# Predicting Interactions Between Full-Scale
Counter-Rotating Vertical-Axis Tidal Turbines Using
Actuator Lines

Mikaël Grondeau, Sylvain S. Guillou

► To cite this version:

Mikaël Grondeau, Sylvain S. Guillou. Predicting Interactions Between Full-Scale Counter-Rotating Vertical-Axis Tidal Turbines Using Actuator Lines. Journal of Marine Science and Engineering, 2025, 13, pp.1382. 10.3390/jmse13081382. [https://doi.org/10.3390/jmse13081382](https://doi.org/10.3390/jmse13081382)

HAL Id: hal-05183345

https://normandie-univ.hal.science/hal-05183345v1

Submitted on 23 Jul 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
Article

# Predicting Interactions Between Full-Scale Counter-Rotating Vertical-Axis Tidal Turbines Using Actuator Lines

Mikaël Grondeau* and Sylvain S. Guillou ID

Laboratoire Universitaire des Sciences Appliquées de Cherbourg, Université de Caen Normandie, LUSAC UR 4253, 60 rue Max-Pol Fouchet, CS 20082, 50130 Cherbourg-en-Cotentin, France; sylvain.guillou@unicaen.fr

* Correspondence: mikael.grondeau@unicaen.fr

## Abstract

As with wind turbines, marine tidal turbines are expected to be deployed in arrays of multiple turbines. To optimize these arrays, a more profound understanding of the interactions between turbines is necessary. This paper employs the Actuator Line Method alongside the Lattice Boltzmann Method and Large Eddy Simulation to develop a numerical model of tidal turbine arrays. It studies a vertical-axis turbine manufactured by HydroQuest/CMN that is equipped with two counter-rotating columns, each comprising two rotors. The ambient turbulence and upstream velocity profiles correspond to the characteristics of a tidal site such as the Alderney Race. Six turbine layouts are modeled: three aligned layouts with three turbines and three staggered layouts with four turbines. The spacing between turbines varies depending on the layout. This study yields several observations regarding array configuration. A minimum distance of 300 m, or $12D_{eq}$, between aligned turbines is necessary for full wake recovery. At shorter distances, the accumulation of velocity deficits significantly decreases the efficiency of the third turbine in the array. Pairs of counter-rotating vortices are observed in the wake of turbines. The evolution of these vortices and their influence on the wake depend greatly on the array configuration. An optimal configuration is observed in which the overall averaged power is not impaired by the interactions.

**Keywords:** vertical-axis tidal turbine; turbine arrays; wake; actuator line; large eddy simulation; lattice Boltzmann method

<figure><img src="image_2.png" /></figure>

check for
updates

Received: 27 June 2025

Revised: 17 July 2025

Accepted: 19 July 2025

Published: 22 July 2025

**Citation:** Grondeau, M.; Guillou, S.S. Predicting Interactions Between Full-Scale Counter-Rotating Vertical-Axis Tidal Turbines Using Actuator Lines. *J. Mar. Sci. Eng.* **2025**, *13*, 1382. https://doi.org/10.3390/jmse13081382

**Copyright:** © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## 1. Introduction

Tidal energy converters, such as tidal turbines, harness the energy of tidal currents and convert it into a usable form of energy, typically electricity. Developing tidal turbines helps enrich the energy mix with a renewable and predictable form of energy. Only in certain specific areas are the tidal currents strong enough to be exploited. To optimize cost and power, tidal turbines are likely to be installed on tidal farms [1]. At present, there is no industrial-scale tidal farm in operation; however, projects involving the deployment of turbines are being implemented. These projects are poised to establish the foundation for the development of industrial farms. This work is part of such a project, the OceanQuest project. OceanQuest's technology is the HydroQuest turbine, a vertical-axis, bottom-mounted device, as shown in Figure 1.

Prior to undertaking a comprehensive examination of the numerous studies on tidal turbines, it is essential to provide a synopsis of the environmental context within which
these turbines operate. Although highly localized, tidal sites can be found on almost every continent on Earth. Most of them are exposed to wind and waves [2,3]. These surface disturbances propagate through the water column and influence the flow. As shown in the analysis of in situ measurements in [4], sea states greatly affect load fluctuations without changing the average value. The bottom roughness also affects the stream and generates a high level of ambient turbulence of up to 20% [5]. The ambient turbulence is site-specific and should be considered with great care when studying the implementation of a tidal site [2]. Turbulence characteristics and vortex size have also been shown to be affected by local roughness and may differ from one place to another within the same site [6–8]. Ambient turbulence influences the behavior of tidal turbines. Experimental and numerical studies have been performed on both vertical-axis tidal turbines (VATTs) and horizontal-axis tidal turbines (HATTs) to assess their effects [1,9–11]. From these investigations, turbulence was found to accelerate wake recovery, reduce the average power coefficient, and increase the load fluctuations. Investigations on tidal turbine arrays should therefore take this phenomenon into account.

In order to study tidal turbines and their environments, three possible approaches are available. The first and most cumbersome is in situ deployment, which has been successfully used to test prototypes and measure tidal flow characteristics [2]. This approach provides extremely valuable data but is expensive and very localized within tidal areas. Flume tank experiments are also used to test various tidal turbine technologies [11,12]. Access to the flow surrounding the turbines facilitates the acquisition of comprehensive and sophisticated statistics on flow and turbine behavior [13]. The main disadvantages of these approaches are the scale effects and the possible unrealistic blockage of the flow by the device. Experimental approaches are often used along with Computational Fluid Dynamics (CFD) approaches. It is acknowledged that numerical methods require experimentation for validation; nevertheless, these methods facilitate the creation of a full-scale model.

CFD VATT modeling techniques can be sorted into different categories based on increasing complexity and computational costs. The least expensive way to model turbines within a CFD simulation is the actuator cylinder approach coupled with the Reynolds-Averaged Navier-Stokes (RANS) solver [14]. This approach can be used to model large-scale arrays but can be inaccurate when complex and unsteady flows are considered. On the other side of the spectrum are blade-resolved simulations [15]. These approaches are the most accurate. Using state-of-the-art blade-resolved CFD, the authors of [15] were able to model two VATT models interacting. Even with access to large HPC centers, the computational cost of blade-resolved CFD is prohibitive when considering an array of more than two turbines. An interesting alternative is the Actuator Line Model (ALM) coupled with Large Eddy Simulation (LES) [16,17]. While less expensive than blade-resolved approaches, ALM-LES enables an unsteady description of the flow and tracking of the blades.

Due to its ability to capture unsteady phenomena, the ALM-LES method has been used to study advanced turbine behavior such as blade fatigue in HATT arrays [18]. When combined with a seabed morphology boundary condition, ALM-LES can model full-scale tidal turbines at actual tidal sites [19]. Regarding VATTs, [20,21] extensively examined the influence of ALM parameters on the accuracy of the model. Their findings are therefore extremely valuable for setting up a robust ALM-LES model. These investigations suggest that ALM-LES is an interesting tool for studying small arrays of full-scale vertical-axis tidal turbines.

Tidal turbine technology is not as well-established as wind turbine technology, and there are almost as many turbine technologies as there are manufacturers. Because these technologies behave differently, studies are required to optimize each design. The VATT
in question here, the HydroQuest tidal turbine, has its own particularities and features four counter-rotating rotors distributed across two columns. The behavior of this specific turbine has been studied in model-scale experiments [11], in which unsteady upstream conditions were considered. A full-scale HydroQuest VATT was studied in [16] under various turbulent upstream conditions. To the best of the authors' knowledge, no study has examined the interaction between multiple VATTs with counter-rotating rotors under realistic upstream conditions. The aforementioned setup includes complex interactions between unsteady phenomena and turbines. In light of the existing literature, ALM-LES appears to be a suitable methodology.

The ALM-LES method is implemented here within the Lattice Boltzmann Method (LBM) to calculate flow. The LBM is an efficient unsteady, weakly compressible CFD method that is well-suited for modeling large areas [8,22]. It also has low dissipation, making it well-suited for modeling the advection and dissipation of wakes. In the case of submerged water flow, simulations are well below the LBM compressibility limits. The LBM code is the PaLaBoS library [23]. ALM-LBM-LES is used in this paper to study full-scale HydroQuest turbines. Ambient turbulence is accounted for by generating a realistic turbulent boundary layer upstream of the turbines using a Synthetic Eddy Method (SEM) [24]. To avoid introducing too many unknown sources of disruption, the free surface and rough bottom are disregarded. Interactions between turbines are studied for six layouts with up to four turbines. The spacing and arrangement of the elements along the streamwise direction vary between cases. The analysis focuses on how the flow and power output are influenced by the relative distances and positions of the turbines. This paper is organized as follows. Section 2 introduces the numerical models, turbine specifications, and arrangements. Section 3 investigates the averaged flow and power, as well as the morphology of the wake. Section 4 presents the conclusions and perspectives.

<figure><img src="image_3.png" /><figcaption>**Figure 1.** HydroQuest/CMN tidal turbine tested at Paimpol-Bréhat (Brittany, France). Reproduced with permission from HydroQuest [25], 2022.</figcaption></figure>

**2. Materials and Methods**

2.1. *Governing Equations*

The LBM is a weakly compressible, unsteady CFD model derived from the Boltzmann
equation [26]. Assuming that the fluid is isothermal and has a low Mach number, it is
equivalent to a second-order finite-volume approach based on the Navier-Stokes equation.
Unlike the latter, the LBM solves the problem at a mesoscopic scale. The behavior of
groups of molecules is statistically represented with distribution functions. A distribution
function is the density of molecules at a specific time and place that have a certain velocity.
Interactions between molecules are modeled with the collision operator $\Omega$, as shown in Equation (1). $f'_i$ represents the post-collision distribution functions. The simulations presented in this paper are run with the open-source LBM library Palabos [23]. Palabos offers a wide range of collision operators depending on the physics at play. For studying tidal turbines with actuator lines, a regularized collision operator is selected [27,28].

$$f'_{i}(\mathbf{x}, t) = \Omega(f_{i}(\mathbf{x}, t)). \qquad (1)$$

This collision operator does not add too much complexity to the scheme and provides enough stability and accuracy for the applications presented here. The stencil for velocity discretizations has 27 directions of propagation and is commonly referred to as D3Q27. Each node of the lattice now has 27 distribution functions ($f_i$), one per direction. The displacement of groups of molecules is performed during the streaming stage, preceded by the collision. Equation (2) shows the streaming equation:

$$f_{i}(\mathbf{x} + c_{i}\Delta t, t + \Delta t) = f'_{i}(\mathbf{x}, t), \qquad (2)$$

where $c_i$ represents the directions of the D3Q27 stencil and $\Delta t$ is the time step. The Large Eddy Simulation model is a constant Smagorinsky model. It is implemented in the LBM with the approach described in [29].

$$S_{\alpha\beta} = \frac{\Pi_{\alpha\beta}^{neq}}{2} \frac{\rho\tau c_s^2 - \sqrt{\rho^2\tau^2c_s^4 + 2C_s^2\|\Pi_{\alpha\beta}^{neq}\|\rho}}{C_s^2\|\Pi_{\alpha\beta}^{neq}\|\rho}. \qquad (3)$$

$$\Pi_{\alpha\beta}^{neq} = \sum_{i=0}^{27} (c_{i\alpha}c_{i\beta}f_{i}^{neq}) - \delta_{\alpha\beta}c_{s}^{2}. \qquad (4)$$

The strain rate tensor $S_{\alpha\beta}$ is computed locally using Equation (3). The variable $\rho$ is the fluid density in lattice units, $\tau$ is the LBM relaxation rate, and the speed of sound in lattice units is $c_s = 1/\sqrt{3}$. The tensor $\Pi^{neq}$ is calculated using Equation (4), where $c_{i\alpha}$ is the $\alpha$th component of the i'th direction of the D3Q27 stencil. The symbol $\delta_{\alpha\beta}$ is the Kronecker symbol, and $f_{i}^{neq}$ represents the non-equilibrium distribution functions. The Smagorinsky coefficient is $C_s = 0.1$.

## 2.2. Actuator Line Model

Blade-resolved approaches require either a grid spacing inferior to $y^+ = 1$ in the boundary layer, or a sophisticated wall model. Even with the latter, it can be quite challenging to predict flow separation [30]. Both require large computational resources for high Reynolds-number simulations. In order to save computational resources and keep the benefit of the unsteadiness, an Actuator Line Model is used here. An ALM represents the blade action on the fluid through a forcing term in the LBM collision equation. An actuator line is placed where the blade should be. These lines comprise multiple point elements. The forcing term is calculated at the location of these elements. ALM input data are the tabulated lift coefficient $C_L$, the tabulated drag coefficient $C_D$, and the fluid velocity interpolated at each element $U_{rel}$. The algorithm by Shamsoddin and Porté-Agel (2016) [31] is implemented. It takes into account the dynamic stall of the blade during rotation. The force is calculated using Equation (5):

$$\frac{f_{AL}}{\rho} = (c\Delta l)(\frac{U_{rel}^2}{2})(C_{LDS}e_L + C_{DLS}e_D), \qquad (5)$$

where $C_{LDS}$ and $C_{DLS}$ are the coefficients modified by the dynamic stall algorithm. $c$ is the chord, and $\Delta l$ is the spacing between two elements. The vectors $e_L$ and $e_D$ form the local
frame of reference. In the dynamic stall algorithm, the derivative of the angle of attack with respect to the azimuth angle of the blade is obtained through a cubic spline interpolation of the average angle of attack. The angle of attack is averaged over 15 revolutions of the turbine. This implementation was validated by Grondeau et al. (2019) [16]. Time-averaging the angle of attack can eliminate variations in incidence caused by turbulence. However, since this study does not focus on load fluctuations, the influence of time-averaging on the averaged power coefficient and wake should be minimal. The model still takes into account instantaneous variations in velocity $U_{rel}$. The ALM-LBM-LES approach was successfully used to model a VATT placed inside a turbulent boundary layer. Three meshes were used to evaluate the influence of grid spacing. The numerical results were found to be in good agreement with the experimental data, especially for the shape of the wake and the velocity deficit, as shown in Figure 2.

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Average turbulent kinetic energy normalized by the upstream velocity. The planes are located 2D downstream of the turbine. On the left is the experimental data from [32], and on the right are results from the ALM-LBM-LES model used in this paper and in [16]. Arrows are used to project the average velocity onto the figure’s plane. A black rectangle represents the turbine contour. Reproduced with permission from [16], published by MDPI in *Energies*, 2019.</figcaption></figure>

To remain consistent with [16], the same parameters are selected. The spacing between two ALM elements along a line is equal to the grid size, $\Delta x = 0.257$ m. To avoid discontinuities and instabilities, the ALM force is distributed around each ALM element using Equation (6), where $\eta(\epsilon, d)$ is the distribution kernel.

$$f_{AL}(x) = \eta(\epsilon, d) f_{AL}. \qquad (6)$$

$$\eta(\epsilon, d) = \frac{1}{\epsilon^3 \pi^{3/2}} \exp\left(-\left(\frac{d}{\epsilon}\right)^2\right). \qquad (7)$$

The distribution kernel is described in Equation (7). The variable $d$ is the distance between an ALM element and a node of the lattice where the force is distributed. The kernel parameter $\epsilon$ is equal to $2\Delta x = 0.514$ m [21]. The fluid velocity $U_{rel}$ is interpolated using the distribution kernel. The mesh is described in Section 2.4.

## 2.3. Turbine Specifications

The present study examines a vertical-axis tidal turbine designed and manufactured by HydroQuest (Grenoble, France)¹ and CMN (Construction Mécanique de Normandie, Cherbourg, France). The turbine is equipped with four rotors, each consisting of three blades, as illustrated in Figure 1. The rotors have straight blades and are of the Darrieus H-
type. The left-hand side and right-hand side rotors are counter-rotating. The blade profile is based on the NACA0018 profile. The specific details regarding the blade's profile and chord remain proprietary and are not disclosed. The lift and drag coefficients are calculated with the XFLR5 software v6.61 ², using the following parameters: $Ma = 0$, $N_{crit} = 9$, and 300 panels. Depending on its azimuthal angle, the turbulence intensity, or its location within the boundary layer profile, the blade experiences varying Reynolds numbers. Consequently, calculations with XFLR5 are executed within the range of Reynolds numbers from $7 \times 10^4$ to $6 \times 10^6$. The rotor characteristics are summarized in Table 1.

<table><caption>**Table 1.** Configuration of the rotor.</caption><thead><tr><td>Quantity</td><td>Value</td><td>Unit</td></tr></thead><tbody><tr><td>Tip-speed ratio λ</td><td>1.5</td><td>~</td></tr><tr><td>Reynolds number based on radius</td><td>$5.9 \times 10^6$</td><td>~</td></tr><tr><td>Radius R</td><td>4.0</td><td>m</td></tr><tr><td>Height of the blades H</td><td>3.8</td><td>m</td></tr></tbody></table>

The rotor's axis, connectors, and blades are modeled using actuator lines. Only the blade ALM model uses the tabulated $C_L$ and $C_D$. The axis and the H-connectors have constant drag. The axis has a drag coefficient of $C_{D_{axis}} = 1$, and the drag of the H-connectors is evaluated with Ansys Fluent (v14.5, Ansys Inc., Canonsburg, PA, USA). The stator boundary condition is enforced at its surface location using the boundary condition from [33]. A free-slip condition is applied. Figure 3 depicts the turbine model. The gravity base is not modeled. The distance between the bottom of the stator and the seabed is 5 m. An equivalent diameter, representative of the turbine width, is defined for the turbine and is $D_{eq} = 25.0$ m.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** HydroQuest turbine model used for the LBM simulation. The blue parts were modeled with the ALM, and the gray parts were modeled with an off-lattice free-slip boundary condition. Reproduced with permission from [16], published by MDPI in *Energies*, 2019.</figcaption></figure>

## 2.4. Turbine Layout and Simulation Setup

In total, 6 turbine layouts are investigated: 3 aligned scenarios and 3 staggered scenarios. The streamwise distance between turbines varies for each scenario. The distance between the turbines is defined from the rear of the upstream turbine stator to the front of the downstream turbine stator. The investigated distances are 50.0 m, 100.0 m, and 150.0 m. The aligned layouts have 3 turbines, and the staggered layouts have 4 turbines. The depth of the domains is constant across all scenarios and is set to a realistic value of
43.12 m, considering a tidal site like the Alderney Race. The domains also have a constant width of approximately 20 $D_{eq}$ = 500 m. The length of the numerical domains depends on the distance between the turbines. The dimensions of the domains are summarized in Table 2, along with identifiers for each layout. For the 3$T_i$ scenarios, the most upstream turbine is designated as the first one, the one in between as the second, and the most downstream one as the third. For the 4$T_i$ scenarios, the most upstream turbine is designated as the first, the turbines in the second row are the second and third, and the most downstream one is the fourth.

<table><caption>**Table 2.** Turbine layouts.</caption><thead><tr><th>Scenario</th><th>Layout</th><th>x-Spacing</th><th>y-Spacing</th><th>x-Length</th></tr></thead><tbody><tr><td>3T<sub>1</sub></td><td>3 aligned</td><td>50 m</td><td>~</td><td>560 m</td></tr><tr><td>3T<sub>2</sub></td><td>3 aligned</td><td>100 m</td><td>~</td><td>660 m</td></tr><tr><td>3T<sub>3</sub></td><td>3 aligned</td><td>150 m</td><td>~</td><td>760 m</td></tr><tr><td>4T<sub>1</sub></td><td>4 staggered</td><td>50 m</td><td>70 m</td><td>560 m</td></tr><tr><td>4T<sub>2</sub></td><td>4 staggered</td><td>100 m</td><td>70 m</td><td>660 m</td></tr><tr><td>4T<sub>3</sub></td><td>4 staggered</td><td>150 m</td><td>70 m</td><td>760 m</td></tr></tbody></table>

As in most LBM simulations, the domain is discretized using Cartesian grids. Each mesh is composed of three grids, with a $\Delta x$ and $\Delta t$ ratio of two between neighboring grids. The grid with the smallest spacing contains the turbine and the close wake. The grid with the intermediate spacing is used to calculate the domain between the turbines and the wake further away from the turbines. The rest of the domain is solved on the coarsest grid. The mesh is illustrated in Figures 4 and 5. The finest grid has a spacing of $\Delta x$ = 0.257 m, which is chosen following the convergence study carried out by Grondeau et al. (2019) [16]. The time step in the finest grid is $\Delta t$ = $5.27 \times 10^{-3}$ s. In the LBM, the time step $\Delta t$ is imposed through the physical speed of sound of the fluid. This usually yields an extremely small time step. When acoustic effects are not considered, it is common practice to artificially decrease the speed of sound in the fluid. In this case, the hydrodynamic results are valid up to a Mach number of 0.3. For the current study, the Mach number computed with the velocity $U_{x\delta}$ is $Ma$ = 0.05, well below the limit of 0.3 required by the LBM. The total number of nodes for each scenario is given in Table 3.

<table><caption>**Table 3.** Computational effort for the ALM-LBM-LES simulations. Time spent refers to the wall-clock time.</caption><thead><tr><th>Scenario</th><th>Nodes</th><th>N Processors</th><th>Time Spent/Revolution</th></tr></thead><tbody><tr><td>3T<sub>1</sub></td><td>29 × 10<sup>6</sup></td><td>280</td><td>1654 s</td></tr><tr><td>3T<sub>2</sub></td><td>35 × 10<sup>6</sup></td><td>280</td><td>1972 s</td></tr><tr><td>3T<sub>3</sub></td><td>39 × 10<sup>6</sup></td><td>280</td><td>2133 s</td></tr><tr><td>4T<sub>1</sub></td><td>42 × 10<sup>6</sup></td><td>420</td><td>1702 s</td></tr><tr><td>4T<sub>2</sub></td><td>46 × 10<sup>6</sup></td><td>420</td><td>1837 s</td></tr><tr><td>4T<sub>3</sub></td><td>50 × 10<sup>6</sup></td><td>420</td><td>1810 s</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>Figure 4. Numerical domain for the 3T₁ scenario discretized into 3 Cartesian grids. Turbines are represented with solid black rectangles. Each cuboid shown contains 14 × 14 × 14 nodes. (Top) Plane of normal z located at the mid-height of the turbine. (Bottom) Plane of normal z located at y = 0 m.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5. Numerical domain for the 4T₃ scenario discretized into 3 Cartesian grids. Turbines are represented with solid black rectangles. Each cuboid shown contains 14 × 14 × 14 nodes. (Top) Plane of normal z located at the mid-height of the turbine. (Bottom) Plane of normal z located at y = 0 m.</figcaption></figure>

The inlet of the numerical domain is a Dirichlet velocity boundary condition. A SEM boundary condition is added to it in order to generate realistic turbulence [34]. The average axial velocity profile is constant in the y-direction and is set to match measurements made in the Alderney Race [35]. It is illustrated in Figure 6 and is calculated as

$$U_x(z) = U_{x\delta} \left(\frac{z}{\delta}\right)^{\frac{1}{n}}, \qquad (8)$$

where z is the distance from the seabed and $U_{x\delta}$ is the average velocity at z = δ. In this study, δ = 10.0 m, $U_{x\delta}$ = 1.475 m.s⁻¹, and n = 6.5. The flow velocity in the Alderney Race varies significantly depending on the location. The value we chose was not measured at the maximum velocity location and is consistent with the first deployment site at Paimpol-Bréhat. The SEM is calibrated to generate turbulence with an intensity that could be
encountered in a real-life tidal site. Figure 7 plots the turbulence intensity measured within the ALM-LBM-LES simulation, 50 m upstream of the most upstream turbine. The sizes of the SEM vortices were set to a constant value of 8 m. In Grondeau et al. (2022) [9], the integral length scale of the turbulence was quantitatively linked to the prescribed size of the SEM vortices. Using a vortex size of 8 m should then produce an integral length scale of approximately 8 m. This value was chosen to match the average integral length scale $L_z$ measured in the Alderney Race [36]. Velocity Dirichlet boundary conditions are applied to the top and side planes of the domain according to the velocity profile depicted in Figure 6. Spalding's boundary layer velocity profile [37] is applied to the boundary located at the bottom of the domain. A constant pressure is imposed at the outlet of the numerical domain.

<figure><img src="image_8.png" /><figcaption>Figure 6. Average inlet velocity profile calculated with Equation (8) and from ADCP measurements [35]. The red dotted lines indicate the turbine footprint and the black dotted line indicates the free surface. Reproduced with permission from [16], published by MDPI in *Energies*, 2019.</figcaption></figure>

$$I_{3D} = f(z)$$

<figure><img src="image_9.png" /><figcaption>Figure 7. Average turbulence intensity measured 50 m upstream of the most upstream turbine in an ALM-LBM-LES simulation. The red dotted lines indicate the turbine footprint and the black dotted line indicates the free surface.</figcaption></figure>

### 3. Results and Discussion

In this section, the results from the simulations of the aligned scenarios ($3T_i$) and the staggered scenarios ($4T_i$) are presented and discussed. The simulations were run for 530 s
before calculating the statistics. This duration is approximately one channel flow-through time for the 3T₃ and 4T₃ scenarios. The statistics were sampled over 244 s, with a frequency of 190 Hz. According to [38], a minimum of 20 turbine revolutions is required to average the data. With a TSR of 1.5, it took the turbine approximately 11 s to complete a single revolution. In 244 s, the turbine completed 21 revolutions, which is more than what is recommended by [38]. For all layouts, the focus is primarily on the wake velocity, then on the power output, and finally on the wake morphology.

## 3.1. Computational Effort

Table 3 summarizes the computational effort required for the six ALM-LBM-LES simulations. All computations were carried out on the CRIANN (Centre Régional d'Informatique et d'Applications du Numériques de Normandie) cluster Myria ³.

## 3.2. Wake Velocity

An instantaneous 3D view of the streamwise velocity for the 3T₃ scenario is shown in Figure 8. Since this scenario has the largest spacing between turbines, it is interesting to note that downstream turbines are located in areas affected by the wake. Moreover, it was qualitatively observed that the ambient vortices had a size on the same order of magnitude as the turbines, reflecting the prescribed SEM vortex length of 8 m. Because of the computational cost of this study, the simulations were not run long enough to compute advanced statistics, such as autocorrelation, which could have provided a quantitative estimation of the actual size of the vortices.

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Instantaneous axial velocity at the turbine mid-height for the 3T₃ scenario in the ALM-LBM-LES simulation.</figcaption></figure>

Investigating how turbine spacing affects the wake can be tedious, as the wake is a 3D, unsteady phenomenon that occupies a large volume. To obtain a global view of the wake with a manageable amount of data, a limited set of quantities was selected. The averaged axial velocity deficit was plotted along the turbine centerline for all scenarios. The velocity deficit is defined as $U_{Def} = (U_{xδ} - U_x)/U_{xδ}$. Figure 9 shows the velocity deficits for the 3Tᵢ scenarios. In all scenarios, wake recovery is not achieved between turbines, and downstream turbines are always located in regions with reduced velocity. Also, the velocity deficit upstream of the third turbine is greater than that upstream of the second turbine. This was observed across all three scenarios. The velocity deficit accumulates with each successive turbine, as also observed by [14]. After approximately 1 $D_{eq}$, the velocity recovery follows a linear trend, with a similar trend observed across all cases and turbines. This behavior is examined in more detail later in this section. The peak velocity deficit at each turbine varies depending on the turbine and the scenario, and no consistent trend was observed (see Figure 9). As the statistical line passes through the interior of the stator, the quantities calculated at the turbine location have little physical meaning.
In the $4T_i$ scenarios, the turbines are not aligned, so two figures are needed to show the velocity deficit along the x-direction. Figure 10 shows $U_{Def}$ for central turbines 1 and 4. The velocity deficit upstream of turbine 4s varies significantly across scenarios. For the $4T_3$ scenario, the wake of turbine 1 almost fully recovered, so its influence on turbine 4 should be minimal. As in the $3T_i$ scenarios, a linear trend was observed in the far wake of each turbine. Figure 11 shows $U_{Def}$ for lateral turbines 2 and 3, and no clear differences between the scenarios are visible. More differences may appear in integral quantities, such as the power coefficient, or in the 3D plots of the wakes examined in the following sections.

For better comparison between arrays, two additional figures are included. Figure 12 shows the wake downstream of the last turbine in the $3T_i$ and $4T_i$ scenarios. As noted previously, all velocity deficit plots exhibit a linear trend after approximately 1 $D_{eq}$. All scenarios are highly similar, with one exception: the velocity deficit downstream of turbine 4 in the $4T_3$ scenario is significantly lower than in the other scenarios. This is consistent with the results in Figure 10. Figure 13 compares the wake of turbine 4s in the $4T_i$ scenarios with the wake of turbine 2 in the $4T_1$ scenario. Indeed, turbine 2 in the $4T_1$ scenario is isolated and almost undisturbed. As seen in Figure 13, the wake of turbine 4 in the $4T_3$ scenario is almost identical to that of an undisturbed turbine.

The results in Figures 9–13 suggest that the spacing between turbines and the number of turbines in the array do not significantly affect the velocity deficit downstream of the array. This remains true up to a critical spacing, beyond which the wake of an upstream turbine can fully recover. In such a scenario, the velocity deficit downstream of the array is similar to that of an isolated turbine. This critical spacing lies between 200 m and 300 m. As shown in Figure 10, the corresponding critical velocity deficit upstream of the turbine falls within the range [0.05, 0.1].

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Average axial velocity deficit at the turbine mid-height and at the y-coordinate of the left rotor column in the $3T_i$ scenarios. The origin is placed at the center of the first turbine in the array. The vertical dashed lines represent the turbine locations.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Average axial velocity deficit at the turbine mid-height and at the y-coordinate of the left rotor column of turbines 1 and 4 in the 4Tᵢ scenarios. The origin is placed at the center of the first turbine in the array. The vertical dashed lines represent the turbine locations.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Average axial velocity deficit at the turbine mid-height and at the y-coordinate of the left rotor column of turbine 2 in the 4Tᵢ scenarios. The origin is placed at the center of the first turbine in the array. The vertical dashed lines represent the turbine locations.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 12.** Average axial velocity deficit at the turbine mid-height and at the y-coordinate of the left rotor column in the wake in the 3Tᵢ and 4Tᵢ scenarios. Profiles have been shifted in the x-direction to superimpose the wakes of the turbines.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 13. Average axial velocity deficit at the turbine mid-height and at the y-coordinate of the left rotor column of turbine 2 in the 4T₁ scenario and turbine 4 in all 4Tᵢ scenarios. Profiles have been shifted in the x-direction to superimpose the wakes of the turbines.</figcaption></figure>

### 3.3. Turbine Power

In this section, the average power coefficients of the turbines in the 3Tᵢ and 4Tᵢ scenarios are investigated. The power coefficients are made dimensionless by dividing the power coefficient of each turbine by that of the first turbine of the corresponding layout, 3T₃ or 4T₃, as shown in Equation (9).

$$C_p^*(T_i, 3T_i) = \frac{C_P(T_i, 3T_i)}{C_P(T_1, 3T_3)}, \quad C_p^*(T_i, 4T_i) = \frac{C_P(T_i, 4T_i)}{C_P(T_1, 4T_3)} \qquad (9)$$

A distinction is made between top rotors and bottom rotors. In the figures below, the power coefficients of the bottom rotors are shown in dull colors, while those of the top rotors are shown in bright colors. The power coefficients for the 3Tᵢ scenarios are shown in Figure 14. Across all scenarios, turbine 1s have the highest Cₚ* values and turbine 3s have the lowest. The loss in Cₚ* for turbine 3s is substantial. All turbine 3s have Cₚ* values approximately 60% lower than those of turbine 1s. This is consistent with the results in Section 3.2, where an accumulation of the velocity deficit for downstream turbines was observed. The top rotors of turbines 1 and 2 have higher Cₚ* values than the bottom rotors. This is likely due to the upstream boundary layer shown in Figure 6. The difference in Cₚ* between the top and bottom rotors is more pronounced for turbine 2s than for turbine 1s. No difference is observed for turbine 3s. Further insight into these two observations is given in Section 3.4, based on the velocity deficit maps. Turbine 1 in the 3T₁ scenario generates slightly less power than turbine 1 in the 3T₂ scenario, which generates less power than turbine 1 in the 3T₃ scenario. This difference could be due to how the array affects flow blockage, but it is also small enough to fall within the method's margin of error.

Figure 15 depicts the Cₚ* values for the turbines in the 4Tᵢ scenarios. The power coefficients of all turbines fall within a narrower range compared to the 3Tᵢ scenarios. The lowest power is observed in the bottom rotor of turbine 4 in the 4T₁ scenario and is around 80%. Turbines 2 and 3 have higher Cₚ* values than turbine 1 and seem to benefit from the staggered configuration. The flow likely accelerates when bypassing the first turbines in the arrays, a known phenomenon in wind and tidal farms [14]. The gain in power is highest in the 4T₃ scenario, with an 8% increase in Cₚ*. No difference between the top and bottom rotors is observed for turbine 4s, except in the 4T₁ scenario. This is likely due to interactions between turbines and the expansion of the wake. Section 3.4 provides a more detailed analysis of this phenomenon.

Lin et al. (2024) [15] modeled two aligned VATTs using blade-resolved LES. By testing various distances between turbines, they observed that the power coefficient of the second turbine was halved when the distance between turbines was 10 diameters. In our case, the
greatest reduction in power occurred in the 3T₁ scenario, where the power coefficient of the second turbine decreased by 40%. In this scenario, the distance between turbines was 50 m, equivalent to two Dₑq or 6.25 D. In addition to technological differences, Ref. [15] used a constant velocity inlet with no turbulence, which was not the case here. Turbulence is known to reduce the wake recovery distance. For example, in [1], the influence of the distance between two aligned HATTs was examined. Two inflow conditions were tested: one with a turbulence intensity of 3% and the other with 15%. With a distance of 12 D between turbines and a turbulence intensity of 3%, the power coefficient of the second turbine was reduced by 50%. However, with a distance of 6 D and a turbulence intensity of 15%, the reduction was only 20%. This could explain why, despite the shorter spacing compared to [15], the power coefficient reduction in our study was smaller.

It is worth noting that turbine 4s in the 4Tᵢ scenarios and turbine 2s in the 3Tᵢ scenarios have similar power coefficients. All are located in the wake of a single turbine. These observations are consistent with the velocity deficit upstream of these turbines, which lies between 10 and 20%.

Dimensionless power coefficient of cases 3Tᵢ

<figure><img src="image_16.png" /><figcaption>**Figure 14.** Dimensionless power coefficients ($C_p^*$) of turbines in the 3Tᵢ scenarios. Bright colors represent the $C_p^*$ values of the top rotors, while dull colors represent those of the bottom rotors.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Figure 15.** Dimensionless power coefficient ($C_p^*$) of turbines in the $4T_i$ scenarios. Bright colors represent the $C_p^*$ values of the top rotors, while dull colors represent those of the bottom rotors.</figcaption></figure>

### 3.4. Wake Morphology

Grondeau et al. (2019) [16] studied the wake of a single HydroQuest turbine, identical to the one modeled here. To observe the shape of the wake, the average flow field was sliced with planes normal to $\vec{x}$ at several downstream distances. Two counter-rotating vortices were observed in the wake of the same turbine using this approach. A similar procedure was used in the $3T_i$ and $4T_i$ scenarios.

#### 3.4.1. Wake Morphology for the Aligned Setup

Starting with the most compact scenario ($3T_1$), the velocity deficit was plotted at two downstream distances for each turbine: 0 $D_{eq}$ (turbine location) and 1 $D_{eq}$. Recall that the spacing between turbines in this case is 50 m, or 2 $D_{eq}$. Figure 16a,b show the results for the first turbine; Figure 16c,d show the results for the second turbine; and Figure 16e,f show the results for the third and most downstream turbine. Looking at Figure 16a,c,e, the influence of upstream turbines on the second and third turbines is apparent. The boundary layer is thicker at the locations of turbines 2 and 3. Figure 16b,d,f depict the velocity deficit in the wakes of the turbines. The influence of upstream turbines on downstream turbines is even more noticeable. Counter-rotating vortices are observed, as reported in [16]. Top vortices have their centers located at the top of the turbine footprint, at about a quarter of the turbine width from the center. The intensity of the top vortices increases for the second and third turbines, modifying the shape of the wake. This stretches the bottom vortices toward the sides, although it has limited influence on the overall wake width, which is relatively similar between the second and third turbines. The most significant effect is the upward stretching of the wake toward the free surface.

The second aligned scenario ($3T_2$), with a spacing between turbines of 100 m, is shown in Figure 17a–f. Differences can still be observed at the turbine locations, and the boundary layer is thicker for downstream turbines. The shapes of the wakes of turbines 1 and 2, as shown in Figure 17b,d, are similar. The main differences are that the wake footprint is larger and the velocity deficit is more pronounced for turbine 2. Only the wake of turbine 3 is stretched significantly upward.
The averaged velocity field for the $3T_3$ scenario, with a spacing between turbines of 150 m, is shown in Figure 18a–f. There is little difference between the boundary layers at the turbine locations. Additionally, the velocity fields shown in Figure 18c,e are almost identical. The wakes of turbines 2 and 3 are no longer mushroom-shaped but instead more rectangular, like that of turbine 1. Between turbines 2 and 3, the wake footprint spreads in the y- and z-directions, and the velocity deficit increases. The intensity of the counter-rotating vortices does not seem to increase between turbines.

The distance between turbines significantly affects the morphology of the wake. For spacings below 150 m, the counter-rotating vortices become more intense with each successive turbine. This greatly affects the shape of the wake by stretching it toward the surface, which can be seen in Figure 19. A 3D wake animation for the $3T_1$ scenario, based on the isocontour of the Lambda-2 criterion, is provided in the Supplementary Materials. Because only three turbines were modeled here, the effect on the array was limited. These vortices have also been described in experimental studies [32,39,40]. As explained in [32], the counter-rotating vortex pairs originate from the $U_z$ velocity gradient in the z-direction (see the original work for more details). Thus, an increase in this gradient for downstream turbines may cause the intensification of the vortices. Across all $3T_i$ scenarios, the wake footprint of the first turbine does not seem to extend beyond the turbine footprint. This is not the case for the wakes of turbines 2 and 3, consequently, turbine 3s are fully embedded in the velocity deficit region, whereas turbine 2s are not. Turbine 1s are only influenced by the upstream boundary layer. So, it is likely that the bottom and top rotors of turbine 3s experience a more uniform vertical velocity profile than those of turbines 1 and 2 due to the mixing caused by the wake. This may explain why the difference in $C_p$ between the top and bottom rotors, as discussed in Section 3.3, vanishes between turbines 2 and 3.
<figure><img src="image_18.png" /><figcaption>(a) 0 $D_{eq}$ downstream of the first turbine.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(b) 1 $D_{eq}$ downstream of the first turbine.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(c) 0 $D_{eq}$ downstream of the second turbine.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>(d) 1 $D_{eq}$ downstream of the second turbine.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(e) 0 $D_{eq}$ downstream of the third turbine.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>(f) 1 $D_{eq}$ downstream of the third turbine.</figcaption></figure>

**Figure 16.** Axial velocity deficit downstream of the turbines in the 3T₁ scenario. Arrows indicate the projected velocity components on the figure planes. Black rectangles denote the turbine contours.
<figure><img src="image_24.png" /><figcaption>(a) 0 $D_{eq}$ downstream of the first turbine.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>(b) 1 $D_{eq}$ downstream of the first turbine.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>(c) 0 $D_{eq}$ downstream of the second turbine.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>(d) 1 $D_{eq}$ downstream of the second turbine.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>(e) 0 $D_{eq}$ downstream of the third turbine.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>(f) 1 $D_{eq}$ downstream of the third turbine.</figcaption></figure>

**Figure 17.** Axial velocity deficit downstream of the turbines in the 3T₂ scenario. Arrows indicate the projected velocity components on the figure planes. Black rectangles denote the turbine contours.
<figure><img src="image_30.png" /><figcaption>(a) 0 $D_{eq}$ downstream of the first turbine.</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>(b) 1 $D_{eq}$ downstream of the first turbine.</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>(c) 0 $D_{eq}$ downstream of the second turbine.</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>(d) 1 $D_{eq}$ downstream of the second turbine.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>(e) 0 $D_{eq}$ downstream of the third turbine.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>(f) 1 $D_{eq}$ downstream of the third turbine.</figcaption></figure>

**Figure 18.** Axial velocity deficit downstream of the turbines in the 3T₃ scenario. Arrows indicate the projected velocity components on the figure planes. Black rectangles denote the turbine contours.

<figure><img src="image_36.png" /></figure>

**Figure 19.** Instantaneous isocontour of the velocity deficit at $U_{Def} = 0.1$ for the 3T₁ scenario. Height Z is projected onto the isocontour. Turbines are shown in black.
### 3.4.2. Wake Morphology for the Staggered Setup

The procedure described at the beginning of Section 3.4 was applied to the $4T_1$ scenarios. For the $4T_1$ scenario, the average velocity deficit is shown in Figure 20 at $1.5D_{eq}$ downstream of the first turbine and $1.5D_{eq}$ downstream of the second and third turbines. This corresponds to $0.5D_{eq}$ upstream of the second, third and fourth turbines. In Figure 20a, the wake of the first turbine does not expand vertically beyond the turbine footprint. The wake is slightly wider than the turbine footprint but does not exceed $2D_{eq}$ in the y-direction. The counter-rotating vortices can be seen. As shown in Figure 20b, the wake of the first turbine creates a strong velocity deficit gradient in the z-direction. The velocity deficit is different between the top and the bottom of the turbine footprint and is estimated to be approximately 30%.

Figure 21a shows the velocity deficit $0.5D_{eq}$ upstream of turbines 2 and 3, or $3.5D_{eq}$ downstream of turbine 1. The wake expansion in the vertical direction is still restricted to the turbine footprint. In the y-direction, the wake expands up to $2D_{eq}$. No counter-rotating vortex is visible. Figure 21b shows the velocity deficit $0.5D_{eq}$ upstream of turbine 4, or $3.5D_{eq}$ downstream of turbines 2 and 3. The wakes of turbines 1, 2, and 3 are distinct from one another. The wake of turbine 1 covers all of turbine 4's projected area. The difference in the velocity deficit between the top and bottom of turbine 4 is below 10%.

Figure 22a,b show the average axial velocity deficit $0.5D_{eq}$ upstream of turbines 2 and 3 and $0.5D_{eq}$ upstream of turbine 4, respectively. This corresponds to $5.5D_{eq}$ downstream of turbine 1 and turbines 2 and 3, respectively. The wake of turbine 1 expands beyond $2D_{eq}$ in the y-direction. As shown in Figure 22b, the wakes of turbines 1, 2, and 3 are still $1D_{eq}$ apart. The wake of turbine 1 is visible and covers most of turbine 4's area, and the velocity deficit is almost uniform.

In Section 3.3, it was noted that in the $4T_2$ and $4T_3$ scenarios, the power coefficients of the top and bottom rotors of turbine 4s were identical. In the $4T_1$ scenario, a difference of 10% was identified. This variation is likely attributable to disparate vertical velocity gradients present at the turbine locations. This gradient was relatively low for turbine 4s in the $4T_2$ and $4T_3$ scenarios, while it was higher in the $4T_1$ scenario. It was also noted in Section 3.3 that turbines within the second row benefited from the blockage produced by turbine 1. This resulted in higher $C_p$ values. As shown in the velocity deficit maps, the wakes of turbine 1s keep expanding in the y-direction. This would explain why the increase in $C_p$ was greater in the $4T_3$ scenario. This was not the case in the $4T_1$ and $4T_2$ scenarios, where the $C_p$ values of turbines 2 and 3 were close but the wake expansion differed. The vertical expansion of all the wakes of the $4T_i$ turbines was limited to the turbine footprints. This outcome was significantly different from that of the $3T_i$ scenarios, where the vertical expansion was amplified for multiple aligned turbines situated at distances less than 150 m from each other.

<figure><img src="image_37.png" /><figcaption>Figure 20. *Cont.*</figcaption></figure>
<figure><img src="image_38.png" /><figcaption>**Figure 20.** Axial velocity deficit downstream of the turbines in the 4T₁ scenario. Arrows indicate the projected velocity components on the figure planes. Black rectangles denote the turbine contours.</figcaption></figure>

<figure><img src="image_39.png" /><figcaption>**Figure 21.** Axial velocity deficit downstream of the turbines in the 4T₂ scenario. Arrows indicate the projected velocity components on the figure planes. Black rectangles denote the turbine contours.</figcaption></figure>

<figure><img src="image_40.png" /><figcaption>**Figure 22.** Axial velocity deficit downstream of the turbines in the 4T₃ scenario. Velocity vectors are projected onto the plane. The turbine footprint is delimited in black.</figcaption></figure>
## 4. Conclusions and Outlook

ALM-LBM-LES simulations of realistic full-scale VATTs arranged into small arrays were performed. A total of six layouts were investigated: three aligned configurations with three turbines (3$T_i$ scenarios) and three staggered configurations with four turbines (4$T_i$ scenarios). The distances between consecutive aligned turbines were 50, 100, 150 m in the 3$T_i$ scenarios, and 100, 200, 300 m in the 4$T_i$ scenarios. All simulations used the same turbulent upstream boundary condition, generating a realistic inflow. The average velocity deficit along the turbine centerline was analyzed, as well as the average power coefficient and the shape of the wake.

It was observed that the centerline velocity deficit downstream of the array was not influenced by the number of turbines and that arrays 3$T_i$, 4$T_1$, and 4$T_2$ exhibited similar wakes. The distance between turbines appeared to have a greater influence. Indeed, the wake in the 4$T_3$ scenario differed significantly from the others. It was suggested that, in this scenario, there was a critical spacing of between 200 m and 300 m that led to a quasi-full recovery of the flow. This resulted in the far wake of the array being identical to that of a single turbine. The influence of different turbulence intensities on the far wake of a single HydroQuest VATT was assessed in [16] using the same numerical approach. The impact on both the velocity deficit and the shape of the wake was observed. Consequently, the observations in this paper may be affected by local variations in turbulence characteristics, such as those caused by seabed morphology.

The impact of the distance between turbines on the average power coefficient was evident. For a distance between turbines greater than 100 m, the second turbine only lost 10% power to the first turbine in the array. The accumulation of the velocity deficit between turbines led to a significant drop in efficiency of around 50% for the third turbines in the 3$T_i$ arrays. Across all scenarios, the mixing in the wake from upstream turbines reduced the velocity gradient in the vertical direction. This translated into smaller differences in $C_p^*$ between the top and bottom rotors of downstream turbines. The lateral 4$T_i$ turbines benefited from the staggered configuration with a maximal increase in $C_p^*$ of 8% compared to the first turbine in the array.

Observing the shape of the wake, it was suggested that the counter-rotating vortices created by the turbine became more intense with each passing turbine for a spacing below 150 m. Among the consequences was a stretching of the wake toward the free surface. When the spacing was equal to or greater than 150 m, this phenomenon was no longer observed.

These observations can be used to design more practical guidelines for these specific arrays:

* To optimize power output and allow for flow recovery, aligned turbines should be kept at least 300 m apart, or 12 equivalent diameters.

* A lateral spacing of 70 m, or 2.8 equivalent diameters, in the lateral direction is enough to prevent the wake from merging.

* A staggered configuration can increase power production due to the blocking effect.

* Because counter-rotating vortices tend to increase the turbine footprint, the evolution of counter-rotating vortex pairs should be predicted.

In this paper, an integral length scale was chosen to match the observed integral length at the Alderney Race tidal site. However, significantly changing this scale would likely affect the results. For example, studies in the wind turbine sector, such as [41], have observed phenomena linking large-scale vortices of turbulent flows to wake meandering. Future studies could focus on the influence of this parameter.
As discussed in the introduction, free-surface waves affect load fluctuations and are therefore important for fatigue predictions. Including extreme events, such as storms, in our model could provide valuable data for designing the turbine's structure.

Understanding the influence of turbulence, layouts, and the fact that turbulence varies within a site, a similar study incorporating the actual seabed morphology of a tidal site would give valuable information on how a small farm would perform. Such a study could focus on how the seabed's morphology locally affects the wake and load fluctuations.

**Supplementary Materials:** The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/jmse13081382/s1, Figure S1: Isocontour of Lambda-2 criterion at -4e-5. Simulation of case 3T₁.

**Author Contributions:** Conceptualization, M.G. and S.S.G.; methodology, M.G. and S.S.G.; software, M.G.; validation, M.G.; formal analysis, M.G.; investigation, M.G.; resources, S.S.G.; data curation, M.G.; writing—original draft preparation, M.G.; writing—review and editing, M.G. and S.S.G.; supervision, S.S.G.; project administration, S.S.G.; funding acquisition, S.S.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** Funding for this work was provided by the Région Normandie, project SEMARIN 17E01557, and by l'Agence de l'Environnement et de la Maîtrise de l'Energie, project OceanQuest 1682C0085.

**Data Availability Statement:** Data are available on request due to restrictions.

**Acknowledgments:** The simulations presented in this paper were carried out at the Centre Régional Informatique et d'Applications Numériques de Normandie. The authors wish to thank HydroQuest for their data.

**Conflicts of Interest:** The authors declare no conflicts of interest.

## Abbreviations

The following abbreviations are used in this manuscript:

VATT Vertical-Axis Tidal Turbine

HATT Horizontal-Axis Tidal Turbine

CFD Computational Fluid Dynamics

RANS Reynolds-Averaged Navier-Stokes

ALM Actuator Line Method

LES Large Eddy Simulation

LBM Lattice Boltzmann Method

SEM Synthetic Eddy Method

## Notes

## References

1. Mycek, P.; Gaurier, B.; Germain, G.; Pinon, G.; Rivaolen, E. Experimental study of the turbulence intensity effects on marine current turbines behaviour. Part II: Two interacting turbines. *Renew. Energy* **2014**, *68*, 876–892. [CrossRef]

2. Thiébaut, M.; Filipot, J.F.; Maisondieu, C.; Damblans, G.; Duarte, R.; Droniou, E.; Chaplain, N.; Guillou, S. A comprehensive assessment of turbulence at a tidal-stream energy site influenced by wind-generated ocean waves. *Energy* **2020**, *191*, 116550. [CrossRef]

3. Guillou, N. Modelling effects of tidal currents on waves at a tidal stream energy site. *Renew. Energy* **2017**, *114*, 180–190. [CrossRef]

4. Moreau, M.; Germain, G.; Maurice, G.; Richard, A. Sea states influence on the behaviour of a bottom mounted full-scale twin vertical axis tidal turbine. *Ocean Eng.* **2022**, *265*, 112582. [CrossRef]

<footnote n="1">¹ https://www.hydroquest.fr/en/ (accessed on 10-06-2025)</footnote>

<footnote n="2">² http://www.xflr5.tech/xflr5.htm (accessed on 19-06-2024)</footnote>

<footnote n="3">³ https://services.criann.fr/services/calcul/cluster-myria/ (accessed on 10-06-2025)</footnote>
5. Milne, I.A.; Sharma, R.N.; Flay, R.G.J.; Bickerton, S. Characteristics of the turbulence in the flow at a tidal stream power site. *R. Soc.* **2013**, *371*, 196. [CrossRef] [PubMed]

6. McVey, J.R.; Kilcher, L.; Thomson, J.; Yang, Z. Turbulence measurements at three potential tidal energy sites in the Salish Sea. *Appl. Ocean Res.* **2025**, *158*, 104561. [CrossRef]

7. Deb, M.; Yang, Z.; Wang, T.; Kilcher, L. Turbulence modeling to aid tidal energy resource characterization in the Western Passage, Maine, USA. *Renew. Energy* **2023**, *219*, 118694. [CrossRef]

8. Mercier, P.; Grondeau, M.; Guillou, S.; Thiébot, J.; Poizot, E. Numerical study of the turbulent eddies generated by the seabed roughness. Case study at a tidal power site. *Appl. Ocean Res.* **2020**, *97*, 102082. [CrossRef]

9. Grondeau, M.; Guillou, S.S.; Poirier, J.C.; Mercier, P.; Poizot, E.; Méar, Y. Studying the Wake of a Tidal Turbine with an IBM-LBM Approach Using Realistic Inflow Conditions. *Energies* **2022**, *15*, 2092. [CrossRef]

10. Clark, T. Turbulence in Marine Environments (TiME): A framework for understanding turbulence and its effects on tidal devices. In Proceedings of the EWTEC, Nantes, France, 6–11 September 2015.

11. Moreau, M.; Bloch, N.; Germain, G.; Maurice, G. Experimental study of the upstream bathymetry effects on a ducted twin vertical axis turbine. *Renew. Energy* **2024**, *224*, 120140. [CrossRef]

12. Moreau, M.; Germain, G.; Maurice, G. Experimental Investigation of Surface Waves Effect on a Ducted Twin Vertical Axis Tidal Turbine. *J. Mar. Sci. Eng.* **2023**, *11*, 1895. [CrossRef]

13. Ahmed, U.; Apsley, D.D.; Afgan, I.; Stallard, T.; Stansby, P.K. Fluctuating loads on a tidal turbine due to velocity shear and turbulence: Comparison of CFD with field data. *Renew. Energy* **2017**, *112*, 235–246. [CrossRef]

14. Gauvin-Tremblay, O.; Dumas, G. Hydrokinetic turbine array analysis and optimization integrating blockage effects and turbine-wake interactions. *Renew. Energy* **2022**, *181*, 851–869. [CrossRef]

15. Lin, X.; Zhang, J.; Zheng, J.; Liu, S. Performance and wake interaction between two aligned vertical axis turbines. *Ocean Eng.* **2024**, *292*, 116478. [CrossRef]

16. Grondeau, M.; Guillou, S.; Mercier, P.; Poizot, E. Wake of a Ducted Vertical Axis Tidal Turbine in Turbulent Flows, LBM Actuator-Line Approach. *Energies* **2019**, *12*, 4273. [CrossRef]

17. Stansby, P.K.; Ouro, P. Modelling marine turbine arrays in tidal flows. *J. Hydraul. Res.* **2022**, *60*, 187–204. [CrossRef]

18. Mullings, H.; Ouro, P.; Stallard, T. Synthesis and characterisation of turbulent flows to predict fatigue loading of tidal turbines in arrays. *J. Fluids Struct.* **2025**, *136*, 104321. [CrossRef]

19. Ouro, P.; Stansby, P.; Macleod, A.; Stallard, T.; Mullings, H. High-fidelity modelling of a six-turbine tidal array in the Shetlands. In Proceedings of the European Wave and Tidal Energy Conference, Bilbao, Spain, 3–7 September 2023; Volume 15. [CrossRef]

20. Mohamed, O.S.; Melani, P.F.; Soraperra, G.; Brighenti, A.; Ferrara, G.; Betti, V.; Balduzzi, F.; Bianchini, A. Development of a computationally efficient CFD model for vertical-axis hydrokinetic turbines: A critical assessment. *Ocean Eng.* **2025**, *324*, 120692. [CrossRef]

21. Melani, P.F.; Balduzzi, F.; Ferrara, G.; Bianchini, A. Tailoring the actuator line theory to the simulation of Vertical-Axis Wind Turbines. *Energy Convers. Manag.* **2021**, *243*, 114422. [CrossRef]

22. Parmigiani, A.; Latt, J.; Begacem, M.B.; Chopard, B. A Lattice Boltzmann Simulation of the Rhone river. *Int. J. Mod. Phys. C* **2013**, *24*, 1340008. [CrossRef]

23. Latt, J.; Malaspinas, O.; Kontaxakis, D.; Parmigiani, A.; Lagrava, D.; Brogi, F.; Belgacem, M.B.; Thorimbert, Y.; Leclaire, S.; Li, S.; et al. Palabos: Parallel Lattice Boltzmann Solver. *Comput. Math. Appl.* **2021**, *81*, 334–350. [CrossRef]

24. Jarrin, N.; Prosser, R.; Uribe, J.C.; Benhamadouche, S.; Laurence, D. Reconstruction of turbulent fluctuations for hybrid RANS/LES simulations using a Synthetic-Eddy-Method. *Int. J. Heat Fluid Flow* **2009**, *30*, 435–442. [CrossRef]

25. HydroQuest. 2022. Available online: https://www.hydroquest.fr/ (accessed on 10 June 2025).

26. Guo, Z.; Shu, C. *Lattice Boltzmann Method and Its Application in Engineering*; World Scientific: Singapore, 2013.

27. Latt, J. Lattice Boltzmann method with regularized pre-collision distribution functions. *Math. Comput. Simul.* **2006**, *72*, 165–168. [CrossRef]

28. Malaspinas, O. Increasing stability and accuracy of the lattice Boltzmann scheme: Recursivity and regularization. *arXiv* **2015**, arXiv:1505.06900. [CrossRef]

29. Malaspinas, O.; Sagaut, P. Consistent subgrid scale modelling for lattice Boltzmann methods. *J. Fluid Mech.* **2012**, *700*, 514–542. [CrossRef]

30. Zhou, D.; Bae, H.J. Sensitivity analysis of wall-modeled large-eddy simulation for separated turbulent flow. *J. Comput. Phys.* **2024**, *506*, 112948. [CrossRef]

31. Shamsoddin, S.; Porté-Agel, F. A large-eddy simulation of vertical axis wind turbine wakes in the atmospheric boundary layer. *Energies* **2016**, *9*, 366. [CrossRef]

32. Rolin, V.; Porté-Agel, F. Experimental investigation of vertical-axis wind-turbine wakes in boundary layer flow. *Renew. Energy* **2018**, *9*, 366.
33. Guo, Z.; Zheng, C.; Shi, B. Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method. *Chin. Phys.* **2002**, *11*, 366-374. [CrossRef]

34. Polleto, R.; Craft, T.; Revell, A. A new divergence free synthetic eddy method for the reproduction of inlet flow conditions for LES. *Flow Turbul. Combust.* **2013**, *91*, 519-539. [CrossRef]

35. Thiébaut, M.; Sentchev, A.; Bailly du Bois, P. Merging velocity measurments and modeling to improve understanding of tidal stream resource in Alderney race. *Energy* **2019**, *178*, 460-470. [CrossRef]

36. Thiébaut, M.; Filipot, J.F.; Maisondieu, C.; Damblans, G.; Jochum, C.; Kilcher, L.F.; Guillou, S. Characterization of the vertical evolution of the three-dimensional turbulence for fatigue design of tidal turbines. *Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.* **2020**, *378*, 20190495. [CrossRef] [PubMed]

37. Spalding, D. A single formula for the law of the Wall. *J. Appl. Mech.* **1961**, *28*, 455-458. [CrossRef]

38. Rezaeiha, A.; Kalkman, I.; Blocken, B. CFD simulation of a vertical axis wind turbine operating at a moderate tip speed ratio: Guidelines for minimum domain size and azimuthal increment. *Renew. Energy* **2017**, *107*, 373-385. [CrossRef]

39. Moreau, M.; Germain, G.; Maurice, G. Experimental performance and wake study of a ducted twin vertical axis turbine in ebb and flood tide currents at a 1/20th scale. *Renew. Energy* **2023**, *214*, 318-333. [CrossRef]

40. Bachant, P.; Wosnik, M. Characterising the near-wake of a cross-flow turbine. *J. Turbul.* **2015**, *16*, 392-410. [CrossRef]

41. Hamilton, N.; Doubrawa, P.; Moriarty, P.; Letizia, S.; Thedin, R. Modal dynamics of wind turbine wake meandering from lidar observations. *Renew. Energy* **2025**, *254*, 123555. [CrossRef]

**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.