\chapter{Rotor fence performance}\label{Chapter2}


This chapter explores the hydrodynamic performance of horizontal axis tidal turbines in a fence arrangement. Firstly, the performance metrics of a single turbine and the concept of channel blockage is introduced. These ideas are then extended to a turbine fence, presenting turbine fence performance metrics and introducing local blockage. Finally, fence performance losses are discussed and potential mitigation methods are presented. 


\section{Performance metrics}

\subsection{Single turbine}\label{CH2:Single_turb}

The performance of a single horizontal axis turbine is typically characterised by its thrust and power coefficient, defined as 
\begin{equation}
    C_t = \frac{T}{\tfrac{1}{2}\rho A_t u_{\infty}^2}, \; \;
    C_p = \frac{P}{\tfrac{1}{2}\rho A_t u_{\infty}^3}
\end{equation}

\noindent where $C_t$ is the thrust coefficient, $C_p$ is the power coefficient, $T$ is the thrust on the turbine, $P$ is the power produced by the turbine, $\rho$ is the density of the fluid, $A_t$ is the swept area of the turbine and $u_{\infty}$ is the free-stream flow velocity normal to the turbine. Turbine performance varies with the tip-speed-ratio ($\lambda$), which is defined as 

\begin{equation}
    \lambda=\frac{\Omega R}{u_{\infty}}
\end{equation}

\noindent where $\Omega$ is the rotational speed of the turbine in radians per second and $R$ is the turbine radius. Typically, $C_t$ monotonically increases with $\lambda$ and $C_p$ initially increases with $\lambda$ to a maximum value ($C_p^*$) at $\lambda^*$ after which performance reduces. A maximum $C_p$ exists due to a reduction in blade torque ($Q_t$) at high speed through increasing drag losses. Power from a turbine is extracted following 

\begin{equation}
    P=Q_t \Omega.
\end{equation}

\noindent This implies that power increases as the turbine rotates faster; however, $Q_t$ is a function of $\Omega$ that decreases at high $\Omega$ due to a sub-optimal angle of attack ($\alpha$) on the blade. Figure \ref{Fig3:VelocityTri} plots the velocity triangle at a radial station ($r$) on a turbine blade and demonstrates how the angle of attack decreases with $\Omega$. 

% improve this plot
\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/numerical/BEM_001.png}
\caption{(a): Rotor discretisation; (b): resolution of lift and drag forces at radius $r$.}
\label{Fig3:VelocityTri}
\end{figure}



Figure \ref{Fig3:VelocityTri} shows that the flow velocity over a turbine blade ($W$) is a function of the velocity in the stream-wise direction ($u_x$) and the velocity in the rotational direction; composed of the flow in the rotational plane ($u_\theta$) and the angular velocity of the section ($\Omega r$). 

\begin{equation}
    W = \left[ \left(\Omega r + u_{\theta} \right)^{2} +u_{x}^{2} \right]^{1/2}
\end{equation}


The flow approaches the turbine blade at an angle $\phi$, which is the sum of $\alpha$ and the blade twist ($\beta$). As a result, a unit span of the section produces a lift force ($F_l$) and a drag force ($F_d$) defined as

\begin{equation}
    F_l = \tfrac{1}{2}\rho W^2 c_l c, \; \;
    F_d = \tfrac{1}{2}\rho W^2 c_d c
\end{equation}

\noindent where $c$ is the chord length, $c_l$ is a lift coefficient, and $c_d$ is a drag coefficient. $c_l$ and $c_d$ are often taken from two-dimensional (2D) aerofoil sections; this simplification requires no flow along the turbine radius and is explored in Section \ref{CH2:BE_model}. Force in the stream-wise ($F_x$) and rotational ($F_{\theta}$) directions at $r$ can be determined by applying a rotation to $F_l$ and $F_d$. This yields 


\begin{equation*}
\begin{aligned}
    F_x = F_l \cos{\phi} + F_d \sin{\phi} \\
    F_{\theta} = F_l \sin{\phi} - F_d \cos{\phi}.
\end{aligned}
\end{equation*}

\noindent The forces can be integrated over the length of the blade (from the radius of the nacelle, $r_{hub}$, to the end of the blade $R$) to determine $T$ and $Q_t$,

\begin{equation*}
    \begin{aligned}
        T = \int_{r_{hub}}^{R} F_x(r) dr \\ 
        Q_t = \int_{r_{hub}}^{R} F_{\theta}(r) r dr
    \end{aligned}
\end{equation*}

\noindent While the relations are simple to express, $T$ and $Q_t$ are complex functions dependant on the inflow properties, the turbine design, and the turbine operation. Consequently, various models exist to predict turbine performance and the impact on the surrounding flow, a selection of models are discussed in Chapter \ref{CH3:Model_turb}


\subsection{Turbine fence}
A turbine fence is an array of turbines which operates in the same plane. This arrangement has been shown to generate greater power than staggered alignments \citep{Draper2014} and the close proximity of turbines in this configuration allows for the use of shared infrastructure within the fence; such as in the Orbital Marine O2 and SR2000 turbine \citep{OrbitalMarinePower2019} and the Sustainable Marine Energy PLAT-I device \citep{SME}.

Alongside the performance metrics presented for individual turbines in Section \ref{CH2:Single_turb} the basin efficiency ($\eta$) is an important characteristic of a turbine fence. Basin efficiency is defined as the ratio between useful power extracted from the flow and total power removed from the flow,

\begin{equation}
    \eta = \frac{\sum P}{\sum P_{removed}}
\end{equation}

\noindent where $\sum P$ is the sum of power produced by all the turbines and $\sum P_{removed}$ is sum of all power removed from the flow, including from parasitic drag and dissipated in wake mixing downstream of the turbines. Parasitic drag includes forces which do not contribute to the power production, such as the force applied from the support structure to the flow. The definition of \cite{Belloni2013} was used to determine $\eta$ throughout this thesis following,

\begin{equation}
    \begin{aligned} 
    \eta = \frac{\sum P}{(\sum T+\sum T_{structure})_x u_{\infty}}.
    \end{aligned}
    \label{Eq:Basin}
\end{equation}

\noindent where $\sum T_x$ is the sum of the turbine thrusts in the stream-wise direction and $\sum T_{structure - x}$ is the sum of the thrust from the support structure in the stream-wise direction. This value only considers the floating structures in this thesis and does not include additional sources of parasitic drag such as from the support towers and other structural support.  



\section{Blockage}\label{Ch2_Blockage}
Pertinent to improving the performance of a turbine is an understanding of the blockage effect. \cite{GARRETT2007} first investigated blockage in a channel demonstrating that an improvement in tidal turbine performance was possible due the volume-flux constraint presented by the channel to the flow through the turbine. A key finding from this study was that the maximum $C_p$ surpassed the well-established Lanchester-Betz limit\footnote{Also referred to as the Betz limit.} for unblocked flows of $16/27$ in blocked channels. It was shown that a new limit of  $(16/27) (1-B_g)^{-2}$ applies to turbines in blocked flow conditions, where $B_g$ is the blockage ratio defined as $\tfrac{\sum A_t}{A_c}$  ($A_c$ is the channel cross-sectional area). This effect has been observed experimentally in the work of \cite{Bahaj2007} who saw a performance improvement when testing a turbine in a cavitation tunnel and more recently by \cite{Ross2020} in their testing of a HAT and VAT in varied blockage conditions.


The result from Garrett and Cummins' studies have been expanded on by various authors, leading to additional definitions of blockage for different area ratios, namely global blockage ($B_g$), array blockage (ratio of frontal area of turbine array, $A_a$, to channel cross section area, $B_a=\tfrac{A_a}{A_c}$) and local blockage (ratio of device area, $A_t$, to the local passage cross-sectional area, $A_l$, $B_l=\tfrac{A_t}{A_l}$). These three definitions are illustrated in Figure \ref{Figure 1.2}. Using these definitions for blockage, \citet{Nishino2012} introduced an approach to analyse an ideal system at both the turbine and the array scale using continuity and momentum relations to maintain compatibility between the two models. Through this theoretical frame work they identified a new limit for power extraction from a long fence of turbines partially blocking an infinitely wide channel ($B_g=0$) of $C_p=0.798$ when $B_l \approx 0.4$, a 34.5\% improvement on the Betz limit of 0.593. A small further uplift in performance, dependent on the Froude number, can be achieved when the effects of free-surface deformation are also considered \citep{Vogel2016}.



A limitation to the models discussed above is the reliance on linear actuator disc theory which is an idealised parameterisation of turbine behaviour that does not account for real rotor properties such as the shape of the blade profile. This detail can be represented by using more refined methods discussed in Chapter \ref{CH3:Model_turb}.


Performance improvements from increased local blockage have been observed numerically by \cite{Schluntz2015}, who explored the effect of local blockage on rotor design. It was shown that higher power output is possible when turbines designed for low blockage conditions are moved closer together, however greater performance is observed when turbines are designed for the higher blockage conditions. Experimentally, \cite{Cooke2016} has demonstrated the effect of local blockage in a flume through the use of thrust measurements on porous disks. More recently, scale turbine testing of two 1.2 m diameter turbines at a 0.25D tip-to-tip spacing by \cite{McNaughton2019} has provided further evidence of the constructive interference effect of local blockage, observing a performance improvement above the single turbine maximum power when placing a second turbine adjacent at a close spacing. 


\begin{figure}[h]
         \center{\includegraphics[width=\textwidth]
        {Figures/Chapter_1/Length_scales_Blockage.png}
        \caption{\label{Figure 1.2} Cross section view of the representative areas for global (green), array (red) and local (blue) blockage, the circles represent individual turbines swept area.}}
\end{figure}



\section{End-losses}

A further limitation of \cite{Nishino2012} was the infinitely long turbine fence assumption where all turbines in the fence see uniform flow; in practise fences are a finite length and performance varies due to flow diversion around the ends of the fence as illustrated in Figure \ref{Fig:Approach_Flow}. The effect of fence length was investigated by \cite{Cooke2016} who used a fence of porous discs in a flume to investigate the effects of spacing and variation in performance along the fence. Their results showed that power could be increased by adjusting the spacing as suggested by \cite{Nishino2012}, however energy losses arising from viscous effects and flow diversion around the ends of the fence limited the power to less than hypothesised in the ideal case. The reduction in power due to flow diversion around turbine fences has also been observed numerically in a RANS-BE model by \cite{Vogel2017}. \cite{Vogel2017} observed a drop in performance along a fence of four turbines ($B_l = 0.196$, $C_p = 0.535$ for the inboard turbines and $C_p = 0.528$ for the outboard turbines), crediting this reduction in performance to flow diverting around the fence, reducing the flow speed through the turbines. 

\begin{figure}
    \centering
    \includegraphics[width=1\textwidth]{Figures/Chapter_1/Approach_flow_With_Y_velocity_annotated.png}
    \caption{Streamlines of flow approaching a turbine fence, highlighting the greater flow diversion around the end turbines over the inner turbines. The contours are the absolute value of the cross-stream flow velocity.}
    \label{Fig:Approach_Flow}
\end{figure}


 
The performance loss at the end of a turbine fence is often referred to as the fence end-loss. This loss is a result of insufficient confinement at the end of the fence allowing flow to bypass the turbines, leading to a performance reduction for the end turbines. While an attempt to include this effect in analytical models has been made by \citet{Cooke2016}, this model does not accurately capture the thrust distribution across the fence. Instead estimates for short array performance require the use of numerical models to resolve the performance variation along the array \citep{Schluntz2014,Vogel2017}.


\section{Methods to improve turbine performance}

Turbine performance can be augmented by either a physical manipulation to the turbine design (physical optimisation) or through a modification to how turbines are operated and oriented (operational optimisation). Within these two classes various methods have been investigated to improve total energy output. This section presents a selection of these methods which are further developed throughout this thesis.


\subsection{Physical optimisation}

The physical optimisation of turbines requires adjusting the design of the rotor (blade profile, twist and solidity) as well as the infrastructure surrounding the rotor (nacelle design and ducting). 


\subsubsection{Turbine design}

The design of a turbine is a loosely constrained problem with many potential blade cross-sections and geometries, making it difficult to define an optimal design. Consequentially, simplifying assumptions are required to constrain the problem and optimise. Typically, this includes constraining the design to a single aerofoil or family of aerofoils such as the MHKF1 \citep{Shiu2012} which has been specifically designed to suit the needs of tidal turbines (high lift-to-drag ratio, low susceptibility to cavitation and structurally efficient) or the NACA-63-XXX family which have previously been used in turbine design \citep{Bahaj2007,Laurens2016,Park2016}. It is also common to transition between profiles \citep{Bahaj2007,Shiu2012} as the optimal profile is a function of the radial position on the blade.

Various methods have been suggested for defining the blade twist and solidity distribution along the span of a turbine blade \citep{Masters2011,McIntosh2011,Malki2013}. While methods vary slightly, they all rely on the blade element method and its underlying assumptions (outlined in Section \ref{CH2:BE_model}) to determine the optimal turbine blade twist and solidity. This method targets an optimal angle of attack, $\alpha$, (angle where the ratio of lift to drag is a maximum) and a target local thrust coefficient, $c_x$, along the blade length. These variables are not independent of each other due to feedback between $\alpha$, which maximises torque ($r F_l \sin{\phi} - r F_d \cos{\phi}$), and $c_x$. A change in $c_x$ directly affects the $F_l$ and $F_d$ over the turbine blades as well as the flow through the rotor. Given this, the target $c_x$ is usually determined from iteration until an optimal $C_p$ is achieved. $c_x$ may also vary over the blade length as regions along the turbine may require different $c_x$ to optimise power production or the target $c_x$ may cause intersecting geometries as large $c_x$ results in a larger chord; in which case target $c_x$ can be reduced. A blade element method is used for the initial design of a turbine in Chapter \ref{Chapter5}. 

For the design of turbine fences, local blockage effects allow the fence to sustain a slightly higher thrust which alters the optimal $c_x$ between turbines within an array. \cite{Cao2019} showed this effect over a four-turbine fence in highly blocked conditions ($B_g=14.1\%$) with the maximum fence power being achieved when the inboard and outboard turbines were designed with a uniform $c_x$ of 3.4 and 3.6. In contrast the optimal $c_x$ for a single turbine in an identical blockage conditions was 3.0.


\subsubsection{Ducting of turbines}

Some designs \citep{EuropeanMarineEnergyCentre2018} of tidal turbines have included ducting around rotors to improve power output by convective acceleration of flow through the throat of the duct. This has previously been considered in the wind industry \citep{Igra1981,Khamlaj2018} in order to improve the mass flow rate through turbines to achieve greater power output per swept area of rotor and has lead to the development of various prototype designs \citep{Wind-works2019,HaloEnergy2019}. Work by \cite{Khamlaj2018} has demonstrated, using a Reynolds-Averaged Navier-Stokes (RANS) model with an embedded blade element (BE) actuator disk model, that the introduction of ducting on wind turbines can marginally improve the total power output ($\sim 2\%$).

Despite an increase in total power output, radially ducted wind and tidal turbines tend to have a reduced $C_p$ when the performance is normalised for the area at the opening of the duct. This has been investigated for tidal applications by \cite{Belloni2013} who, like \cite{Khamlaj2018}, used RANS-BE simulations to investigate the performance of ducted turbines. \cite{Belloni2013} observed an overall reduction in $\eta$ with the introduction of ducting due to the higher thrust it imposed onto the flow. This study was limited as it only analysed one type of radially symmetric ducting around a turbine drawn from observations of a proposed commercial devices. Work in this area by \cite{Fleming2016} produced similar results to \cite{Belloni2013} when investigating a variety of duct shapes. Using an actuator disc model and taking advantage of radial symmetry, 2D simulations were carried out on a total of 8 duct profile shapes of equal length to the turbine diameter, $D$. Like \cite{Belloni2013}, the results indicated that the $C_p$, normalised using frontal area of the duct, was reduced for all turbines when compared to the unducted case. Despite the decreased performance \cite{Fleming2016} observed an overall increase in power output due to the higher flow through the ducted turbine. Additionally, the shape of the ducting was important to improve $\eta$, as the thrust applied to the flow varied with ducting profile. Fleming found that a thick cambered profile was a superior design as the camber in the profile allowed for the convective acceleration of flow to the rotor plane and the thick side wall reduced overall viscous losses. Large viscous losses were observed in the case of thin cambered profiles due to large re-circulation zones forming on the outer surface of the duct as the flow separated. 

Ducting of tidal turbines has also been shown to improve the power output in yawed flow conditions up to angles of 20$^\circ$ due to better alignment of the flow with the rotor plane inside the duct \citep{Belloni2013a}. While turbine performance is improved, $\eta$ is reduced as the thrust applied to the duct increases due to asymmetric flow separation from the ducting. Although the total load is higher on the structure and turbine, \cite{Belloni2013a} were able to demonstrate that a duct removed almost all of the azimuthal variation in loading on the rotor when compared to the bare turbine.

Ducting of turbines is further explored in this thesis in Chapter \ref{Chapter3} at the turbine fence scale.

\subsection{Operational optimisation}

Adjustment to turbine orientation and operation has been shown to improve performance. Operational improvements are advantageous for energy providers as they provide additional power output at little to no additional investment in infrastructure.  


\subsubsection{Differential speed controls}
A common way to adjust the power output from a rotor is to modify the rotational speed. Rotors are designed for a specific $\lambda$ where power production is optimised for target inflow conditions. Within a turbine fence, local flow interactions between turbines and bypass flows around the fence's end alter the inflow conditions from the solitary design case for each turbine. Consequentially, the rotational speed needs to be modified to locally optimise power for each turbine or to balance the force distribution along the fence. This has been studied experimentally by \cite{McNaughton2021} who demonstrated that differential speed controls are an effective control strategy to balancing the thrust between inboard and outboard turbines on a turbine fence, exposing them to similar loading over their life cycles. This is explored further in Chapter \ref{Chapter5}.




\subsubsection{Yawing of turbines}

An operational control common to both wind and tidal turbines is yaw. Yaw adjustments are often required to ensure that the turbine is positioned normal to the flow, in wind this is due to the continually changing inflow direction and for tidal, the reversal of the tide. The operation of a single tidal turbine yawed relative to the inflow produces less power than when it is normal to the flow due to the reduced mass flux through the turbine. In addition there is increased unsteady loading as a result of blade advancing and retreating effects \citep{Galloway2011}. Yawing a turbine imposes a lateral force onto the flow, which could be used to constrain the inflow and wake expansion through a turbine fence. Furthermore, outboard turbine yaw may compensate for the flow diversion around the fence end. This control strategy is investigated in Chapter \ref{Chapter5}.







% \section{Turbine performance}
% \subsection{Single turbine performance}
% \subsection{Turbine fence performance}
% \section{Blockage}
% \subsection{Global blockage}
% \subsection{Local blockage}
% \section{End constraints}
% \subsection{End losses}




