Towards a Generalized Tidal Turbine Wake Analytical Model for Turbine Placement in Array Accounting for Added Turbulence ^†

Abstract

This study seeks to establish a comprehensive model for estimating both the velocity deficit and turbulence intensity within a tidal turbine farm across various layout configurations. The model incorporates a spectrum of ambient turbulence intensity ranging from 5% to 20%, a rotor diameter-to-depth ratio between 20% and 60%, and a rotor thrust coefficient that varies from 0.64 to 0.98. The influence of added turbulence is factored into the evaluation of the velocity deficit within the farm. Consistent with findings from prior research, the results indicate that in a tidal farm consisting of 16 turbines, a staggered array configuration yields 21% more power compared to a rectilinear array. This staggered setup benefits from enhanced flow acceleration and greater spacing between turbines, which facilitates improved wake recovery. The findings suggest that the farm’s dimensions can be optimized by reducing lateral spacing in the rectilinear array and longitudinal spacing in the staggered array without compromising efficiency. Such reductions in farm size can lead to decreased cable expenses and create opportunities for future expansion. For the tidal turbines in shallow water regions, the ratio of rotor diameter to depth is shown to affect the power generated by the turbines. The power produced in the farm decreases with an increase in the rotor diameter-to-depth ratio due to the limited wake expansion along the vertical plane. The efficiency of a tidal farm can be increased by high ambient turbulent intensity, sufficient turbine spacing, and low rotor diameter-to-depth ratio. These factors improve the wake recovery to allow more energy to be extracted by a downstream turbine. This low-computational model can be useful in studying the wake interaction of tidal turbine parks in different configurations.

1. Introduction

In the past ten years, the tidal stream energy sector has achieved notable success in the deployment and testing of full-scale tidal stream turbines (TSTs) at specialized testing locations, with individual units capable of generating up to 1 MW [1]. TST is seeing a convergence in technology, and the majority of devices that are being developed are horizontal-axis turbines (HAT). For example, IRENA [2] reported that as of 2020, more than 80% of the underwater active tidal stream devices are HAT. This industrial preference for HAT results from mature technology of aesthetically similar wind turbines, economics, and its high efficiency. The next step towards commercialization is the investigation of the turbine wake interaction in the tidal array. Wake is a region of disturbed flow behind a turbine that is associated with a decrease in flow velocity and an increase in turbulence. The wake induced in the downstream turbine can reduce the output power and undermine the structural integrity of the turbine blades.

Multiple studies have been dedicated to investigating the wake effect in tidal farms with an emphasis on velocity deficits. For instance, an experiment by Nobel et al. [3] reported an increase in power extraction by three scaled turbines in a staggered array. This results from flow acceleration between the closely spaced turbine upstream. The experiment in Mycek et al. [4] shows the performance of the downstream turbine in tandem configuration is affected at low ambient turbulence. Furthermore, the effect of rotor diameter-to-depth (DH) ratio has been studied [5]. The limited wake expansion along the vertical plane may also affect the power produced by the turbine downstream. While the reduced models provide notable results, they do not correctly replicate the complexity of full-scale tidal turbines.

Furthermore, various researchers have conducted high-fidelity numerical simulations of wake interactions within the tidal park, employing different turbine models, notably the Actuator Disc Method (ADM) and the Blade Element Momentum Theory (BEMT) [6,7,8]. These studies also show higher energy production occurs at a minimal lateral spacing around 2–3 D while maximizing the longitudinal spacing. The minimal lateral spacing can accelerate the flow for the downstream turbine, whereas the large longitudinal spacing ensures flow recovery. For this reason, the staggered array produces higher energy than the rectilinear array with the same number of turbines in a particular site [9]. Although the numerical simulation provides excellent results, the computational cost for large-scale farms is expensive.

For this reason, researchers have devised a low computational analytical/empirical model to estimate the turbine wake. These models are developed from self-similar flow characteristics and are mainly used for estimating the far wake region. Over time, the analytical models have evolved from a simple top-hat model [10] to more accurate artificial intelligence (AI)-based empirical models [11,12] to estimate the velocity deficit in the wind farm. These analytical models require calibration for highly turbulent shallow water tidal turbines [13,14,15]. The calibration becomes necessary in high turbulent tidal currents with a limited channel depth compared to the wind turbines in the atmospheric boundary layer. Semi-empirical turbulence models for wind turbines have been developed to estimate the far wake of tidal turbines by calibrating experimental data and high-fidelity numerical simulations. These models, though providing approximate values, yield relatively good results in the far wake region. A detailed review of tidal turbine wake characteristics has been conducted by [16].

An innovative analytical model addressing the velocity deficit of a tidal turbine has been developed by [17] and subsequently validated through experiments involving a three-bladed tidal turbine conducted by [18]. Additionally, refs. [13,19] introduced a semi-empirical model to describe the velocity deficit in the wake of the tidal turbine. This model employs the Jensen equation for predicting velocity deficit, while proposing an exponential law for wake expansion, in contrast to Jensen’s linear approach. Moreover, ref. [20] formulated an expression to estimate the far-wake velocity deficit in shallow channels, based on an experimental study with a rotor diameter-to-depth ratio of 60%. The DH ratio refers to the ratio of rotor diameter to channel depth. Furthermore, ref. [14] applied empirical models from wind turbines to refine coefficients using numerical data from tidal turbines across various turbulent intensities. It is important to note that these velocity deficit models primarily consider the ambient turbulence of the flow. Note that these analytical models allow only for the representation of mean velocity without regard to the turbulence effect. For a single turbine model, the effect of turbulence wake may not affect the performance of the turbine; however, it can significantly affect the energy production of downstream turbines in a park [6,21].

Turbine interaction in a farm has been shown to increase turbulence intensity both experimentally and numerically. For more accurate estimation, the effective turbulence should be considered when evaluating the turbulence in the turbine farm. The effective turbulence considers the ambient turbulence present in the flow, the additional turbulence generated by the turbine, and the turbulence resulting from interactions among turbines within a farm. Recent investigation using AI has been carried out for turbine placement in tidal farms, including [22,23].

For this reason, the authors proposed an added turbulence model for a full-scale tidal turbine in the article [24] and implemented it in a farm model [25] The present paper constitutes an extension of the conference paper [25]. In this study, we proposed a generic empirical model to estimate the power production in tidal turbine farms. This model for both velocity deficit and turbulent intensity accounts for a range of rotor diameter-to-depth (DH) ratios from 20% to 60%, a range of realistic ambient turbulence (5% to 20%) measured at tidal sites, and a range of rotor thrust coefficients (0.64 to 0.98) available for commercial tidal turbines. This paper aims to provide a low computational model to estimate the power production in tidal farms in realistic sites under ideal conditions. The rest of the paper is organized as follows: Section 2 presents the methodology to obtain the numerical result and the generic empirical model. The tidal farm model is introduced in Section 3, along with its analysis of the results and discussion in Section 4. Finally, the paper ends with concluding remarks.

Turbulence in Tidal Farm

The turbulence in the turbine farm increases due to the kinetic energy extraction by upstream turbines and the turbine interaction in the farm. The effective turbulence in a tidal farm can be expressed as follows:

$$I_{eff}^2=I_0^2+{\left(I_{+,i}+I_{+,ij}\right)}^2$$

(1)

where $I_0$ is the ambient turbulence, $I_{+}$ is the individual added turbulence from the turbine $T_i$, and $I_{+,ij}$ is the added turbulence on turbine $T_i$ interacting with turbines $T_j$. To the best of our knowledge, there is no precise definition of the added turbulence due to turbine interaction. Shamshirband et al. [26] classify turbulence models into four categories, as follows:

• Added turbulence model—calculated for the wake after a single turbulence;
• Added turbulence model—calculated for all nearby turbines;
• Total turbulence model—calculated for the wake after a single turbulence;
• Total turbulence model—calculated for all nearby turbines.

A study conducted by [27] indicates that the influence of adjacent turbines is crucial for predicting the turbulence intensity of a specific turbine. In reference [28], a model is presented to estimate the turbulence intensity within the wake of a wind turbine array. For a single turbine, Crespo’s turbulence relation [29] is employed to determine the additional turbulence, after which the local turbulence intensity is derived using the general turbulence expression applicable to the wake. The turbulence contribution resulting from the wake interaction of the nearest turbine is defined by reference [28] as follows:

$$I_{+j}=\max \left({\displaystyle \frac{A_{w}4}{\pi d_0^2}}{I}_{+ij}\right)$$

(2)

where $I_{+j}$ is the contribution of added turbulence by turbine j. $A_w$ is the wake area of turbine $T_j$ intersecting turbine $T_i$, and $I_{+,ij}$ is the added turbulence intensity induced by the turbine i at the turbine j. Recently, Ref. [30] proposed a similar model for multiple turbines. The individual added turbulence is calculated from [31] and a calibration term to account for the wake interaction with the closest turbine as suggested by [27]. The interaction term is considered to represent half of the added turbulence at the tip. This value can be positive in cases of full overlap or negative in instances of partial overlap, attributed to a diminished velocity shear layer in the overlapping area. This relationship is articulated in Equation (3):

$$I_{+,ij}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{I}_{+,i}{\cos }^2\left({\displaystyle \frac{\pi r_i}{D}}\right)\hfill & \mathrm{full} \mathrm{overlap}\hfill \\ -{\displaystyle \frac{1}{2}}{I}_{+,i}{\sin }^2\left({\displaystyle \frac{\pi \left(y-y_i\right)}{D_{w,i}}}\right){\cos }^2\left({\displaystyle \frac{\pi (z-H)}{L_{w,ij}}}\right)\hfill & \mathrm{partial} \mathrm{overlap}\hfill \\ 0\hfill & \mathrm{else}\hfill \end{array}\right.$$

(3)

where $r_i$ is the spanwise distance from the wake center of $T_i$, and $L_{w,ij}$ is the vertical length of the intersected wake region of $T_i$ and $T_j$.

2. Methodology

2.1. Numerical Model

The numerical model’s data serves as a benchmark for developing the generic empirical model. The actuator disc approach is commonly employed in the analysis of wind and tidal turbine farms, owing to its low computational expense and reasonably good accuracy in the far wake region. This method has been validated in the authors’ prior research [24], which is supported by established experimental data.

2.1.1. Actuator Disc Model

The tidal turbine rotor is modeled using a uniform actuator disc approach in OpenFOAM. This Actuator Disc Model (ADM) simulates the energy extraction process of the turbine by facilitating a momentum transfer that results in a pressure differential across the disc. The thrust force applied on the disc region is evaluated as Equation (4):

$$T={\displaystyle \frac{1}{2}}\rho KA{U}_d^2$$

(4)

where $\rho$ is fluid density, K is the resistance coefficient, A is the frontal surface area, and $U_d$ is the velocity at the disc location. This approach of simulating tidal turbines proposed by Harrison et al. [32] is used extensively in the literature for studies attentive to far wake region [7,8,33]. Details of the numerical model developed and its validation can be found in the previous work of the authors [24].

2.1.2. Numerical Domain

Figure 1 shows a full-scale tidal turbine considered in a 50 m depth channel similar to the Alderney Race. The rotor diameter-to-depth ratios of 20%, 40%, and 60% are considered to cover the range of commercial energy extraction by a turbine in a shallow water regime. The corresponding turbine diameter is given in

Table 1. Rotor size at different diameter-to-depth ratios.

	D/H	20%	40%	60%
H (m)
50		10	20	30

.

2.2. Empirical Model

In this section, we propose a generalized low-computational model to estimate the velocity deficit and turbulence intensity in the wake of a turbine. The detailed development of this method is presented in the authors’ previous works [24,34].

2.2.1. Wake Radius Model

Previously, the authors in ref. [24] proposed a model for estimating the wake radius of the turbine at different ambient turbulence for a DH40 (i.e., 40% rotor diameter-to-depth ratio). However, in shallow water, the ratio of the rotor diameter to the water depth influences the radius of the turbine wake, as illustrated in Figure 2. The wake radius is determined by the Full-Width Half Maximum (FWHM) of a standard Gaussian function, as detailed by the authors in [24]. The generalized wake radius is expressed as Equation (5):

$$\begin{array}{c}{r}_w=a{\left({\displaystyle \frac{x-x_0}{D}}\right)}^b\\ \begin{array}{cc}a=2.36+1.834{\displaystyle \frac{I_0}{D/H}},\hfill & b=0.27{\left({\displaystyle \frac{D}{H}}\right)}^{-0.275}\hfill \end{array}\end{array}$$

(5)

where x, $x_0$, and D are the downstream distance, the turbine position downstream, and the rotor diameter, respectively. The terms a and b are coefficients depending on $I_0$, the ambient turbulence, and $D/H$, the rotor diameter-to-depth ratio.

2.2.2. Velocity Deficit Model

To determine the wake velocity of the tidal turbine, the wake radius model is integrated into the Jensen model [10]. While the Jensen model’s top-hat shape profile provides an estimate of the average velocity within the wake, numerical data reveal that the minimum wake velocity follows a Gaussian distribution. In this study, we suggest employing a Gaussian model directly to assess the velocity deficit in the wake, eliminating the need for a correction coefficient as recommended by Lo Brutto et al. [35]. The velocity deficit is subsequently represented by Equation (6):

$${\displaystyle \frac{\Delta U}{U_{\infty }}}=\left[{\displaystyle \frac{\left(1-\sqrt{1-C_T}\right)}{{\left(\frac{r_w}{r_0}\right)}^2}}\right]\times \exp \left(-{\displaystyle \frac{{\left(y_0-y\right)}^2}{r_w^2}}\right)$$

(6)

where $y_0$ and $z_0$ are the turbine coordinates along y, the spanwise and z, the vertical directions, respectively. $C_T$ is the thrust coefficient, $r_w$ is the wake radius downstream, and $r_0$ is the rotor radius. The exponential function provides the Gaussian shape profile along the lateral plane.

2.2.3. Turbulence Intensity Model

The turbulence intensity in turbine wake is expressed by Quarton and Ainsle [36] for the wind turbine case as follows:

$${\mathrm{I}}_{\mathrm{w}}={\left({\mathrm{I}}_0^2+{\mathrm{I}}_{+}^2\right)}^{0.5}$$

(7)

where $I_w$ is the total turbulence in the wake, $I_0$ is the ambient turbulence, and $I_{+}$ is the added turbulence by the rotor. Assuming the ambient turbulence is stable in the flow, it is, therefore, sufficient to propose an added turbulence model to evaluate the turbulence in the flow.

An added turbulence model of a full-scale turbine has been developed by the authors. For a generic model, the added turbulence is expressed in Equation (8):

$$\begin{array}{c}{I}_{+}=c{\left({\displaystyle \frac{x-x_0}{D}}\right)}^{-d}\times \exp \left(-{\displaystyle \frac{\left.{\left(y_0-y\right)}^2\right)}{r_w^2}}\right)\\ \mathrm{c}=0.407\left({\displaystyle \frac{D}{H}}\right)C_T^{4.83}+0.179,d=0.681I_0+0.472\end{array}$$

(8)

where $x_0$, $y_0$, and $z_0$ are the turbine coordinates along x, the streamwise, y, the spanwise, and z, the vertical directions, respectively. c and d are coefficients relating to non-dimensional rotor diameter-to-depth ratio $(D/H)$, and the ambient turbulence $I_0$. $C_T$ is the thrust coefficient, $r_w$ is the wake radius downstream, and $r_0$ is the rotor radius.

The generic model shows acceptable results in the far wake for the velocity deficit and turbulence intensity compared to experimental data at different rotor DH ratios, as shown in Figure 3. Thus, the generalized model can be applied in the turbine park with rather good results in the far wake.

3. Tidal Farm

3.1. Wake Interaction in Tidal Farm

To accurately estimate the wake of turbines in a tidal farm, the analytical model must consider the interactions between the wakes of multiple turbines. To achieve this, single turbine models use the superposition principle to estimate the combined effect of overlapping wakes in the farm. There are basically three types of wake interaction in tidal farms namely; full overlap (or tandem), partial overlap, and no interaction, as shown in Figure 4. However, a turbine may be affected by a mixed wake on the farm.

The wake overlap area is calculated as follows:

$$\begin{array}{c}{A}_{overlap}=\left\{\begin{array}{cc}0,\hfill & if{r}_w+r_0\le y\hfill \\ A_0,\hfill & if{r}_w-r_0\ge y\hfill \\ A_{partial},\hfill & otherwise\hfill \end{array}\right.\end{array}$$

(9)

where $r_w$ and $r_0$ are the radius of the wake and turbine, respectively, y is the lateral distance between the turbines, and $A_{partial}$ is the intersecting area between the wake area $A_w$ and the rotor swept area $A_0$, as shown in Figure 5. The wake intersection area $A_{partial}$ is calculated using Equation (10) as proposed by [38].

$$A_{overlap}=r_w^2\left({\theta }_w-{\displaystyle \frac{\sin \left(2{\theta }_w\right)}{2}}\right)+r_0^2\left({\theta }_r-{\displaystyle \frac{\sin \left(2{\theta }_r\right)}{2}}\right)$$

(10)

where ${\theta }_w$ and ${\theta }_r$ are the angles of the wake intersection arc and rotor intersection arc, respectively, and can be respectively expressed as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\theta }_w={\cos }^{-1}\left({\displaystyle \frac{r_w^2+y^2-r_0^2}{2y{r}_w}}\right),{\theta }_r={\cos }^{-1}\left({\displaystyle \frac{r_w^2-y^2-r_0^2}{2y{r}_0}}\right)\end{array}\end{array}$$

(11)

To combine the cumulative wake interactions on downstream turbines, various superposition methods have been proposed in the literature to estimate wake–turbine interactions. For example, Refs. [28,30] utilize the Linear Sum of Squares (LSS) method to evaluate the wake in wind farms. The LSS method was first adopted by [39], drawing an analogy to the superposition of neighboring plume concentrations, which conserves pollutant concentration due to linearity. LSS sums the velocity deficit of individual turbines to conserve the total momentum deficit in the wake. Another superposition method used by researchers [13,40] to evaluate the wake interaction is the Root Square method (RSM). The RSM aims to conserve the kinetic energy deficit. These superposition models are largely empirical, with no rich theoretical justification. In this study, the Root Square Method is used to evaluate the velocity deficit and turbulence intensity in the form of kinetic energy deficit and turbulence energy for the interacting turbines.

3.2. Description of Generic Model in Tidal Farm

In ref. [24], the authors proposed a model to estimate the added turbulence of a full-scale tidal turbine. Using the ambient turbulence, the wake turbulence is reconstructed and results in good estimation in comparison with numerical data. The wake turbulence model is used to evaluate the wake expansion taking into account the local added turbulence effect in the wake. A Jensen–Gaussian velocity deficit model is proposed to estimate the velocity deficit in the wake of the full-scale turbine. This model is generalized to account for a range of rotor thrust coefficients, rotor diameter-to-depth ratio, and ambient turbulence. In addition, with our new model, we estimate the turbulent intensity of the wake at different hydrodynamic conditions. For a tidal farm application, the wake interaction between turbines is evaluated using trigonometric relation and estimated by superimposing the wake interacting effect to evaluate the velocity deficit and the turbulence intensity in a tidal farm. The generic velocity deficit and turbulent intensity in the tidal farm are expressed in Equations (13)–(16). In these equations, the added turbulence intensity is first estimated accounting for the wake interaction in the tidal farm as expressed in Equation (13). Using the added local added turbulence in the farm, the effective turbulence is evaluated in Equation (12). Further, the wake radius expansion in Equation (14) is calculated using the effective turbulence in the wake. This ensures the use of approximate local turbulence at the turbine locating. To evaluate the velocity deficit, the wake radius model is used in the Jensen–Gaussian model expressed in Equation (15), and the kinetic energy deficit relation in Equation (16) accounting the wake interaction is used to estimate the velocity deficit in the tidal farm for turbine $T_i$ to $T_N$, where N is the number of turbines.

$$I_{eff}=I_0^2+I_{+,j}^2$$

(12)

where $I_{eff}$ is the effective turbulence expressed in Equation (7) as the sum of ambient turbulence $I_0$ and added turbulence $I_{+}$. The added turbulence to account for the turbine interaction in the tidal farm is expressed as Equation (13):

$$\begin{array}{c}{I}_{+,j}^2={\displaystyle \sum _{i=1}^N}{\left(e{\left({\displaystyle \frac{x_{ij}-x_i}{D}}\right)}^{-f}\times \exp \left(-{\displaystyle \frac{\left.{\left(y_{ij}-y_i\right)}^2\right)}{r_w^2}}\right)\right)}^2{\displaystyle \frac{A_{overlap}}{A_0}}\\ \mathrm{e}=0.407\left({\displaystyle \frac{D}{H}}\right)C_T^{4.83}+0.179,f=0.681I_0+0.472\end{array}$$

(13)

where $x_i$, $x_{ij}$, $y_i$, $y_{ij}$ is the longitudinal and lateral position of turbine $T_i$ and interacting turbine $T_{ij}$ upstream, respectively. D is the rotor diameter. The terms e and f are coefficients depending on the ambient turbulence $I_0$ and the rotor diameter-to-depth ratio ($D/H$).

The wake radius model is evaluated below:

$$\begin{array}{c}{r}_w=g{\left({\displaystyle \frac{x-x_0}{D}}\right)}^h\\ \begin{array}{cc}g=2.36+1.834{\displaystyle \frac{I_{eff}}{D/H}},\hfill & h=0.27{\left({\displaystyle \frac{D}{H}}\right)}^{-0.275}\hfill \end{array}\end{array}$$

(14)

The velocity deficit model is calculated as follows:

$${\displaystyle \frac{\Delta U}{U_{\infty }}}=\left[{\displaystyle \frac{\left(1-\sqrt{1-C_T}\right)}{{\left(\frac{r_w}{r_0}\right)}^2}}\right]\times \exp \left(-{\displaystyle \frac{{\left(y_0-y\right)}^2}{r_w^2}}\right)$$

(15)

$$\begin{array}{c}{\left(1-{\displaystyle \frac{u_j}{U}}\right)}^2={\displaystyle \sum _{i=1}^N}{\left(1-{\displaystyle \frac{u_{ij}}{U}}\right)}^2{\displaystyle \frac{A_{overlap}}{A_0}}\end{array}$$

(16)

where y are the turbine coordinates along the spanwise direction. $C_T$ is the thrust coefficient, $r_w$ is the wake radius downstream, and $r_0$ is the rotor radius. The exponential function provides the Gaussian shape profile along the lateral plane. A schematic description of the generic empirical model is shown in Figure 6.

3.3. Description of Cases Studied

The first case concerns a farm with 4 turbines in tandem for a DH ratio of 20% with an upstream turbulence rate of 10% and speeds of the order of those of the Raz Blanchard (France). The turbine diameter is 20 m. The results of the analytical model are compared with the results of a simulation of a farm using Actuator Discs. This case highlights the good behaviour of the model in the most restrictive farm configuration. In particular, on the longitudinal profile of axial velocity and turbulence. The second case compares the power produced by two tidal farm configurations: staggered and rectiliear layouts. A turbulence of 10% is also imposed. The results are also compared with a CFD model using actuator discs. The aim here is to quantify the error of the analytical model compared with the CFD simulation in terms of power over two different configurations. The third case focuses on the farm with staggered positioning and explores the effect of the DH ratio on the power produced. The fourth case looks at the effects of ambient turbulence on energy production for a fixed DH ratio of 40%. An in-line positioning is adopted here because it is more constraining. In the fifth case, the farm with in-line positioning is used to study the effects of turbine control, which here corresponds to a choice of the thrust coefficient Ct. The four lines of turbines have different Ct values, leading to different power outputs. Finally, the sixth case considers the layout of a 40-turbine farm obtained by optimization in a previous study. In this application, several turbulence intensities and inlet speeds are considered.

4. Analysis and Discussion

4.1. Turbine Array in Tandem Configuration

A simple tidal farm comprising four turbines in in-line configuration is studied. The rotor diameter-to-depth (DH) ratio of 20% is considered in the simple array. The ambient turbulence used in this case is 10%, whereas the Betz thrust coefficient $C_{T,Betz}$ is used throughout this chapter unless otherwise stated. Turbines are designated as $T_i$ where i is the turbine’s position in the farm. For tandem configuration (full overlap), the downstream turbines are fully immersed in the wake of the upstream turbines, as shown in Figure 4a. The wake interaction is evaluated as expressed in Equation (9). Figure 7 shows the empirical model provides a good centerline estimation of the normalized velocity and turbulent intensity compared to the numerical data. At $S_x=5D$, the downstream turbine is subjected to higher incoming turbulence compared to the upstream turbine. An incremental increase in turbulent intensity is observed due to cumulative added turbulence effects. The power produced by turbines downstream is much lower compared to the upstream turbine due to limited turbine spacing. The contour in Figure 8 shows that as the turbine spacing doubles, the wake largely recovers to upstream condition, leading to extraction of higher power by the downstream turbines. The incoming turbulence of turbine $T_2$ decreases from 15.3% to 12.5% when the turbine spacing is double.

The wake effect is higher when the turbine spacing $S_x$ is small, causing an increase in the incoming turbulence intensity of the downstream turbine. In a channel at 10% ambient turbulence, the results show that the turbine downstream can detect up to 50% higher turbulence due to the successive wake effect at small turbine spacing. However, Figure 7 presents the maximum turbulence along the centerline (Gaussian profile), and the mean turbulence across the rotor is lower as indicated in Figure 8. The turbine spacing is shown to affect the power production of the turbine downstream, which will affect the overall efficiency of the farm. For $I_0=10\%$, the turbine spacing of $S_x=7D$ allows sufficient wake recovery and acceptable added turbulence on the downstream turbine. However, as the wake recovers faster at high ambient turbulence due to strong mixing and diffusion [41,42] in the wake, the turbine spacing may be affected by the ambient turbulence. The effect of ambient turbulence is presented in Section 4.4.

4.2. Power Production in Different Array Configuration

The power produced by a turbine is calculated as Equation (17):

$$P_i={\displaystyle \frac{1}{2}}\rho C_{p}A{U}_i^3$$

(17)

where $\rho$ is the water density, $C_P$ is the power coefficient, A is the cross-sectional area evaluated as $\pi D$/4, and $U_i$ is the incoming velocity for turbine $T_i$.

In this section, the two basic tidal farm configurations, rectilinear and staggered 20 m diameter turbine at DH40, are analyzed. The farm comprising N turbines ($N=16$) in an ideal channel is illustrated in Figure 9. The turbines are assumed to perform at Betz operating conditions. The turbine spacing is $S_x=7D$ and $S_y=4D$, respectively, to ensure flow recovery and benefits from accelerated flow from the upstream turbines. In the ideal farm, the ambient turbulent intensity of 10% and a constant mean flow of 2.8 m/s are used, similar to the flow in the Alderney Race [43]. However, the variability of the power curve with current velocity is not considered in the algorithm.

Figure 10 illustrate a comparison between numerical and analytical results across various array configurations. The turbines positioned upstream generated the highest power output, as they were unaffected by wake interactions. Conversely, the downstream turbine exhibited reduced power production due to the influence of wake interactions within the farm. This observation aligns with the numerical results presented in Figure 10.

For the rectilinear configuration, the turbine downstream is fully in the wake of the upstream turbines. The wake interaction is evaluated as presented in Figure 4b. The rectilinear array (Figure 10a,c) identifies each turbine row independently as having no interaction between turbine rows, whereas the staggered configuration shows the wake interaction in both longitudinal and lateral directions, as indicated in Figure 10b,d. The numerical and empirical power produced by each turbine is compared in Figure 11. In the numerical model, the power is evaluated at a location 1.5D away from the turbine before the pressure jump begins to develop.

The power extracted by a turbine is calculated as follows by Equation (17). The empirical model underpredicts the numerical power in the farm by 2.5%. A comparison of power produced by each turbine in the rectilinear and staggered array using the empirical model is presented in Figure 12. It is noted that the upstream turbine produces identical power output irrespective of the turbine configuration. For the rectilinear array, the power produced from turbine $T_1$–$T_4$ decreases due to the cumulative wake effect. The downstream turbine is affected by the wake, therefore producing less power. Similarly, for a staggered array, the turbine downstream produces less power due to the wake effect; however, the power produced is significantly higher than a similar turbine in a rectilinear array. For instance, a turbine $T_3$ with a double turbine spacing in a staggered array produces 23.6% more power than a corresponding $T_3$ in a rectilinear array. The wide turbine spacing allows a substantial recovery velocity deficit in the flow.

The cumulative power produced by N turbines in the farm is evaluated as Equation (18):

$$P_{farm}=\sum _{i=1}^N{P}_i$$

(18)

The total power produced in the rectilinear and staggered farm is 22.1 MW and 28.1 MW, respectively. For an identical farm size, this study shows the staggered array produces 6 MW power higher than the rectilinear configuration, which is equivalent to 21.4%. The increase in power in staggered configuration is largely due to turbine spacing. The longitudinal turbine-to-turbine spacing is double, with staggered configuration allowing sufficient flow recovery downstream. This is consistent with previous tidal farm studies [6,7,8]. The added turbulence effect is higher in the rectilinear configuration, as presented in Figure 13. The results show the added turbulence effect is significant in a rectilinear array as a consequence of limited turbine spacing. The turbulence intensity of the turbine downstream in the staggered array is largely recovered due to sufficient turbine spacing. The farm efficiency, ${\eta }_{farm}$, is defined as the ratio of the total power output from all turbines in a farm, $P_{farm}$, to the maximum power, $P_{farm}^{max}$, that would be produced if they operated under unperturbed conditions.

$${\eta }_{farm}={\displaystyle \frac{P_{farm}}{P_{farm}^{max}}}$$

The farm efficiencies for the rectilinear and staggered array are 69.1% and 87.8%, respectively. This increase in efficiency is solely due to the turbine configuration in the farm.

It is noted that in the rectilinear configuration, the lateral spacing $S_y$ between turbine rows can be reduced without affecting the efficiency of the farm. The question remains, what is the minimal turbine spacing that barely diminishes the output power? The lateral spacing $S_y$ is reduced from $4D$ to $2D$ while maintaining the longitudinal spacing of $7D$, as shown in Figure 14. The cumulative power in the farm remains 22.1 MW for the different lateral spacing. However, a wake interaction effect is spotted between the turbine rows. In reality, there may be some slight differences as the model evaluates the power using the centerline velocity instead of the average velocity across the rotor. In the staggered array, Figure 15 compares the velocity contour for longitudinal spacing $S_x$ reduction from $7D$ to $5D$ (i.e., the effective in-line spacing is reduced from $14D$ to $10D$). The total power extracted from the farm dropped from 28.1 MW to 27.1 MW. The 3.1% reduction in efficiency can be tolerated in regard to the reduction in the farm size. Reducing the farm size whilst maintaining efficiency is essential because it will drastically reduce cable costs and will present an opportunity for future expansion of the farm.

4.3. Effect of Rotor DH Ratio in Tidal Farm

The effect of the rotor DH ratio on tidal farms is also examined. The DH ratio represents the rotor diameter relative to the channel depth. Rotor wake recovery is influenced by bypass flow. Figure 16 illustrates that at a low rotor diameter-to-depth ratio, wake expansion and interaction are greater. The power generated by downstream turbines decreases as the DH ratio increases, as shown in Figure 17. The total farm power, $P_{farm}$, at DH20, DH40, and DH60 is 29.8 MW, 28.1 MW, and 26.9 MW, respectively. At a low rotor diameter-to-depth ratio (i.e., DH20), the wake recovery process is faster because the flow in the bypass region is sufficient to mix the low velocity at the rotor’s core with the ambient flow in the bypass region. Conversely, at a high DH ratio (i.e., DH60), the velocity deficit along the rotor is significant compared to the free stream flow in the bypass region, thus delaying wake recovery to upstream conditions. This indicates that for identical turbine sizes, the power extracted can vary with the channel diameter-to-depth ratio.

4.4. Effect of Ambient Turbulence on Tidal Farm

To investigate the effect of ambient turbulence on the power produced in a pilot farm, we studied a rectilinear array farm at DH40 with inter-device spacing $S_x=7D$ as presented at 5%, 10%, 15%, and 20% ambient turbulence. Figure 18 shows an increase in wake expansion as a result of the increase in ambient turbulent intensity. At $S_y=4D$, the in-line turbines act independently due to sufficient lateral spacing. The power extracted by individual turbines in the row is presented in Figure 19. The results indicate a decrease in power by virtue of the cumulative wake effect along in-line turbines. The power produced by the upstream turbine is identical for all ambient turbulence because the incoming flow for the upstream turbine is stable and not affected by any perturbation in the flow. Similarly, the power production is identical for each row at a given ambient turbulence. The continuous decrease in power for in-line turbines in a row at a given ambient turbulence results from the wake effect. However, at higher ambient turbulence, the wake recovery is faster, therefore producing relatively higher power, as shown in

Table 2. Power extracted by turbines at different ambient turbulence in a rectilinear array.

Power (MW)
	${\mathit{I}}_{\mathbf{0}}$	5%	10%	15%	20%
${\mathit{T}}_{\mathit{N}}$
$T_1$		2.04	2.04	2.04	2.04
$T_2$		1.10	1.30	1.39	1.45
$T_3$		0.95	1.18	1.26	1.32
$T_4$		0.87	1.12	1.19	1.24
$P_{row}$		4.96	5.64	5.88	6.05
${\eta }_{row}$		60.80%	69.20%	72.10%	74.10%

. To clarify, turbine $T_2$ records a 24% increase in power due to increasing turbulent intensity from 5% to 20%. This increase in power extracted is noted for each turbine downstream. The wake recovery is accelerated by increasing ambient turbulence, thus producing more power. The farm efficiency increases with an increase in ambient turbulence as a result of increased wake recovery, as shown in Table 2.

In essence, the ambient turbulence can affect power production by downstream turbines. A low turbulent tidal site will require large turbine spacing to establish flow recovery and reduce the added turbulence effect on the downstream turbine. In contrast, a highly turbulent site can benefit from accelerated flow recovery permitting smaller turbine spacing in the array. The predicted power extracted by the rotor in the first row remains identical for different ambient turbulence because the incoming flow is equal. There is no added turbulence effect upstream for the isolated turbine to hasten the flow recovery.

4.5. Effect of Command Strategies on Farm Power

This section presents the effect of the thrust coefficient on the turbine array. The turbine diameter is 20 m, and the rotor diameter-to-depth ratio is 40%. We consider a rectilinear tidal farm consisting of 16 turbines with 4 turbines in each row, as shown in Figure 20. The first row named, $R_1$, has identical Betz thrust of coefficient for all turbine $T_1$ to $T_4$ (see

Table 3. Power extracted by turbines for several command strategies in a rectilinear array.

	${\mathit{T}}_{\mathit{N}}$	${\mathit{C}}_{\mathit{T}}$	$\mathit{U}$ (m/s)	${\mathit{P}}_{\mathit{i}}$ (MW)
	$T_1$	0.89	2.78	2.00
	$T_2$	0.89	2.39	1.27
	$T_3$	0.89	2.32	1.16
	$T_4$	0.89	2.27	1.10
Row1				5.53
	$T_1$	0.70	2.78	1.83
	$T_2$	0.70	2.51	1.35
	$T_3$	0.70	2.46	1.27
	$T_4$	0.70	2.44	1.23
Row2				5.68
	$T_1$	0.75	2.78	1.90
	$T_2$	0.89	2.49	1.43
	$T_3$	0.89	2.34	1.20
	$T_4$	0.89	2.29	1.12
Row3				5.65
	$T_1$	0.70	2.78	1.83
	$T_2$	0.75	2.51	1.40
	$T_3$	0.85	2.44	1.34
	$T_4$	0.89	2.35	1.21
Row4				5.78

). Similarly, second-row $R_2$ turbines have identical thrust coefficients ($C_T$ = 0.7). Turbine $T_1$ in the third row $R_3$ has a thrust coefficient of 0.75, whereas the rest of the turbines in the row have identical Betz thrust coefficients (see Table 3). The turbines in $Ro{w}_4$ have different thrust coefficients in ascending order, as shown in Table 3.

Figure 21a shows the power production by the turbine presented in Table 3. The total power produced in the turbine row is comparable. The upstream turbine $T_1$ produces the highest power, as it is not affected by the upstream turbine wake. The turbine $T_1$ in $Ro{w}_1$ with the highest $C_T$ produces the maximum power, whereas the turbine in $Ro{w}_3$ with minimum $C_T$ produces the least power, as shown in Table 3.

However, the power produced in $Ro{w}_2$ with a lower turbine thrust coefficient is 2.64% higher than $Ro{w}_1$ with $C_{T,Betz}$. Increasing the thrust coefficient is interpreted as increasing the drag, causing a momentum exchange. The velocity recovery of turbines with lower thrust coefficient is faster leading to higher upstream velocity for the downstream turbines. Similarly, comparing $Ro{w}_1$ and $Ro{w}_3$, an increase of 0.12 MW in $Ro{w}_3$ is observed due to the lower thrust coefficient in turbine $T_1$. The low thrust coefficient of turbine $T_1$ allows faster wake recovery of the downstream turbines. The turbines in $Ro{w}_4$ produce the highest power of 5.78 MW in the tidal farm. The turbine thrust coefficient affects the wake of the downstream turbine. Figure 21b shows the recommended strategy in the tidal farm is turbines arranged with increasing thrust coefficient (i.e., $Ro{w}_4$). However, the overall increase in power due to the variation in thrust coefficient is limited in a small tidal array. This effect can be significant in a large array and, therefore, should be considered in optimizing the tidal turbine farm.

4.6. Application on Large Tidal Farm

Lo brutto et al. [13] use Particle Swarm Optimization to evaluate the power production in tidal turbine farms. The best turbine configuration for uni-directional flow is from the optimization presented in Figure 22. The tidal farm comprises a 10 m diameter turbine in a channel with a rotor diameter-to-depth ratio of 20%. The low DH ratio will allow more wake expansion for the downstream turbine. The minimum turbine spacing is set as 5D in both longitudinal and lateral directions. The average turbine spacing in the longitudinal and lateral directions in the optimized farm is 7.16D and 14.06D, respectively. The turbine spacing is sufficiently large to ensure the wake recovery for a downstream turbine.

Figure 23, Figure 24 and Figure 25 presents the contour of the normalized velocity and turbulent intensity at different velocities. At cut-in speed, the power produced in the farm is very low, as turbines do not produce electricity below the cut-in speed.

Table 4. Summary of power extracted at different hydrodynamic conditions in an optimized tidal turbine farm.

	${\mathit{I}}_0$		5%	10%	15%
$\mathit{U}$ (m/s)
1.0		$P_{farm}$ (MW)	0.773	0.872	0.894
2.0		$P_{farm}$ (MW)	6.187	6.976	7.151
3.0		$P_{farm}$ (MW)	20.881	23.545	24.135
		${\eta }_{farm}$ [%]	79.13	89.22	91.46

shows an increase in ambient turbulence increases the power extracted by the turbine. The power produced in the farm increases significantly with an increase in upstream velocity from 2 m/s to the rated speed of 3 m/s. This large increase in power for a given turbulent intensity is attributed to the increase in the incoming velocity in the farm. The power extracted is proportional to the cube of the velocity (P ∝ $U^3$). For example, at 5% ambient turbulence, $P_{farm}$ increases by 14.7 MW when the upstream velocity increases from 2.0 m/s to 3.0 m/s, as shown in Table 4. For a given incoming velocity, an increase in ambient turbulence increases the power produced in the tidal farm. The tidal farm attains a maximum efficiency of 91.46% when the ambient turbulence is 15%.

5. Conclusions

This study proposes a generic empirical model to predict the velocity deficit and turbulence intensity in turbine farms. The new model accounts for the effects of wake-added turbulence intensity and wake interactions within the farm. Benefiting from the turbine arrangement, the staggered array generates approximately 6 MW more power than the rectilinear array under the same conditions. This increase in power is attributed to sufficient wake recovery in the staggered configuration. Comparisons with different turbine spacings have shown that both rectilinear and staggered arrays can benefit from reduced lateral and longitudinal spacing without compromising farm efficiency. Additionally, the results indicate that in shallow water, the channel depth can impact overall power production in a farm. An increase in power is observed at DH20, resulting from sufficient wake expansion at a low rotor diameter-to-depth ratio. The power produced in the farm also increases with ambient turbulence in the flow. An increase in turbulence enhances recovery, which may allow harnessing more power. To sum up, our study shows that the efficiency of a tidal farm can be increased by (1) high ambient turbulent intensity, (2) sufficient turbine spacing, and (3) low rotor diameter-to-depth ratio. These factors improve the wake recovery to allow more energy to be extracted by a downstream turbine. This generic model can provide insight into the power production in tidal farms under different conditions. For the perspective of future work, the wake of velocity deficit and turbulence intensity shows dependence on a number of parameters. Therefore a multi-objective function optimization algorithm like the Particle Swarm Optimization (PSO) is required for turbine placement to maximize the power production in tidal turbine farms.