figure 16, is the convergence of the perturbation pattern to the vortex. There is a slight dispersion of modes and especially their orientation at $x/c = 12$, yet it is noteworthy that the analyses at the downstream position $x/c = 24$ yields exact similar modes and orientation. This suggests that the vortex dynamics filters out the fine-scale details of both the near-wing flow and the inflow turbulence, and evolves toward an intrinsic dynamics that depends only weakly on the initial conditions, including those associated with vortex formation and the incoming turbulent environment.

## 5 Conclusion

This experimental study investigates the influence of inflow turbulence on wingtip vortex dynamics at a chord-based Reynolds number of 5000. Freestream turbulence is systematically varied using passive grids, with turbulence intensities ranging from 1.4% to 8.2%.

The results highlight two primary mechanisms by which the freestream flow affects the vortex dynamics: first, by directly interacting with the vortex core as it develops downstream; and second, by modifying the separated flow at the wing upper surface, which governs the initial vortex formation. The increase in freestream turbulence

intensity leads to a significant reduction in flow separation on the wing, thereby modifying the strength and unsteadiness of the wingtip vortex. Vortex meandering amplitude increases with both freestream turbulence and downstream distance. However, the relationship is modified by the low Reynolds separated flow effect at the wing upper surface. While freestream turbulence directly excites the meandering of the vortex, at the same time, the wake unsteadiness reduces as separation diminishes. The combination of these opposing effects likely explains why the vortex displacement amplitude does not scale linearly with turbulence intensity. The frequency response of the vortex motion closely reflects the inflow preferred frequency-implying that these inflow frequencies lie within the vortex receptivity bandwidth while the extent of wing flow separation strongly influences the amplitude of the vortex motion.

Proper Orthogonal Decomposition further reveals that the dominant coherent structures are associated with vortex displacement modes, while higher modes correspond to vortex core deformations. Far downstream, all configurations converge toward identical modal structures and alignment, suggesting a universal behavior of vortex dynamics regardless of initial vortex formation conditions.

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