# The effect of freestream turbulence on wing-tip vortex meandering and deformation

Marie Couliou, Srikar Yadala, Girish K. Jankee, Ingrid Neunaber, R. Jason Hearst

► To cite this version:

Marie Couliou, Srikar Yadala, Girish K. Jankee, Ingrid Neunaber, R. Jason Hearst. The effect of freestream turbulence on wing-tip vortex meandering and deformation. International Journal of Heat and Fluid Flow, 2025, 117, pp.110013. 10.1016/j.ijheatfluidflow.2025.110013. hal-05264745

HAL Id: hal-05264745

https://hal.science/hal-05264745v1

Submitted on 17 Sep 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# The effect of freestream turbulence on wing-tip vortex meandering and deformation ☆

M. Couliou<sup>a, b, ∗</sup>, S. Yadala<sup>b</sup>, G.K. Jankee<sup>b</sup>, I. Neunaber<sup>b, c</sup>, R.J. Hearst<sup>b</sup>

## ARTICLE INFO

Dataset link: https://doi.org/10.18710/HLVDJ
7

**Keywords:**

Freestream turbulence

Active grid

Wing-tip vortex

Vortex meandering

Vortex deformation

## ABSTRACT

The effect of freestream turbulence (FST) on a wing-tip vortex was investigated at a chord-based Reynolds number of $10^5$. Experiments were conducted at four FST intensities, generated by an active grid, 0.30%, 1.84%, 7.70%, and 13.25%, with integral scales ranging from 0.5 to 2.6 times the wing chord length. Stereoscopic particle image velocimetry measurements document the effects of FST on the meandering motion of the wing-tip vortex in the near field and middle field of a NACA0012 wing, i.e., planes $x/c = [2, 5, 12]$ downstream from the trailing edge of the wing. Conditional averaging based on reentring the coordinate system on the vortex center has been used to eliminate the influence of vortex motion. When the analysis was conditionally averaged on the vortex's core position, a reduction in vortex strength with increasing FST was observed, along with a slight increase in diffusion for the highest FST cases (FST levels > 6%). Snapshot proper orthogonal decomposition (POD) analysis on the coherent component of the streamwise vorticity field revealed two dominant modes associated with meandering displacement for all FST cases. POD analysis further reveals turbulence accelerates the onset of spatial modes associated with vortex deformation, which typically emerge in the far wake ($x/c \ge 36$) under non-turbulent conditions but appear in the near wake ($x/c = 5$) when FST is present. The vortex deformation modes contribute more significantly to the total entrophy in turbulent cases in the near wake than in the far wake under non-turbulent conditions.

## 1. Introduction

An airfoil, when operating as a lifting wing, generates lift by creating a vorticity sheet that rolls up near its tips into three-dimensional structures, commonly referred to as wing-tip vortices. These vortices play a key role in the aerodynamic performance of the wing and influence its wake where they persist. Numerous studies, e.g., Sarpkaya and Daly (1987), have focused on the evolution of wing-tip vortices, acknowledging their susceptibility to a multitude of factors including Reynolds number, angle of attack, ground proximity, buoyancy, wind shear and freestream turbulence. The impact of turbulence on wing-tip vortices has broad implications across diverse fields, such as aeronautics and wind energy.

In aeronautics, wake vortex dynamics have been extensively studied since the 1970s, particularly in the context of ensuring safe aircraft separation during take-off and landing (Hallock and Holzäpfel, 2018). Comprehensive overviews by Hünecke (2001), Gerz et al. (2002) and Breitsamter (2011) detail these efforts, examining the nature and

behavior of wake vortices, their decay, and the risks associated with wake encounters. This extensive research has shaped regulatory frameworks and operational guidelines, with major contributions from programs such as C-Wake, AWIATOR, and FARWake. These studies incorporated in-flight measurements in which ambient — or freestream — turbulence was both present and monitored (Schmitt et al., 2007), offering early insights into its influence on wing-tip vortex behavior within the broader context of traffic safety.

Ambient or freestream turbulence (FST) is indeed a crucial factor for any flying engine. Commercial aircraft encounter atmospheric boundary layer effects during landing and take-off phases, as well as high-altitude turbulence while cruising. Understanding the behavior of wing-tip vortices in the presence of freestream turbulence is therefore essential for optimizing aerodynamic efficiency and mitigating their downstream effects. At cruising altitudes, aircraft and their resulting wing-tip vortices can face turbulent environments. An example is the clear-air turbulence (CAT), caused by strong wind shear near jet

<footnote><sup>a</sup> DAAA, ONERA, Institut Polytechnique, Meudon, 92190, France</footnote>

<footnote><sup>b</sup> Department of Energy & Process Engineering, Norwegian University of Science & Technology, Trondheim, NO-7491, Norway</footnote>

<footnote><sup>c</sup> FLOW, Department of Engineering Mechanics, KTH Royal Institute of Technology, Stockholm, SE-100 44, Sweden</footnote>

<footnote>☆ This article is part of a Special issue entitled: 'TSFP13' published in International Journal of Heat and Fluid Flow.</footnote>

<footnote>\* Corresponding author.
E-mail address: marie.couliou@onera.fr (M. Couliou).</footnote>
streams or rapid temperature gradients (Clem, 1955), that can generate gust velocities exceeding 5–10 m/s. Recent research (Prosser et al., 2023) indicates that climate change may intensify CAT, increasing its frequency and severity globally. This underscores the growing need to understand how turbulence influences the formation, evolution, and dissipation of wake structures and wing-tip vortices.

Turbulence also plays a significant role in urban environments, where unmanned aerial vehicles (UAVs) operate under challenging conditions. In these environments, the turbulence intensity, $T_i = u_{rms}/\bar{u}$ (where $\bar{u}$ is the mean velocity and $u_{rms}$ is the standard deviation of the velocity), can reach 10%–20% with length scales up to ten times the chord length of the aircraft (Watkins et al., 2006). In the case of wind turbines, turbulence naturally present in the atmospheric boundary layer and in the wakes of upstream turbines has great impact, increasing fatigue loads and reducing efficiency and lifespan (e.g., Vermeer et al., 2003; Vita et al., 2020; Stevens and Meneveau, 2017). At the same time, FST improves the recovery of wind turbine wakes, inter alia through an earlier breakdown of the tip vortices that shed from the rotor blades and are transported downstream on a helical path (e.g., Wu and Porté-Agel, 2012; Aliferis et al., 2019; Neunaber, 2020; Li et al., 2020; Jooss et al., 2023; Gambuzza and Ganapathisubramani, 2023; Hillestad et al., 2024). In a more general context, it is reasonably well documented that increased levels of FST can decrease wake periodicity and recovery length (or time) for disks (Rind and Castro, 2012; Vinnes et al., 2023), cylinders (Eames et al., 2011), and wall-mounted cubes (Castro and Robins, 1977; Hearst et al., 2016a). These examples illustrate the wide-ranging impact of turbulence, from aviation to wind energy, highlighting the importance of studying its effects on both near-field and downstream flow phenomena.

Both Reynolds number and FST play pivotal roles for the dynamics of the flow around an airfoil. While the Reynolds number influences near-airfoil flow features such as separation and attachment, FST can significantly alter the flow separation characteristics and consequently airfoil performance (Devinant et al., 2002). For example, the influence of turbulence on airfoil aerodynamics has been extensively studied over a wide range of Reynolds numbers. Turbulence effects are primarily associated with stall behavior, often leading to an increase in the stall angle or the maximum lift coefficient (McKeough and Graham, 1980; Mish and Devenport, 2001; Gilling et al., 2009; Damiola et al., 2023). Hoffmann (1991) observed delayed stall and increased lift on a rigid NACA0015 airfoil subjected to turbulent inflow conditions. Similarly, Vita et al. (2020) and Li and Hearst (2021) reported that FST can suppress laminar separation bubbles and promote flow attachment on DU96-W-180 and NREL S826 airfoils, respectively. These modifications to boundary layer behavior and lift characteristics also impact the downstream wake, particularly the strength and coherence of tip vortices, which are closely linked to the state of separated flows over the wing (Zhang et al., 2020). Building on these insights, this study will focus on the downstream wake behavior, with particular emphasis on the dynamics of wing-tip vortices in FST.

As mentioned earlier, wing-tip vortices develop near the tips of the airfoil when the vortex sheet that initially forms the wake rolls up. A characteristic feature of these trailing vortices is the motion of their axes, commonly referred to as vortex meandering. Since the 1980s, many researchers have observed and documented this phenomenon (e.g., Sarpkaya and Daly, 1987; Devenport et al., 1996; Chow et al., 1997), noting that vortex meandering causes an apparent diffusion in the velocity field because averaging is performed on a coherently moving structure; Devenport et al. (1996) proposed an analytical method to account for this effect in pointwise measurements. Our understanding of the exact mechanisms underlying vortex meandering remains incomplete; however, there is a consensus in the literature that FST exerts a significant influence on this phenomenon.

FST not only influences near-airfoil flow but also extends beyond the immediate vicinity of the wing, affecting both the wake and the dynamics of the wing-tip vortex. Heyes et al. (2004) studied the effect

of grid-generated turbulence with $T_i$ ranging from 1% to 5%. The authors used particle image velocimetry (PIV) to quantify and remove the effect of meandering directly from the snapshots. They found that the meandering amplitude increases with the freestream turbulence intensity, increases linearly with the streamwise distance, and decreases linearly with the vortex circulation. Bailey and Tavoularis (2008) later introduced a two-point technique to directly reconstruct vortex velocity profiles from the measurements. Van Jaarsveld et al. (2011) used PIV to study the effect of freestream turbulence on the decay of a single vortex and a vortex pair and showed that the vortex decay scales with turbulence intensity; Bailey et al. (2018) specifically confirmed this scaling for a wing-tip vortex.

Additionally, turbulence deforms the vortex core, altering its strength, coherence, and azimuthal velocity distribution (Bailey et al., 2018). Several studies have demonstrated that the deformation of the vortex and its eventual breakdown are key to understanding wake behavior under turbulent conditions. For instance, Bailey et al. (2006), Bailey and Tavoularis (2008) observed that increasing freestream turbulence alters the azimuthal velocity profiles and accelerates vortex decay, with significant effects on the vortex core deformation. Similarly, Ben Miloud et al. (2020) showed that turbulence induces not only greater meandering amplitude but also deformation in the vortex core, contributing to faster wake mixing and breakdown. These findings align with the theoretical framework proposed by Bölle (2024), who highlighted the role of turbulence-induced instabilities in modifying vortex coherence and accelerating its dissipation.

Although turbulence has been shown to significantly affect vortex motion, its influence on vortex deformation remains less thoroughly explored. In addition, the impact of high levels of freestream turbulence (i.e., > 10%) on vortex motion and deformation has not been studied as most previous studies have used passive grids, which limit the achievable turbulence intensity. This leads to two research questions, namely, *What is the influence of high levels of freestream turbulence on the vortex motion?* and *What is the impact of freestream turbulence on the deformation of a vortex?* We will address the research questions through the following four steps. First, we will use an active grid to generate high levels of FST (up to 13%); second, we will employ high-speed PIV to capture the motion of the wing-tip vortex at different downstream distances from the wing; third, the impact of different turbulence levels on the motion of the vortex will be characterized; and fourth, we will characterize the impact of turbulence levels on the deformation of the vortex using proper orthogonal decomposition. Our results ultimately demonstrate that the FST triggers an earlier onset of instabilities contributing to the meandering and deformation of the tip vortex.

## 2. Experimental set-up

### 2.1. Wind tunnel and wing

The experiments were conducted in the large-scale closed-loop wind tunnel at the Department of Energy and Process Engineering at the Norwegian University of Science and Technology (NTNU, Trondheim, Norway). The wind tunnel has a test-section with dimensions of 2.71 m × 1.8 m × 11.1 m (width × height × length). The wind tunnel can be equipped with an active turbulence grid positioned 0.8 m downstream of the convergent section, allowing for the generation of highly turbulent flows upstream of the wing which will generate the wing-tip vortices. The specific active grid used in this facility is described in detail by Kildal et al. (2023). It spans the entire cross-section of the test section and consists of 90 shaft assemblies, with each assembly being individually operated by an integrated stepper motor. The grid has a mesh length of $M = 0.1$ m and was operated using various “fully random” sequences (Hearst and Lavoie, 2015), in which top-hat distributions of the rotational velocity, rotation period and acceleration
<figure><img src="image_3.png" /><figcaption>Fig. 1. Photographs captured during the experimental campaign. (a) The wing model in the wind tunnel while the active grid is in operation. (b) The wing-tip vortex rolling up (REF case). (c) The tip vortex far downstream passing through the laser sheet (REF case).</figcaption></figure>

can be set independently for each shaft. In this study, only the rotational velocity was varied between cases as this parameter was shown to have the greatest impact on the produced turbulence levels (Hearst and Lavoie, 2015). The specifics of the actuation protocols and their outputs are discussed further in Section 3.

The wing-tip vortices were generated by a rectangular wing, with a NACA0012 symmetric airfoil profile, previously used in similar studies (Jacquin et al., 2007; Bölle et al., 2023) — whose aerodynamic properties have been extensively characterized (Chiu et al., 2023; Yonemoto et al., 2008; Pranesh et al., 2019; Yu et al., 2024) and whose wake behavior has been the subject of numerous experimental investigations (Devenport et al., 1996; Chow et al., 1997; Heyes et al., 2004; Bailey et al., 2006; Bailey and Tavoularis, 2008; Ben Miloud et al., 2020; Bölle et al., 2023). The wing has a chord length of $c = 0.125$ m and a span of $b = 0.5$ m. Because of the large size of the wind tunnel, the total model blockage was very low (0.2%). The wing was mounted to a sting with a NACA0015 profile with a chord length of 0.045 m and a length of 0.506 m which in turn was mounted to an extruded aluminium profile attached to the wind-tunnel floor. The airfoil is set at an angle of attack of 9°, consistent with the configuration used by Bölle et al. (2023). The wing was placed such that its trailing edge was at mid-height of the wind tunnel. Boundary-layer transition over the airfoil was triggered by zig-zag shaped tape with 6 mm width and 0.205 mm thickness, placed 6.5 mm (0.052c) downstream of the leading edge. Some photographs captured during the experimental campaign of the wing model in the wind tunnel, the resulting wing-tip vortex rolling up, and the vortex far downstream, are shown in Fig. 1a, Fig. 1b and Fig. 1c, respectively.

The coordinate system used in this study is shown in Fig. 2. The x, y, and z axes were aligned with the streamwise (width of the wind tunnel), and vertical (height of the wind tunnel) directions of the wind tunnel reference frame, respectively. The corresponding velocity components are represented by $[u, v, w]$. The origin of the coordinate system is set at the left trailing-edge tip of the airfoil, which is positioned at an angle of attack of $\alpha = 9^\circ$, the only angle of attack used in the present study, 0.25 m away from the transverse centerline and 0.945 m above the floor of the wind tunnel; $x = 0$ is located 6.778 m = 67.78M downstream of the active grid. A second coordinate system denoted [ŷ, ŷ] is defined based on the instantaneous center of the

left wing-tip vortex and will also be used in Section 4.3 to analyze how FST alters vortex characteristics, independently of vortex wandering. In the following sections, time-averaged quantities are represented by $\overline{\cdots}$ while fluctuations are denoted by $\cdots'$. The notation $\dot{\cdots}$ refers to time-averaged quantities in the ($\hat{y}, \hat{z}$) coordinate system, which are sometimes referred to as conditionally-averaged quantities. $\langle \cdots \rangle$ represents azimuthal averaging.

## 2.2. Hot-wire measurements

Hot-wire anemometry was used to measure the turbulence statistics and flow homogeneity in the wind tunnel without the wing model. A Dantec 55P11 single hot-wire probe, controlled by a Dantec Stream-Line Pro constant temperature anemometer, was employed. The probe consisted of a tungsten sensing element with dimensions of 1.25 mm in length and 5 µm in diameter. The temperature correction approach of Hultmark and Smits (2010) was used. To measure the turbulence statistics of the incoming flows, the probe was fixed in close proximity to a Pitot-static tube and a temperature probe on the four degree-of-freedom wind tunnel traverse. At $x = 0$, hot-wire samples were acquired for 600 s, which corresponds to 62400 flow turn-overs based on the mean velocity and chord length, and for 120 seconds at the other streamwise positions, which corresponds to 12480 of the same flow turn-overs. The hot-wire was sampled at a nominal acquisition frequency of 75 kHz, with an analogue low-pass filter set to 30 kHz, after ensuring that all relevant frequencies are in this range. Assuming small-scale isotropy (cf., Hearst et al. (2012)), an iterative digital low-pass filtering method (Mi et al., 2005, 2011) was used with a digital filtering frequency of $f_c = 1.1 f_k$, where $f_k$ is the Kolmogorov frequency, also reported in Table 1, to reduce the remaining high frequency noise. The Kolmogorov frequency was estimated as $f_K = \bar{u}/(2\pi\eta)$ in the same iterative process, where $\bar{u}$ is the mean advection velocity and $\eta$ is the Kolmogorov microscale estimated with a 7th-order centered-difference scheme and converted to space using Taylor's hypothesis (Hearst et al., 2012).
<figure><img src="image_4.png" /><figcaption>Fig. 2. A schematic of the experimental set-up (not to scale, top view).</figcaption></figure>

<caption>**Table 1**</caption>

Summary of the turbulence properties in the wind tunnel without the wing. Statistics are at a distance of 6.778 m = 54.2c = 67.78M downstream of the grid, i.e., at the wing trailing edge location, (x, y, z) = (0, 0, 0).

<table><thead><tr><th>Inflow case</th><th>Ω<sub>g</sub> ± ω<sub>g</sub> [Hz]</th><th>Grid Mode</th><th>ū [m/s]</th><th>T<sub>i</sub> [%]</th><th>L<sub>11</sub>/c</th><th>Re<sub>λ</sub></th><th>f<sub>k</sub> [Hz]</th></tr></thead><tbody><tr><td>REF</td><td>–</td><td>(no grid)</td><td>12.96</td><td>0.30</td><td>–</td><td>–</td><td>(638)</td></tr><tr><td>A</td><td>–</td><td>(static)</td><td>12.93</td><td>1.83</td><td>0.48</td><td>62</td><td>4525</td></tr><tr><td>B</td><td>7 ± 2</td><td>(vertical)</td><td>13.00</td><td>7.59</td><td>1.76</td><td>370</td><td>8011</td></tr><tr><td>C</td><td>2 ± 1</td><td>(all)</td><td>12.58</td><td>13.05</td><td>2.56</td><td>580</td><td>10222</td></tr></tbody></table>

## 2.3. Stereoscopic particle image velocimetry measurements

Cross-stream stereoscopic PIV (SPIV) oriented along the yz-plane at different downstream locations, $x/c$, was employed to obtain quantitative velocity data. The set-up is shown in Fig. 2. Di-Ethyl-Hexyl-Sebacate (DEHS) droplets produced by a pressurized aerosol generator served as seeding. The mean physical diameter of this seeding was 1 µm. A Photonics DM 100–532 DH 10 mJ dual pulse Nd:YAG high-speed laser emitting green light at a wavelength of 532 nm illuminated the particles. The laser head was placed below the wind tunnel’s glass floor. Using an appropriate combination of optics, the laser beam was shaped into a ≈ 1 mm thick light sheet which entered the test section from below. The size of the field of view (FOV) was limited by the maximum extent to which the laser sheet could be stretched, despite using a very powerful laser source. Imaging of the particles was carried out using two Phantom V2012 1 MP high-speed cameras featuring a 1280 × 800 px² CMOS sensor. These were equipped with Scheimpflug adapters and Sigma 180 mm macro lenses to capture the necessary field of view (FOV). The cameras were placed on either side of the laser sheet as shown in Fig. 2.

The cameras and the laser were synchronized using a LaVision PTU-X controller and the LaVision DaVis 10.2 software suite. Imaging was carried out in double-frame mode with a time separation of 60 µs between an image pair. The acquisition frequency was 101 Hz and 12,442 image pairs were captured per test case at every tested streamwise station. We note that the lowest resolvable frequency is limited by the total acquisition time of about 123 seconds, corresponding to approximately 0.008 Hz. Therefore, the low-frequency contribution of the broadband vortex meandering dynamics (Bölle, 2024) may be underestimated due to limited temporal resolution. Image pre-processing, calibration and vector-field computation was also performed using Davis 10.2. Cross-correlation was carried out using multiple passes starting from a 96 × 96 px² interrogation window down to 48 × 48 px² window for the final pass with an overlap of 50%. This resulted in a spacing of 4.5 mm between adjacent vectors in both the y- and z-directions. There are many factors that contribute to the uncertainty associated with SPIV measurements, such as optical setup, seeding particles, background noise, calibration errors, cross-correlation algorithm, etc. Taking into account these factors and following the methodology

from Bhattacharya et al. (2017), calculations showed a 1.5% bias uncertainty in the velocity measurements for an average particle displacement of 7 pixels, considering that the sub-pixel uncertainty is approximately ±0.1 pixels (Raffel et al., 2018). The vector spacing is 4.5 mm = 0.036c, which is sufficient for a first-order estimate of the vortex core velocity profile, yielding approximately 6 vectors in each direction across the core in both the y- and z-directions.

For the cases with the active grid (A, B and C), the wing’s trailing edge was 66.53M downstream of the grid as mentioned in Section 2.1. The PIV setup was translated to three downstream positions corresponding to $x/c = 2, 5$ and $12$. The same was done with the REF case representing low-turbulent inflow. However, taking advantage of the long streamwise homogeneity due to the clean incoming flow condition in case REF, extra far-field measurements were acquired by moving the wing upstream in the wind tunnel while keeping the PIV FOV fixed at $x = 7.40$ m. The wing was moved to $x = [3.78; 2.28; 0.78]$ m which enabled the PIV field to be located at $x/c = 24, 36$, and $48$ from the trailing edge of the wing for REF (only). For each downstream position, the PIV FOV was centered on the time-averaged position of the left-wing vortex.

## 3. Flow conditions

The experiments were conducted at a mean velocity of $U_∞ = 13$ m/s, corresponding to a chord-based Reynolds number of $Re_c = U_∞c/v = 1.04 \times 10^5$ (v being the air kinematic viscosity). Four test cases were generated for this study: REF, A, B, and C. The active grid was removed for REF, producing a relatively clean incoming flow. For case A, the active grid was in place but kept static and fully open, producing a turbulent flow with low intensity. For case B, only the vertical axes of the active grid were randomly spun with rotational velocities $Ω_g ± ω_g$ distributed in the range $7 ± 2$ Hz, and case C had all active grid axes spinning randomly within the range $2 ± 1$ Hz producing the highest turbulence intensity. The sequences used follow the “fully random” protocol of Hearst and Lavoie (2015).

A preliminary study of flow homogeneity in the empty wind tunnel (no wing) was performed with hot-wire anemometry to understand the incoming flow conditions produced for the various test cases. The
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Streamwise evolution of the freestream turbulence intensity, $T_i$, downstream for $-1 \le x/c \le 15$ at $(y, z) = (0, 0)$.</figcaption></figure>

turbulence intensity, $T_i$, is evaluated as $u_{rms}/\bar{u}$ calculated from the hot-wire data. Table 1 summarizes the main turbulence flow conditions for the four test cases measured at a distance of 6.778 m = 67.78 M = 54.22c downstream of the inlet and 0.25 m left from the centerline, i.e., where the wing's left trailing tip was placed when it was present. The table includes values for the mean velocity, the turbulence intensity, the streamwise integral length scale, $L_{11}$, obtained from integrating the autocorrelation function of the streamwise velocity fluctuations to the first zero-crossing, and the Taylor microscale-based Reynolds number, $Re_λ = λu_{rms}/v$. The Taylor microscale $λ$ is given by $λ = \sqrt{u_{rms}^2/(\partial u'/\partial x)^2}$ and was calculated by assuming small-scale isotropy and estimating the gradients using Taylor's hypothesis and a 7th-order centered-difference scheme as described by Hearst et al. (2012). The produced turbulence intensities were 0.30%, 1.83%, 7.59%, and 13.05% for cases REF, A, B, and C, respectively. The integral scale normalized by the chord length is $L_{11}/c = 0.48, 1.76$, and 2.56 for cases A, B, and C, respectively.

The homogeneity of the background flows along the vortex streamwise evolution was evaluated by hot-wire measurements along a streamwise line located at the left trailing tip (0.25 m left from the centerline). It was measured in an empty wind tunnel from the position where the leading edge of the wing was to 15c downstream of the trailing edge, i.e., from 66.53M to 86.53M downstream of the active grid. The streamwise scan plotted in Fig. 3 shows that the turbulence intensity $T_i$ remained around 0.23% for REF, decayed from 1.84% to 1.58% for A, from 7.91% to 6.81% for B, and from 13.65% to 10.94% for C. The variation of turbulence is thus 2.5% for the most turbulent case (case C, $T_i = 13.05$%) along the streamwise direction between $x/c = 0$ and 15. Such a result is expected as grid-generated turbulence decays in space in a wind tunnel (e.g., Comte-Bellot and Corrsin (1966)) and is reported here for completeness.

The turbulent velocity spectra obtained from the hot-wire measurements for all cases are shown in Fig. 4 in dimensional frequency-space up to the respective Kolmogorov frequencies in Fig. 4a and wavenumber-space normalized by dissipative scales in Fig. 4b. The REF case is only plotted in Fig. 4a up to its filter frequency because its turbulence level is too low ($T_i = 0.3\%$) to be meaningfully represented by the normalization in Fig. 4b. In Fig. 4b, the non-dimensional wavenumber spectra are plotted with the dissipative scaling using the Kolmogorov microscale, $\eta$. This normalization is used such that all turbulent spectra collapse at the small scales and differ at the large scales, following canonical expectations. A $(\kappa\eta)^{-5/3}$ line is added for reference. The absence of low-frequency peaks in the spectra indicates that the active grid forcing does not preferentially energize a particular frequency or set of frequencies, which is the desired outcome of this actuation approach. As the turbulence intensity increases, the inertial range

grows, representing an increase in the separation of scales, as expected from previous active grid measurements (Larssen and Devenport, 2011; Hearst and Lavoie, 2015) and theory for increasing $Re_λ$ (Pope, 2000).

# 4. Effect of FST on wing-tip vortex proprieties

## 4.1. Time-averaged properties of the vortex

The vorticity $\omega_x = \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}$ is used to identify the wing-tip vortex. The contour maps of the time-averaged streamwise vorticity $\bar{\omega}_x$ for the different tested cases are presented in Fig. 5. To perform a relative comparison, the maximum vorticity at the vortex center, $\bar{\omega}_{x,max}$, was used for vorticity normalization.

First, we examine the contours of the normalized averaged axial vorticity $\bar{\omega}_x/\bar{\omega}_{x,max}$ at $x/c = 2$ for the case REF (Fig. 5a). The high concentration of vorticity near the wing tip highlights the wing-tip vortex, serving as a baseline for comparison. The wake shear layer around the vortex concentration area is well-defined in all cases which is consistent with the fact that the vortical shear layer is still rolling up into well-defined concentrated vortices at the measurement plane five chord lengths downstream of the wing. When transitioning from the reference case (REF) to case C (i.e., with increasing FST), we observe a spatial spreading of the region with the highest vorticity, i.e., where $\bar{\omega}_x/\bar{\omega}_{x,max}$ is close to unity. Specifically, in cases B (Fig. 5c) and C (Fig. 5d), the vorticity contours appear more diffused compared to the reference case without a grid (REF), covering a larger area around the vortex core. This diffusion might indicate a weakening of the coherent vortex core, likely due to enhanced mixing or instabilities induced by the higher levels of FST in these cases. Alternatively, it could reflect increased unsteadiness or wandering of the vortex core, such that the time-averaged velocity field appears more diffused due to spatial smearing.

Next, vortex kinematics are discussed. In cases REF and A, the vortex appears to remain stationary, as it can be well defined using time-averaged vorticity. We can therefore also analyze its trajectory using this quantity. Focusing on the reference case (REF), the streamwise evolution of the normalized time-averaged vorticity is analyzed for $x/c = 2, 5$ and 12 (Fig. 5a, Fig. 5e Fig. 5i). At $x/c = 2$, the maximum vorticity is located at $y/c = 0.02$, while at $x/c = 12$, it is located at $y/c = -0.25$ indicating a vortex descent caused by induced flow effects. For case A, the vortex descent is slower. At $x/c = 2$, the maximum vorticity is located at $y/c = 0$, while at $x/c = 12$, it is located at $y/c = -0.06$. The vortex descent $\Delta_z/c$ is thus approximately 0.27 for REF compared to 0.06 for A over a streamwise length $\Delta_x/c = 10$. This indicates that even a small increase in FST (from $T_i = 0.3\%$ to 1.84%) affects the vortex trajectory significantly. The same analysis is not performed for cases B and C, as the vorticity contours in these cases are more diffuse, making it less straightforward to interpret the vortex trajectory.

The diffusion of the time-averaged vorticity field is particularly pronounced in the most turbulent cases ($T_i = 7.59\%$ and $T_i = 13.05\%$, respectively case B and C), primarily because the vortex exhibits significant motion over time. This motion leads to spatial smearing in the averaged field, making it challenging to identify the vortex structure directly. To address this, the vortex motion is explicitly evaluated and the vortex center detected in the following section.

## 4.2. Vortex meandering

The initial step in the post-processing of the PIV field is the identification of the vortex center. This detection process is carried out in two phases. Initially, we determine the instantaneous extremum position $(y_m, z_m)$ where $m$ represents the location of the maximum axial vorticity, $\omega_x$; $(y_m, z_m)$ is restrained to the vertices of the PIV mesh. Next, a criterion is applied to determine whether the detected maximum corresponds to a coherent vortex or is merely due to surrounding turbulence
<figure><img src="image_6.png" /><figcaption>Fig. 4. Velocity spectra for all cases plotted in (a) dimensional frequency-space and (b) wavenumber-space normalized by dissipative scales. The REF case is only plotted in (a) because its turbulence level is too low to be meaningfully represented by the normalization in (b).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Contours of the normalized time-averaged axial vorticity $\bar{\omega}_x/\bar{\omega}_{x,max}$. Each column corresponds to a given turbulence case (REF, A, B and C) and each row corresponds to a measurement location with $x/c = 2$ (a–d), $x/c = 5$ (e–h), $x/c = 12$ (i–l). The dashed line indicates the trailing edge of the wing.</figcaption></figure>

or measurement noise. To identify tip vortices reliably, a threshold is introduced: only snapshots in which the maximum axial vorticity $\omega_x$ exceeds half the maximum vorticity of a reference case (REF) are retained. Snapshots that do not meet this criterion are discarded. To refine this localization, we then calculate $x_\omega$, the barycenter of the axial vorticity over a reduced region $\Omega_m$ centered about $x_m$. The integration area, $\Omega_m$, is taken as a disk of radius 0.12c for all the tests. This region was found to be large enough to contain the core of the vortex while excluding the freestream flow. The vortex center is found to move with different amplitude depending on the location of the PIV plane and the level of FST. For the highest FST (case C, $T_i = 13.05\%$) and at $x/c = 12$, the vortex moves so extensively that the vortex center is frequently not detected because it lies outside the field of view. Fig. 6a summarizes the percentage of instances within the measurement time that the tip vortex is located outside the field of view. In half the instances for case C at $x/c = 5$ and case B at $x/c = 12$ the vortex center cannot be detected, while three-quarters of the instances do not have an identifiable vortex core for the most turbulent case (case C,

$T_i = 13.05\%$) at the downstream position $x/c = 12$. This indicates that these configurations (case B at $x/c = 5$, and case C at $x/c = 5$ and 12) should be interpreted with caution, as the vortex is sometimes, but not systematically, located outside of the observable region. Nevertheless, this observation itself is a result in that it clearly demonstrates that the meandering amplitude is a function of FST.

The vortex motion is quantified using the meandering amplitudes. To determine the meandering amplitudes in the transverse and span-wise directions, denoted $\sigma_y$ and $\sigma_z$, respectively, the standard deviations of the time series for the vortex axis displacements from the mean position, $\delta_y$ and $\delta_z$, along the transverse and vertical directions are calculated. The dependence of these amplitudes on the incoming turbulence intensity is presented in Fig. 6b. As the vortex is convected to $x/c = 5$, the meandering amplitudes are larger than those at $x/c = 2$, indicating that the area covered by the scatter of the instantaneous vortex center positions increases, particularly for the highest FST levels ($T_i = 7.59\%$ and $T_i = 13.05\%$). This behavior is consistent with the observations of Bailey et al. (2018) (where their highest $T_i$ was
<table><caption>(a)</caption><thead><tr><th></th><th>x/c = 2</th><th>x/c = 5</th><th>x/c = 12</th></tr></thead><tbody><tr><th>REF</th><td>0%</td><td>0%</td><td>0%</td></tr><tr><th>A</th><td>0%</td><td>0%</td><td>0%</td></tr><tr><th>B</th><td>6.6%</td><td>13.4%</td><td>57.4%</td></tr><tr><th>C</th><td>23.1%</td><td>56.0%</td><td>76.1%</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>Fig. 6. (a) Percentage of instances where the tip vortex lies outside of the field of view. (b) Variation of meandering amplitudes $\sigma_i/c$ with incoming turbulence intensity, $T_i$.</figcaption></figure>

5%) and Ben Miloud et al. (2020) (where their highest $T_i$ was 6%), who noted that the meandering amplitudes strongly increased in both directions with increasing FST.

For the highest FST — i.e. $T_i = 7.59\%$ and $T_i = 13.05\%$ — cases of the present study, which exceed the 6% achieved in previous studies (Bailey et al., 2018; Ben Miloud et al., 2020), a preferential direction in the y-direction is observed: $\sigma_y$ is three times larger than $\sigma_z$ at both x-planes ($x/c = 2$ and $x/c = 5$). This observed preferential meandering direction, corresponding to an angle of approximately 19° with respect to the y-axis (transverse direction), is notably smaller than the 48° typically associated with the classical Crow instability, which develops farther downstream and is predicted by a vortex filament model (Crow, 1970). While the increasing meandering could be associated with the growth of the displacement mode — potentially indicating the onset of a Crow-like instability — the reduced angle suggests a modification of the underlying mechanism, possibly driven by asymmetric or localized perturbations. In particular, the early onset of this meandering — well upstream of the characteristic Crow instability development length — could be triggered by a perturbation in the symmetry plane of the wing, such as that induced by the wing/mast junction. This interpretation aligns with the sensitivity to localized disturbances highlighted by Brion et al. (2007). However, since such a structural perturbation is present in all configurations, it cannot by itself explain the observed behavior. Notably, the preferential meandering direction emerges only in cases with $T_i > 6\%$, suggesting that background turbulence in the wake plays a critical role. One hypothesis is that elevated turbulence levels enhance the receptivity of the vortex pair to initial perturbations, thereby contributing both to the early onset and the altered orientation of the instability. Although the ratio may vary in reality due to missing portions of the vortex — specifically, at $x/c = 12$, where 57.4% of the vortex lies outside the field of view for case B and 76.1% for case C (see Fig. 6a) — this does not diminish the relevance of the discussion, as the meandering amplitudes will remain significantly high in case.

### 4.3. Conditional averaged properties of the vortex

The vortex characteristics are examined in greater detail at the streamwise plane $x/c = 5$, which provides an optimal compromise for capturing the vortex: it is sufficiently far downstream to facilitate vortex formation, yet not too far to cause excessive vortex motion and loss of the vortex from the field of view which becomes particularly problematic in the two most turbulent cases ($T_i = 7.59\%$ and $T_i = 13.05\%$, cases B and C, respectively) for $x/c = 12$ as shown in Fig. 6a. To investigate the alterations in vortex characteristics induced by FST, without interference from vortex wandering, the velocity fields, and

thus the vorticity fields, have been averaged across the frames after adjusting each instantaneous field to be recentered with the vortex at the origin, i.e., $(\hat{y}, \hat{z}) = (0, 0)$. The contours of the normalized conditionally averaged streamwise vorticity $\hat{\omega}_x/\hat{\omega}_{x,max}$ are plotted in Fig. 7(a–d). The conditionally averaged vortex size and shape appear to remain fairly constant as FST increases. From these conditionally averaged axial vorticity fields, the maximal streamwise vorticity $\hat{\omega}_{x,max}c/U_\infty$ is extracted and is found to be 9.86, 11.13, 8.15, and 8.0 for cases REF, A, B, and C, respectively. From cases A to C, $\hat{\omega}_{x,max}c/U_\infty$ decreases as FST increases suggesting a diffusion of the vortex with increasing FST in agreement with Pentelow (2014) and Ben Miloud et al. (2020). Interestingly, the maximum streamwise vorticity across all cases is not measured for case REF but for the low turbulence case ($T_i = 1.84\%$, case A). It is worth noting that this phenomenon has also been observed in other flows interacting with ambient turbulence. Similar behavior has been reported for surface waves interacting with turbulence (Smeltzer et al., 2023), as well as in the flow around a wall-mounted cube immersed in a turbulent boundary layer (Hearst et al., 2016b). A hypothetical explanation for this behavior in the present configuration — namely, the wing-tip vortex generated by a lifting wing — is provided in the following paragraph.

The azimuthal velocity is determined by using the velocity components $v$ and $w$ in the Cartesian coordinate system and then converting them into polar coordinates: $u_\theta = w \cos(\theta) - v \sin(\theta)$. The conditional and azimuthal average of the vortex azimuthal velocity $\langle \hat{u}_\theta \rangle$ was evaluated as a function of the radial distance $r$ with $r = 0$ being the center of the vortex and defined as the barycenter of vorticity of the conditionally averaged vorticity field. Fig. 7e shows the evolution of the $\langle \hat{u}_\theta \rangle$ profile as a function of $r$ and for the different FST levels. This profile is used to characterize the vortex size as the vortex core radius, i.e., the localization of maximum $\langle \hat{u}_\theta \rangle$. The vortex core radius remains constant around 0.09c for all cases. This aligns with the work of Dghim et al. (2021) where they note that when the meandering correction is applied, increasing FST did not have much effect on the conditionally averaged vortex radius. Another observation from Fig. 7e is that the peak of $\langle \hat{u}_\theta \rangle$ is maximum for the low turbulence case ($T_i = 1.84\%$, Case A) and then decreases as FST increases. To explain why the maximum conditional vorticity and peak azimuthal velocity are observed in case A, we hypothesize that the incoming turbulence modifies the flow over the wing, likely by promoting flow attachment due to the suppression of laminar separation. This would lead to an increase in lift, consistent with previous findings in the literature (e.g., Hoffmann (1991), Vita et al. (2020), Li and Hearst (2021), Damiola et al. (2023)), and consequently strengthen the tip
<figure><img src="image_9.png" /><figcaption>Fig. 7. Contours of the normalized conditionally averaged streamwise vorticity $\omega_x/\omega_{x,max}$ for case REF (a), A (b), B (c), and C (d). Normalized azimuthal core conditionally averaged velocity profiles for each turbulent case at $x/c = 5$ (e)</figcaption></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Contours of the standard deviation of the conditioned vorticity $\omega_x$ for cases REF (a), A (b), B (c), and C (d).</figcaption></figure>

vortex. While direct measurements of lift or pressure would be valuable to confirm this scenario, the explanation we propose aligns with established experimental observations. This change in flow dynamics may lead to an increase in the strength of the vortex, as turbulence can enhance shear layer interactions and promote more stable vortex formation (Bailey et al., 2018; Ben Miloud et al., 2020).

Finally, the standard deviation of the conditioned vortex time-series for the four tested cases is shown in Fig. 8. No significant variations are observed for case REF (Fig. 8a) and case A (Fig. 8b). In cases B (Fig. 8c) and C (Fig. 8d), an elliptical shape in the standard deviation of the vortex is observed. This suggests that the vortex may be deforming. To investigate this further, modal decomposition is employed in the next section.

## 5. Proper orthogonal decomposition analysis

To investigate the effect of the FST on the wing-tip vortex dynamics further, proper orthogonal decomposition (POD) is used. The snapshot formulation of POD proposed by Sirovich (1987) is employed for this purpose. Given that capturing the full vortex dynamics is intended, snapshot-POD was applied on the time-series of the unconditioned streamwise vorticity field here. As vorticity is Galilean-invariant, it can be preferable for identification of coherent structures in the flow. A good description of applying POD to vorticity fields and its comparison with the more common operation on the velocity fields is discussed by Kostas et al. (2005). In the current work, the fluctuating component of the streamwise vorticity $\omega'_x$ was decomposed using snapshot POD in the following manner:

$$ \omega'_{x}(y, z, t) = \sum_{n} \Lambda_{n} \Theta_{n}(t) \Psi_{n}(y, z), \quad (1) $$

where $\Psi_{n}(y, z)$ are a set of empirical eigenfunctions (spatial POD modes) that contain information regarding the spatial organization of various fluctuations with the corresponding temporal coefficients represented by $\Theta_{n}(t)$, $n$ being the mode number. $\Lambda_{n}$ are the corresponding eigenvalues which, in the current computation, are representative of the contribution of a particular mode to the total entrophy of the flow. It is worth noting that the modes are sorted in descending order of their entrophy fraction ($\Lambda_{n}/\sum \Lambda_{n}$) and henceforth, any mention of mode number corresponds to its rank based on its relative contribution to the total entrophy.

### 5.1. The reference case from near- to far-field

As detailed in Section 2.3, PIV measurements were acquired at $x/c = 5, 12, 24, 36$ and $48$ downstream of the wing's trailing edge for the clean incoming flow case ($T_i = 0.30\%$, case REF). POD analysis is first carried out on these measurements as this will also serve as the benchmark to compare the effect of FST with. The spatial organization of fluctuations corresponding to the first four POD eigenmodes at these streamwise stations is shown in Fig. 9 and the entrophy fractions of the first 10 modes at the different streamwise stations are shown in Fig. 10.

At all five streamwise stations shown in Fig. 9, the first modes contain two opposite regions of vorticity fluctuations, forming a nearly circular vortex dipole centered around the mean vortex center. In the second mode at $x/c = 5$ (Fig. 9b), a weak imprint of the wing's wake is also visible. Barring this, the second POD modes at all streamwise stations have a similar structure as the first mode. This indicates that these two modes are paired and orthogonally rotate together. These mode-pairs are helical displacement modes representing a Kelvin wave with an azimuthal wavenumber of $m = 1$ as described by Fabre et al. (2006) (also see Dghim et al., 2021). The presence of this dipole represents the transverse velocity components being non-zero at the vortex center which results in vortex meandering (Fabre et al., 2006; Edstrand et al., 2016; Dghim et al., 2021).

The third and fourth modes show differences at the measured streamwise locations. At $x/c = 5$, coherent structures are seen just below the wing which relates to the wing's wake. At $x/c = 12$, the entrophy fraction of these modes is about 1%, which is significantly lower than the first two modes. The same is true for the fourth mode at $x/c = 24$. POD modes with lower energy (or entrophy, in the case of vorticity-based analysis) represent less coherent and less dynamically significant structures, making it reasonable to disregard them in order to focus on the dominant flow features. Thus, these modes are not
<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Spatial eigenfunctions from the POD computations $\Psi_n$, for case REF at (a–d) $x/c = 5$, (e–h) $x/c = 12$, (i–l) $x/c = 24$, (m–p) $x/c = 36$, and (q–t) $x/c = 48$.</figcaption></figure>

considered for the current analysis. From $x/c = 24$ onward, the third mode contains a central structure and one structure on either side of it but with opposite polarity. These resemble deformation modes resulting from the amplification of elliptical instabilities with an azimuthal wavenumber of $m = 2$ within the vortex tube (see Lacaze et al., 2007). The fourth modes at $x/c = 36$ and 48 also represent deformation modes with an azimuthal wavenumber of $m = 2$, but the organization of the coherent structure resembles a vortex quadripole. The effect of this mode on the base flow is to deform the vortex core into an ellipse (Fabre et al., 2006).

Having connected the mode shapes to various dynamics previously reported in literature, their relative importance is assessed via their entrophy fraction, presented in Fig. 10. At $x/c = 5$, the entrophy fraction of first two modes representing vortex meandering is just ≈ 4% of the total energy. It quickly rises to about 40% at $x/c = 12$ and 24. For the last two streamwise stations it increases to about 60% of the total entrophy. This is a direct indication of the vortex meandering amplitude increasing as one moves downstream. As such, this is in good agreement with previous observations (Bölle et al., 2023). At $x/c = 5$, the entrophy fraction of the third and fourth modes is very similar to the first two modes indicating the strong influence the wing’s wake still has on the flow field. At $x/c = 12$, the entrophy fraction of these modes is about 1%, which is significantly lower than the first two modes. Interestingly, from $x/c = 24$ onward, the entrophy fraction of these two modes starts increasing compared to the previous two streamwise locations. Their contribution to the total entrophy of the flow also increases relative to the first two modes, along the streamwise direction as shown in Fig. 10b. This suggests that while the displacement modes

$(n = 1$ and 2) remain the most dominant, the vortex deformation modes $(n = 3$ and 4) emerge and are no longer negligible as one moves downstream.

## 5.2. Effect of freestream turbulence

Having determined a baseline with case REF, the spatial organization of the cases with incoming turbulence is now assessed. This analysis is carried out at $x/c = 5$ for all three turbulence cases and additionally at $x/c = 12$ for case A. The first four POD spatial modes of the FST cases are presented in Fig. 11 and the entrophy fraction of the first 10 modes is shown in Fig. 12a. In case A, a vortex dipole is observed downstream of the wing-tip in the transverse plane in the two leading POD modes at both streamwise locations. These modes represent the meandering motion of the vortex as discussed with the REF case and also observed in the literature (Fabre et al., 2006; Edstrand et al., 2016; Dghim et al., 2021). In comparison to the REF case, differences arise when their entrophy fractions are considered. Together, these modes of case A contribute about 50% of the total entrophy to the flow at both these streamwise locations. This is significantly higher compared to the REF case at $x/c = 5$ where these modes constituted only ≈ 4%. The entrophy fractions are more comparable at $x/c = 12$, but these modes contribute ≈ 40% in the REF case, which is still lower compared to the current case A with turbulence. This directly correlates to the increase in the meandering motion with the addition of turbulence which was discussed in Section 4.1.

A second difference arises from the second mode onwards at $x/c = 5$, when looking at the coherent structures present. Notably, the signature of the wing’s wake, which was apparent in the second mode of the
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** (a) Enstrophy fraction of the first 10 POD eigenmodes for cases REF at $x/c = 5, 12, 24, 36, 48$. (b) Enstrophy contribution for mode 3 onwards, compared to the sum of modes 1 and 2.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Spatial eigenfunctions from the POD computations, $\Psi_n$. Case A at $x/c = 5$ (a–d) and $x/c = 12$ (e–h) are presented. At $x/c = 5$, cases B (i–l) and C (m–p) are also shown.</figcaption></figure>

REF case (Fig. 9b) does not appear for the turbulence cases (cases A, B, C; see Fig. 11b, Fig. 11j, and Fig. 11n, respectively). In fact, coherent structures related to the wing's wake are not observed in the third and fourth modes either, see Fig. 11c–d, Fig. 11k–l, and Fig. 11(o–p). While it is not presented for brevity, the imprint of the wing's wake is observed only after the seventh mode and contributes less than 1% to the total enstrophy. The addition of incoming turbulence diffuses the wake of an airfoil more rapidly in space as is also evidenced in the studies of Bailey et al. (2006) and Maldonado et al. (2015); this phenomenon has also been documented for other flows, e.g., cubes (Castro and Robins, 1977; Hearst et al., 2016a), wind turbines (Aliferis et al., 2019; Li et al., 2020; Jooss et al., 2023; Gambuzza and Ganapathisubramani, 2023), amongst others.

At both streamwise locations, the third and fourth modes represent vortex deformation and resemble the corresponding modes in the REF case at $x/c = 36$ (see Fig. 9o and p). Based on the organization of the

coherent structures within, we hypothesize that the third mode could be related to the amplification of elliptical instabilities (Lacaze et al., 2007) and the fourth mode is a vortex quadripole (Fabre et al., 2006). The enstrophy contribution of these two modes relative to the first two modes is $(\Lambda_3 + \Lambda_4)/(\Lambda_1 + \Lambda_2) = 21.5\%$ and $32.8\%$, respectively, at $x/c = 5$ and 12, which is higher than the deformation modes in the REF case even at $x/c = 48$ in fact (= 16%) as shown in Fig. 12b. This observation strongly suggests that FST has a significant effect on the vortex deformation modes. Not only are the deformation modes observed as early as $x/c = 5$, but also their contribution to the total enstrophy of the flow is more significant than that at $x/c = 48$ in the REF case.

At $x/c = 5$, the coherent structures in the first four modes of case B (Fig. 11i–l) mirror case A but span a larger spatial area. For the displacement modes ($n = 1$ and 2), this implies an increase in the meandering amplitude which agrees well with what was observed in
<figure><img src="image_15.png" /><figcaption>Fig. 12. (a) Enstrophy fractions of the first 10 POD modes of the various test cases with FST. (b) Enstrophy contribution for mode 3 onwards, compared to the sum of modes 1 and 2.</figcaption></figure>

Fig. 6b. For case C (Fig. 11m-p), the features in the spatial modes remain consistent with cases A and B, but over an even larger spatial area. It is worth noting that in case C the tip vortex was outside the field of view for 56% of the acquired frames as listed in Fig. 6a. So, the result of this POD computation must be considered carefully as it only represents the 44% of the time that the vortex was located near the wing-tip. Nevertheless, it is interesting that the features do agree to a certain extent with cases A and B. Finally, the enstrophy fractions of the first four modes in cases B and C (see Fig. 12a) are very low compared to case A at this streamwise location and similar to that in the REF case. It is hypothesized that for these high turbulence cases, other features, like deformations, become more prominent due to the turbulence effects and in turn distribute the total enstrophy to a larger number of modes. In contrast, for case A, the vortex is mostly affected by the meandering, resulting in the first two modes showing greater dominance.

To summarize, independently of the level of turbulence, the two first POD modes are displacement modes and are always associated with the meandering motion of the vortex. The addition of turbulence significantly diminishes the effect of the wing's wake, which is prominent in the REF case at $x/c = 5$. This is evidenced by the disappearance of the wake signature in the leading POD modes, which is clearly seen in the REF case (see Fig. 9b-d), but absent in all turbulent cases (A, B, and C; see Fig. 11), including the low turbulence case A ($T_i = 1.83\%$), indicating that the coherent wake structures are disrupted by the incoming FST. In contrast, as primarily observed with the elliptical shape in the standard deviation of the vortex (see Fig. 8), the POD analysis confirmed that FST accelerates the onset and relative importance of the vortex deformation modes. While in the REF case, these modes do not appear until $x/c = 24$, with the addition of even the lowest amount of turbulence, deformation modes have a prominent effect on the flow field as early as $x/c = 5$. While it is not presented here, these modes were observed even two chord lengths downstream of the wing.

## 6. Conclusions

SPIV measurements combined with the use of an active grid enabled us to document the effects of freestream turbulence on the meandering motion of a wing-tip vortex in the near field and middle field downstream of a wing. The temporal tracking of the vortex center shows that the amplitude of vortex meandering increases proportionally with freestream turbulence (FST), corroborating well with previous studies. Notably, a novel finding is that at the highest FST levels ($T_i > 6\%$), a preferential orientation in the meandering motion towards one direction is observed. It is found that in the transverse direction ($\sigma_y$)

the amplitude is three times larger than in the spanwise direction ($\sigma_z$) for the highest $T_i$ cases ($T_i = 7.59\%$ and $T_i = 13.05\%$). This direction corresponds to an angle of approximately 19° with respect to the y-axis (transverse direction), smaller than the classical Crow instability angle which develops farther downstream, suggesting a modified mechanism possibly influenced by localized perturbations and enhanced by elevated background turbulence. Despite the vortex being outside the field of view for case C for a significant period of the acquisition, the meandering would remain pronounced, highlighting the critical role of turbulence in both the early onset and altered orientation of the instability.

The statistical analysis, conditioned on the vortex core position, reveals a general reduction in vortex strength with increasing FST, accompanied by a modest increase in diffusion. However, an exception is observed from the REF case to case A, where an increase in vortex strength is detected. We hypothesize that the increase of turbulence from case REF ($T_i = 0.30\%$) to case A ($T_i = 1.84\%$) likely alters the flow over the wing and promotes flow attachment by suppressing laminar separation, enhancing shear layer interactions and contributing to the formation of more stable vortex structures (Bailey et al., 2018; Ben Miloud et al., 2020). Consequently, case A exhibits the highest maximum conditional vorticity, peak azimuthal velocity, and strongest meandering modal energies among all cases (REF, A, B, and C).

Employing Proper Orthogonal Decomposition (POD), the motion of the wing-tip vortex is discerned independently of precise center detection and despite potentially limited resolution within the vortex core. The POD analysis enabled us to enlighten some effects of FST on the vortex. FST accelerates the establishment of the vortex or at least its dynamics. Indeed, in the reference case REF, the signature of the wake remains prominent at $x/c = 5$; however, this is no longer the case for cases A, B, and C. While the vortex displacement mode remains dominant, turbulence appears to induce the rise of vortex deformation modes associated with either the propagation of Kelvin waves or helical deformation modes. These deformation modes contribute to the onset of vortex instabilities (Leweke and Le Dizès, 2016). For the REF case without freestream turbulence, POD analysis spanned $5 \le x/c \le 48$. The analysis reveals that spatial modes associated with vortex deformation typically emerge in the far wake ($x/c \ge 36$) under non-turbulent conditions. Interestingly, these deformation modes are spatially identical to those induced by turbulence in the near wake ($x/c = 5$) but their contribution to the total enstrophy of the flow is more significant in the turbulence cases in the near wake than in the far wake in the REF case.

For the turbulence cases, the faster apparition of both motion and deformation modes reinforces the conclusion that turbulence will accelerate the onset of vortex instabilities. Crow (1970) identified the
dominant wake vortex instability as the linear growth of large-scale perturbations leading to vortex pairing, without considering freestream turbulence. Our POD results show that turbulence might accelerate the development of instabilities but also alters their modal structure. With this finding, we can hypothesize that the vortex may be destroyed faster in the turbulent cases than in the REF case. Since the ratio of enstrophy contribution between motion and deformation is changed, we can also hypothesize that instead of a reconnection of vortices due to their motion (i.e., Crow instabilities), other instabilities may arise, where both motion and deformation act as key ingredients. Future investigations farther downstream would be useful to assess this. Although the present study uses a simplified wing model, the characteristic wing-tip vortex it generates is representative of those observed in transport aircraft and other applications such as wind turbines, making our findings broadly relevant to understanding vortex dynamics in various freestream turbulence conditions.

## CRediT authorship contribution statement

**M. Couliou:** Writing – original draft, Visualization, Resources, Project administration, Investigation, Funding acquisition, Formal analysis, Conceptualization.
**S. Yadala:** Writing – original draft, Investigation, Formal analysis.
**G.K. Jankee:** Writing – review & editing, Investigation.
**I. Neunaber:** Writing – review & editing, Investigation.
**R.J. Hearst:** Writing – review & editing, Supervision, Resources, Project administration, Funding acquisition, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

This work is co-funded by the mobility program France-Norway PROGRAMME ÅSGARD 2023 (2023-SCIEN11-ASG07-657.2) from Institut Français de Norvège and by internal research projects funded by the Scientific Direction of ONERA.

## Data availability

The data that support the findings of this study are openly available at DataverseNO: https://doi.org/10.18710/HLVDJ7.

## References

Aliferis, A.D., Jessen, M.S., Bracchi, T., Hearst, R.J., 2019. Performance and wake of a Savonius vertical-axis wind turbine under different incoming conditions. Wind. Energ. 22 (9), 1260-1273. http://dx.doi.org/10.1002/we.2358.

Bailey, S., Pentelow, S., Ghimire, H., Estejab, B., Green, M., Tavoularis, S., 2018. Experimental investigation of the scaling of vortex wandering in turbulent surroundings. J. Fluid Mech. 843, 722-747. http://dx.doi.org/10.1017/jfm.2018.180.

Bailey, S., Tavoularis, S., 2008. Measurements of the velocity field of a wing-tip vortex, wandering in grid turbulence. J. Fluid Mech. 601, 281-315. http://dx.doi.org/10.1017/S0022112008000694.

Bailey, S., Tavoularis, S., Lee, B., 2006. Effects of free-stream turbulence on wing-tip vortex formation and near field. J. Aircr. 43 (5), 1282-1291. http://dx.doi.org/10.2514/1.19433.

Ben Miloud, K., Dghim, M., Fellouah, H., Ferchichi, M., 2020. Free-stream turbulence interaction with a wing-tip vortex. J. Wind Eng. Ind. Aerodyn. 206, 104211. http://dx.doi.org/10.1016/j.jweia.2020.104211.

Bhattacharya, S., Charonko, J.J., Vlachos, P.P., 2017. Stereo-particle image velocimetry uncertainty quantification. Meas. Sci. Technol. 28, http://dx.doi.org/10.1088/1361-6501/28/1/015301.

Bölle, T., 2024. On the linear response theory of vortex meandering and its statistical verification in experiments. J. Fluid Mech. 997, A38. http://dx.doi.org/10.1017/jfm.2024.606.

Bölle, T., Brion, V., Couliou, M., Molton, P., 2023. Experiment on jet-vortex interaction for variable mutual spacing. Phys. Fluids 35 (1), 015117. http://dx.doi.org/10.1063/5.0127634.

Breitsamter, C., 2011. Wake vortex characteristics of transport aircraft. Prog. Aerosp. Sci. 47 (2), 89-134. http://dx.doi.org/10.1016/j.paerosci.2010.09.002.

Brion, V., Sipp, D., Jacquin, L., 2007. Optimal amplification of the crow instability. Phys. Fluids 19, 111703. http://dx.doi.org/10.1063/1.2793146.

Castro, I., Robins, A., 1977. The flow around a surface-mounted cube in uniform and turbulent streams. J. Fluid Mech. 79 (2), 307-335. http://dx.doi.org/10.1017/S0022112077000172.

Chiu, T.Y., Tseng, C.C., Chang, C.C., Chou, Y.-J., 2023. Vorticity forces of coherent structures on the NACA0012 aerofoil. J. Fluid Mech. 974, A52. http://dx.doi.org/10.1017/jfm.2023.815.

Chow, J., Zilliac, G., Bradshaw, P., 1997. Mean and turbulence measurements in the near field of a wingtip vortex. AIAA J. 35 (10), 1561-1567. http://dx.doi.org/10.2514/2.1.

Clem, L.H., 1955. Clear-air turbulence near the jet-stream maxima. Bull. Am. Meteorol. Soc. 36 (2), 53-60. http://dx.doi.org/10.1175/1520-0477-36.2.53.

Comte-Bellot, G., Corrsin, S., 1966. The use of a contraction to improve the isotropy of grid-generated turbulence. J. Fluid Mech. 25 (4), 657-682. http://dx.doi.org/10.1017/S0022112066000338.

Crow, S.C., 1970. Stability theory for a pair of trailing vortices. AIAA J. 8 (12), 2172-2179. http://dx.doi.org/10.2514/3.6083.

Damiola, L., Siddiqui, M.F., Runacres, M.C., De Troyer, T., 2023. Influence of free-stream turbulence intensity on static and dynamic stall of a NACA 0018 aerofoil. J. Wind Eng. Ind. Aerodyn. 232, 105270. http://dx.doi.org/10.1016/j.jweia.2022.105270.

Devenport, W., Rife, M., Liapis, S., Follin, G., 1996. The structure and development of a wing-tip vortex. J. Fluid Mech. 312, 67-106. http://dx.doi.org/10.1017/S0022112096001929.

Devinant, P., Laverne, T., Hureau, J., 2002. Experimental study of wind-turbine airfoil aerodynamics in high turbulence. J. Wind Eng. Ind. Aerodyn. 90 (6), 689-707. http://dx.doi.org/10.1016/S0167-6105(02)00162-9.

Dghim, M., Ben Miloud, K., Ferchichi, M., Fellouah, H., 2021. Meandering of a wing-tip vortex in a grid-generated turbulent flow. Phys. Fluids 33 (11), http://dx.doi.org/10.1063/5.0057083.

Eames, I., Jonsson, C., Johnson, P.B., 2011. The growth of a cylinder wake in turbulent flow. J. Turbul. 12, N39. http://dx.doi.org/10.1080/14685248.2011.619985.

Edstrand, A.M., Davis, T.B., Schmid, P.J., Taira, K., Cattafesta III, L.N., 2016. On the mechanism of trailing vortex wandering. J. Fluid Mech. 801, R1. http://dx.doi.org/10.1017/jfm.2016.440.

Fabre, D., Sipp, D., Jacquin, L., 2006. Kelvin waves and the singular modes of the lamb-oseen vortex. J. Fluid Mech. 551, 235-274. http://dx.doi.org/10.1017/S0022112005008463.

Gambuzza, S., Ganapathisubramani, B., 2023. The influence of free stream turbulence on the development of a wind turbine wake. J. Fluid Mech. 963, A19. http://dx.doi.org/10.1017/jfm.2023.302.

Gerz, T., Holzäpfel, F., Darracq, D., 2002. Commercial aircraft wake vortices. Prog. Aerosp. Sci. 38 (3), 181-208. http://dx.doi.org/10.1016/S0376-0421(02)00004-0.

Gilling, L., Sørensen, N., Davidson, L., 2009. Detached eddy simulations of an airfoil in turbulent inflow. In: 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. p. 270. http://dx.doi.org/10.2514/6.2009-270.

Hallock, J.N., Holzäpfel, F., 2018. A review of recent wake vortex research for increasing airport capacity. Prog. Aerosp. Sci. 98, 27-36. http://dx.doi.org/10.1016/j.paerosci.2018.03.003.

Hearst, R., Buxton, O., Ganapathisubramani, B., Lavoie, P., 2012. Experimental estimation of fluctuating velocity and scalar gradients in turbulence. Exp. Fluids 53, 925-942. http://dx.doi.org/10.1007/s00348-012-1318-0.

Hearst, R.J., Gomit, G., Ganapathisubramani, B., 2016a. Effect of turbulence on the wake of a wall-mounted cube. J. Fluid Mech. 804, 513-530. http://dx.doi.org/10.1017/jfm.2016.565.

Hearst, R.J., Gomit, G., Ganapathisubramani, B., 2016b. Effect of turbulence on the wake of a wall-mounted cube. J. Fluid Mech. 804, 513-530. http://dx.doi.org/10.1017/jfm.2016.565.

Hearst, R., Lavoie, P., 2015. The effect of active grid initial conditions on high Reynolds number turbulence. Exp. Fluids 56 (185), http://dx.doi.org/10.1007/s00348-015-2052-1.

Heyes, A., Jones, R., Smith, D., 2004. Wandering of wing-tip vortices. Department of Mechanical Engineering, Imperial College London.

Hillestad, J.N., Yadala, S., Neunaber, I., Li, L., Hearst, R.J., Worth, N.A., 2024. Volumetric visualization of vanishing vortices in wind turbine wakes. Phys. Rev. Fluids 9, L052701. http://dx.doi.org/10.1103/PhysRevFluids.9.L052701.

Hoffmann, J., 1991. Effects of freestream turbulence on the performance characteristics of an airfoil. AIAA J. 29 (9), 1353-1354. http://dx.doi.org/10.2514/3.10745.

Hultmark, M., Smits, A., 2010. Temperature corrections for constant temperature and constant current hot-wire anemometers. Meas. Sci. Technol. 21, 105404. http://dx.doi.org/10.1088/0957-0233/21/10/105404.

Hünecke, K., 2001. Wake vortex control-a challenge for large transport aircraft. Air & Space Eur. 3 (3-4), 209-213. http://dx.doi.org/10.1016/S1290-0958(01)90097-8.
Jacquin, L., Molton, P., Loiret, P., Coustols, E., 2007. An experiment on jet-wake vortex interaction. In: 37th AIAA Fluid Dynamics Conference and Exhibit. p. 4363. http://dx.doi.org/10.2514/6.2007-4363.

Jooss, Y., Hearst, R.J., Bracchi, T., 2023. Influence of incoming turbulence and shear on the flow field and performance of a lab-scale roof-mounted vertical axis wind turbine. J. Renew. Sustain. Energ. 15, 063302. http://dx.doi.org/10.1063/5.0170059.

Kildal, O., Li, L., Hearst, R., Petersen, Ø., Øiseth, O., 2023. On the use of an active turbulence grid in wind tunnel testing of bridge decks. J. Wind Eng. Ind. Aerodyn. 233, 105331. http://dx.doi.org/10.1016/j.jweia.2023.105331.

Kostas, J., Soria, J., Chong, M.S., 2005. A comparison between snapshot POD analysis of PIV velocity and vorticity data. Exp. Fluids 38, 146–160. http://dx.doi.org/10.1007/s00348-004-0873-4.

Lacaze, L., Ryan, K., Le Dizes, S., 2007. Elliptic instability in a strained batchelor vortex. J. Fluid Mech. 577, 341–361. http://dx.doi.org/10.1017/S0022112007004879.

Larssen, J., Devenport, W., 2011. On the generation of large-scale homogeneous turbulence. Exp. Fluids 50, 1207–1223. http://dx.doi.org/10.1007/s00348-010-0974-1.

Leweke, T., Le Dizès, C.H., 2016. Dynamics and instabilities of vortex pairs. Annu. Rev. Fluid Mech. 48, 507–541. http://dx.doi.org/10.1146/annurev-fluid-122414-034558.

Li, L., Hearst, R., 2021. The influence of freestream turbulence on the temporal pressure distribution and lift of an airfoil. J. Wind Eng. Ind. Aerodyn. 209, 104456. http://dx.doi.org/10.1016/j.jweia.2020.104456.

Li, L., Hearst, R., Ferreira, M., Ganapathisubramani, B., 2020. The near-field of a lab-scale wind turbine in tailored turbulent shear flows. Renew. Energ. 149, 735–748. http://dx.doi.org/10.1016/j.renene.2019.12.049.

Maldonado, V., Castillo, L., Thormann, A., Meneveau, C., 2015. The role of free stream turbulence with large integral scale on the aerodynamic performance of an experimental low Reynolds number S809 wind turbine blade. J. Wind Eng. Ind. Aerodyn. 142, 246–257. http://dx.doi.org/10.1016/j.jweia.2015.03.010.

McKeough, P., Graham, J., 1980. The effect of mean loading on the fluctuating loads induced on aerofoils by a turbulent stream. Aeronaut. Q. 31 (1), 56–69. http://dx.doi.org/10.1017/S0001925900008829.

Mi, J., Deo, R.C., Nathan, G.J., 2005. Fast-convergence iterative scheme for filtering velocity signals and finding Kolmogorov scales. Phys. Rev. E 71 (066304), http://dx.doi.org/10.1103/PhysRevE.71.066304.

Mi, J., Xu, M., Du, C., 2011. Digital filter for hot-wire measurements of small-scale turbulence properties. Meas. Sci. Technol. 22 (12), 125401. http://dx.doi.org/10.1088/0957-0233/22/12/125401.

Mish, P., Devenport, W., 2001. Mean loading effects on the surface pressure fluctuations on an airfoil in turbulence. In: 7th AIAA/CEAS Aeroacoustics Conference and Exhibit. p. 2211. http://dx.doi.org/10.2514/6.2001-2211.

Neunaber, I., 2020. Turbulence of wakes. In: Stoevesandt, B., Schepers, G., Fuglsang, P., Yuping, S. (Eds.), Handbook of Wind Energy Aerodynamics. Springer International Publishing, Cham, pp. 1–34. http://dx.doi.org/10.1007/978-3-030-05455-7_45-1.

Pentelow, S., 2014. Wing-tip vortex structure and wandering. (Ph.D. thesis). Université d'Ottawa, http://dx.doi.org/10.20381/ruor-3740.

Pope, S., 2000. Turbulent Flows. Cambridge University Press, http://dx.doi.org/10.1088/0957-0233/12/11/705.

Pranesh, C., Sivapragasam, M., Deshpande, M., Narahari, H., 2019. Negative lift characteristics of NACA 0012 airfoil at low Reynolds numbers. Sädhanä 44 (1), 21. http://dx.doi.org/10.1007/s12046-018-1008-6.

Prosser, M.C., Williams, P.D., Marlton, G.J., Harrison, R.G., 2023. Evidence for large increases in clear-air turbulence over the past four decades. Geophys. Res. Lett. 50 (11), http://dx.doi.org/10.1029/2023GL103814, e2023GL103814.

Raffel, M., Willert, C., Scarano, F., Kähler, C., Wereley, S., Kompenhans, J., 2018. Particle Image Velocimetry: A Practical Guide, second ed. Springer, Switzerland, http://dx.doi.org/10.1007/978-3-319-68852-7.

Rind, E., Castro, I., 2012. On the effects of free-stream turbulence on axisymmetric disc wakes. Exp. Fluids 53, 301–318. http://dx.doi.org/10.1007/s00348-012-1288-2.

Sarpkaya, T., Daly, J., 1987. Effect of ambient turbulence on trailing vortices. J. Aircr. 24 (6), 309–404. http://dx.doi.org/10.2514/3.45459.

Schmitt, N.P., Rehm, W., Pistner, T., Zeller, P., Diehl, H., Navé, P., 2007. The AWIATOR airborne LIDAR turbulence sensor. Aerosp. Sci. Technol. 11 (7–8), 546–552. http://dx.doi.org/10.1016/j.ast.2007.03.006.

Sirovich, L., 1987. Turbulence and the dynamics of coherent structures. I. Coherent structures. Quart. Appl. Math. 45, 561–571. http://dx.doi.org/10.1090/qam/910462.

Smeltzer, B.K., Römcke, O., Hearst, R.J., Ellingsen, S.Å., 2023. Experimental study of the mutual interactions between waves and tailored turbulence. J. Fluid Mech. 962, R1. http://dx.doi.org/10.1017/jfm.2023.280.

Stevens, R.J., Meneveau, C., 2017. Flow structure and turbulence in wind farms. Annu. Rev. Fluid Mech. 49 (1), 311–339. http://dx.doi.org/10.1146/annurev-fluid-010816-060206.

Van Jaarsveld, J.P.J., Holten, A.P.C., Elsenaar, A., Trieling, R.R., Van Heijst, G.J.F., 2011. An experimental study of the effect of external turbulence on the decay of a single vortex and a vortex pair. J. Fluid Mech. 670, 214–239. http://dx.doi.org/10.1017/S0022112010005197.

Vermeer, L., Sørensen, J., Crespo, A., 2003. Wind turbine wake aerodynamics. Prog. Aerosp. Sci. 39 (6), 467–510. http://dx.doi.org/10.1016/S0376-0421(03)00078-2.

Vinnes, M., Neunaber, I., Lykke, H.M., Hearst, R., 2023. Characterizing porous disk wakes in different turbulent inflow conditions with higher-order statistics. Exp. Fluids 64 (2), 25. http://dx.doi.org/10.1007/s00348-022-03565-9.

Vita, G., Hemida, H., Andrianne, T., Baniotopoulos, C., 2020. The effect of the integral length scale of turbulence on a wind turbine airfoil. J. Wind Eng. Ind. Aerodyn. 204, 104235. http://dx.doi.org/10.1016/j.jweia.2020.104235.

Watkins, S., Milbank, J., Loxton, B.J., Melbourne, W.H., 2006. Atmospheric winds and their implications for microair vehicles. AIAA J. 44 (11), 2591–2600. http://dx.doi.org/10.2514/1.22670.

Wu, Y.T., Porté-Agel, F., 2012. Atmospheric turbulence effects on wind-turbine wakes: An LES study. Energies 5 (12), 5340–5362. http://dx.doi.org/10.3390/en5125340.

Yonemoto, K., Takato, K., Ochi, H., Fujie, S., 2008. Kutta condition violation in two-dimensional NACA0012 airfoil at low Reynolds number. In: 26th AIAA Applied Aerodynamics Conference. p. 6399. http://dx.doi.org/10.2514/6.2008-6399.

Yu, M., Hrynuk, J.T., Booth, D.T., Poudel, N., 2024. Freestream turbulence effects on low Reynolds number NACA 0012 airfoil laminar separation bubble and lift generation. Aerosp. Sci. Technol. 149, 109145. http://dx.doi.org/10.1016/j.ast.2024.109145.

Zhang, K., Hayostek, S., Amitay, M., He, W., Theofilis, V., Taira, K., 2020. On the formation of three-dimensional separated flows over wings under tip effects. J. Fluid Mech. 895, A9. http://dx.doi.org/10.1017/jfm.2020.248.