# POD analysis of regular waves interacting with large scale flow structures in the water column

Philippe Druault, Benoît Gaurier, Grégory Germain

► To cite this version:

Philippe Druault, Benoît Gaurier, Grégory Germain. POD analysis of regular waves interacting with large scale flow structures in the water column. Ocean Engineering, 2026, 352, pp.124601. 10.1016/j.oceaneng.2026.124601. hal-05521059

HAL Id: hal-05521059

https://hal.sorbonne-universite.fr/hal-05521059v1

Submitted on 20 Feb 2026

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 CC BY 4.0 - Attribution - International License
<figure><img src="image_2.png" /></figure>

Research paper

# POD analysis of regular waves interacting with large scale flow structures in the water column

Philippe Druault a,*, Benoît Gaurier b, Grégory Germain b

## ARTICLE INFO

### Keywords:

Wave-turbulence separation

Proper orthogonal decomposition

Large-scale flow structures

Water column.

## ABSTRACT

To accurately characterize prospective tidal energy sites where vortices and wavy environment coexist, it is essential to gain a deeper understanding of wave-vortex interactions. Given the complexity and multi-scale nature of these coupling processes, we focus on the interaction between regular surface waves and vortices developing in the water column. Based on Particle Image Velocimetry measurements conducted in a flume tank, two flow configurations are studied: monochromatic waves propagating either with or against the current, each interacting with large-scale flow structures in a high Reynolds number turbulent shear flow. These flow configurations are characterised by a negative mean velocity vertical gradient. In this work, the Proper Orthogonal Decomposition is implemented to extract the wave velocity field from raw velocity measurements. After validating this wave-turbulence decomposition, the results show that the wave propagation modifies the integral time scale of the free flow, the vertical mixing and its associated energy transfer. The energetic vertical exchanges are primarily governed by the free surface proximity, the kinetic energy levels in the free flow without waves and the wave propagation direction. Waves lead to limit the vertical mixing by reducing the production of the kinetic energy. A quadrant analysis also reveals the instantaneous flow distortion under the wave action, especially when waves propagate against the current.

## 1. Introduction

Ocean currents affect a lot of various environmental mechanisms like sediment transport, marine biodiversity and forces exerted on seabed structures. They also represent a clean source of proper renewable energy by exploiting the kinetic energy of the tidal streams. In tidal energy sites, marine currents exhibit a high level of turbulence that contain many different temporal and spatial scales that interact with each other. It is well admitted that some large scale energetic flow structures coming from the irregular sea floor or from human activities such as in the wake of submerged rigid structures are embedded in these currents. These large scale flow structures play an important role in the transport and mixing of the primary production and living organisms (Mcwilliams, 2016) but also in the fatigue of the devices (Druault et al., 2025; Mullings et al., 2023; Ouro and Stoesser, 2019). Additionally, the wavy environment constitutes another source of increasing turbulence in the water column. Understanding the flow mechanisms in the water column is then essential for the development of tidal energy converters. Previous works have notably analyzed the wave impact on the sea floor boundary layer flow and on sediment transport (Soulsby et al., 1993;

Zhang et al., 2022) but also on the distortion of the turbulent flow near the water surface showing an enhancement of the turbulence near the surface (Teixeira and Belcher, 2002). The distortion of the flow has been previously modelled using a rapid-distortion model reproducing the interaction of weak turbulence with a monochromatic surface water wave (Teixeira and Belcher, 2002). The effect of wave orbital component on the turbulent flow has been also previously analyzed in (Viviano et al., 2018) studying the interaction between waves and gravity currents or in (Xuan et al., 2020) studying the wave-Langmuir turbulence interaction. In this work, we focus on the wave effect in the water column far from the water surface, that corresponds to the depth at which the tidal energy converters are located (Furgerot et al., 2020; Mullings et al., 2023). In such a location, some large scale flow structures can be convected in the current (Furgerot et al., 2020; Mercier et al., 2020) and they may greatly affect the tidal turbine behavior (Druault and Germain, 2022; Mullings et al., 2023). The purpose of this work is then to study the unsteady wave-vortex interactions in such a case. This differs from wave-turbulence interactions previously considered in presence of Langmuir turbulence (Xuan et al., 2020) or of ambient grid turbulence (Smeltzer et al., 2023).

<footnote>ªSorbonne Université, CNRS, Institut Jean Le Rond d'Alembert, F-75005, Paris, France</footnote>

<footnote>ᵇIfremer, Marine Structure Laboratory, 150 quai Gambetta, 62200, Boulogne-sur-mer, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: philippe.druault@sorbonne-universite.fr (P. Druault).</footnote>
Roy et al. (2017, 2018) demonstrated that the flow structures are primarily influenced by the wave frequency interacting with the current, regardless of their relative orientation. Indeed, in case of a monochromatic waves, it is well admitted that the velocity spectra in the water column exhibit a prominent frequency peak at the wave frequency. Magnier et al. (2022) have investigated the effect of waves following the current on the large scale energetic structures developing in the wake behind a wall-mounted, bathymetric cylindrical obstacle. The study shown that the most significant changes in the wake occurred at the lowest wave frequencies, notably marked by localized increases in kinetic energy. Additionally, the large-scale flow structures formed downstream of a wall-mounted square cylinder were examined under the influence of waves moving against the current (Magnier et al., 2023; Saouli et al., 2022). While the overall wake structure remained broadly consistent, the large-scale flow patterns were instantaneously altered, and both the distribution and magnitude of turbulent kinetic energy differed from free flow field without waves. It was also observed that the orbital velocities of the waves instantaneously altered the nature of the flow structure as well as the associated Probability Density Function (PDF) of the fluctuating velocity field (Shounda et al., 2025). Recently, the large scale flow structures developing in the wake of a rigid structure and submitted to waves following or against the current have been analyzed by comparison with the same free flow without waves (Gaurier et al., 2025). It was shown that the waves mainly affect the Turbulent Kinetic Energy (TKE) level, the integral length scales and the PDF of the fluctuating velocity fields. Recent numerical simulations further revealed that the instantaneous behavior of tip vortices behind a tidal turbine is strongly affected by waves following the current (Ouro et al., 2024). Specifically, the tip vortex experiences vertical oscillations that are synchronized with the wave frequency during its downstream advection. However, despite these dynamic interactions, the average characteristics of the turbine's wake appear to be less significantly impacted (Zhang et al., 2021), although the waves intensify and modify the anisotropy of the turbulent flow (Zhang et al., 2023).

To go further, a great challenging task consists in extracting the wave contribution from the raw available velocity field to better emphasize the unsteady wave-turbulent interactions and its associated energetic transfers in the water column. Generally speaking, for wave-turbulence flow application, the triple decomposition of instantaneous unsteady velocity signal $V(X, t)$ is used:

$$V(X, t) = \bar{V}(X) + V'(X, t) = \bar{V}(X) + V_r(X, t) + V_w(X, t) \quad (1)$$

where an overbar indicates the time average operator. $V'(X, t)$ is the fluctuating velocity component satisfying $\bar{V}'(X, t) = 0$ and is decomposed into a turbulent component $V_r(X, t)$ and a wave component $V_w(X, t)$.

A wide range of methods have already been used to extract the wavy component from raw velocity signals. Note that these methods are not applicable to every velocity field database, limitations may arise, in particular, those coming from the data acquisition frequency. In the case of a periodic wave component, the phase average procedure as initially introduced by (Hussain and Reynolds, 1970) can be used to isolate the wave component from the raw velocity signal, as illustrated in (Xuan et al., 2020). Similarly, applying a sine least mean squares to the wave frequency enables the extraction of the phase average wave component (Gaurier et al., 2025; Perez et al., 2020). Bricker and Monismith (2007) have proposed a wave-turbulence decomposition performed in the spectral frequency domain, assuming that wave and turbulence do not interact. Note that a simple bandpass filter centred around the wave frequency peak can not be directly applied since wave and turbulence time scales overlap in the spectral domain. Following the pioneering work (Qiao et al., 2016), several authors have recently proposed to use the Empirical Mode Decomposition (EMD) to extract the orbital component of the wave from velocity time series recorded with ADCP (Acoustic Doppler Velocimeter) (Bian et al., 2020; Tan and Venugopal, 2024). An Empirical Orthogonal Function (EOF) analysis based on ADCP measurements has been also proposed for a simultaneous assessment of

turbulence and waves at highly-energetic tidal sites (Togneri et al., 2021). Such a method has been applied to ADV (Acoustic Doppler Velocimetry) measurements obtained in a towing tank (Perez et al., 2020). When compared to linear wave theory, the Wavelet method highlights superior robustness, although some limitations were mentioned. More recently, an improved Synchro-squeezed Wavelet Transform (SWT) has been evaluated against several existing wave-turbulence decomposition methods (Zhao et al., 2025). Applied to in-situ ADV velocity measurements near the free surface, the effectiveness of the improved SWT method was established, particularly in well estimating the energy dissipation rate. The Dynamical Mode Decomposition has also been recently proposed as a promising tool for wave-turbulence separation (Chávez-Dorado et al., 2025). The robustness of the proposed method has been demonstrated across various velocity datasets and favourably compared to other decomposition methods. These existing wave separation methods each have their own specific features and their applicability strongly depends on the nature of the available database as well as on prior knowledge of the water surface wave properties. For instance, methods such as SWT, Fourier filtering and EMD primarily rely on an efficient adaptive time-frequency processing method. It has also been recently mentioned that the resulting wave energy content, including quantities like the Reynolds stress, is not adequately addressed (Zhao et al., 2025). Assessing the wave induced increase in kinetic energy is crucial for tidal site evaluation. For further information, the reader may refer to the very recent review of wave-turbulence decomposition methods (Chávez-Dorado et al., 2025; Zhao et al., 2025). Moreover, none of them was tested in presence of convective energetic large scale flow structures where the choice of the input parameters used to distinguish the wave and turbulent fields become particularly challenging, especially the extraction of the energetic contribution of the wave field. In the present work, a new way is proposed to extract the wave field based on the energetic and coherence properties of the turbulent flow field. The objective of this work is then to use Proper Orthogonal Decomposition (POD) that has not yet been tested to the author's knowledge, for separating turbulence and wave component in the water column in presence of large scale convected flow structures. Since POD provides an energetic optimal flow decomposition, it is expected to offer a novel approach of isolating the energetic contribution associated with the wave field.

The paper is organised as follows. The description of the experimental database is done in Section 2. Then the POD procedure description for extracting wave from the turbulence is proposed in Section 3. The validation of the filtering process is also given in this third section with a first analysis of the wave and turbulent parts obtained following this new method. In Section 4, an analysis of the wave-turbulence interaction is presented by characterising the energetic exchanges between both contributions and by regarding the flow event modifications across the water column.

## 2. Experimental set-up

The present work deals with the experimental database obtained in the wave and current flume tank of IFREMER at Boulogne-sur-Mer (France) (see Fig. 1) and previously reported in (Gaurier et al., 2025).

The $(x, y, z)$ cartesian coordinate system is used and it corresponds to the streamwise, transverse and vertical direction respectively. In this basis, the velocity vector is expressed as follows $V = ue_x + ve_y + we_z$.

At the inlet, a uniform streamwise velocity field $U_\infty = 0.8m/s$ with a turbulence intensity of 1.5% is specified. Two flow configurations are considered by moving the wavemaker between the inlet or the outlet of the working section of the tank (see Fig. 1). It permits to generate waves following the current (denoted case A) or against the current (denoted case B). As detailed in (Gaurier et al., 2025), the presence of wavemaker (case A) or wave absorber (case B) at the inlet section leads to generate a wake downstream that falls down slowly into the water column. The submergence depths of the wavemaker and wave absorber are $D_g = 50cm$ and $D_a = 35cm$ respectively. Based on a mean
<figure><img src="image_3.png" /><figcaption>Fig. 1. Tank configurations for cases A (top) and B (bottom). The instrumentation is identical regardless of the direction of wave propagation.</figcaption></figure>

submergence depth $D \sim 0.4m$, the flow Reynolds number is then $Re = U_\infty D/\nu = 3.2 \times 10^5$ with $\nu$ the kinematic viscosity of water.

In France, the Aldernay Race which represents one of the most tidal-stream energy potential along the French coast, is characterized by a quite complex fully unsteady turbulent flow with a high level of turbulence (Mullings et al., 2023). This behavior results from intricate interactions between surface gravity waves and large scale flow structures that develop in the water column and originate from irregular seabed bathymetry. These coherent flow structures are characterized by a high level of kinetic energy with a low frequency signature. Similar large scale flow structures with comparable properties can be also observed in the wake of the wind turbine support structures as well as in the wake of tidal turbine. The present study does not aim to fully reproduce the turbulent flow conditions at this specific tidal turbine site. Instead, its objective is to evaluate a new method for separating wave and turbulence velocity fields in the presence of large scale flow structures. In such a way, the large scale flow structures developing in the wake of the wavemaker or the wave absorber share key features such as comparable kinetic energy levels and low frequency signatures with those observed in the water column under in-situ conditions.

As the coupling effects between waves and large-scale flow structures correspond to a complex multi-scale process, especially for a broad wave frequency range, one proposes to limit the complexity of the wave-turbulence interaction by preliminary considering some regular waves.

To approach sea-wave conditions (Magnier et al., 2023), a regular wave with a similar frequency $f_w \approx 0.45Hz$ is imposed for both flow configurations. In case A, the waves follow the current while in case B they are opposed to the current. The waves remain largely two-dimensional in both cases, with very limited transverse (y) side effect.

## 2.1. Velocity and free surface measurements

Instantaneous velocity fields are measured thanks to Particle Image Velocimetry (PIV). PIV measurements are performed within a vertical plane $(x, z)$ as indicated by a red rectangle in Fig. 1. This plane is deliberately located at the streamwise center of the tank. In this area, the turbulent wake generated either by the wavemaker or the damping beach is fully developed in the water column. Note also that the basin is sufficiently deep to ensure that the developing bottom boundary layer does not interact with the wake flows within the PIV measurement plane.

The PIV sampling frequency is $f_{PIV} = 15Hz$ and $N_t = 2700$ instantaneous velocity components $(u, w)$ are obtained in a vertical $(x, z)$-plane of $(N_x, N_z) = (41, 79)$ points, with a spatial discretisation of 11.6mm in each direction. The x-axis origin is taken at the centered position of the PIV measurement plane. The z-axis origin is fixed at the bottom of the tank and the PIV measurement plane is centred in the vertical water depth.

The wave properties (amplitude, wavelength, frequency) are measured using 3 wave probes: a Kenek servo-type wave probe (model

<caption>Table 1</caption>

Wave characteristics generated in the flume tank. $A_w$ and $\lambda$ are the wave amplitude and wavelength respectively. $h$ is the water depth in the basin.

<table><thead><tr><th>Case</th><th>$f_w$ [Hz]</th><th>$A_w$ [mm]</th><th>$\lambda$ [m]</th><th>$h/\lambda$</th></tr></thead><tbody><tr><td>$A_{wave}$</td><td>0.45</td><td>42</td><td>9.9</td><td>0.2</td></tr><tr><td>$B_{wave}$</td><td>0.45</td><td>69</td><td>3.9</td><td>0.5</td></tr></tbody></table>

SHT3-30E) and 2 resistive wave probes. The dynamic Kenek wave probe is marked with the letter A in Fig. 1 and the two other probes are installed on positions B and C (see Fig. 1). The distances between these points are $[AB] = 52cm$ and $[BC] = 70cm$ respectively. The wave probes acquisition is synchronised with the PIV system that triggers the measurements. Details about the wave surface measurement and calibration are given in (Gaurier et al., 2025). The standard deviation of the temporal wave signal $\sqrt{s}/2$ is 0.076m and 0.100m for cases A and B respectively. This value is the average of the standard deviations of the three available wave signals. Table 1 indicates the main wave characteristics in each case. The wave energy injected in the water column is then of a similar order in both flow configurations, even if some differences exist: the wavelength in case A are 2.5 larger than the one in case B while the amplitudes of the waves in case A are approximately 1.5 smaller than in case B.

## 2.2. Mean flow characteristics

A short description of the mean features for the two flow configurations is provided. The reader is referred to (Gaurier et al., 2025) for a deeply analysis.

The classical Reynolds decomposition is first applied to the instantaneous measured velocity field as follows: $u(x, z, t) = \bar{U}(x, z) + u'(x, z, t)$ and $w(x, z, t) = \bar{W}(x, z) + w'(x, z, t)$ where an overbar indicates the time average operator and $(u', w')$ corresponds to the instantaneous fluctuating velocity field.

A mask is first applied on both upper corners of the PIV measurement plane to eliminate shadow effects. The two mean velocity components: $\bar{U}(x, z)$ and $\bar{W}(x, z)$ are shown in Fig. 2. In both cases, the mean streamwise velocity presents a shear velocity with a maximum value at the bottom part of the measurement plane. As both wavemaker and wave-absorber have different submergence depths, the shear velocity in A flow configuration covers the whole vertical extent of the PIV plane while it is only present in the bottom part in B flow configuration. The vertical profile of $\bar{W}(x, z)$ is globally similar in both flow configurations: with a negative very low amplitude by comparison with the ones of $\bar{U}(x, z)$ and it remains globally constant along the vertical extent of the PIV plane.
<figure><img src="image_4.png" /><figcaption>Fig. 2. Mean streamwise and vertical velocity components: $\overline{U}$ (left) and $\overline{W}$ (right).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. Root Mean Square of the Reynolds tensor components: $\sqrt{u'^2}$ (left) and $\sqrt{w'^2}$ (right).</figcaption></figure>

Fig. 3 displays the Root Mean Square of the normal Reynolds stress components: $\sqrt{u'^2}$ and $\sqrt{w'^2}$. The main contribution to the turbulent kinetic energy $k_{2D} = 1/2(\overline{u'^2} + \overline{w'^2})$ is concentrated in the upper part of the measurement plane where the mean shear velocity is significant. As expected, in case B a low turbulence intensity level is observed for $z < 1$, corresponding to the region of nearly uniform mean flow. Note that in the upper part ($z > 1.2$), the amplitudes of $\overline{u'^2}$ and $\overline{w'^2}$ are found to be similar for both flow configurations. This behavior is directly linked to the presence of waves that increases the vertical velocity fluctuations by comparison with free flow without waves (Gaurier et al., 2025).

Fig. 4 displays the vertical profile (at $x_0 = 0$) of the standard deviation $\sqrt{u'^2(z)}$ and $\sqrt{w'^2(z)}$ normalized by the standard deviation of the wave surface signal $\sqrt{s'^2}$. This permits to link the wave surface energy and the flow energy content. Clearly, when waves propagate with the current (case A), the large scale flow structures developing behind the wave maker contain a high level of kinetic energy, generally exceeding the standard deviation of the water wave fluctuations. In contrast, for case B the level of the flow kinetic energy within the water column is lower than the wave surface energy.

### 3. POD process for wave-turbulence decomposition

In this section, the POD method is first described before being applied as a filter. The filtering process is then validated through mean and statistical analyses.

#### 3.1. Proper orthogonal decomposition application

POD is an efficient statistical method providing an optimal decomposition for representing the flow variable energy (here the energy is related to the 2D turbulent kinetic energy, $\overline{u'^2} + \overline{w'^2}$). It enables to extract the energetic coherent events embedded in the turbulent flow. A coherent event is defined as the structure that exhibits the largest mean square projection of the velocity field (Holmes, 2012). This maximization leads to a Fredholm integral eigenvalue problem which consists in determining the discrete orthogonal eigenfunctions of a correlation tensor. In the present case, since the number of available snapshots ($N_t = 2700$) exceeds the number of grid points in the PIV plane ($N_x \times N_z = 41 \times 79$), the snapshot version of the POD (Sirovich, 1987) is employed. In this formulation, the correlation tensor corresponds to the spatial average correlation tensor that accounts for both velocity components. The resulting discrete temporal POD modes denoted $a^{(n)}(t)$, with $n$ varying from 1 to $N_{\text{mod}} = N_t$, are mutually orthogonal: $\overline{a^{(n)}(t)a^{(m)}(t)} = \lambda^{(n)}\delta_{nm}$ with $\delta$ the Kronecker symbol and $\lambda^{(n)}$ the $n^{th}$ POD eigenvalue. The eigenvalues are ordered in descending order as $\lambda^{(1)} > \lambda^{(2)} > \dots > \lambda^{(N_{\text{mod}})} > 0$ and each eigenvalue represents the energy associated with its corresponding mode. In this POD mode basis, each instantaneous fluctuating velocity field ($u', w'$) can be expressed as a linear combination of POD eigenfunctions:

$$u'(\mathbf{X}, t) = \sum_{n=1}^{N_{\text{mod}}} a^{(n)}(t)\Phi_{u}^{(n)}(\mathbf{X}) \text{ and } w'(\mathbf{X}, t) = \sum_{n=1}^{N_{\text{mod}}} a^{(n)}(t)\Phi_{w}^{(n)}(\mathbf{X}), \quad (2)$$
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Superimposition of the standard deviation of the fluctuating streamwise (filled markers) and vertical (empty markers) velocity components normalized with the standard deviation of the wave surface signal. Left: Case A. Right: Case B.</figcaption></figure>

where $\Phi_u^{(n)}(X)$ and $\Phi_{uw}^{(n)}(X)$ are the random spatial coefficients of projection associated with the fluctuating $(u', w')$ velocity components respectively. They satisfy $\Phi_i^{(n)}\Phi_i^{(m)} = \delta_{nm}$ with $i=u,w$.

In each flow configuration, the POD flow decomposition provides a linear approximation of the dominant energetic feature of the flow, characterized by both temporal and spatial evolution of each mode.

To investigate the POD mode energy content, the cumulative energy content of the POD modes $K_1^N = \sum_{n=1}^N \lambda^{(n)} / \sum_{n=1}^{N_{\text{mod}}} \lambda^{(n)}$ is computed. Fig. 5 displays the evolution of $K_1^N$ as a function of $N$. On the same figure (right-hand axis), the energy content defined as $K_2^n = \lambda^{(n)} / \sum_{n=1}^{N_{\text{mod}}} \lambda^{(n)}$ is also plotted against $n$. The x-axis is limited to 40 to highlight the energy repartition of the first POD modes. Case B reaches higher energy levels faster than case A. The first POD eigenvalue for cases A and B accounts for 24% and 12% of the total energy, respectively. Moreover, recovering 80% of the total energy requires considering 9 and 4 modes in cases A and B respectively. When waves propagate with current (case A), the main kinetic energy is more evenly distributed over the first POD modes. This is due to the highly energetic wake flow generated by the wake-maker, which contains coherent structures developing across the PIV plane. For case B, the results are somewhat different as the first two POD modes each contain more than 30% of the total kinetic energy, indicating that the present flow configuration is more coherent than that of case A. This behavior is indeed an artifact of the experimental set up and can be explained by two physical mechanisms. First, The POD energy convergence in the present flow configuration is primarily governed by the large scale energetic flow structures embedded in the turbulent shear flow. Thus, as state in the article, the energetic wake flow generated by the wake-maker (case A) is more highly turbulent than that generated by the wave-damper (case B). Furthermore, in case A, coherent structures are present over nearly the entire vertical height of the PIV measurement plane whereas in case B, they remained confined to the upper part of the PIV plane. The second mechanism, which is closely related to the first, concerns wave propagation in the water column. In the present flow configuration, the wave coherent field is more prominent in case B especially at the lower part of the measurement plane, beneath the region influenced by large scale flow structures. Consequently, the POD mode energy content is not driven solely by wave motions but also by the contribution of the large scale flow structures. The resulted POD energy convergence therefore depends directly on the experimental set up and the vertical location of the experimental measurement plane.

Figs. 6 and 7 represent the first five spatial POD coefficients $\sqrt{\lambda^{(n)}\Phi_u^{(n)}(X)}$ and $\sqrt{\lambda^{(n)}\Phi_{uw}^{(n)}(X)}$ computed in cases A and B respectively. In case B, the first two POD spatial modes (related to $u$ and $w$) exhibit significantly higher amplitudes compared to the subsequent modes. Such an observation is somewhat less pronounced in the other case for waves propagating with current, as the energetic flow structures are more balanced between the first POD modes within the common observation area. Furthermore, when for modes with $n > 2$, the bottom part

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Cumulative energy contribution of the first 40 POD modes: $K_1^N = \sum_{n=1}^N \lambda^{(n)} / \sum_{n=1}^{N_{\text{mod}}} \lambda^{(n)}$ expressed as a function of $N$.</figcaption></figure>

of the plane ($z < 1.2m$) exhibits very small amplitude values, especially in case B since the dominant flow structures are concentrated in the upper part of the measurement plane.

The frequency analysis of the first seven temporal modes reveal significant insights (Fig. 8). The main part of the energy is always contained in the low frequency range, corresponding to the energetic flow structure. However, some frequency peaks emerged for several POD modes at the wave frequency ($f_w = 0.45Hz$). This indicates that the propagating waves in the water column represent a highly coherent event within the flow. By zooming the representation around this wave frequency, one observes that the first POD modes do not behave similarly: the wave frequency peak does not emerge in every mode. The first POD modes are therefore simultaneously associated with the large scale energetic wake flow structure and the propagating coherent waves. The distribution of these two events across the modes is then directly conditioned by the characteristics of the flow. Since POD is a global method accounting for the whole available velocity database, the first two POD modes capture the spectral signatures of both the wake flow (dominant in the upper part) and the waves (more distinguishable in the lower part), especially in case B. As discussed previously (see Fig. 5), in case A the energetic content and the spatial coherence are more evenly distributed over the first POD modes leading to recover the wave spectral signature in more POD modes.

It should be noted that applying POD using only the streamwise velocity component or on both $(u, w)$ along a vertical line lead to a markedly different energy repartition. Clear identification of the wave's frequency peak can not be achieved in the first POD modes in such an application. In the present flow configurations, using the whole available velocity database effectively highlights the wave contribution, owing to the inclusion of the vertical velocity component, especially in the
<figure><img src="image_8.png" /><figcaption>Fig. 6. CaseA. From left to right: First five spatial POD coefficients: $\sqrt{\lambda^{(n)}}\Phi_u^{(n)}(\mathbf{X})$ (top line) and $\sqrt{\lambda^{(n)}}\Phi_w^{(n)}(\mathbf{X})$ (bottom line). A similar colormap is used in each graph.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Case B. From left to right: First five spatial POD coefficients: $\sqrt{\lambda^{(n)}}\Phi_u^{(n)}(\mathbf{X})$ (top line) and $\sqrt{\lambda^{(n)}}\Phi_w^{(n)}(\mathbf{X})$ (bottom line). A similar colormap is used in each graph.</figcaption></figure>

bottom region of the plane, where large scale flow structures are either absent or possess very low energy.

## 3.2. Wave-turbulence POD filtering process

This method has been intensively applied to turbulent flow to extract the large scale coherent energetic flow structures thanks to the first POD modes. POD has been also used as filtering-decomposition method to extract the acoustic wave embedded in the turbulent flow containing large scale flow structures (Druault et al., 2013) by exploiting the wavenumber-frequency characteristics of an acoustic wave. Following this idea, we propose to exploit the separation of time and space variables to filter the wave component from the fluctuating velocity component. Indeed, as discussed in the previous section, the wave frequency signature is clearly indicated in the first POD modes. To extract the wave counterpart of the raw velocity field, the following procedure is then conducted:

1. Perform a Fourier analysis of each POD mode: $\hat{a}^{(n)}(f) = |\hat{a}^{(n)}(f)|\exp(+j\alpha^{(n)}(f))$ where $j^2 = -1$ and the phase of each POD mode takes the form:

$$ \alpha^{(n)}(f) = \arctan \left( \mathfrak{F}(\hat{a}^{(n)}(f)) / \mathfrak{R}(\hat{a}^{(n)}(f)) \right) $$

where $(\mathfrak{F}, \mathfrak{R})$ indicate the imaginary/real part of a complex number.

2. Select the temporal POD modes that have to be filtered by only retaining the POD modes $\hat{a}^{(n)}(f)$ satisfying:

$$ |\hat{a}^{(n)}(f_w)| \ge K|\hat{a}^{(n)}(f_w + \Delta f)| \text{ and } |\hat{a}^{(n)}(f_w)| \ge K|\hat{a}^{(n)}(f_w - \Delta f)| $$

The spectral analysis of the modes allows us to set the parameters $(K, \Delta f)$ to 2 and 0.03Hz respectively. A discussion on the choice of the threshold $K$ is done at the end of this section.

Based on these criteria, POD mode numbers $N = N_A = [1, 2, 3, 5]$ and $N = N_B = [1, 2]$ are retained for cases A and B respectively.

3. Linearly interpolate each of $N$ temporal modes in the frequency domain around the wave peak $[f_w - \Delta f : f_w + \Delta f]$. An illustration of
this step in each flow configuration is presented for the first POD mode in Fig. 9. The filtered temporal modes are denoted $\hat{a}_t^{(n)}$.

4. Return to physical space by applying the inverse Fourier Transform. The modes associated with the turbulent part of the velocity field become $a_t^{(n)}(t) = \Re(\hat{a}_t^{(n)}(f)\exp(+j\alpha^{(n)}(f))$ for $n \in N$, and for $n \notin N$ the modes remain unchanged.

5. Reconstruct the instantaneous turbulent velocity field as follows:

$$u_t'(\mathbf{X}, t) = \sum_{n \in N} a_t^{(n)}(t) \Phi_u^{(n)}(\mathbf{X}) + \sum_{n \notin N} a_t^{(n)}(t) \Phi_u^{(n)}(\mathbf{X}) \quad (3)$$

$$w_t'(\mathbf{X}, t) = \sum_{n \in N} a_t^{(n)}(t) \Phi_u^{(n)}(\mathbf{X}) + \sum_{n \notin N} a_t^{(n)}(t) \Phi_u^{(n)}(\mathbf{X}) \quad (4)$$

with $N$ equal to $N_A$ or $N_B$ as a function of the test case.

The fluctuating instantaneous wave components ($u'_t, w'_t$) are then deduced by computing the difference $u'_t = u' - u'_t$ and $w'_t = w' - w'_t$.

Such a decomposition satisfies the following properties: $\overline{u'_t} = 0, \overline{w'_t} = 0, \overline{u'_t} = 0$ and $\overline{w'_t} = 0$. However, since there is an overlapping frequency range between turbulence and wave components, their cross correlation is non zero (see §4).

Note that for the present POD application, only the first 10 POD modes were considered for the mode selection in the wave field extraction. Higher order POD modes each contain less than 1% of the total energy and only a limited number of them satisfies the required conditions. Their contribution to the results is therefore negligible and it has no effect on the results. Furthermore, a sensitivity analysis with respect to the parameter $K$ was conducted using values of $K = 1.5, 2$ and $2.5$. For both cases (waves following or opposing the current), the Kinetic Energy associated with the extracted wave field $\frac{1}{2}(\overline{u'_t}^2 + \overline{w'_t}^2)$ was determined along the vertical line $x_0$. When using $K = 2$ or $K = 2.5$, the same POD modes are selected, resulting in identical vertical profiles of the wave-induced kinetic energy. When $K = 1.5$ is used, two additional modes are included (modes 4 and 7 for case A and modes 4 and 9 for case B) compared to the $K = 2$ selection. Only minor differences are observed between the results obtained with $K = 1.5$ and $K = 2$. Based on

this analysis, a value of $K = 2$ is retained for both cases in the remainder of this study.

Such a FFT filtering process could be performed at each grid point. However, such an application would require a manual inspection of the spectra associated with each velocity component at each grid point. This process would be fastidious and highly dependent of the subjective visual judgment. Moreover, the characteristics of the velocity spectra vary with the grid position as well as with the properties of the background turbulent flow and its spectral content. The width of the spectrum around the wave frequency is more or less narrow, making it difficult to specify a single frequency band and a unique $K$ value, suitable for each spectrum.

### 3.3. POD Filtering process validation

In this section, the POD wave-turbulence decomposition is successfully examined through instantaneous, mean and statistical analyses.

#### 3.3.1. Instantaneous analysis

The turbulent and wave velocity components are first compared instantaneously with the raw velocity measurements (see Figs. 10 and 11). These components are extracted at two selected locations: $(x_0 = 0, z_1 = 1.52)$ and $(x_0 = 0, z_2 = 0.88)$ located at the top and bottom part of the measurement plane. For both test cases, the top location corresponds to a vertically sheared current with large-scale flow structures, while the bottom location corresponds to a more uniform flow, especially in case B. The graph also includes the temporal wave signal recorded by one of the three wave probes. Note that the location of the wave probe is not exactly above the vertical line at $x_0$ but slightly decays downstream (see Fig. 1), justifying the slight difference between the phase alignment of the wave component and the free surface signal. The passage of the large scale structures is clearly indicated at $z_1$ location in both cases, where large velocity fluctuations exhibit an almost periodical signal at a lower frequency than $f_w$.

The signature of these structures is also visible at the bottom for case A, even if the amplitude of the velocity fluctuations are much smaller than at the top. In contrast, at the bottom position for case B, the velocity fluctuations are dominated by the wavy component. On the other

<figure><img src="image_10.png" /><figcaption>Fig. 8. Spectral analysis of the first seven POD temporal modes $|a_t^{(n)}(f)|$ with $n$ varying from 1 to 7. Left: Case A. Right: Case B.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9. Illustration of the spectral filtering process for the first POD temporal modes $|a_t^{(1)}(f)|$ (black) and $|a_t^{(n)}(f)|$ (red). Note that both spectra are identical apart around the wave frequency. Left: Case A. Right: Case B.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Case A. Comparison of instantaneous fluctuating streamwise (left column) and vertical (right column) velocity components (expressed in m/s) extracting at the top ($x_0 = 0, z_0 = 1.52$) and bottom ($x_0 = 0, z_0 = 0.88$) of the measurement plane. The fluctuating temporal wave signal (in blue and expressed in m) is superimposed onto the fluctuating velocity fields.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Case B. Comparison of instantaneous fluctuating streamwise (left column) and vertical (right column) velocity components (expressed in m/s) extracting at the top ($x_0 = 0, z_0 = 1.52$) and bottom ($x_0 = 0, z_0 = 0.88$) of the measurement plane. The fluctuating temporal wave signal (in blue and expressed in m) is superimposed onto the fluctuating velocity fields.</figcaption></figure>

hand, as expected, the amplitude of both wave components decreases with depth and the two components $u'_w$ and $w'_w$ are out of phase. Interestingly, regardless of the direction of wave propagation, the wave vertical component consistently exhibits higher amplitudes than those of the streamwise component. In the presence of convective vortices, the injected energy due to the surface wave motion (opposing or following the current) is primarily observed in the vertical velocity component.

Figs. 12 and 13 present the normalized PDF histograms of the raw $u'$, turbulent $u'_i$ and wave $u'_w$ fluctuating velocity fields. These PDF are computed along the vertical line at $x_0$ within the water column for both velocity components. An asymmetric distribution appears in the PDF of the raw velocity field, reflecting the presence of the large scale convective flow structures and the wave propagation through the water column. When analysing the PDF of the turbulent components, one can clearly identifies the large scale flow structures that correspond to asymmetric peaks corresponding to strong fluctuations. This effect is more pronounced in case A where these flow structures are more energetic and broader spatially. Around $z = 0.7$m, a Gaussian velocity

distribution is observed in both cases with a very low level of fluctuations, as the current is mainly uniform and of low energetic content. The PDFs of wave velocity fields exhibit a classical bimodal distribution, with two peaks symmetrically distributed around zero. In case B, the peaks of both turbulent and wave component PDFs are of a similar magnitude. The peaks of the fluctuating vertical wave component follows a quasi-linear vertical evolution in close agreement with the linear wave theory in an uniform current. Near the surface, the mean shear current slightly modifies the vertical evolution of the $u'_w$ PDF peaks. In case A, the wave scattering effect due to the convective mean vertically sheared flow is also clearly visible between $z = 0.8$m to $z = 1.6$m.

### 3.3.2. Highest amplitudes of the wave orbital velocities across the water column

By exploiting instantaneous wave components, the highest amplitudes of each fluctuating wave component in the water column are accessed. This is done by time averaging the maximum amplitude of the wave signal in each case. Fig. 14 represents the mean amplitudes of
<figure><img src="image_14.png" /><figcaption>Fig. 12. Case A. Histogram of the fluctuating velocities along the vertical line $x_0$. Left to right: Raw measurements, turbulent component and wave component. Top (streamwise velocity component), Bottom (vertical velocity component). A same colormap is used in each graph.</figcaption></figure>

both streamwise and vertical wave components (denoted $u_{orb}$ and $w_{orb}$) along the vertical direction (at $x_0$). Present results are superimposed to those determined thanks to a different approach using a sinusoidal fitting procedure based on a least squares method (Gaurier et al., 2025). The results closely match those previously obtained. Some slight differences only appear at the top when waves propagate against the current (case B).

The vertical wave (orbital) component exhibits a significant amplitude, especially in B flow configuration, which contributes to modify the mixing within the water column as further discussed in Section 4.2.

## 4. Analysis of the wave-turbulence energetic exchanges

The wave-turbulence energy exchanges are of particular importance for understanding how the energy of the wave surface motion is vertically transferred across the water column. Particularly valuable are the modifications of the integral time scale, the energy transfers involving Turbulence Kinetic Energy (TKE), strongly connected to the intensity of turbulence and more specifically to the TKE budget. These aspects are examined in the following section which is concluded with a quadrant

analysis aimed at investigating how wave action modifies flow events across the water column.

### 4.1. Wave effects on the flow integral time scales

As done previously to validate the wave-turbulence separation (Chávez-Dorado et al., 2025), a statistical analysis is now performed by first analysing the following temporal correlation:

$$R_v(x_0, z, \tau) = \overline{v'(x_0, z, t)v'(x_0, z, t+\tau)} \quad (5)$$

where $v'$ corresponds to the streamwise fluctuating components of the raw, turbulent or wave velocity fields. The correlation levels of the wave components ($R_{u,w}$ and $R_{w,u}$) are significantly lower than those derived from raw velocity fields. Consequently, to better visualise the temporal evolution of the correlations, Fig. 15 represents all correlation values normalised with $R_u(\tau = 0)$ and obtained at two specific heights: $z = 1.4m$ and $z = 1.02m$. The wave temporal correlations exhibit oscillations corresponding to the harmonic wave oscillations at $f_w$. This wave signature is also visible in the raw temporal correlation. In contrast, the turbulence correlation no longer shows this periodic behavior, indicating that the temporal periodicity associated with the wave motion has been effectively removed. Furthermore, the correlation coefficient provides
<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Case B. Histogram of each fluctuating velocity along the vertical line $x_0$. Left to right: Raw measurements, turbulent component and wave component. Top (streamwise velocity component), Bottom (vertical velocity component). A same colormap is used in each graph.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 14.** Vertical profile of the time averaged highest amplitude of the wave streamwise (left graph) and vertical (left graph) components computed from the present POD method (red markers). Superimposition to those computed from a sinusoidal fitting procedure based on a least squares method (Gaurier et al., 2025) (black markers).</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Fig. 15. Normalized temporal correlation $R_v(\tau) = \frac{R_v(x_0, z, \tau)}{R_v(x_0, z, \tau=0)}$ with $v = u', u'_t, u'_w$ computed at two selected z positions: $z = 1.4$ m (top) and $z = 1.02$ m (center). Left: Case A. Right: Case B.</figcaption></figure>

insight into the characteristic temporal scales of the flow, which is defined as the time to the first zero crossing or a fixed threshold. The turbulence correlation $R_{u_t}$ allows the recovering of the integral time scales representative of the free flow without wave (Gaurier et al., 2025). It is also observed that raw velocity field has a shorter correlation length compared with the turbulent flow alone, confirming previous analyses that waves propagating in the water column greatly affect the integral scales of the flow (Gaurier et al., 2025; Magnier et al., 2023).

## 4.2. Turbulence kinetic energy exchanges

Using wave-turbulence POD decomposition, the normal Reynolds tensor components: $\overline{u'^2}$ and $\overline{w'^2}$ are expressed as follows:

$$ \overline{u'^2} = \overline{u_t'^2} + \overline{u_w'^2} + 2\overline{u_t'u_w'} \quad \text{and} \quad \overline{w'^2} = \overline{w_t'^2} + \overline{w_w'^2} + 2\overline{w_t'w_w'} \quad (6) $$

To study the contribution of each term, the following quantities are determined:

$$ R_{tt}^{u} = \frac{\overline{u_t'^2}}{\overline{u'^2}}; \quad R_{ww}^{u} = \frac{\overline{u_w'^2}}{\overline{u'^2}} \quad \text{and} \quad R_{tw}^{u} = \frac{2\overline{u_t'u_w'}}{\overline{u'^2}} \quad (7) $$

$$ R_{tt}^{w} = \frac{\overline{w_t'^2}}{\overline{w'^2}}; \quad R_{ww}^{w} = \frac{\overline{w_w'^2}}{\overline{w'^2}} \quad \text{and} \quad R_{tw}^{w} = \frac{2\overline{w_t'w_w'}}{\overline{w'^2}} \quad (8) $$

Figs. 16 and 17 display the vertical distribution (at the streamwise position $x_0$) of each quantity for cases A and B respectively. For wave following current (A case), as expected, the turbulent contribution $R_{tt}^u$ and $R_{tt}^w$ accounts for more than 85% of the total energy. A slightly smaller value is observed near the bottom area for the streamwise Reynolds component reflecting the low energetic content of the turbulent current in this region. At this position, the wave kinetic energy $\overline{u_w'^2}$ becomes slightly more significant. A different behavior is obtained when the waves propagate against the current (case B). In this case, the wave kinetic energy parameters $R_{ww}^u$ and $R_{ww}^w$ reach more than 20% and 40% of the total energy respectively, across the water column. The

total (raw) kinetic energy is more evenly distributed between the turbulent and wave contributions, especially for the vertical component. For waves against the current, the surface water wave forcing is considerably stronger, leading to promote the kinetic energy exchange in the water column. This is also correlated to the low energy content of the free wake flow in that case, which amplifies the wave effect. Thus, the wave-turbulent interactions ($R_{tw}^u$ and $R_{tw}^w$) are far from negligible in this last flow configuration. In both cases, near the top position ($z > 1.4$), the shear components $\overline{u_t'u_w'}$ and $\overline{w_t'w_w'}$ increase as the energy associated with both wave components reaches its maximum near the water-air interface. To better highlight the energetic coupling between the wave and turbulent components, Fig. 18 shows the vertical profiles of $\overline{u_t'u_w'}$ and $\overline{w_t'w_w'}$ for both flow configurations. For waves following the current (case A), the shear term $\overline{u_t'u_w'}$ remains weak and nearly constant along the vertical direction. Conversely, the wave-turbulence interaction becomes much more significant when waves propagate against the current. This is remarkable considering that in this case, the wavelength of the surface waves is 2.5 times shorter than in case A while the wave amplitude is only about 1.5 times higher (see Table 1). Based on the proposed POD wave-turbulence separation and the orthogonality of POD modes, these cross-terms express as follows:

$$ \overline{u_t'u_w'} = \sum_{n \in N} \Phi_u^{(n)}(\mathbf{X}) \overline{a_t^{(n)}(t)}(a^{(n)}(t) - a_t^{(n)}(t)) \quad (9) $$

$$ \overline{w_t'w_w'} = \sum_{n \in N} \Phi_w^{(n)}(\mathbf{X}) \overline{a_t^{(n)}(t)}(a^{(n)}(t) - a_t^{(n)}(t)) \quad (10) $$

These terms arise from the overlapping frequency range between wave and turbulent fields. They are of higher amplitudes in the shear flow area ($z > 1.2$ m) where both the turbulent activity and the wave contributions are strongest. The magnitude of these cross-terms is then directly correlated to the spectral filtering applied to the N POD modes. These modes associated with either wave or turbulent fields are not mutually orthogonal themselves, thereby traducing the interaction between wave and turbulent fields.
<figure><img src="image_18.png" /><figcaption>Fig. 16. Case A. Vertical profiles (at $x_0$) of the wave and turbulence contributions (expressed in percentage value) in the normal Reynolds components: $\overline{u'}^2$ (left graph, Eqs. 7) and $\overline{w'}^2$ (right graph, Eqs. 8).</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 17. Case B. Vertical profiles (at $x_0$) of the wave and turbulence contributions (expressed in percentage value) in the normal Reynolds components: $\overline{u'}^2$ (left graph, Eqs. 7) and $\overline{w'}^2$ (right graph, Eqs. 8).</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 18. Vertical profiles (at $x_0$) of $\overline{u'_t u'_w}$ and $\overline{w'_t w'_w}$ in cases A (Left) and B (Right).</figcaption></figure>

As a synthesis, the kinetic energy transfer across the water column is enhanced when waves propagate against the current but these vertical exchanges greatly also depend on the water surface proximity and the kinetic energy level of the free flow without waves.

### 4.3. Analysis of the production term in the TKE budget

In order to study the vertical transfer of kinetic energy across the water column, the TKE budget is now considered with a focus on the production of the TKE that is the transfer of energy from the average movement to the turbulent and wave motions. As two dimensional experimental databases are only available, the production term is expressed as follows (Ikhennicheu et al., 2020):

$$ - \left( \overline{u' u'} \frac{\partial \overline{U}}{\partial x} + \overline{w' w'} \frac{\partial \overline{W}}{\partial z} + \overline{u' w'} \frac{\partial \overline{U}}{\partial z} + \overline{w' w'} \frac{\partial \overline{W}}{\partial z} \right) \quad (11) $$

In each flow configuration, the large scale flow structures dominate over wave induced fluctuations. Thus, the computation of the production terms mainly reflects the TKE production associated with these

large scale flow structures. Consequently, although the PIV mesh grid resolution is likely insufficient, it is believed that the mean velocity gradients and hence the production terms are not underestimated to the extent typically expected for coarse PIV meshgrid. However, further investigations employing finer spatial resolution will be necessary to confirm it.

The vertical profiles (extracted at $x_0$) of each term are represented in Fig. 19 for both cases. When waves propagate with the current (case A), for $z < 1.3$ m, the production of TKE is mainly driven by $\overline{u' w'} \frac{\partial \overline{U}}{\partial z}$ which is associated with the turbulent coherent structures. The peak of production is then reached around $z = 1.2$ m corresponding to the large scale flow structure passage. In the upper part ($z > 1.3$ m), the production is more balanced between each term of Eq. 11 as the influence of the waves increases and modifies each Reynolds tensor component individually.

When waves propagate against the current (case B), the production term exhibits a pronounced peak around $z = 1.35$ m corresponding to the main zone of the TKE production, $\overline{u' w'} \frac{\partial \overline{U}}{\partial z}$ driven by the turbulent flow structures. Approaching the water surface, the production becomes more evenly shared among the various terms.
<figure><img src="image_21.png" /><figcaption>Fig. 19. Vertical profiles of the production term and each of its contribution (see Eq. 11). Left: case A. Right: case B.</figcaption></figure>

To go further, the main contribution of the production $\overline{u'w'} \frac{\partial \overline{U}}{\partial z}$ is now decomposed as follows accounting for the wave-turbulence separation:

$$ \underbrace{-u'w'}_{P} \frac{\partial \overline{U}}{\partial z} = \underbrace{-u'_{t}w'_{t}}_{P_1} \frac{\partial \overline{U}}{\partial z} \underbrace{-u'_{w}w'_{w}}_{P_2} \frac{\partial \overline{U}}{\partial z} \underbrace{-u'_{t}w'_{t}}_{P_3} \frac{\partial \overline{U}}{\partial z} \underbrace{-u'_{w}w'_{w}}_{P_4} \frac{\partial \overline{U}}{\partial z} \quad (12) $$

The vertical transport of momentum in the water column is then quantified by analysing each term separately (see Fig. 20). For wave propagating with current, the main contribution of the shear stress is the turbulent part ($P_1$ term) which is almost identical to the term $P$, apart at the top position where wave effects become non negligible due to the water surface proximity. Outside, the large scale coherent structures dominates the wave effect. For waves against the current (case B), the amplitudes of $P_1$ are always quite higher than those of $P$ across the water column. The production of turbulence that is the vertical transport of momentum is then balanced by the wave effect, especially through $\overline{u'_{w}w'_{w}} \frac{\partial \overline{U}}{\partial z}$ and $\overline{u'_{w}w'_{t}} \frac{\partial \overline{U}}{\partial z}$. Since the fluctuating kinetic energy of the wave-turbulent flow originates from both the wave and turbulent components in comparable proportions (see Fig. 17), the shear stress is similarly balanced between these two contributions. This suggests that even if the influence of wave energy is stronger, contributing more significantly to the kinetic energy of the turbulent flow (see Fig. 17), the presence of waves in this case tends to attenuate vertical mixing within the water column. As previously mentioned (see Section 4.1), the large scale flow structures become distorted across the water column under the wave action and their associated integral time scale is reduced. This distortion arises primarily from the mean flow and especially from the action of the wave orbital motion as it has been previously shown in Langmuir turbulence at moderate Reynolds number (Xuan et al., 2019). In the present flow configuration where $\frac{\partial \overline{U}}{\partial z} < 0$ (Fig. 2), the highest amplitudes of both wave components satisfy $\frac{\partial \overline{U_{orb}}}{\partial z} > 0$ and $\frac{\partial \overline{U_{orb}}}{\partial z} \ge 0$ (see Fig. 14). The convective vortex strained by the wave orbital velocities are then shortened, thereby diminishing the vertical mixing. Conversely, it has been shown that in a free mean flow satisfying $\frac{\partial \overline{U}}{\partial z} > 0$, waves propagating against the current increase both the integral time scale and the vertical mixing (Magnier et al., 2023). In such conditions, wave effects are expected to enhance the energy transfer from the average motion to the turbulent flow.

## 4.4. Quadrant analysis

To investigate the flow structures distortion within the water column, a quadrant analysis is conducted. Such an analysis (Wallace, 2016) allows the determination of four flow events related to the sign of the fluctuating velocity components: $Q_1$ ($u' > 0$ and $w' > 0$), $Q_2$ ($u' < 0$ and $w' > 0$), $Q_3$ ($u' < 0$ and $w' < 0$) and $Q_4$ ($u' > 0$ and $w' < 0$). At each $z$ position and for all velocity fields (raw, turbulent and wave), the proportion of each event is successively determined. The vertical profiles presented in Figs. 21 and 22 are spatially averaged over the streamwise

locations $x \in [-0.08 : 0.08]$ for which instantaneous velocity fields are available throughout the entire vertical extent of the PIV measurement plane. In the following figures, $Q_1$, $Q'_1$ and $Q'_t$ ($\omega$ with $i$ varying from 1 to 4) correspond to the quadrant proportion associated with raw, turbulent and wave velocity fields respectively. The results for raw and turbulent fields are superimposed on the same graph (left hand side in each figure) using solid and dashed lines, respectively. The quadrant analysis of the wave velocity field is displayed on the right hand side using dotted lines.

When waves propagate with current (Case A), each event related to the wave field contributes similarly in the water column, accounting for approximately 25% each. Consequently, when comparing results from raw and turbulent velocity fields, the flow event repartition is slightly modified. The most significative difference lies in the reduction of $Q_4$ events. Globally, waves act to reduce the event dispersion while keeping an event distribution in the raw velocity field similar to the turbulent velocity one. Fig. 23 shows the repartition of the events at a specific height ($z = 1m, x = 0$), highlighting that the amplitudes of the fluctuating wave components are considerably lower than those related to the turbulent field. This can be related to the previous comparison of the velocity fluctuation levels between water surface and raw velocity signals (see Fig. 4). In this case, the waves slightly modify the energetic convective events $Q_1$ and $Q_4$, and their overall influence on the event distribution remains limited.

In contrast, significant differences appear when waves propagate against the current. The wave field exhibits a clear distinction between events. In the upper part of the water column ($z > 1.2m$) $Q_1$ and $Q_3$ events each represent about 28% while $Q_2$ and $Q_4$ each account for 22% (Fig. 22). The gap between these pairs of events increases as $z$ decreases. In this case, waves lead to preferentially produce inward and outward interactions within the water column and their effect on each event becomes much more pronounced. Indeed, the amplitude of the wave fluctuations approaches that of the turbulent field, leading to significantly modify the convective flow structures. A representative illustration is provided in Fig. 24 where the instantaneous fluctuating velocity components extracted at a specific location ($z = 1.3m, x = 0$) are represented following the quadrant analysis. The distribution of the flow events in both raw and turbulent fields exhibit different features. The results obtained for the turbulent field are quite similar to those observed in a similar free wake flow without waves (Gaurier et al., 2025). Waves induce a change in momentum direction evidences by a great decrease in $Q_3$ events. The raw fluctuating velocity field becomes more uniformly distributed among the different flow events. Waves propagating against the current thus tend to homogenise the repartition of flow events while increasing the amplitude of the raw velocity fluctuations. At this specific location, the amplitudes of the Reynolds tensor components determined for the raw, turbulent and wave velocity fields are summarized in Table 2. At this position, the TKE related to turbulent and wave fields represent 60% and 35% of the raw TKE, respectively. The sum of the turbulent and wave contributions (105%) exceeds 100% of the raw TKE. This
<figure><img src="image_22.png" /><figcaption>Fig. 20. Vertical profiles of the production term $-\overline{u'w'} \frac{\partial \overline{U}}{\partial z}$ and each of its contribution $P_i$ with $i = 1, 4$ (see Eq. 12). Left: case A. Right: case B.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. 21. Case A. Left: Vertical profiles of the quadrant proportion associated with the raw ($Q_i$, $i = 1:4$, solid line) and turbulent ($Q'_i$, $i = 1:4$, dashed line) velocity. Right: Vertical profiles of the quadrant proportion associated with the wave velocity field ($Q''_i$, $i = 1,4$).</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 22. Case B. Left: Vertical profiles of the quadrant proportion associated with the raw ($Q_i$, $i = 1:4$, solid line) and turbulent ($Q'_i$, $i = 1:4$, dashed line) velocity. Right: Vertical profiles of the quadrant proportion associated with the wave velocity field ($Q''_i$, $i = 1,4$).</figcaption></figure>

<caption>**Table 2**</caption>

Case B. Amplitudes of the Reynolds components of the raw, turbulent and wave fields at ($z = 1.3m, x = 0$).

<table><thead><tr><th></th><th>u&#773;<sup>2</sup></th><th>w&#773;<sup>2</sup></th><th>k</th><th>u'w'</th></tr></thead><tbody><tr><td>Raw (u', w')</td><td>0.0088</td><td>0.0094</td><td>0.0091</td><td>-0.0019</td></tr><tr><td>Turbulent (u', w')</td><td>0.0062</td><td>0.0047</td><td>0.0055(60%)</td><td>-0.0023</td></tr><tr><td>Wave (u', w')</td><td>0.0027</td><td>0.0037</td><td>0.0032(35%)</td><td>0.0003</td></tr></tbody></table>

excess is directly correlated to the wave-turbulence cross-terms as discussed previously (see Eqs. 9–10 and Fig. 18). It is then confirmed that the flow reorganisation leads to increase the Turbulent Kinetic Energy, accompanied by a reduction of the shear stress component as discussed in previous sections.

In previous sections, it was demonstrated that the wave propagation across the water column impacts the integral time scales, the TKE, the turbulence production and the instantaneous flow events. A prelim-

inary attempt is now carried out to better understand how convective eddies are distorted under the wave effect in the water column. This is done by investigating the temporal evolution of flow event repartitions for the raw, turbulent and wave velocity fields, evaluated separately at the same specific location ($z = 1.3m, x = 0$, see Fig. 25). Only case B results are presented since they are the most representative. The harmonic wave field displays a regular temporal pattern of flow event distribution across quadrants from $Q_4$ to $Q_1$. Since the wave dominant $Q_1$ and $Q_3$ flow events in the wave field have some amplitudes comparable to those found in the turbulent field, the raw velocity field is instantaneously affected. Indeed, the time evolution of the raw flow events shows that the occurrence of $Q_1$ events is directly correlated with those of the wave fields. The $Q_4$ events (sweeps) of the turbulent velocity field corresponding to low amplitude vertical velocity components, are then directly influenced and modified throughout the water column. The convective flow structures seem to be altered along the vertical direction of wave propagation. A more comprehensive assessment of such distortions would require additional experimental data, particu-
<figure><img src="image_25.png" /><figcaption>**Fig. 23.** Case A: Repartition of events at (z = 1m, x = 0). Left to right: Raw velocity field, turbulent velocity and wave fields.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Fig. 24.** Case B: Repartition of events at (z = 1.3m, x = 0). Left to right: Raw velocity field, turbulent velocity and wave fields.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>**Fig. 25.** B case: Time evolution ([80 : 160]s) of the flow events related to each flow contribution (raw, turbulent and wave fields) extracted at (z = 1.3, x = 0): Q₁-black, Q₂-blue, Q₃-red, Q₄-green.</figcaption></figure>

larly with a larger measurement domain. This distortion seems to be reinforced when waves propagate against the current. Such a preliminary result can be related to previous work (Mouchref et al., 2024) where quadrant analysis revealed the modification of the flow events in a wind turbine wake submitted to surface waves. Further experiments involving additional wave-turbulence flow configurations with varying wave amplitudes and frequencies are required to gain deeper insight into the wave-turbulence mechanisms originating from these quadrant event modifications.

## 5. Conclusion

To properly characterize future tidal energy sites where large scale flow structures coexist with a wavy environment, it is essential to gain a deeper understanding of wave-vortex interactions in the water column. By analysing experimental PIV database obtained in a flume tank, a new

post-processing method based on Proper Orthogonal decomposition is implemented to extract the wave velocity field from the high energetic complex turbulent flows. When regular waves interact with large scale flow structures, the proposed wave-turbulence decomposition prove effective for at least two reasons: i) the vertical turbulent component is of the same order of magnitude as the wavy vertical component; ii) wave propagation dominates the turbulent part within a specific area, corresponding here to the bottom part of the measurement field.

Present wave-turbulence decomposition has been validated through the analysis of the properties of the extracted wave fields. The wave effects become more significant when the levels of velocity fluctuations in the flow approach those of the water surface fluctuations. For instance, in case B, the wave action extends deeper into the water column and is more intense, being further amplified when waves propagate against the flow and also when waves amplitude is high. The wave effects on the energetic vertical exchanges are primarily governed by three factors: the water surface proximity, the levels of kinetic energy in the free flow without waves and the propagation direction of the surface waves. These effects increase when waves propagate against the current, especially through an enhanced contribution of the wave contribution to the total flow kinetic energy.

The effect of the sign of the mean vertical velocity gradient on the vertical mixing has been also identified. For the present flow configurations, a negative vertical gradient is observed in the flow: $\frac{\partial \bar{U}}{\partial z} < 0$. Thus, knowing that the vertical gradient of the vertical wave orbital velocities is always positive, regardless of the wave propagation direction, the waves tend to reduce vertical mixing by limiting the kinetic energy vertical transfer. Present results highlight the importance of accounting for the orbital wave velocity when characterizing vertical wave-turbulence energy transfers.

Furthermore, when the level of the turbulent kinetic energy becomes comparable to that of the surface wave fluctuations, the waves greatly alters the vortices. Some preliminary results based on quadrant analysis emphasise the flow event modifications across the water column. This permits to explain that wave propagation modifies the integral time scale of the free flow across the water column, which is confirmed by the analysis of the temporal correlation of the wave and turbulent velocity signals. To gain deeper insight into the mechanisms driving wave-vortex
distortions in such a high Reynolds number turbulent shear flows, access to fully three dimensional numerical or experimental database would be beneficial.

To conclude, current work provides an improved understanding of the wave-turbulence interactions in a high Reynolds number turbulent shear flow, especially in terms of vertical energy exchanges between wave and large scale flow structures. Although the results in the present paper are based on regular waves, the conclusions can still be helpful to the numerical modelling of the wave-turbulence interaction. Furthermore, present results have important implications for the characterization of the tidal turbine farm sites especially at sites where both large scale flow structures and wavy environment coexist. Accounting for wave turbulence interactions prior to analyzing each contribution separately appears essential, as it would permit to better anticipate and predict unsteady tidal turbine power fluctuations, ultimately aiding in the selection of optimal tidal turbine farm sites.

In the present study, POD is proposed as an alternative wave-turbulence separation method for turbulent flows where waves are embedded in a highly complex shear flow containing large scale flow structures. The main advantage of POD in this context lies in its ability to provide an optimal energetic flow decomposition. Present method, like other existing approaches, is strongly dependent on the available database. For instance, it depends not only on the time/frequency content but also on the spatial domain (spatial extent and spatial discretization). Indeed, POD is a global method whose effectiveness is demonstrated here by taking into account the turbulent velocity fields within a measurement plane where both large scale flow structures and waves are present either separately or fully embedded within one another. A comparative analysis with other existing wave-separation methods would then be beneficial even if up to now, no universal criterion currently exists for accurately estimating wave energy content in highly complex turbulent shear flows and thus for objectively validating wave-turbulence separation approaches.

One potential improvement to the present POD approach would be to perform the decomposition in frequency domain instead of the time domain, that permits to avoid the mode mixing as previously noted when considering the EMD method (Bian et al., 2020) and observed here in the POD time flow decomposition. In the presence of large scale flow structure fluctuations, spectral POD would certainly meet the same constraint due to the overlapping frequency range between turbulence and wave motions. However, a similar additional filtering process could still be applied to the spectral modes.

From a general point of view, using a linear decomposition to separate turbulent and wave velocity fields implicitly assumes that the two components are linearly independent. However, this assumption does not hold in practice. That explains why after applying the POD method, certain POD modes have to be filtered to try to reproduce favorably each part of the signal. Then, two steps are at least required to separate wave and turbulent contributions, which are intrinsically coupled. Wave and turbulent motions coexist in realistic current flow and they are associated with very different time and space length scales as well as with different energy distribution across broadband spectra. Then, present method, as applied here, would not be directly suitable for irregular waves with a broadband spectrum. Nevertheless, an extension of the present method is feasible. If the wavenumber frequency spectrum of the wave field in the water column is known, it becomes possible to combine POD with FFT based filtering method to separate wave and turbulence across the full wavenumber frequency domain. A similar approach has been already applied in the context of acoustic-hydrodynamic separation (Druault, 2017) based on the knowledge of the wavenumber frequency spectrum of the acoustic component. This approach requires a dataset with sufficient wavenumber and frequency resolution to support such an analysis. Present results need then to be complemented by testing numerous other flow configurations with varying amplitude and frequency wave following or opposing the current, and possibly irregular waves with a broad wave frequency range.

## CRediT authorship contribution statement

**Philippe Druault:** Writing – original draft, Writing – review & editing, Visualization, Software, Investigation, Formal analysis; **Benoît Gaurier:** Writing – review & editing, Data curation, Conceptualization; **Grégory Germain:** Writing – review & editing, 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.

## Acknowledgements

This study was supported by funding from the Hauts-de-France state-region contracts (CPER) IDEAL (2021–2028) and by the French Agence Nationale de la Recherche through the Verti-Lab project (ANR-23-LCV1-0009-01).

## References

Bian C., Liu X., Zhou Z., Chen Z., Wang T., Gu Y., 2020. Calculation of winds induced bottom wave orbital velocity using the empirical mode decomposition method. J. Atmos. Ocean. Technol. 37, 889–900.

Bricker J.D., Monismith S., 2007. Spectral wave-turbulence decomposition. J. Atmos. Oceanic Technol. 24, 1479–1487.

Chávez-Dorado J., Scherl I., Dibenedetto M., 2025. Wave and turbulence separation using dynamic mode decomposition. J. Atmosp. Oceanic Techn. 42, 509–526.

Druault P., 2017. Analysis of the acoustic/hydrodynamic separation in an aeroacoustic 2d plane mixing layer flow. In: 23rd AIAA/CEAS Aeroacoustics Conference.

Druault P., Gaurier B., Germain G., 2025. Impact of varying turbulent flow conditions on the tidal turbine blade load fatigue. Renewable Energy 123370.

Druault P., Germain G., 2022. Prediction of the tidal turbine power fluctuations from the knowledge of incoming flow structures. Vol. 252. Ocean Eng.

Druault P., Hekmati A., Ricot D., 2013. Discrimination of acoustic and turbulent components from aeroacoustic wall pressure field. J. Sound Vib. 332, 7257–7278.

Furgerot L., Sentchev A., Bois P. B.D., Lopez G., Morillon M., Poizot E., Mér Y., Benis A.C., 2020. One year of measurements in alderney race: preliminary results from database analysis. Philos. Trans. Ser. A., Math. Phys. Eng. Sci. 378.

Gaurier B., Druault P., Germain G., 2025. Wave effects on large scale turbulent flow structures propagating in the water column. Marine Energy Res. 2, 10020.

Holmes P., 2012. Turbulence, coherent structures, dynamical systems and symmetry. Cambridge university press.

Hussain A., Reynolds W., 1970. The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 41, 241–258.

Ikhennicheu M., Druault P., Gaurier B., Germain G., 2020. Turbulent kinetic energy budget in a wall-mounted cylinder wake using PIV measurements. Ocean Eng. Vol. 210.

Magnier M., Gaurier B., Germain G., Druault P., 2022. Analysis of the wake of a wide bottom-mounted obstacle in presence of surface wave following tidal current. In: Trends in Renewable Energies Offshore, pp. 151–159.

Magnier M., Saouli Y., Gaurier B., Druault P., Germain G., 2023. Modification of the wake of a wall-mounted bathymetry obstacle submitted to waves opposing a tidal current. Ocean Eng. 288, 116087.

Mcwilliams J.C., 2016. Submesoscale currents in the ocean. Proc. R. Soc. A: Math. Phys. Eng. Sci. 472.

Mercier P., Ikhennicheu M., Guillou S., Germain G., Poizot E., Grondeau M., Thiébot J., Druault P., 2020. The merging of kelvin-helmholtz vortices into large coherent flow structures in a high reynolds number flow past a wall-mounted square cylinder. Ocean Eng., 204, 107274.

Mouchref C., Viggiano B., Fercak O., Bossuyt J., Ali N., Meneveau C., Gayme D., Cal R., 2024. Wave-phase dependence of reynolds shear stress in the wake of fixed-bottom offshore wind turbine via quadrant analysis. J. Renew. Sustain. Energy 16.

Mullings H., Draycott S., Thiébot J., Guillou S., Mercier P., Hardwick J., Mackay E., Thies P., Stallard T., 2023. Evaluation of model predictions of the unsteady tidal stream resource and turbine fatigue loads relative to multi-point flow measurements at raz blanchard. Energies 16 (20), 7057.

Ouro P., Mullings H., Christou A., Draycott S., Stallard T., 2024. Wake characteristics behind a tidal turbine with surface waves in turbulent flow analyzed with large-eddy simulation. Phys. Rev. Fluids 9.

Ouro P., Stoesser T., 2019. Impact of environmental turbulence on the performance and loadings of a tidal stream turbine. Flow Turb. Comb. 102, 613–639.

Perez L., Gossu R., Grinham A., Penesis I., 2020. Evaluation of wave-turbulence decomposition methods applied to experimental wave and grid-generated turbulence data. 218. Ocean Eng.

Qiao F., Yuan Y., Deng J., Dai D., Song Z., 2016. Wave-turbulence interaction-induced vertical mixing and its effects in ocean and climate models. p. 374.

Roy S., Debnath K., Mazumder B., 2017. Distribution of eddy scales for wave current combined flow. Appl. Ocean Res. 63, 170–183.
Roy S., Samantaray S., Debnath K., 2018. Study of turbulent eddies for wave against current. Vol. 150. Ocean Eng.

Saouli Y., Magnier M., Germain G., Gaurier B., Druault P., 2022. Experimental characterisation of the waves propagating against current effects on the wake of a wide bathymetric obstacle. In: JH2022-18èmes Journées de l'Hydrodynamique.

Shounda J., Roy K., Barman K., Debnath K., Mazumder B., 2025. Coherent structures of turbulent eddies in a wave-current co-existing flow. Water Waves.

Sirovich L., 1987. Turbulence and the dynamics of coherent structures. i. coherent structures. Q. Appl. Math. 45, 561–571.

Smeltzer B., Romcke O., Hearst R., Ellingsen S., 2023. Experimental study of the mutual interactions between waves and tailored turbulence. J. Fluid Mech. 962.

Soulsby R., Hamm L., Klopman G., Myrhaug D., Simons R., Thomas G.P., 1993. Wave-current interaction within and outside the bottom boundary layer. Coastal Eng. 21, 41–69.

Tan T., Venugopal V., 2024. Characterisation of turbulence at sites with coexisting waves and currents: An analysis by empirical mode decomposition. Vol. 313. Ocean Eng.

Teixeira M., Belcher S., 2002. On the distortion of turbulence by a progressive surface wave. J. Fluid Mech. 458, 229–267.

Togneri M., Masters I., Fairley I., 2021. Wave-turbulence separation at a tidal energy site with empirical orthogonal function analysis. Ocean Eng.

Viviano A., Musumeci R.E., Foti E., 2018. Interaction between waves and gravity currents: description of turbulence in a simple numerical model. Environ. Fluid Mech. 18, 117–148.

Wallace J.M., 2016. Quadrant analysis in turbulence research: history and evolution. Annual Rev. Fluid Mech. 48, 131–158.

Xuan A., Deng B., Shen L., 2019. Study of wave effect on vorticity in langmuir turbulence using wave-phase-resolved large-eddy simulation. J. Fluid Mech. 875, 173–224.

Xuan A., Deng B., Shen L., 2020. Numerical study of effect of wave phase on reynolds stresses and turbulent kinetic energy in langmuir turbulence. J. Fluid Mech. 904.

Zhang X., Simons R., Zheng J., Zhang C., 2022. A review of the state of research on wave-current interaction in nearshore areas. Ocean Eng.

Zhang Y., Zhang Z., Zheng J., Zhang J., Zheng Y., Zang W., Lin, X., Fernandez-Rodriguez E., 2021. Experimental investigation into effects of boundary proximity and blockage on horizontal-axis tidal turbine wake. Ocean Eng.

Zhang Z., Zhang Y., Zheng Y., Zhang J., Fernandez-Rodriguez E., Zang W., Ji R., 2023. Power fluctuation and wake characteristics of tidal stream turbine subjected to wave and current interaction. Energy 264, 126185.

Zhao X., Ji D., Long L., Yang Z., Yang Z., Liu D., Lorke A., 2025. Wave-turbulence decomposition and turbulence parameterization in aquatic wave environments using improved synchrosqueezed wavelet transform (iswt). Ecological Informatics.