
\chapter{End-wall constrained turbines}\label{Chapter3}

This chapter investigates the effects of constraining a turbine fence with an end-wall to reduce bypass flow and improve efficiency. 

Section \ref{Ch3:numerical} details the numerical model used to analyse the fence and design the end-wall. Section \ref{Ch3:experimental} presents the testing methodology, including the manufacture of the prototype design and the experimental test facility. Experimental results with and without an end-wall are compared with the numerical results in this section. Limitations of the experimental and numerical study are discussed in Section \ref{CH3:model_limits} and finally, a summary of the results is outlined in Section \ref{Ch3:conclusions}.

\section{Overview}

As discussed in Section \ref{Ch2_Blockage}, the local blockage effect, first analysed by \cite{Nishino2012}, shows that decreasing the inter-turbine spacing ($s$) can result in an increase in turbine power production. This result has been verified both numerically \citep{Schluntz2015,Vogel2017} and experimentally \citep{Cooke2016,McNaughton2019}. While these results reaffirm the key findings from \cite{Nishino2012}, a deviation from the theoretical framework arises at the end of the turbine fences due to flow diversion around the ends of the fence, decreasing outboard turbine performance. This effect is not captured in \cite{Nishino2012} due to the assumption of a ''long fence'', but is observed in the experiments of \cite{Cooke2016} and the numerical analysis of \cite{Vogel2017}. Some improvements to the theory based modelling for shorter turbines fences were made by \cite{Nishino2013} who considered a 2-scale approach to model shorter fences. However, this work did not include consideration for fence end effects. 

While long fences may be desirable commercially, shorter fences are more likely to be designed as technology develops. Within shorter fences, the losses resulting from flow diversion around the outer turbines may represent a more significant percentage of the total fence power and, therefore, should not be ignored when optimising overall performance. The work presented in this chapter aims to further understand the end effect by investigating flow diversion at varied inter-turbine spacing for a short fence of 4 turbines. Following this, a turbine fence end-wall is examined both numerically and experimentally as a possible mitigation strategy for end losses. 

It is anticipated that the support structure in floating tidal fences, specifically the buoyancy aids, can be designed to re-align flow and improve turbine power output. This would take the form of a turbine fence end-wall, which provides a physical barrier to reduce the flow diversion around the ends of a fence (as observed by \cite{Cooke2016} and \cite{Vogel2017}). End-walls may improve performance by confining the flow in a similar way to ducting around a single turbine. The impact of single-turbine ducting has been investigated previously \citep{Belloni2013,Fleming2016}. These studies concluded that placing ducting around a turbine's perimeter improved the power density through the rotors; however, this improvement was not as effective as increasing its diameter to match the ducting opening size. Unlike the single turbine ducting, which requires additional material to be added to the turbine, this study focuses on providing confinement at the turbine fence scale by utilising existing buoyancy aids for a floating tidal turbine platform. Positioning an end-wall adjacent to the end turbine imposes a lateral force that helps channel flow that would otherwise have been diverted around the end of the fence through the fence. 


This study presents the design and experimental testing of a turbine fence end-wall for a four turbine fence composed of 1.2 m diameter ($D$) turbines (modelled as two turbines with a symmetry plane). These turbines were designed for deployment in a four turbine fence configuration by \cite{Cao2018} and manufactured and first tested by \cite{McNaughton2019}. The turbines were instrumented to measure thrust, torque, speed and angular positions, with each turbine blade instrumented to collect edge-wise and flap-wise bending moments. The end-wall was instrumented to collect stream-wise and cross-stream loads; however, due to a strain gauge's failure in the stream-wise direction, only cross-stream loads are presented. 



\section{Numerical analysis}\label{Ch3:numerical}

A four turbine fence was analysed at varied inter-turbine spacing ($s$) using the RANS-BE model detailed in Section \ref{CH2:BE_model}. Following this, a parametric study was undertaken to determine an end-wall geometry which improved the total power output from the fence.


\subsection{Methods}\label{CH3:Method1}

The turbine analysed in this study was that of \cite{Cao2018}, which uses a FX-84-W-140 aerofoil profile with a thickened trailing edge. The aerofoil two-dimensional (2D) lift and drag coefficients were supplied by Cao and used to model the turbines with an embedded BE actuator disk (AD) method.

\subsubsection{Numerical Model}
The incompressible RANS equations were solved numerically using the computational fluid dynamics (CFD) software ANSYS Fluent 15.0 \cite{ANSYSInc.2013}. The $k-\omega$ shear stress transport (SST) turbulence closure scheme of \cite{Menter1994} was implemented. This modelling approach blends the $k-\omega$ model, which performs well in the near-wall region, with the $k-\epsilon$ in the far-field. The model also implements a $k$ limiter function which is well suited to adverse pressure gradients, such as in the wake of a turbine. This approach has previously been used to model flow over an aerofoil profile with a favourable comparison to observations \citep{Menter1994} and to simulate flow through a tidal rotor \citep{Liu2015,Kolekar2015,Cao2019}.

A blade element actuator disk model was embedded into the RANS solver through an in-house subroutine developed by \cite{McIntosh2011}. This utilises the fan internal boundary condition available in Fluent, with the pressure drop and downstream swirl velocity determined by solving the blade element equations. The sub-routine sampled the simulated flow field in each cell within the rotor's swept area and used the blade element approach to incorporate the hydrofoil properties (in the form of the 2D lift and drag coefficients $C_l$ and $C_d$) of the turbine blades to determine the pressure drop and swirl velocity. The Glauert tip correction model \citep{Glauert} was applied to reduce the loading near the blade tip.

A structured computational mesh was developed using ANSYS ICEM \citep{Inc.2012}. The cross-section of the rectangular numerical domain matches the cross-section of the SSPA towing tank facility where physical experiments were conducted (Section \ref{Ch3:experimental}; 10 m wide (8.33$D$), 5 m deep (4.17$D$) and the domain is 36 m long (30$D$) in the stream-wise direction as illustrated in Figure \ref{fig3:Domain}. Two turbines of 1.2 m diameter were placed perpendicular to the flow 12 m downstream (10$D$) of the inlet and towards the side of the tank. A uniform velocity profile with inflow speed 0.8 m/s and turbulence intensity of 8\% was specified at the velocity inlet; turbulence intensity decayed to approximately 5\% at the rotor plane. A zero-gradient pressure outlet was specified on the downstream boundary. Symmetry boundary condition were applied to the side boundaries to simulate the effect of a fence of 4 turbines in a 20 m (16.67$D$) wide channel, halving the computational cost. Consequently, the global blockage for this configuration is 4.5\%.



% The numerical analysis of the turbine fence was completed using a steady RANS solver in ANSYS Fluent 15.0 \citep{ANSYSInc.2013} with a user-defined function (UDF) to interface with a Blade Element (BE) model \citep{McIntosh2011}. The mesh used in this study was composed of structured hexahedrons developed in the commercial software ANSYS ICEM \citep{Inc.2012}. The computational domain is a rectangular prism matching the SSPA towing tank facility's cross-section where physical experiments were conducted (Section \ref{Ch3:experimental}). The domain was 10 m wide, 5 m deep and 36 m long in the stream-wise direction, as illustrated in Figure \ref{fig3:Domain}. Two turbines of 1.2 m diameter ($D$) were placed perpendicular to the flow 10$D$ downstream of a velocity inlet boundary and 20$D$ upstream of a pressure outlet boundary. A symmetry boundary condition was applied to the side boundaries to simulate the effect of a fence of four turbines in a 20 m wide channel. The turbine used was that of \cite{Cao2019}, which was constructed from a Wortmann FX-84-W-140 profile with a truncated trailing edge to achieve improved performance with Reynolds number $O(10^5)$. The global blockage (ratio of total turbine cross-sectional area to channel cross-sectional area) is 4.5\%. A 0.8 m/s flow speed with a uniform profile was specified at the velocity inlet; turbulence input parameters were adjusted to achieve an approximate turbulence intensity of 5\% at the rotor plane. 


\begin{figure}[ht]
    \centering
    \includegraphics[width=0.8\textwidth]{Figures/Chapter_3/numerical/model_domain.png}
    \caption{Computational domain and boundary conditions for the numerical model, $D$ is the turbine diameter (1.2 m) and $s$ is the inter-turbine spacing. Inner (red) and outer (blue) turbines are represented by disks.}
    \label{fig3:Domain}
\end{figure}

\subsubsection{Mesh convergence}

% - Mesh Convergence study
The mesh convergence study followed that of \cite{Cao2019} with the same turbine design modelled within an identically parameterised structured mesh. The structured mesh was constructed using a total of 15 million hexahedral elements (27 million elements when the end-wall was included) with the rotor consisting of 82 elements in the radial direction and 164 around the azimuth. The mesh growth rate was limited to 1.05 across the rotors and in the stream-wise direction. Mesh refinement was investigated in the zone between turbines, as this spacing varied in the study. The element size growth rate was fixed at 1.05 and the maximum element size in this region was varied. 

The convergence study results, presented in Table \ref{Tab_MCS}, illustrate close agreement between all three resolutions with the element size of 0.0083$D$ and 0.0125$D$ having less than 0.163\% relative difference in the estimates for $C_p$. A smaller relative error of less than 0.051\% was observed for the turbine thrust coefficient, $C_t$, defined as the ratio between the total thrust exerted by the turbine to the undisturbed momentum flux through the turbine's swept area. Due to the close agreement, a maximum element size of 0.0125$D$ was selected for the inter-turbine spacing.


% A mesh convergence study was conducted, extending the analysis of \cite{Cao2019} by examining an identically parameterised turbine mesh with modifications in the inter-turbine region and around the end-wall to analyse the new parameters introduced in this work. The maximum element size between the turbines was altered to determine the dependency of $C_p$ and $C_t$ on this parameter. The final structured mesh contained 15 million hexahedral elements (27 million when the end-wall was included). The rotors were constructed with 82 elements in the radial direction and 164 around the azimuth. A fixed growth rate of 1.05 was specified in all directions in the rotor vicinity; this was relaxed to 1.1 towards the boundaries.

% The convergence study results, presented in Table \ref{Tab_MCS}, illustrate close agreement between all three resolutions with the element size of 0.0083$D$ and 0.0125$D$ having less than 0.163\% relative error separating the estimates for $C_p$; a smaller relative error of less than 0.051\% was observed for $C_t$. Given the close agreement, 0.0125$D$ was selected as the maximum inter-turbine spacing element size.

\begin{table}[]
\label{Tab_MCS}
\centering
\caption{Effect of maximum element size on $C_p$ and $C_t$ between two 1.2 m diameter turbines with 0.25D spacing.}

\begin{threeparttable}



\begin{tabular}{ccccc}
\hline
\multicolumn{1}{l}{Max element size ($D$)} & \multicolumn{1}{l}{$C_p$} & \multicolumn{1}{l}{$C_t$} & \multicolumn{1}{l}{\% $C_p$ Error $^*$} & \multicolumn{1}{l}{\% $C_t$ Error $^*$} \\  \hline
0.0083                                & 0.580                     & 1.199                     & -                                  & -                                  \\
0.0125                                & 0.579                     & 1.199                     & -0.163\%                           & -0.051\%                           \\
0.0167                                & 0.576                     & 1.196                     & -0.691\%                           & -0.245\%                           \\ \hline

\end{tabular}

\begin{tablenotes}
    \item [$^*$] Highest resolution model is used as the baseline for error calculations.
\end{tablenotes}
\end{threeparttable}
\end{table}


Mesh independence around the end-wall was also examined. As part of this study, both the non-dimensional wall distance, $y^+$, and the maximum stream-wise element length, $L_{el}$, were varied. $y^+$ is defined by

\begin{equation}
    y^+ = \frac{u_* y}{\nu}
\end{equation}

\noindent where $y$ is the distance from the wall $u_*$ is the friction velocity defined as 
\begin{equation}
    u_* = \sqrt{\frac{\tau_w}{\rho}}
\end{equation}

\noindent where $\tau_w$ is the wall shear stress. $y^+$ is an important criterion for resolving the loading on the end-wall as the $k-\omega$ SST turbulence model is used in this study. This model implements a log-law wall function to simulate the boundary layer, this requires the $y^+$ as an input. Further discussion of this model can be found in Section \ref{Ch2:turbulence}.

Convergence was determined based on the wall's 2D lift coefficient at mid-rotor depth following:


\begin{equation}
\label{CL_int}
\begin{aligned} 
C_{L} = \frac{\oint (p\widehat{n}\cdot \widehat{k}) d \mathbf{S}}{\frac{1}{2}\rho u_{\infty}^{2} L_{c}}
\end{aligned}
\end{equation}


\noindent where $\widehat{n}$ is the surface normal unit vector, $\widehat{k}$ is the unit vector in the direction of lift which is perpendicular to the flow, $\mathbf{S}$ is the curve around the perimeter of the hydrofoil, $u_{\infty}$ is the free-stream velocity and $L_{c}$ is the chord length. As shown in Figure \ref{Fig3:Mesh_converges}, meshes with $y^+$ between 100-225 converge to similar values. The smallest $y^+$ tested diverges from the other results, likely due to the scale of resolution introducing cells that overlap with the buffer layer between the viscous sublayer and the flow \cite{ANSYSInc.2013}. Accordingly, a $y^+$ of 100 and a $L_{el}/L_c$ of 0.056 were adopted for this study.


\begin{figure}[h]
\centering
\includegraphics[width=0.7\textwidth]{Figures/Chapter_3/numerical/Cl_Y_pl.png}
\includegraphics[width=0.7\textwidth]{Figures/Chapter_3/numerical/Cl_L.png}
\caption{Model mesh convergence with non-dimensional wall normal distance $y^+$ at 50\% chord, fixed $L_{el}/L_c=0.056$, (a) and the minimum element length to chord length ratio of $L_{el}/L_c$, fixed $y^+=100$ (b).}
\label{Fig3:Mesh_converges}
\end{figure}


\subsubsection{Root bending moment extraction from RANS-BE model}\label{S3_BERBM}

Individual blade root bending moments were calculated from the RANS-BE model by applying the blade element equations to the converged velocity field. This process first involved identifying the velocity components and radial positions of each node within the structured mesh over the turbine swept area. At each blade angle ($\theta$) and radial location ($r$) the velocity components from the RANS-BE model were transformed into normal, radial and azimuthal components following,

\begin{equation}
\begin{bmatrix} u_{norm}\\u_{r}\\u_{\theta}\end{bmatrix} = 
\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} u_{x}\\u_{y}\\ u_{z}\end{bmatrix}
\end{equation}

\noindent where $u_x$, $u_y$ and $u_z$ are the velocities in the x, y and z directions from the RANS-BE model and $u_{norm}$, $u_r$ and $u_{\theta}$ are the transformed velocities normal to the turbine plane, in the radial direction and in the azimuth direction. The blade element equations from Section \ref{CH2:BE_model} were rearranged to solve for the angle of attack ($\alpha$) and velocity magnitude ($W$). Using this information $C_l$ and $C_d$ from the 2D hydrofoil data set could be extracted and transformed into stream-wise and rotational thrust coefficients, $C_x$ and $C_t$,

\begin{equation}
    \begin{aligned}
    C_x = C_l \cos(\phi) + C_d \sin(\phi), \\ C_t = C_l \sin(\phi) - C_d \cos(\phi)
    \end{aligned}
\end{equation}

\noindent where $\phi$ is the sum of the angle of attack and the blade twist angle ($\phi=\alpha +\beta$). Forces in the stream-wise and rotational directions on each element were calculated using
\begin{equation}
    \begin{aligned}
    F_x = \frac{1}{2} \rho W^2 C_x c(r), \\
    F_{\theta} = \frac{1}{2} \rho W^2 C_t c(r).
    \end{aligned}
\end{equation}

\noindent The bending moments about the hub were determined in the edge-wise ($M_{EW}$) and the flap-wise ($M_{FW}$) direction by integrating the loading along an annular segment following 

\begin{equation}
    M_{EW} = \int_{r_{hub}}^{R}F_{\theta} rdr, \quad M_{FW} = \int_{r_{hub}}^{R}F_xrdr .
\end{equation}

\noindent This process was repeated for each azimuthal segment in the turbine swept area to extract the variation in root bending moment with the azimuthal position. Finally, the root bending moments were non-dimensionalised by

\begin{equation}
    C_{FW} = \frac{N_b M_{FW}}{\tfrac{1}{2} \rho u_{\infty}^2 A_t R}, \quad
    C_{EW} = \frac{N_b M_{EW}}{\tfrac{1}{2} \rho u_{\infty}^2 A_t R}
\end{equation}

\noindent where $R$ is the turbine radius. 

\subsubsection{Model iteration methodology} \label{CH3:iteration_method}

Due to shifting inter-turbine spacing and varied end-wall designs, the tip-speed-ratio ($\lambda$) that achieves the highest $C_p$ ($\lambda^*$) changed across configurations. Therefore $\lambda^*$ must be accurately calculated to facilitate the comparison of turbine performance across configurations. To limit this study's scope, inboard and outboard turbines are operated at the same $\lambda$ (synchronised controls).

$\lambda^*$ was determined by running a set of three simulations at different $\lambda$ close to the expected optimal value based on previous studies by \cite{Cao2019}. A quadratic curve was then fit to the average turbine $C_p$ output from the model and $\lambda$. The $\lambda$ corresponding to the maximum $C_p$ from this curve was used for the next simulation. The results from this iteration were compared to the approximated value from the quadratic curve; if the error between the values was less than 0.01\%, this $\lambda$ was adopted as $\lambda^*$. If this criterion was not met, the new iteration results were used to fit a curve to the three largest values of $C_p$, and the search process was repeated. This methodology helped determine the optimal operating point in the 100+ simulations examined as part of this study, taking advantage of the parallel computing resources available on the ARCUS cluster.  


\subsection{Effect of inter-turbine spacing on tidal fence performance}


The effect of inter-turbine spacing was initially investigated to characterise the performance improvement from increased local blockage and quantify the end losses. Based on this analysis, a spacing was selected for further examination and physical testing. 


\subsubsection{Turbine performance}\label{sc3:unducted_perf}

$\lambda^*$ and the corresponding turbine $C_p$ and $C_t$  were found for 6 inter-turbine spacings ($s$) ranging between 0.25$D$ and 2.333$D$. Fence-averaged results from these simulations are presented in Figure \ref{Fig3:Unducted_perf}. As the fence spacing is decreased from $s/D=2.333$ ($B_l = 5.7\%$) to $s/D=0.25$ ($B_l = 15.1\%$), there is a monotonic increase in power production and thrust when $\lambda$ is held constant. A further increase in average $C_p$ of 5.6\% between the largest and smallest spacings was realised by adjusting the operating point of the turbines slightly. The corresponding increase in $C_t$ and $\lambda^{*}$ was 6.0\% and 4.5\%, respectively. This complements \cite{Nishino2012}, who demonstrated for a fixed global blockage ($B_g$) that an increase in $B_l$ can improve the fence's performance. The increased performance arises from the greater pressure gradients (thrust) that can be sustained across the rotor plane as turbine spacing is reduced.



% $\lambda^*$ and the corresponding turbine $C_p$ and $C_t$ were determined using the methodology outlined in Section \ref{CH3:iteration_method} for a total of 7 inter-turbine spacings ($s$) ranging between 0.125$D$ and 2.333$D$. The fence averaged results from these simulations are presented in Figure \ref{Fig3:Unducted_perf}. The results show that as the fence spacing is decreased from 2.333$D$ ($B_l = 5.7\%$) to 0.125$D$ ($B_l = 16.8\%)$ there is a monotonic increase in power production and thrust when $\lambda$ is held constant. This complements \cite{Nishino2012}, demonstrating for a fixed global blockage ($B_g$), an increase in $B_l$ can improve the fence's performance. An improvement in $C_p$ at the optimal operating point of 6.02\% was achieved by decreasing the spacing from 2.333$D$ to 0.125$D$. The corresponding increase in $C_t$ and $\lambda^{*}$ was 6.68\% and 5.13\%, respectively. These results suggest that higher turbine thrusts can be sustained at reduced turbine spacing, allowing greater pressure gradients to be maintained across the rotor plane resulting in higher performance. 



\begin{figure}
\centering
\includegraphics[width=\textwidth]{Figures/Chapter_3/numerical/b_Unducted_Optimal_PowerCoefficent_TSR_curve.png}
\includegraphics[width=\textwidth]{Figures/Chapter_3/numerical/b_Unducted_Optimal_ThrustCoefficent_TSR_curve.png}
\caption{Variation in fence averaged $C_p$ (top) and $C_t$ (bottom) with $\lambda$ as a function of inter-turbine spacings for a fixed global blockage of 4.5\%. The optimal operating point, $\lambda^*$, is marked on each curve with a black cross.}
\label{Fig3:Unducted_perf}
\end{figure}


\begin{table}[h]
\centering
\caption{Turbine $C_p$ and $C_t$ at $\lambda^*$ for varied inter-turbine spacing. }
\label{Table:Inter_turbine_spacing}
\begin{tabular}{|cc|ccc|ccc|}
\hline
\multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Spacing \\ {[}$D${]}\end{tabular}}} & \multirow{2}{*}{\textbf{$\lambda^{*}$}} & \multicolumn{3}{c|}{\textbf{$\mathbf{C_{p}}$}}              & \multicolumn{3}{c|}{\textbf{$\mathbf{C_{t}}$}}              \\ \cline{3-8} 
                                                                                     &                                         & \textbf{Outer} & \textbf{Inner} & \textbf{Average} & \textbf{Outer} & \textbf{Inner} & \textbf{Average} \\ \hline
% 0.125                                                                                & 6.56                                    & 0.557          & 0.570          & 0.563            & 1.116          & 1.126          & 1.121            \\ \hline
0.250                                                                                & 6.52                                    & 0.553          & 0.569          & 0.561            & 1.108          & 1.119          & 1.114            \\ \hline
0.500                                                                                & 6.49                                    & 0.548          & 0.564          & 0.556            & 1.099          & 1.111          & 1.105            \\ \hline
0.750                                                                                & 6.45                                    & 0.543          & 0.559          & 0.551            & 1.090          & 1.102          & 1.096            \\ \hline
1.000                                                                                & 6.41                                    & 0.539          & 0.554          & 0.547            & 1.081          & 1.093          & 1.087            \\ \hline
2.000                                                                                & 6.27                                    & 0.530          & 0.538          & 0.534            & 1.054          & 1.060          & 1.057            \\ \hline
2.333                                                                                & 6.24                                    & 0.528          & 0.534          & 0.531            & 1.048          & 1.053          & 1.051            \\ \hline
\end{tabular}
\end{table}


$C_p$ and $C_t$ at $\lambda^*$ for inner and outer turbines over a range of inter-turbine spacings is presented in Table \ref{Table:Inter_turbine_spacing}. This table highlights that as spacing is reduced, $C_p$ and $C_t$ increase, so to does the disparity between inner and outer turbines (Figure \ref{Unducted_Power}). The inner turbine achieves a slightly higher $C_p$ and $C_t$ than the outer turbine due to the greater flow confinement. Wake expansion downstream of the inner turbines is laterally constrained by the presence of adjacent turbines and their wakes, allowing larger pressure gradients to be maintained and thus improving the power output. This lateral constraint is only present for the inboard region on the outer turbines leading to end losses. This resulting asymmetry can be quantified by a symmetry factor between the rotors defined as:

\begin{equation}
    \label{Symm_fac}
    \begin{aligned} 
    SF_{\Phi} = \frac{\Phi_{inner}-\Phi_{outer}}{\Phi_{inner}}
    \end{aligned}
\end{equation}

\noindent where $SF_{\Phi}$ is the symmetry factor for the parameter $\Phi$. Positive values of this metric indicate a bias towards the inner turbines.


Figure \ref{fig3:unducted} plots the variation in turbine symmetry factor as a function of inter-turbine spacing. Similar treads in the symmetry factor for both $C_p$ and $C_t$ are present, with $\Phi_{C_p}$ being approximately 2.6 times larger than $\Phi_{C_t}$. The smallest asymmetry between the turbines was observed at the largest spacing of $s/D$=2.33, although there was still a 1.5\% bias towards the inner turbine's performance due to the differing local blockage of the inner and outer turbines. 


\begin{figure}[h]
\centering
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_TSR_Spacing.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_Symmetry_Factor.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_PowerCoefficent_Spacing.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_ThrustCoefficent_Spacing.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_alpha_Spacing.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_FlowRate_Spacing.png}

\caption{$\lambda^*$ (a), $\Phi$ (b), $C_p$ (c), $C_t$ (d), $\alpha$, span-wise averaged between 0.3R-0.85R, (e) and turbine flow rate (f) variation with inter-turbine spacing at $\lambda^*$.}
\label{fig3:unducted}
\end{figure}

A maximum was observed for $\Phi$ at $s/D$=0.75, with the inner turbine producing 2.9\% more power and 1.1\% more thrust. As turbine spacing decreases from $s/D$=0.75, the difference in power and thrust along the fence is slightly reduced. Examining the mass flow rate in Figure \ref{fig3:unducted}f provides some explanation of the changing power distribution along the fence. Moving the turbines closer together improves the mass flux of the inner turbine due to the increased local blockage. This is not the same for the outer turbine as at larger spacings ($s/D>0.75$) the fence end-losses are greater than the constructive interference between the turbines, this relationship is reversed as the turbines get closer together and the inner turbine mass flux increases. Examining the flow yaw angle in Figure \ref{fig3:yaw_plots} supports these results, with reduced yaw angles on the inner turbine as they are positioned closer together. Interplay between a constructive interaction of turbines and fence end-losses are responsible for the varied response of flux at decreased spacing for the outer turbine as highlighted in Figure \ref{fig3:yaw_plots}. This figure shows an increased yaw over the outermost region of the outer turbine as the spacing decreases from $s/D$=2.33 to $s/D$=0.75, which outweighs the improvement on the inner most region due to improved local blockage. The yaw over the inner edge of the outer turbine continues to reduce as the turbines move closer together and the turbine flux increases for $s/D<0.75$.

It should be noted that, while increasing the mass flux increases the power available for extraction, the efficiency of power extraction is not constant. This is because increased mass flux is, in general, going to increase the angle of attack ($\alpha$) on the blade due to a higher stream-wise velocity through the turbine. A correction to $\alpha$ can be partially made by increasing $\lambda$, reducing $\alpha$ linearly along the radius of the blade. It is, however, unlikely that this correction is going to perfectly compensate for the spacing adjustment. This highlights the importance of designing turbines for the expected local blockage conditions as further improvements are possible by correcting the blade twist distribution as the spacing is adjusted, this has been explored in the work of \cite{Vogel2019} who looked at different turbine designs for varied local blockages. Figure \ref{fig3:unducted}e plots the effect of spacing on angle of attack averaged over the primary power producing region of the blades (0.3R-0.85). The optimal angle of attack for the aerofoil used in this study was 6$^\circ$, close to the average $\alpha$ at the maximum spacing 2.33$D$. It can be noted that decreasing the spacing decreases the angle of attack after a correction to $\lambda$ was made to extract the optimal power. This distance from $\alpha=6$ highlights a potential for further performance improvements with altered designs at closer spacings.


% The effect of spacing on angle of attack averaged over the primary power producing region of the blades (0.3R-0.85) is presented in Figure \ref{fig3:unducted}e. The optimal angle of attack for the aerofoil used in this study was 6$^\circ$, close to the average $\alpha$ at the maximum spacing 2.33$D$. It can be noted that decreasing the spacing decreases the angle of attack. This is due to a change in the flow over the blades resulting from closer spacing, increased thrust and a higher $\lambda^*$; altering the required twist distribution along the blade's length. \cite{Vogel2017} determined that a change to blade pitch angle is needed when moving from a single turbine to a four turbine fence (increased local blockage) with a spacing of $s/D$=1. The results examined here agree with the work of \cite{Vogel2019}, however this study provides an interesting contrast to the prior work as the inter-turbine spacing (and hence local blockage) is varied instead of the turbine design (which was altered in \cite{Vogel2019}). As a result of the sub-optimal $\alpha$ a decline in performance might be expected. However, counterbalancing this effects is a change in flow rate with inter-turbine spacing which improves the flow velocity through the inner turbine as spacing decreases from $s/D$=2.33 to $s/D$=0.25. This is due to greater confinement by the outboard turbine improving flow alignment normal to the rotor plane. Examining the flow yaw angle in Figure \ref{fig3:yaw_plots} verifies this result with reduced yaw angles on the inner turbine as they are positioned closer together. There is decreases flow velocity through the outer turbine as spacing moves from $s/D$=2.33 to $s/D$=0.75, this then increases from $s/D$=0.75 to $s/D$=0.25. Interplay between a constructive interaction of turbines and fence end-losses are responsible for this varied response of flux at decreased spacing. Figure \ref{fig3:yaw_plots} shows an increased yaw over the outermost region of the outer turbine as the spacing decreases from $s/D$=2.33 to $s/D$=0.75, this outweighs the improvement on the inner most region due to improved local blockage. The yaw over the inner edge of the turbine continues to reduce as the turbines move closer together and the turbine flux increases for $s/D$<0.75. Given the definitions of $C_t$ and $C_p$ these values are $\propto u^2$ and $u^3$. As a result, an increase in the flow rate will improve the turbine's performance and may outweigh the decrease in performance due to sub-optimal angles of attack. This interplay between constructive and non-constructive interference effects and the change in angle of attack causes a maximum of $\Phi$ to exist in Figure \ref{fig3:unducted}. 



% The shape of the $\Phi$ plot can be explained by jointly examining the angle of attack and flow rate curves (Figure \ref{fig3:unducted} e and f). Firstly, examining the angle of attack, it can be noted that:
% \begin{itemize}
%     \item decreasing the spacing decreases the angle of attack. This is due to a change in the flow field resulting from closer spacing, increased thrust and a higher $\lambda^*$, altering the required twist distribution along the blade's length. A similar observation is made in the work of \cite{Vogel2017} who determined that a change to blade pitch angle is needed when moving from a single turbine to a four turbine fence with a spacing of $s/D$=1. The results examined here agree with the work of \cite{Vogel2019}, however this study provides an interesting contrast to the prior work as the inter-turbine spacing (and hence local blockage) is varied instead of the turbine design (which in \cite{Vogel2019} was optimised for different local blockages).
%     \item the angle of attack is no longer at the optimum ($\alpha^*=6^\circ$) at the closer spacing resulting in a less efficient section. 
% \end{itemize}
% \noindent As a result of the above performance is expected to diminish. However, counterbalancing these effects is a change in flow rate (which is $\propto u$) with inter-turbine spacing which:
% \begin{itemize}
    % \item improves the flow velocity through the inner turbine as spacing decreases from $s/D$=2.33 to $s/D$=0.25; this is due to greater confinement by the outboard turbine improving flow alignment normal to the rotor plane. Examining the flow yaw angle in Figure \ref{fig3:yaw_plots} verifies this result.
    % \item decreases the flow velocity through the outer turbine from $s/D$=2.33 to $s/D$=0.75, then increase the flow velocity from $s/D$=0.75 to $s/D$=0.25. This is a result of the interplay between a non-constructive interference between the rotors on the outer edge of the outer turbine (greater yaw and decreased flow normal to the turbine) and a constructive interference on the inner edge of the turbine (less yaw and improve flow normal to the turbine). 
    % \item decreases the flow velocity when $s/D$ $<$0.25. As the spacing is reduced below $s/D$=0.25, the flow rate decreases due to increased flow diversion around the fence (Figure \ref{fig3:yaw_plots}).
% \end{itemize}
% \noindent Given the definitions of $C_t$ and $C_p$ these values are $\propto u^2$ and $u^3$. As a result, an increase in the flow rate will improve the turbine's performance and may outweigh the decrease in performance due to sub-optimal angles of attack. The interplay between constructive and non-constructive interference effects and the change in angle of attack causes a maximum of $SF$ to exist. 

\begin{figure}[h]
\centering
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/yaw/052_Slices_yaw_angle.png} %2.33D
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/yaw/049_Slices_yaw_angle.png} %0.75D
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/yaw/048_Slices_yaw_angle.png} %0.5D
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/yaw/047_Slices_yaw_angle.png} %0.25D

\caption{Yaw angle at the turbine plane for $s/D$=2.33 (top left), $s/D$=0.75 (top right), $s/D$=0.50 (bottom left) and $s/D$=0.25 (bottom right). Positive yaw is to the right.}
\label{fig3:yaw_plots}
\end{figure}



% The above is sufficient to explain why the performance improves with decreased spacing for the inboard turbine. However, it is not sufficient when explaining why the performance of the outer turbine is improved. Given that the angle of attack is sub-optimal and the flow rate through the turbines decreases (relative to the largest $s/D$), it would be expected that $C_p$ and $C_t$ both decrease with smaller spacings. It is necessary to recognise that the turbine $\lambda^*$ increases at smaller spacing, increasing the local blade velocity, which is sufficient to increase turbine power provided that blade torque does not decrease significantly. 

% To investigate this, the azimuthal variation becomes important. Figure \ref{fig3:velocity_normal_turbine} plots the velocity magnitude at the turbine plane for $s/D$=2.33 and $s/D$=0.25. An increase in the velocity is observed on the outer portion of the outer turbine. This increase is further from the centre of the rotor leading to a higher torque, improving power generation with a lower overall flow rate and sub-optimal angle of attack. 

% \begin{figure}
% \centering
% \includegraphics[width=\textwidth]{Figures/Chapter_3/numerical/sD=2.333-uu_infty[-].png} %2.33D
% \includegraphics[width=\textwidth]{Figures/Chapter_3/numerical/sD=0.25-uu_infty[-].png} %0.25D

% \caption{Velocity magnitude at the turbine plane for $s/D$=2.333 (top) and $s/D$=0.25 (bottom). The outer-most turbines are on the left and inner-most are on the right).}
% \label{fig3:velocity_normal_turbine}
% \end{figure}



% As the turbine separation decreases from s/D=2.33 to s/D=0.75 there is an increase in the inner turbine flow rate due to greater confinement by the outboard turbine, a corresponding decrease is observed for the outboard turbine due to flow diverting around the end of the fence, this can be verified by examining the yaw angle in Figure \ref{fig3:yaw_plots}.
% At s/D=0.75 the minimum outboard turbine flow rate is reached, leading to higher $SF$. 

% As the turbines move closer together to s/D=0.25 the bypass flow from the inboard turbine positively interacts with the outboard turbine and there is an increase in flow rate for all turbines. At the closest spacing, s/D=0.125, the flow rate appears to start decreasing again due to both the increased flow diversion around the fence (Figure \ref{fig3:yaw_plots}) and the close spacing forcing the blade to a sub-optimal angle of attack ($\alpha$). 

% The change in $\alpha$ over this range results in $C_p$ growing at a smaller rate and $C_t$ increasing at a greater rate with respect to the decrease in spacing. This is consistent with the work of \cite{Vogel2017} who determined that a change to blade pitch angle was required when moving from a single turbine to a 4 turbine fence with spacing of s/D=1. The results examined here agree with the work of \cite{Vogel2019}, however this study provides an interesting contrast to the prior work as the inter-turbine spacing (and hence local blockage) is varied instead of the turbine design (which in \cite{Vogel2019} was optimised for different local blockages).



\subsubsection{Unsteady loading on turbine blades}\label{sc3:unsteady_unduct}

Due to the non-uniform confinement between the inner and outer turbines and the local flow area's aspect ratio (nacelle of the turbine is 0.83$D$ below the surface and 3.33$D$ from the bottom of the domain), there is a variation in the blade root bending moments in the RANS-BE model. Figure \ref{fig3:Flap_bending_spacings} plots the flap-wise ($C_{FW}$) and edge-wise ($C_{EW}$) root bending moment coefficients with inter-turbine spacing using the method described in Section \ref{S3_BERBM}. The plot demonstrates that decreasing the inter-turbine spacing increases the root bending moment in both directions, with the inner turbine having greater bending loads than the outer turbine. 


\begin{figure}[h]
\centering
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/turb_2_Flap.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/turb_2_Edge.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/turb_1_Flap.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/turb_1_Edge.png}

\caption{Azimuthal variation in flap-wise (top) and edge-wise (bottom) root bending moment coefficients with inter-turbine spacing for the outer (left) and inner (right) turbines. The red dots indicate the azimuthal position of maximum root bending moment. The illustrations at the top of each plot show the location of the blade (in bold), with blue and red representing the outer and inner turbines. }
\label{fig3:Flap_bending_spacings}
\end{figure}


For the inner turbine flap-wise moments are typically highest just after the blade passes the inboard section of the fence. The cause of this is the increased bypass flow speed from constructive interference with the counter-rotating turbine on the other side of the symmetry plane. A similar observation can be made for the outer turbine at the same angular position; however, the loading is a factor of constructive interference from the co-rotating inner turbine. For the outer turbine, there is also a reduction (relative to the average root bending moment at a specified spacing) in the root bending moment as the blade sweeps the 90$^\circ$ to 200$^\circ$ portion of the rotation; this is a combined effect of flow diversion around the fence (Figure \ref{fig3:yaw_plots}) and underneath the fence (Figure \ref{fig3:Unduc_pitch}). As turbines spacing decreases, there is a growing disparity between the blade loading on the inner and outer turbines. 


\begin{figure}
\centering
\includegraphics[width=0.99\textwidth]{Figures/Chapter_3/numerical/047_Slices_pitch_angle.png}
\caption{Pitch angle at the turbine plane for $s/D$=0.25 with turbines operating at $\lambda^*$. Positive pitch it up.}
\label{fig3:Unduc_pitch}
\end{figure}


Using the results from the $s/D$=0.25 simulation at $\lambda^*$, the stream-wise velocity ($u_x$) through the rotor plane was extracted and decomposed into an annular averaged ($\left<{u_{x}}\right>$) and a deviatoric component ($u_{x}'$) such that $u_x= \left<{u_{x}}\right>+u_{x}'$. This metric indicated the angular and radial position where flow (as a proxy for loading on the blades) is changed; in contrast, root bending moments only indicated the angular position of a change in load. The results of the velocity decomposition ($u_x$, $\left<{u_{x}}\right>$, $u_{x}'$) are presented in Figure \ref{fig3:UnducDeviatoric_analysis} for the inboard and outboard rotors. This figure highlights that there is a significantly reduced flow rate ($\left<{u_{x}}\right>$) through the outer turbine when compared with the inner turbine. Examining $u_{x}'$ demonstrates the cause for the lower flow rate at the outer turbine is due to lower $u_{x}'$ at the outermost section of the outer turbine. Higher $u_{x}'$ for the inboard and outboard regions of the inner turbine and the inboard region of the outer turbine highlight the constructive interference effects between the turbines. At the outboard section of the outer turbine, bypass flow is yawed around the rotor, reducing the flow rate in the $u_x$ direction (this can be observed in Figure \ref{fig3:yaw_plots}). 


\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{Figures/Chapter_3/numerical/047_Slices_x_velocityturbines.png}
\caption{Total (top), azimuthally averaged (middle) and deviatoric (bottom) components of the stream-wise velocity at the outer (left) and inner (middle) turbines as well as the difference, $\Delta$, of the inner turbine's velocity component minus the outer turbine's velocity components turbine (right) for $s/D$=0.25 spaced rotors operating at $\lambda^*$. The free-stream flow direction is into the page.}
\label{fig3:UnducDeviatoric_analysis}
\end{figure}

% Figure \ref{fig3:Unduc_RBM} plots the flap-wise and edge-wise root bending moment for the inboard and outboard turbines at s/D=0.25. These results demonstrate that the bending moment experienced by the inner turbine is almost always greater than the outer turbine in both the edge-wise and flap-wise directions. The exception to this is the outboard section of the inner turbine where interference between the turbines decreases the root bending moment in the flap-wise direction. Over a rotation of the turbine, the peak loading in the flap-wise direction for both turbines appears to occur just after to the blades pass the innermost portion of the rotation at 300$^\circ$. The minimum loading position changes between inboard and outboard turbines with the inboard turbine having a minimum load just prior to the turbine blade passing the outboard turbine and the outboard turbine having a minimum loading half way between the outermost portion of a rotation and the bottom of the rotation. Figure \ref{fig3:Unduc_pitch} demonstrates that there is high pitch angle as the blade passes the bottom of the fence indicating flow bypassing underneath the fence, this is analogous to what was observed in the yaw angle at the end of the fence in Figure \ref{fig3:yaw_plots}. The combined effect of the yaw and pitch angles reduce the flow through the outboard turbine and provide an explanation for why the minimum flap-wise root bending moment is located at the 135$^\circ$ position. 


% \begin{figure}
% \centering
% \includegraphics[width=0.72\textwidth]{Figures/Chapter_3/numerical/RBM_BE_FW.png}
% \includegraphics[width=0.72\textwidth]{Figures/Chapter_3/numerical/RBM_BE_EW.png}
% \caption{Flap-wise (top) and edge-wise (bottom) root bending moment for inboard (red) and outboard (blue) turbines operating at $\lambda^*$ at s/D=0.25. The three illustrations at the top of each plot show the location of the blade (in bold) in the domain, with blue and red representing the outboard and inboard turbines.}
% \label{fig3:Unduc_RBM}
% \end{figure}





\subsubsection{Conclusions}

Analysis of a fence of four 1.2 m diameter turbines at varied inter-turbine spacings was undertaken. Decreasing the spacing between the turbines from $s/D$=2.33 to $s/D$=0.25 was shown to increase the fence's performance by 5.6\%. A smaller improvement in the outboard turbine's performance (relative to the inboard turbine) was observed as the turbines were placed closer together, with the greatest imbalance occurring at $s/D$=0.75. This spacing produced the largest relative difference in loading between the inboard and outboard turbines due to fence end-losses being larger than the constructive interference from local blockage. As the spacing decreased further, the flow through the turbine increased due to improved local blockage. This result suggests that there are spacing’s which give higher imbalances between the inboard and outboard turbine loading. These spacings may not necessarily be the closest, indicating that designers may wish to consider inter-turbine spacing if trying to balance fatigue along a fence. 

Based on the results presented in Figure \ref{fig3:unducted}, the minimum inter-turbine spacing of $s/D$=0.25 produces the fence's maximum power output. As a result, this inter-turbine spacing was selected for further investigation.

The performance improvements observed here are based solely on adjusting $\lambda$ and inter-turbine spacing; further enhancement of the power output could be achieved by designing the turbine for the specific blockage conditions as shown by \cite{Schluntz2015}. The rotors used in this model were designed for a global blockage ($B_g$) of 0.142 and local blockage of 0.377 \citep{Cao2019}; however, blockage conditions analysed in this study are far lower than this, with a $B_g$ of 0.045 and $B_l$ of between 0.057-0.168. The optimal shape of the turbine blades is expected to change as the global blockage is reduced by a factor of three. Furthermore, the operating conditions examined in this study restricted $\lambda$ to be the same for all turbines along the fence. It is expected that, given the different flow conditions between the inboard and outboard rotors, altering $\lambda$ along the fence would improve performance.


% An inter-turbine spacing of s/D=0.250 was selected to examine the variation in loading of a rotor blade for a complete rotation. It was found that at this spacing both flap-wise and edge-wise root bending moments were greater for the inboard turbine except in the outboard region of the inner turbine where interference effects resulted in a drop in flap-wise bending moment.




\subsection{Numerical design of turbine fence end-wall}

A parametric study of end-wall shape was carried out to determine whether an end-wall could be used to improve flow alignment and reduce the loss in outer turbine performance. Several end-wall camber and thickness were investigated for a 1$D$ long elliptical cross-section placed adjacent to the outer turbine at half the inter-turbine spacing (i.e., 0.125$D$ tip-to-wall spacing). A parabolic camber was applied to the ellipse's major axis. The bottom of the 2 m (1.67$D$) deep end-wall was rounded using a semi-circular section.  For each end-wall design considered, $\lambda^*$ was determined following the method outlined in Section \ref{CH3:iteration_method}. Three end-wall thicknesses (0.05$D$, 0.10$D$ and 0.15$D$) and five cambers (-4\%, -2\%, 0\%, 2\%, 4\%) were tested, these cross-section profiles are presented in Figure \ref{fig3:wall_sections}.

\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/numerical/Sections_sidewall.png}
\caption{End-wall cross-section profiles for thicknesses of 0.05$D$ (top), 0.10$D$ (middle) and 0.15$D$ (bottom) and cambers of 0\% (orange), 2\% (blue) and 4\% (green).}
\label{fig3:wall_sections}
\end{figure}


\subsubsection{Effect of end constraint on turbine performance}

Four metrics were used to quantify and end-walls performance, $C_p$, $C_t$, $\Phi_X$ and basin efficiency ($\eta$). $\eta$ is a metric for the efficiency with which power is extracted, defined as the ratio between the power extracted from the flow ($P$) and the total power removed from the flow (from the turbine, wake mixing, support structure etc.). The formulation of \cite{Belloni2013} was used to determine $\eta$ such that 

\begin{equation}
    \begin{aligned} 
    \eta = \frac{P}{(T_{turbine}+T_{structure})_{x}u_{\infty}}
    \end{aligned}
    % \label{Eq:Basin}
\end{equation}

\noindent where $T_{turbine}$ and $T_{structure}$ are the thrust in the stream-wise direction ($x$) from the turbine and support structure (such as the nacelle and end-wall), respectively. The performance of the fence with respect to these metrics is examined in this section. 

\subsubsection{Fence power}

Fence power at $\lambda^*$ for each end-wall profile, scaled by the optimal without end-wall fence power, is presented in Figure \ref{fig3:ducted_power}. Values greater than 1 indicate an improvement in fence power output. High positive camber (curvature of the end-wall nearest to the turbines is convex) and thicker end-walls produced greater power, with an increase of 3.95\% for the highest end-wall camber and thickness (4\% camber and 15\% thickness). The performance of both turbines was improved, with a 5.59\% increase in outer turbine power and a 2.36\% increase for the inner turbine. While a slight improvement in fence performance can be credited to increased global blockage from the thicker end-wall, the primary cause for performance gain is the end-wall improving flow alignment. The end-wall increases the mass flow rate through the turbines (4.6\% and 2.8\% increase for outer and inner turbines respectively) and improves the angle of attack on the turbine blades towards the blade tips. Figure \ref{fig3:Yaw_sidewall} plots the impact on the flow yaw due to the presence of the end-wall. A significant reduction in flow diversion around the turbine fence is observed adjacent to the end-wall's interior face. There are also minor improvements in the innermost turbine's flow alignment.


\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{Figures/Chapter_3/numerical/PowerCoefficent_Geom.png}
\caption{Fence averaged power with an end-wall at $\lambda^*$ scaled by the without end-wall fence averaged power for $s/D$=0.25. The dashed line plots the region where the with and without end-wall power is the same.}
\label{fig3:ducted_power}
\end{figure}


\begin{figure}
    \centering
    % \includegraphics[width=0.75\textwidth]{Figures/Chapter_3/numerical/whole_yaw_angleDifference_plotsBase.png}
    % \includegraphics[width=0.75\textwidth]{Figures/Chapter_3/numerical/whole_yaw_angleDifference_plotsDucted.png}
    \includegraphics[width=0.95\textwidth]{Figures/Chapter_3/numerical/whole_yaw_angleDifference_plotsDifference.png}
    \caption{Change in yaw angle magnitude with the inclusion of an end-wall for a fence with $s/D$=0.25 turbine spacing operating at $\lambda^*$ ($\lambda^*=6.52$ without an end-wall and $\lambda^*=6.37$ with an end-wall). Change in yaw angle magnitude is determined as the yaw angle with an end-wall minus the yaw angle without; a negative change in yaw angle magnitude indicates the flow is closer to normal to the turbine plane.}
    
    % Flow yaw angle relative to the turbine plane for a fence with s/D=0.25 turbine spacing operating at $\lambda^*$ without a sidewall (a), with a sidewall of 15\% thickness and 4\% camber (b) and the difference, determined as the yaw angle with side wall minus the yaw angle without (c). Free-stream flow direction is into the page.
    
    \label{fig3:Yaw_sidewall}
\end{figure}


\subsubsection{Fence thrust}

Across the parameter space investigated for the end-wall geometry, fence averaged thrust decreased relative to the no end-wall case (Figure \ref{fig3:ducted_thrust}); this is partly due to the lower $\lambda^*$ of fences with end-walls. Improved flow alignment and mass flow rate due to the end-wall allows the outer turbine to operate in more uniform flow. As a result, $\lambda^*$ is reduced as is the thrust on the turbines. For the end-wall that achieved the highest $C_p$, the smallest reduction in thrust to the without end-wall case was observed; average turbine $C_t$ decreased by 0.67\%. 


\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{Figures/Chapter_3/numerical/ThrustCoefficent_Geom.png}
\caption{Fence averaged thrust with an end-wall at $\lambda^*$ scaled by the without an end-wall fence averaged power for $s/D$=0.25.}
\label{fig3:ducted_thrust}
\end{figure}


\subsubsection{Fence basin efficiency}

Basin efficiency brings together the two performance metrics (power and thrust) discussed previously, relating them as the efficiency of power extraction from the flow. Figure \ref{fig3:basin_eff} plots the basin efficiency over the set of end-wall profiles investigated. Although turbine power is improved and thrust is reduced with the end-wall, the overall efficiency of energy extraction does not improve due to the parasitic drag from the end-wall and associated energy dissipation in wake remixing. $\eta$ decreased, relative to the without an end-wall case, for all end-walls examined due to the thrust contribution of the structure. Despite this, the impact on $\eta$ was relatively small with $\eta_{ducted}/\eta_{unducted}$ remaining close to unity. A maximum basin efficiency with the end-wall of 99.93\% the without end-wall value was achieved for a 2\% camber and 10\% thickness end-wall. The modest change in $\eta$ is a result of the end-wall having a relatively small stream-wise thrust component, as shown in Figure \ref{fig3:wall_loads}. The stream-wise wall loading is a function of both thickness and camber, whereas the lateral wall loading is a function of camber and largely independent of thickness. Overall the lateral force is significantly larger than the force in the stream-wise direction; this results from the exchange of momentum as the flow is realigned towards the free-stream direction by the end-wall. 


\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{Figures/Chapter_3/numerical/BasinEfficiency_Geom.png}
\caption{Fence basin efficiency ($\eta$) with an end-wall at $\lambda^*$ scaled by the without an end-wall fence basin efficiency for $s/D$=0.25.}
\label{fig3:basin_eff}
\end{figure}



\begin{figure}
\centering
\includegraphics[width=0.99\textwidth]{Figures/Chapter_3/numerical/Sidewall_force.png}
\caption{Force coefficients, normalised by the free-stream momentum flux through a turbine, acting on the end-wall in the stream-wise (left) and cross-stream (right) direction for a variety of wall cambers and thicknesses, with the turbines operating at $\lambda^*$. Positive stream-wise thrust is in the free-stream flow direction and positive cross-stream thrust is away from the centre of the fence.}
\label{fig3:wall_loads}
\end{figure}

Increasing camber is correlated with increasing the stream-wise thrust due to the larger frontal area presented to the flow and separation occurring on the more highly cambered profiles. Separation is more prevalent with thinner profiles due to the sharper curvature at the leading edge resulting in greater separation and increased thrust; this is demonstrated by the sharp increase in the stream-wise thrust curve with camber for the 5\% thickness end-wall in Figure \ref{fig3:wall_loads}. This highlights the importance of end-wall thickness to avoid flow separation which was also observed by \cite{Fleming2016} in their analysis of various turbine ducting shapes. Figure \ref{fig3:Ducting_Separation}b and Figure \ref{fig3:Ducting_Separation}e plot the flow field at hub height around a profile of 4\% camber at 5\% and 15\% thickness, respectively. The plot shows flow separation at the leading edge of the thinner end-wall and delayed flow separation for the thicker wall, supporting the observations of \cite{Fleming2016}.

% Figure \ref{fig3:wall_loads} plots the stream-wise and cross-stream loading on the end-wall geometries. An increasing relation was observed for stream-wise thrust with increased thickness and camber; this is a consequence of the larger cross-section area projected in the flow direction as these variables increase. Flow separation occurred on the higher cambered and the thinner profiles (Figure \ref{fig3:Ducting_Separation}) due to the sharper curvature at the leading edge. This caused a rapid increase in stream-wise thrust with camber for these profiles (as can be observed for the 5\% thickness end-wall in Figure \ref{fig3:wall_loads}) highlighting the importance of thickness to avoid flow separation. Figure \ref{fig3:Ducting_Separation}b and Figure \ref{fig3:Ducting_Separation}e plot the flow field at hub height around a profile of 4\% camber at 5\% and 15\% thickness, respectively. The plot shows flow separation at the leading edge of the thinner end-wall and delayed flow separation for the thicker wall. A similar finding was made by \cite{Fleming2016} in their analysis of a variety of turbine ducting profiles. 


Figure \ref{fig3:Ducting_Separation} also shows an improvement in flow symmetry about the centre-line of the turbine with the 4\% camber end-wall. An increase in bypass flow speed is observed at the outboard section of the fence when the end-wall is in place, closer resembling the inner side's bypass flow field. The improved bypass flow speed is a function of the lateral confinement, which is observed to be smaller when the end-walls camber is lower (Figure \ref{fig3:Ducting_Separation}d, Figure \ref{fig3:Ducting_Separation}e and Figure \ref{fig3:Ducting_Separation}f). A similar bypass flow is observed between matched camber profiles (but varied thickness), suggesting that lateral confinement is not strongly influenced by end-wall thickness. This is confirmed by Figure \ref{fig3:wall_loads}, which shows similar lateral force across profile thicknesses. 


\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/047_sidewall_velocity_magnitudelong_section.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/176_sidewall_velocity_magnitudelong_section.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/180_sidewall_velocity_magnitudelong_section.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/189_sidewall_velocity_magnitudelong_section.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/172_sidewall_velocity_magnitudelong_section.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/numerical/178_sidewall_velocity_magnitudelong_section.png}
\caption{A plan view of the outboard turbine and side wall at the hub height showing the non-dimensional flow speed $|u|/u_{\infty}$ field for operation without an end-wall (a), an end-wall of 4\% camber and 5\% thickness (b), -4\% camber and 5\% thickness (c), 0\% camber and 15\% thickness (d),  4\% camber and 15\% thickness (e) and -4\% camber and 15\% thickness (f).}
\label{fig3:Ducting_Separation}
\end{figure}



\subsubsection{Fence symmetry factor}

The symmetry factor was used to analyse the thrust and power distribution along the fence, where positive values indicate higher inner turbine loading, and negative values indicate higher outer turbine loading. As observed for the fence without an end-wall in Section \ref{sc3:unducted_perf}, the $\Phi C_p$ and $\Phi C_t$ are offset by a constant value; for conciseness, only $\Phi C_p$ is plotted in Figure \ref{fig3:symfac_cp_wall}. As can be observed in this plot, most end-wall profiles continue to bias the inner turbine; however, the thickest (15\%) and most positively cambered (4\%) profile biased the outboard turbine by 0.4\% .The ability to make the flow more uniform across the fence with an end-wall means that approaches which design turbines for uniform blockage ratios, e.g. \cite{Schluntz2015,Cao2019} can also be applied to shorter fences of turbines.


\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{Figures/Chapter_3/numerical/symfac_cp_Geom.png}
\caption{$\Phi C_p$ over a range of end-wall cambers and thickness at $\lambda^*$ for a turbine spacing of $s/D$=0.25.}
\label{fig3:symfac_cp_wall}
\end{figure}


\subsubsection{End-wall profile selection}\label{CH3:FenceChoice}

An end-wall of stream-wise length $D$, 15\% thickness and positively cambered by 4\% was determined to be the best candidate geometry for the four turbine fence with 0.25$D$ tip-to-tip spacing based on the performance metrics examined in this section. When the end-wall is at a 0.125$D$ tip-to-wall spacing, it provides sufficient lateral constraint to increase the power and decrease turbine thrust; this is quantified in Table \ref{Tab3:Sym}. The wall profile selected was both the thickest and most heavily cambered of the geometries analysed. It is anticipated that thicker and more heavily cambered profiles will not further improve performance due to the increased stream-wise load they experience (decreasing $\eta$) and increased bias of outboard turbine load and power production (negative $\Phi C_t$ and $\Phi C_p$). 


\begin{table}[h]
\centering
\caption{$C_p$ and $C_t$ at $\lambda^*$ for a 4 turbine fence at a spacing of $s/D$=0.25 with and without an end-wall of 1$D$ length, 4\% camber and 15\% thickness.}
\begin{tabular}{lllll}
                & \multicolumn{2}{l}{No End-wall} & \multicolumn{2}{l}{With End-wall} \\ \hline
                & $C_p$          & $C_t$          & $C_p$           & $C_t$           \\ \hline
Inner Turbine   & 0.569          & 1.119          & 0.582           & 1.107           \\
Outer Turbine   & 0.553          & 1.108          & 0.584           & 1.108           \\ \hline
Average         & 0.561          & 1.114          & 0.583           & 1.108           \\ \hline
Symmetry Factor & 2.72\%         & 1.04\%         & -0.35\%         & -0.08\%         \\ \hline
\end{tabular}
\label{Tab3:Sym}
\end{table}



\subsubsection{Unsteady loading on turbine blades}

Following the methodology used in Section \ref{sc3:unsteady_unduct} differences in the azimuthally averaged ($\left<{u_{x}}\right>$) and deviatoric components ($u_{x}'$) of the stream-wise velocity through inboard and outboard turbines are presented in Figure \ref{fig3:Deviatoric_analysis_sidewall}. The difference in total velocity ($u_x$) suggests that the outer turbine's proximity to the end-wall could introduce some unsteady blade loading as the blade passes. Comparing the difference in azimuthally averaged velocity between the inner and outer turbines ($\Delta\left<{u_{x}}\right>$) from Figure \ref{fig3:UnducDeviatoric_analysis} (no end-wall) and Figure \ref{fig3:Deviatoric_analysis_sidewall} (with end-wall) shows a reduced disparity in turbine flux. This indicates that the end-wall has reduced the bypass losses and forced more flow through the outer turbine's swept area.


\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{Figures/Chapter_3/numerical/189_Slices_x_velocityturbines.png}
\caption{Total (top), azimuthally averaged (middle) and deviatoric (bottom) components of the stream-wise velocity at the inner (left) and outer (middle) turbines as well as the difference, $\Delta$, of the inner turbine's velocity component minus the outer turbine's velocity components turbine (right) for $s/D$=0.25 spaced rotors operating at $\lambda^*$ with a end-wall. The free-stream flow direction is into the page.}
\label{fig3:Deviatoric_analysis_sidewall}
\end{figure}


Figure \ref{fig3:flow_sidewall} demonstrates the influence of the end-wall on the stream-wise flow through the turbine plane. As can be observed in this plot, flow is accelerated between the end-wall and the turbine to be faster than the bypass flow between the inner and outer turbines; this demonstrates the end-wall's effectiveness in reducing end losses and also highlights potential for increased loading over this region. Alongside the increase in flow speed, the end-wall increases the flap-wise blade root bending moment as the blade passes the end-wall (Figure \ref{fig3:ducted_RBM}). However, the average load and load amplitude over a rotation is smaller than the without end-wall case (Table \ref{tab3:RBM_loading}). In the edge-wise direction, the outer turbine loading appears to be offset from the without end-wall case, this may be due to the increased mass flux through the majority of the turbine swept area and the change in $\lambda^*$ required when the end-wall is installed.

\begin{figure}
    \centering
    \includegraphics[width=0.7\textwidth]{Figures/Chapter_3/numerical/turbs_x_velocityDifference_plotsBase.png}
    \includegraphics[width=0.7\textwidth]{Figures/Chapter_3/numerical/turbs_x_velocityDifference_plotsDucted.png}
    \includegraphics[width=0.7\textwidth]{Figures/Chapter_3/numerical/turbs_x_velocityDifference_plotsDifference.png}
    \caption{Stream-wise flow velocity through the turbine plane for a fence with $s/D$=0.25 turbine spacing operating at $\lambda^*$ without an end-wall (a), with an end-wall of 15\% thickness and 4\% camber (b) and the difference, determined as the flow velocity with end-wall minus the flow velocity without (c). Free-stream flow direction is into the page.}
    \label{fig3:flow_sidewall}
\end{figure}



% - inner turbine
The change in loading around the azimuth for the inner turbine is similar to that of the outer turbine in the edge-wise direction, with constant changes in loading at all angular positions. For the flap-wise direction this shift is negative, a result of the lower $\lambda^*$ relative to the without end-wall case. The increased mass flux and change in $\lambda^*$ also appears to increase the edge-wise root bending moment of this turbine. 

% - Comment on fatigue 
The impact of the end-wall also reduced the difference in magnitude between the root bending moment for outer and inner turbines, meaning throughout the life cycle of the fence, similar fatigue damage will occur between the turbines. This is useful for improving the structure's design life estimate as there is balanced degradation of the turbines across the fence. 



% Following the methodology used in Section \ref{sc3:unsteady_unduct} differences in the azimuthally averaged ($\left<{u_{x}}\right>$) and deviatoric components ($u_{x}'$) of the stream-wise velocity through inboard and outboard turbines are presented in Figure \ref{fig3:Deviatoric_analysis_sidewall}. The difference in total velocity suggests the outer turbine's proximity to the end-wall could introduce some unsteady blade loading as it passes. The increased flow velocity indicates that the end-wall has reduced the bypass losses and forced more flow through the rotor's swept area. Figure \ref{fig3:flow_sidewall} shows the effect of the end-wall on stream-wise velocity, demonstrating a localised increase in velocity adjacent to the wall, increasing the unsteady loading on the rotor. 
%% The change in velocity is observed by comparing the $u_{x}'$ component of the flow through the outer turbine for the turbine with and without end-wall (Figure \ref{fig3:UnducDeviatoric_analysis} and Figure\ref{fig3:Deviatoric_analysis_sidewall}, respectively). 


% The edge-wise and flap-wise root bending moments are presented in Figure \ref{fig3:ducted_RBM}. Overall the end-wall slightly reduced the root bending moment in the flap-wise direction and slightly increased the edge-wise bending; this is consistent with the results of the integrated coefficients $C_t$ and $C_p$. The inclusion of the end-wall introduced a secondary peak in the azimuthal variation of outboard turbine flap-wise root bending moment as it passed the wall, supporting the observation made previously that increased flow speed in this region increases the loading. The amplitude of the loading was reduced (Table \ref{tab3:RBM_loading}), and the azimuthal variation more closely aligned with the inboard turbine due to the end-wall. Consequently, throughout the fence's life cycle, similar fatigue damage would be expected on the turbines, simplifying the scheduling of maintenance and improving the structure's design life as there is balanced degradation of the turbines across the fence. 



\begin{table}[]
\centering
\caption{Comparison of root bending moment between the inner and outer turbines with and without a end-wall}
\label{tab3:RBM_loading}
\begin{tabular}{|c|c|lll|lll|}
\hline
\multirow{2}{*}{Bending Direction}                                                            & \multirow{2}{*}{Turbine} & \multicolumn{3}{c|}{Without end-wall}                                         & \multicolumn{3}{c|}{With End-wall}                                            \\ \cline{3-8} 
                                                                                              &                          & \multicolumn{1}{c}{min} & \multicolumn{1}{c}{max} & \multicolumn{1}{c|}{mean} & \multicolumn{1}{c}{min} & \multicolumn{1}{c}{max} & \multicolumn{1}{c|}{mean} \\ \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Flap-wise \\ root bending ($C_{FW}$)\end{tabular}} & Inner                    & 0.736                   & 0.764                   & 0.748                     & 0.723                   & 0.753                   & 0.737                     \\
                                                                                              & Outer                    & 0.726                   & 0.756                   & 0.741                     & 0.729                   & 0.750                   & 0.738                     \\
                                                                                              & Difference               & 0.010                   & 0.008                   & 0.007                     & -0.006                  & 0.004                   & -0.001                    \\ \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Edge-wise \\ root bending ($C_{EW}$)\end{tabular}} & Inner                    & 0.540                   & 0.571                   & 0.552                     & 0.551                   & 0.588                   & 0.566                     \\
                                                                                              & Outer                    & 0.523                   & 0.550                   & 0.537                     & 0.552                   & 0.584                   & 0.567                     \\
                                                                                              & Difference               & 0.017                   & 0.021                   & 0.015                     & 0.000                   & 0.005                   & -0.001                    \\ \hline
\end{tabular}
\end{table}




\begin{figure}
\centering
\includegraphics[width=0.72\textwidth]{Figures/Chapter_3/numerical/RBM_BE_FW_wall.png}
\includegraphics[width=0.72\textwidth]{Figures/Chapter_3/numerical/RBM_BE_EW_wall.png}
\caption{Flap-wise (top) and edge-wise (bottom) root bending moment for inboard (red) and outboard (blue) turbines operating at $\lambda^*$ at $s/D$=0.25. Solid lines represent the without end-wall case and the dotted line represent the with end-wall case. The three illustrations at the top of each plot show the location of the blade (in bold) in the domain, with blue and red representing the outboard and inboard turbines.}
\label{fig3:ducted_RBM}
\end{figure}





 \subsubsection{Conclusions}

A parametric study of a cambered elliptical cross-section end-wall of length $D$, thickness (5\%, 10\% and 15\% length) and camber (-4\%, -2\%, 0\% 2\% and 4\% length) was examined for installation at the ends of a four turbine long fence with an inter-turbine spacing of $s/D$=0.25. This study showed an end-wall could increase the fence's power generating capability by up to 3.95\%, achieving this through improved flow alignment (especially over the outboard section of the fence). A slight reduction in turbine thrust was also observed owing, in part, to a decrease in $\lambda^*$. Consequently, the impact on basin efficiency (which included parasitic power removed from the flow by the end-wall) was a minor reduction of between 0.5\% and 3\% relative to the no end-wall case. 

The end-wall reduced the difference in root bending moments between the inboard and outboard turbines bringing the averaged loading closer together and balancing the turbine fatigue across the rotors in the fence. Whilst this work is promising in terms of improved performance, further investigation into integrating the end-wall into the BE turbine blade design methodology may further improve performance. 

%% Put a chapter split here
\section{Experimental model}\label{Ch3:experimental}
%%% HERE
Complementary to the numerical study presented in Section \ref{Ch3:numerical}, an experimental study of a fence of four turbines constrained by an end-wall is presented in this section. Due to limitations on the number of tests over the testing program, only one end-wall wall geometry (length $D$, 15\% thickness and 4\% camber) and inter-turbine spacing ($s/D$=0.25) was considered. 

Details on the manufacture of the turbines used in this experiment can be found in \cite{Cao2019} and results from initial testing at the FloWave test facility in Edinburgh can be found in \cite{McNaughton2019}. The focus of the previous work by \cite{Cao2019} and \cite{McNaughton2019} has been on constructive interference through the exploitation of the local blockage effect. This work focuses on fence end losses, using the experimental results from  \cite{McNaughton2019} first to quantify the fence end losses before new experimental results are presented for the end-wall constrained fence. 


\subsection{Experimental methodology}

The methodology presented in this section is relevant to the towing test tanks with an end-wall; further detail on the testing procedure and results for the without end-wall towing tank tests can be found in \textbf{JAMES NEW PAPER}. A total of 10 tests were conducted across 7 $\lambda$ ranging between 5.7 and 7.1 for the turbine fence with end-wall. Duplicate testing at 3 $\lambda$ close to $\lambda^*$ were run to confirm the results and quantify the uncertainty.

\subsubsection{Experimental facility}

The towing tank testing was completed at the SSPA test facility in Gothenburg, Sweden. The computational domain (discussed in Section \ref{CH3:Method1}) matched the 10 m wide, 5 m deep and 240 m long towing tank in which two turbines of 1.2 m ($D$) diameter were placed. Turbines were arranged as per Figure \ref{fig3:exp_domain}, spaced at 0.25$D$ (0.3 m) tip-to-tip with the turbine end-wall placed at a distance of 0.125$D$ tip-to-wall from the outboard turbine. The inboard turbine was spaced at a distance of 0.125$D$ from the tank end-wall to take advantage of symmetry, emulating a fence of four turbines. 

The turbines were instrumented with thrust and torque transducers, and the individual blades measured flap-wise and edge-wise root bending moments with strain gauges. Additional strain gauges were fastened to the end-wall to record both stream-wise and cross-stream loading. However, due to instrument failure, only the cross-stream loading data is available. Tests were conducted at a speed of 1 m/s with the towing carriage accelerating at a rate of 0.05 $\mathrm{m}/\mathrm{s}^2$, providing a minimum of 90 seconds of usable data for each run. A minimum settling time of 10 minutes was observed between tests.

\begin{figure}
\centerline{
\includegraphics[width=0.7\textwidth]{Figures/Chapter_3/experiment/Domain_experiment.png}
}
\caption{Diagram of the SSPA towing tank experimental setup. $D$ is the turbine diameter of 1.2 m and $s$ is the inter-turbine spacing of 0.25$D$. The red and blue circles are the swept area of the inboard turbine and outboard turbines respectively and the dotted line is the turbine fence end-wall.}
\label{fig3:exp_domain}
\end{figure}


\subsubsection{Prototype manufacture}

The 4\% camber, 0.15$D$ thick end-wall from Section \ref{CH3:FenceChoice} was selected for construction. The bottom of the end-wall was positioned 2 m (1.67$D$) below the water surface and the edges were rounded to reduce drag. The turbines were submerged to a hub depth of 1 m and thus were symmetrically placed about the end-wall mid-depth.

Loading on the end-wall was estimated from the RANS-BE model to be 300N in the cross-stream direction and 10N in the stream-wise direction at $\lambda^*$; these loads were used for the structural design of the wall. A conservative assumption that the load acted at the end-walls unsupported tip was used to estimate bending on the end-wall and support structure (Figure \ref{fig3:sidewall_install}). Checks for shear, axial and bending loads were performed at critical locations in the frame to determine its capacity. This design's critical loading case occurred during installation if it were suspended by a single one of the two connecting members. In this scenario, one of the steel frame's connecting plates was 91\% of its capacity. To avoid this loading condition, the wall was carefully rigged to distribute the 3600 N self-weight as it was placed into the water. The purple rigging equipment is visible in Figure \ref{fig3:sidewall_install}.

The wall was manufactured using a Computer Numerical Controlled router (CNC) to shape it into four segments. The segments were formed by layering marine grade plywood sheets and cutting the excess material to the appropriate size. Hollow sections on the wall's inner faces allowed a steel frame and lead weights to be inserted. 80 kg of lead weights were added to offset the structure's buoyancy so that it was approximately neutrally buoyant when assembled. The wooden surface was coated in marine grade epoxy, and boltholes, which held the wood to the steel frame, were filled with sealant. While every effort was made to ensure a smooth finish, the nature of the wooden material and epoxy resulted in some surface roughness on the end-walls exterior, which increased the skin friction drag on the structure. The steel frame was constructed from 3 mm thick 50 mm square hollow sections with strain gauges attached to the top of the supports to measure the force acting in the cross-stream direction. Images of the partially assembled end-wall and the end-wall installed adjacent to the fence are presented in Figure \ref{fig3:sidewall_install}.


\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Figures/Chapter_3/experiment/sidewall_manfactured.png}
\includegraphics[width=0.75\textwidth]{Figures/Chapter_3/experiment/Sidewall_side_view.png}
\includegraphics[width=0.75\textwidth]{Figures/Chapter_3/experiment/sidewall_front.jpg}
\caption{Partially assembled end-wall showing the steel frame, voids for lead weights and the locations of the strain gauges (top), installed end-wall and support structure (middle) and font view of the installed end-wall in the turbine fence (bottom).}
\label{fig3:sidewall_install}
\end{figure}


\subsubsection{Processing of root bending moment from strain gauge data}\label{SC3:RBM_expr_method}

A methodology for processing the phase averaged root bending moment was developed to average the azimuthal variation in bending moment across the blades and produce consistent results with the thrust and torque transducer data. The thrust and torque transducer data is preferenced when comparing the signals due to the external calibration of this piece of equipment.

\noindent \underline{Zeroing} 

Blade root bending moment data were first processed by removing the zero offsets from the strain gauges in the root bending moment sensors and the thrust and torque transducer to account for the change in the gauge zero from variations in temperature and other environmental factors over the experiment. Zero offsets were recorded before each test run.

\noindent \underline{Azimuthal Averaging} 

Next, an azimuthal average was calculated for each blade and each bending direction by averaging the root bending moment in 1-degree bins. A blade averaged value was also derived from the thrust and torque transducer data following
\begin{equation}
        M_{FW}^* = \frac{T}{N_b}r_T, \quad
        M_{EW}^* = \frac{Q_t}{N_b}
\end{equation}
\noindent where $M_{FW}^*$ and $M_{EW}^*$ are the blade averaged flap-wise and edge-wise root bending moment from the thrust and torque transducer, $Q_t$ is the torque, and $r_T$ is the distance from the centre of the turbine to the line of action on the blade in the thrust direction. $r_T$ was derived numerically from the blade resolved simulations of \cite{Cao2019} to be 0.4 m. 

\noindent \underline{Scaling} 

Each strain gauge was scaled such that
\begin{equation}
    SG_{new} = SG_{raw} C
\end{equation}
\noindent where $SG_{new}$ is the new strain gauge value, $SG_{raw}$ is the phase averaged strain gauge data, and $C$ is the factor to match the mean of the strain gauge data to the thrust and torque transducer. The factor ($C$) is the ratio of $M_{FW}^*$ or $M_{EW}^*$, dependant on the direction, and the mean strain gauge value over all three turbine blades. This process ensures the data set is internally consistent using the average load measured from the thrust and torque transducer to adjust the blade measurements. 


\noindent \underline{Blade Averaging} 

Finally, the root bending moment is averaged over the phase-offset blades loads to arrive at a blade and azimuthally averaged root bending moment following
\begin{equation}
    M_{j} = \sum_{N_b}^{i=1} \frac{SG_{new_i,j}(\theta_i)}{N_b} 
\end{equation}
\noindent where $M_{j}$ is the blade and azimuthally averaged root bending moment, the subscript $j$ denotes the edge-wise and flap-wise directions, $SG_{new_i}$ is the azimuthally averaged strain gauge data for blade $i$ which has been phase corrected for the blade offset ($\theta=0^\circ$ for blade 1, $\theta=120^\circ$ for blade 2, and $\theta=240^\circ$ for blade 3). An additional quantity $M_{j}^-$ was defined to compare directly against $M_{FW}^*$ and $M_{EW}^*$ by using $\theta=0$ for all blades. 

\noindent \underline{Self Weight Correction}

A correction for the blade self-weight and buoyancy was applied to the edge-wise root bending moments 

\begin{equation}
    M'_{EW} = M_{EW} - g r_B (m_B - V_B \rho) \sin\theta
\end{equation}
\noindent where $M'_{EW}$ is the edge-wise root bending moment corrected for blade self-weight and buoyancy, $g$ is the acceleration due to gravity, $r_B$ is the distance to the centre of mass of the blade, $m_B$ is the mass of the blade, and $V_B$ is the volume of the blade. The blade's geometric properties ($r_B$ and $V_B$) were derived from a SolidWorks of the model developed by \cite{McNaughton2019}.


\noindent \underline{Non-dimensionalised Parameters}

The root bending moment was non-dimensionalised following 

\begin{equation}
    C_{FW} = \frac{N_b M_{FW}}{\tfrac{1}{2} \rho u_{\infty}^2 A_t R}, \quad
    C_{EW} = \frac{N_b M'_{EW}}{\tfrac{1}{2} \rho u_{\infty}^2 A_t R}.
\end{equation}




% The methodology presented in the above assumes that each blade makes an equal contribution to the loading on the thrust and torque transducer, this is to say that all the blades are aligned in the same way, fixed with the same rigidity to the support and free from defects. These assumptions appear to be valid for the blue turbine as demonstrated in Figure \ref{fig3:expr_RBM_proc}. This figure shows the raw and processed strain gauge as well as $M_{FW}^*$ and $M_{SG}^*$. A slightly larger offset was applied to the raw signal from blade 1 than blades 2 and 3, this may have been due to zero drift of this strain gauge from the start of the day when the zero measurement was recorded. The reconstructed signal from the strain gauges $M_{SG}^*$ and the bending moment from the thrust and torque transducer $M_{FW}^*$ align closely, showing three distinct peaks when the signals are combined corresponding to a blade passing peak loading and an adjacent blade about to enter the peak loading phase. 

% In the case of the red turbine $M_{FW}^*$ and $M_{SG}^*$ do not align as closely as shown in Figure \ref{fig3:expr_RBM_proc_error}. The signal in the thrust and transducer appears to be dominated by a once-per-revolution frequency and does not have the 3 distinct peak which were observed for the blue turbine case. This could suggest that the load contribution from one blade is slightly higher than the others which could cause this result. The data does appear to suggest that the amplitude of loading for blade 1 is slightly larger than the r 


% \begin{figure}
% \centering
% \includegraphics[width=0.75\textwidth]{Figures/Chapter_3/experiment/Example_processing_RBM.png}
% \caption{Left: Raw data from the flap-wise root bending moment strain gauges (orange, blue and green) and $M_{FW}^*$ from the thrust and torque transducer data.
% Right: Processed strain gauge data and 5 degree moving average of $M_{FW}^*$ (red) and $M_{SG}^*$ (blue)}
% \label{fig3:expr_RBM_proc}
% \end{figure}


% \begin{figure}
% \centering
% \includegraphics[width=0.75\textwidth]{Figures/Chapter_3/experiment/Example_processing_RBM_Red_correction.png}
% \caption{An example of the post processing methodology identifying increased loading on blade 1 (orange) and reduced loading on blade 3 (green).}
% \label{fig3:expr_RBM_proc_error}
% \end{figure}



\subsection{Performance of turbines without end constraints}\label{sc3:exper_turb_nowall}

This section presents the experimental results of \textbf{JAMES PAPER} under a shifted perspective which focuses on fence end losses to provide context and justification for placing an end-wall adjacent to the end turbine discussed in Section \ref{sc3:expr_wall}. All results presented in this section were obtained with a constant tow speed of 0.8 m/s.

\subsubsection{Performance parameters}

Fence averaged $C_p$ and $C_t$ from the experimental work of \textbf{JAMES PAPER} are presented in Figure \ref{fig3:exper_perf}. \textbf{James's} results confirm \cite{Nishino2013} the observation that closer turbine spacings produce greater power output. However, as discussed when examining numerical models for turbine spacing in Section \ref{sc3:unducted_perf}, this performance improvement is not shared uniformly along the turbine fence due to end losses. Instead, the increase in fence power is primarily driven by an improvement in inboard turbine performance; this is confirmed by $SF_{C_p}$ and $SF_{C_t}$ with bias being shifted towards the inboard turbine with decreased spacing. 

Also of note from this data set, at larger turbine spacings (s/D=0.72 and s/D=1), the inboard turbine generates less power than the outboard turbine; this result is surprising given the higher local blockage of the inboard turbine. A reduction of inboard turbine power may be due to the invalid assumption of the tank wall as a symmetry plane. In the experiment, fluid and the tank wall do not move at the same speed due to the bow wave and wake of the turbine fence inducing a relative motion between them, forming a boundary layer and altering the inflow condition. This is considered in further detail in Section \ref{CH3:model_limits}.


% The symmetry factor ($SF$) was used to compare performance of the inboard and outboard rotors. In general both $SF_{C_p}$ and $SF_{C_t}$ were observed to shift the bias towards the inboard turbine as the spacing was decreased. This is consistent with the numerical results which credited this phenomenon to an increase in local blockage of the inboard turbine and a greater fence end loss with decreased turbine spacing. In general, $SF_{C_p}$ and $SF_{C_t}$ decreased with increasing $\lambda$, suggesting a bias towards the outboard turbine. This is likely due to the increased thrust on the turbines at higher $\lambda$ reducing the mass flow rate through the turbines. As the inner turbine operates at a higher thrust, this reduction in mass flow rate is expected to occur at lower $\lambda$ than the outer turbine causing the symmetry factor to shift.






\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Inboard_Turbine_C_p.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Inboard_Turbine_C_t.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Outboard_Turbine_C_p.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Outboard_Turbine_C_t.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Array_Averaged_C_p.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Array_Averaged_C_t.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Symmetry_Factor_C_p.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Symmetry_Factor_C_t.png}
\caption{$C_p$ and $C_t$ performance curves from experimental testing of a tidal turbine fence at varied inter-turbine spacing of $s/D=1$ (green) $s/D=0.72$ (purple) and $s/D=0.25$ (blue). Results for the inboard turbine are plotted in the top row, outboard in the top-middle, fence averaged on the bottom-middle and symmetry factor in the bottom row. The wait time between tests is between 10 and 20 minutes and duplicate data has been averaged.}
\label{fig3:exper_perf}
\end{figure}


\subsubsection{Root bending moments}\label{sc3:rbm_exper_spacing}

Phase averaged root bending moment data is presented in Figure \ref{fig3:exper_rbm_spacing} for $s/D=1$, $s/D=0.72$ and $s/D=0.25$. The phase averaged root bending moment was calculated following the methodology outlined in Section \ref{SC3:RBM_expr_method}. Due to the triplicate data from the blade-mounted strain gauges, there is a smooth phase-averaged signal across the entire rotation for all gauges.

Figure \ref{fig3:exper_rbm_spacing} provides further detail on both local blockage and end losses by examining the edge-wise root bending moment (blade torque). The inboard turbine has a steady increase in $C_{EW}$ with decreased spacing at all azimuthal locations; this leads to an improvement in $C_p$. The cause of this improvement is due to various factors associated with local blockage (improved flow speed, wake constraint, inflow angle) and is examined in further detail in \textbf{JAMES}. Contrasting this result is the outboard turbine signal, which only improves as it passes the inboard turbine (270$^\circ$). At the outboard location (90$^\circ$), the root bending moment appears to be unaffected by the change in spacing, demonstrating a growing end loss with decreased spacing. These two findings are consistent with the results from Section \ref{sc3:unducted_perf} and provide further justification for why an end-wall may be a practical solution for improving turbine performance, especially at close turbine spacing.  



% Figure \ref{fig3:exper_rbm_spacing} demonstrates that there is a larger variation (difference between maximum and minimum moment) in loading over a rotation as the inter-turbine spacing is decreased, especially for the outboard turbine. The maximum flap-wise root bending moment typically occurs at the 255$^\circ$ position, corresponding to the inner rotor's blade passing the symmetry plane and the outer rotor's blades passing the inner rotor. A second peak in the loading is observed at 60$^\circ$ for the inboard turbine when the inter-turbine spacing is decreased from s/D=0.72 to s/D=0.25. This suggests an increased bypass flow over this portion of the rotation resulting from the higher local blockage. A small dip at 20$^\circ$ is observed for the inboard turbine at the s/D=0.25. This result is consistent with previous observations of tower passing effects which have been observed to have a phase lag of approximately 15$^\circ$ by \cite{Payne2018VariationPosition} and \cite{Ahmed2017FluctuatingData}.

% Edge-wise root bending moments appear to have less variation over a rotation than the flap-wise direction. Greater root bending moments are observed as the inter-turbine spacing is decreased, leading to improved power output (Figure \ref{fig3:exper_perf}). The peak power production region is a flat peak which spans over the 90$^\circ$ to the 180$^\circ$ portion of the rotation. The phase averaged signal of the inboard and outboard turbines at s/D=0.25 are very similar indicating that flow correction from a sidewall is not required to balance the edge-wise bending signals at this spacing. A noticeable drop off in power production of the rotors was observed at 280$^\circ$ as blades pass the symmetry plane and the inner turbine (for the inboard and outboard turbines respectively). It is hypothesised that the increased bypass flow speed over this region has moved the local angle of attack of the turbine blade to a sub-optimal position, resulting in increased drag (higher flap-wise bending) and decreased lift (reduced edge-wise bending moment). 


\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/experiment/rbm_varied_spacing_C_{FW}_.png}
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/experiment/rbm_varied_spacing_C_{EW}_.png}

\caption{Phase averaged flap-wise (top) and edge-wise (bottom) root bending moments at varied inter-turbine spacing of $s/D=1$ (green) $s/D=0.72$ (purple) and $s/D=0.25$ (blue) for a fixed $\lambda$ of 6.44. The inboard and outboard turbine data is represented by the dots and circles respectively. The wait time between tests is between 10 and 20 minutes and duplicate data has been averaged; blade self weight and buoyancy has been controlled for in the edge-wise root bending moment signal. The three illustrations at the top of each plot show the location of the blade (in bold) in the domain, with blue and red representing the outboard and inboard turbines in these illustrations.}
\label{fig3:exper_rbm_spacing}
\end{figure}



\subsection{Performance of turbines with end-wall constraint}\label{sc3:expr_wall}

The performance of the turbine fence with an end-wall is examined in this section. Unlike the results presented in Section \ref{sc3:exper_turb_nowall}, the towing speed for these tests was 1 m/s (instead of 0.8 m/s). Higher tow speeds were used to ensure Reynolds independence of the results as discussed in \textbf{Ref to James paper}. Consequentially there is a slight difference between the without end-wall results presented in this section at s/D=0.25 and those from Section \ref{sc3:exper_turb_nowall}. The results here were collected alongside \textbf{REF TO JAMES PAPER} and are the authors own original work. 


\subsubsection{Performance parameters}


Table \ref{tab3:comp_perf} presents a comparison between performance metrics from the RANS-BE model and experimental study; this table highlights the significant difference in $C_p$ (up to 21.3\%) and $C_t$ (up to 6.1\%) between the models. The cause for this disparity is primarily credited to the limitations of the RANS-BE model and experimental error (discussed further in Section \ref{CH3:model_limits}). Despite this, scaled values of $C_p$ and $C_t$ allow comparison between the data sets. 

\begin{table}[]
\centering
\caption{A comparison of turbine performance metrics $C_p$ and $C_t$ at $\lambda^*$ for a 4 turbine fence at a spacing of s/D=0.25 from a RANS-BE model and an experimental study.}
\label{tab3:comp_perf}
\begin{tabular}{cc|rrr|rrr|}
                               &            & \multicolumn{3}{c|}{Without end-wall}                                                & \multicolumn{3}{c|}{With end-wall}                                                   \\ \cline{3-8} 
                               & \textbf{}  & \multicolumn{1}{c}{RANS-BE} & \multicolumn{1}{c}{Exper} & \multicolumn{1}{c|}{Error} & \multicolumn{1}{c}{RANS-BE} & \multicolumn{1}{c}{Exper} & \multicolumn{1}{c|}{Error} \\ \hline
                               & $\lambda*$ & 6.52                        & 6.44                      & 1.2\%                      & 6.37                        & 6.44                      & -1.1\%                     \\ \hline
\multirow{2}{*}{Inner Turbine} & $C_p$      & 0.569                       & 0.484                     & 17.6\%                     & 0.582                       & 0.480                     & 21.3\%                     \\
                               & $C_t$      & 1.119                       & 1.058                     & 5.8\%                      & 1.107                       & 1.053                     & 5.1\%                      \\ \hline
\multirow{2}{*}{Outer Turbine} & $C_p$      & 0.553                       & 0.479                     & 15.6\%                     & 0.584                       & 0.486                     & 20.2\%                     \\
                               & $C_t$      & 1.108                       & 1.044                     & 6.1\%                      & 1.108                       & 1.048                     & 5.7\%                     
\end{tabular}
\end{table}



% Paragraph 1 introduce the results you can see
Figure \ref{fig3:comp_wall_perf} plots the experimental and numerical model results for $C_p$ and $C_t$. This figure validates the numerical model's observation that the outboard turbine's performance is improved when the end-wall is in place. However, the experimental result's improvement is not as significant as that identified in the numerical model. Experimental results showed a 1.5\% increase in outboard turbine power at $\lambda^*$ and a 0.8\% decrease for the inboard turbine to achieve an overall improvement of 0.5\% when the end-wall is in place. The improved performance is achieved at approximately the same $\lambda$ and fence averaged thrust as the without an end-wall case. At $\lambda^*$, the end-wall installation caused the outboard turbine thrust to increase and the inboard turbine thrust to decrease, bringing both values closer to the mean fence thrust and reducing $SF_{C_t}$.

Inboard turbine $C_p$ was found to decrease with the installation of an end-wall, contradicting the model prediction that the inboard turbine's power would increase alongside the outboard turbine. This observation was consistent over the range of $\lambda$ tested and highlights a potential limitation of using the RANS-BE model for this application. Given the experimental result, a wall placed sightly further away may have been more effective in balancing turbine power. Despite this, at the end-wall-turbine spacing of 0.125$D$, the thrust was balanced between the turbines. This is critical considering that thrust loading is typically higher than torque loading, making it a necessary input to maintenance scheduling and fatigue estimations. By reducing the disparity in thrust loading between the turbines (as achieved from the end-wall installation), turbine degradation is better balanced along the fence, improving maintenance scheduling. 



\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Inboard_Turbine_C_pC_p_star.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Inboard_Turbine_C_tC_t_star.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Outboard_Turbine_C_pC_p_star.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Outboard_Turbine_C_tC_t_star.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Array_Averaged_C_pC_p_star.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Array_Averaged_C_tC_t_star.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Symmetry_Factor_C_p.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Symmetry_Factor_C_t.png}

\caption{$C_p$ (left) and $C_t$ (right) performance  curves  comparing between experimental testing (orange) and the RANS-BE model (blue) for  a  tidal turbine with (dashed line) and without (solid line) an end-wall. The results of the inboard turbine are presented on the top row, the outboard turbine in the top middle row and the fence averaged in the bottom middle row. The symmetry factors for $C_p$ and $C_t$ are plotted on the bottom row. $C_p$ curves have been scaled by the without end-wall fence averaged maximum power ($C_p^*$) and the $C_t$ curves have been scaled by the fence averaged thrust at $C_p^*$ ($C_t^*$). For the experimental results the wait time between tests is between 10 and 20 minutes and duplicate data has been averaged.}
\label{fig3:comp_wall_perf}
\end{figure}



% Paragraph 3 reason for this result
The experimental result are further understood by considering how the turbine performance is improved. Typically, this is through increasing the mass flux through the turbine or achieving a better angle of attack along the blades; the opposite is also true and worse performance comes from the reduced mass flux and worse angles of attack. For a reduced mass flux a drop in $C_t$ and $C_p$ is expected ($C_p\propto u^3$ and $C_t \propto u^2$). Given this relation, the 0.8\% drop in power for the inboard turbine is expected to be accompanied by a 0.5\% drop in thrust; this is close to the measured drop in thrust, which is just under 0.5\%. In contrast, the 1.5\% increase in power for the outboard turbine is expected to have a 1.0\% increase in thrust; instead, only a 0.4\% increase in thrust was observed. This result suggests that the improvement in inboard turbine performance is not strictly connected to the mass flux and may be a function of an improved angle of attack of the blade; this is explored further in Section \ref{sc3:exp_rbm_wall} with the analysis of root bending moment data.



% Paragraph 4 discuss the limitation/causes for error

In contrast to the above, the RANS-BE results suggested that the mass flux through both turbines increased with the end-wall installation. This resulted in a higher thrust on the outboard turbine and over-predicted the change in thrust with the end-wall, causing greater confinement (and therefore power generating capacity) of the inboard turbine and an over prediction of inboard turbine power. In practice, the use of RANS-BE models with yawed flows passing around the ends of turbine fences is not strictly valid as there is a shear profile in the flow; this is discussed in further detail in Section \ref{CH3:model_limits}.



% Overall the inclusion of a sidewall in the experiment resulted to a modest increase in $C_p$ of 0.5\% with the performance improvement of the outboard turbine compensating for the decline in performance of the inboard turbine. In contrast the RANS-BE model saw an improvement in performance along the entire fence leading to an increase in fence averaged $C_p$ of 3.95\%. This result suggests that the RANS-BE model may not be able to capture the exact change in integrated parameters $C_p$ and $C_t$ when the sidewall is installed. However it is a useful tool for investigating relative change in these parameters when a sidewall is installed. As can be observed in the symmetry factor plots in Figure \ref{fig3:comp_wall_perf} both the experimental data and RANS-BE model capture a downwards shift in bias from the inboard turbine to the outboard turbine when the sidewall is installed. 





% \begin{figure}
% \centering
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_with_wall_C_p.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_with_wall_C_t.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_with_wall_Symmetry_Factor_C_p.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/experiment/perf_curve_with_wall_Symmetry_Factor_C_t.png}
% \caption{$C_p$ (top left) and $C_t$ (top right) performance curves from experimental testing of a tidal turbine fence with (dashed line with crossed markers) and without (dotted line with circle markers) a sidewall for s/D=0.25. Results for the inboard, outboard and fence averaged turbines are red, blue and black respectively. $SF_{C_p}$ (bottom left) and $SF_{C_t}$ (bottom right) for the with (dashed line with crossed markers) and without (dotted line with circle markers) sidewall cases are also presented. The wait time between tests is between 10 and 20 minutes and duplicate data has been averaged.}
% \label{fig3:exper_wall_perf}
% \end{figure}






\subsubsection{Root bending moments}\label{sc3:exp_rbm_wall}

% Paragraph 1 - Experimental observations

Phase averaged root bending moments at $\lambda^*$ from the experimental and the RANS-BE model are presented in Figure \ref{fig3:comp_rbm_wall} following the methodology outlined in Section \ref{SC3:RBM_expr_method}. The end-walls influence in the experimental results is localised around the wall at the 90$^\circ$ blade position on the outboard turbine. At this location, there is a significant increase in $C_{EW}$ and $C_{FW}$ as a result of flow realignment from the end-wall. This introduces a second peak into the loading signal, bringing the azimuthal variation closer to the inboard turbine. Comparing the with and without end-wall $C_{EW}$ curves, power is increased in the 180$^\circ$ portion closest to the end-wall, and the remaining portion looses performance. The improvement is greater than the losses leading to a higher overall turbine $C_p$; however, this highlights a potential limitation of an end-wall as there are performance reducing effects that may reduce net turbine power. The data also supports the hypothesis that the performance improvement comes from a locally improved angle of attack (as the performance improvement is localised around the end-wall) rather than an increase in the turbine's mass flow rate. 

More subtle changes are observed for the inboard turbine with a marginal reduction in $C_{FW}$ just after the blade passes the outboard turbine at 135$^\circ$ and a uniform decrease in $C_{EW}$. The uniform decrease in $C_{EW}$ supports the idea presented in the previous section that the mass flux through this turbine is reduced; however, this result is small and well within the range of experimental error and strain gauge drift.  




\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Outboard_C_{FW}_.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Inboard_C_{FW}_.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Outboard_C_{EW}_.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Inboard_C_{EW}_.png}

\caption{Comparison of RANS-BE (blue) and experimental (orange) results at $\lambda^*$. Phase averaged flap-wise (top) and edge-wise (bottom) root bending moments are presented for outboard (left) and inboard (right) turbines. Solid lines represent the without end-wall case and dashed lines are used for the with end-wall data sets. Experimental results for the edge-wise root bending moment have controlled for blade self weight and buoyancy and a 5$^\circ$ moving average is applied to improve the clarity of the result. The three illustrations at the top of each plot show the location of the blade (in bold) in the domain, with blue and red representing the outboard and inboard turbines in these illustrations.}
\label{fig3:comp_rbm_wall}
\end{figure}


% Paragraph 2 - Explain the impact of the wall - performance improvement comming from xx


% Paragraph 3 talk about the RANS-BE results

Overall the RANS-BE model overestimated the loading on the turbines and the loading change due to the end-wall. This is quantified in table \ref{tab3:comp_rbm}, which shows the error in the average root bending moments between the RANS-BE model and the experimental data. The RANS-BE model's error was between 3.9\% and 4.4\% in the flap-wise direction and 10.8\% and 18.9\% in the edge-wise direction. A factor contributing to this poor agreement is the different $\lambda^*$ the RANS-BE model predicted when the end-wall is introduced. The RANS-BE model predicted a reduction in $\lambda^*$ leading to lower thrust on the fence, altering the inflow profile and compounding the error in the power production. No change in $\lambda^*$ was observed in the experiment, perhaps due to the jagged shape of the performance curve around $C_p^*$ making it tough to identify $\lambda^*$ correctly. Figure \ref{fig3:comp_rbm_wall_same_TSR} plots the azimuthal variation in root bending moment for rotors operating at a constant $\lambda$=6.44. This produces a closer agreement with the experimental data; however, there is still a significant difference between the root bending moments. The cause of this error is discussed further in Section \ref{CH3:model_limits}. Despite the offset to the mean root bending moment, the RANS-BE model captures the approximate amplitude of the variations in root bending moment over a rotation, albeit with inaccurate phasing (especially for the outboard turbine).

\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Outboard_C_{FW}_same_tsr.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Inboard_C_{FW}_same_tsr.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Outboard_C_{EW}_same_tsr.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Inboard_C_{EW}_same_tsr.png}

\caption{Comparison of RANS-BE (blue) and experimental (orange) results \textbf{at the same $\lambda$=6.44}. Phase averaged flap-wise (top) and edge-wise (bottom) root bending moments are presented for outboard (left) and inboard (right) turbines. Solid lines represent the without an end-wall case and dashed lines are used for the with an end-wall data sets. Experimental results for the edge-wise root bending moment have controlled for blade self-weight and buoyancy and a 5$^\circ$ moving average is applied to improve the clarity of the result. The three illustrations at the top of each plot show the location of the blade (in bold) in the domain, with blue and red representing the outboard and inboard turbines in these illustrations.}
\label{fig3:comp_rbm_wall_same_TSR}
\end{figure}


Comparing the without end-wall flap-wise root bending moment between the RANS-BE model and experimental data plotted in Figure \ref{fig3:comp_rbm_wall}, both data sets have a clear peak for inboard and outboard turbines. The maximum root bending moment was recorded at 260$^\circ$ in the experiment, whereas the RANS-BE predicted this to be at 300$^\circ$. In both cases, this loading coincided with passing an adjacent turbine; passing the symmetry plane for the inboard turbine and passing the inboard turbine for the outboard turbine. A secondary local maximum in the flap-wise root bending moment signal was observed for the inboard turbine as it passed the outboard turbine in both the RANS-BE model and the experimental model. Like the larger peak, a phase offset of 40$^\circ$ was observed between the data sets.


% The edge-wise root bending moment has worse agreement between the RANS-BE and the experimental data than the flap-wise. For the without sidewall case a 13\% higher root bending moment is predicted by the RANS-BE model with higher torque at all azimuthal locations (leading to higher $C_p$). The experimental results show the maximum loading coincides with the blade passing the inboard section of the fence at 300$^\circ$. This point is offset in the RANS-BE model by 90$^\circ$ for the outboard turbine and 60$^\circ$ for the inboard turbine. 

The introduction of an end-wall appears to have a different response in the RANS-BE model and the experiment. In the RANS-BE model, a noticeable positive offset in edge-wise root bending moment and a negative offset in the flap-wise root bending moment was observed (due to lower $\lambda$) alongside a secondary peak in the outboard turbine loading. While there is little evidence of an offset in the experimental data, an additional loading peak is present. Like the loading peaks without the end-wall, this was observed to be out of phase by 40$^\circ$ between the data sets.




\begin{table}[]
\centering
\caption{A comparison of mean root bending moments over a rotation for a fence of 4 turbines operating at $\lambda^*$ and s/D=0.25 with and without a end-wall.}
\label{tab3:comp_rbm}
\begin{tabular}{cc|rrr|rrr|}
                               &            & \multicolumn{3}{c|}{Without End-wall}                                                & \multicolumn{3}{c|}{With End-wall}                                                   \\ \cline{3-8} 
                               &            & \multicolumn{1}{c}{RANS-BE} & \multicolumn{1}{c}{Exper} & \multicolumn{1}{c|}{Error} & \multicolumn{1}{c}{RANS-BE} & \multicolumn{1}{c}{Exper} & \multicolumn{1}{c|}{Error} \\ \hline
                               & $\lambda^*$ & 6.52                        & 6.44                      & 1.2\%                      & 6.37                        & 6.44                      & -1.1\%                     \\ \hline
\multirow{2}{*}{Inner Turbine} & $\overline{C_{FW}}$   & 0.748                       & 0.720                     & 3.9\%                      & 0.737                       & 0.718                     & 2.6\%                      \\
                               & $\overline{C_{EW}}$   & 0.085                       & 0.075                     & 12.6\%                     & 0.089                       & 0.075                     & 18.9\%                     \\ \hline
\multirow{2}{*}{Outer Turbine} & $\overline{C_{FW}}$   & 0.741                       & 0.710                     & 4.4\%                      & 0.738                       & 0.712                     & 3.6\%                      \\
                               & $\overline{C_{EW}}$   & 0.082                       & 0.074                     & 10.8\%                     & 0.089                       & 0.076                     & 17.9\%                    
\end{tabular}
\end{table}








% \begin{figure}
% \centering
% \includegraphics[width=0.9\textwidth]{Figures/Chapter_3/experiment/rbm_wall_exper_C_{FW}_.png}
% \includegraphics[width=0.9\textwidth]{Figures/Chapter_3/experiment/rbm_wall_exper_C_{EW}_.png}

% \caption{Phase averaged flap-wise (top) and edge-wise (bottom) root bending moments with (open circles) and without (dots) a sidewall for s/D=0.25 at $\lambda=6.44$. Results for the inboard and outboard turbine are red and blue respectively. The wait time between tests is between 10 and 20 minutes and duplicate data has been averaged; blade self weight and buoyancy has been controlled for in the edge-wise root bending moment signal. The three illustrations at the top of each plot show the location of the blade (in bold) in the domain, with blue and red representing the outboard and inboard turbines in these illustrations.}
% \label{fig3:exper_rbm_wall}
% \end{figure}






\subsubsection{End-wall loading}

Due to a strain gauge's failure during the end-wall installation, additional processing was required to determine the end-wall loading from the experimental results. The data available was from one of two strain gauges measuring bending moment in the cross-stream direction at the connection between the end-wall and the supporting beam. A linear elastic frame model was developed in SPACE GASS 12.85 to determine the relationship between bending moment at this member and cross-stream loading. A factor to convert the bending moment to a cross-stream force from this strain gauge was determined by applying a unit load in the cross-stream direction at the centre of pressure (determined from the RANS-BE model) and scaling the load to match the measured bending moment. The results of the analysis are presented in Figure \ref{fig3:comp_sidewall_load}.

The end-wall cross-stream loading presented in Figure \ref{fig3:comp_sidewall_load} corresponds to the confinement force constraining the wake expansion and re-aligning the inflow with the free-stream. A linear relationship between rotor $\lambda$ and cross-stream loading was observed with greater $\lambda$ corresponding to higher end-wall loading. Increased loading with $\lambda$ is expected due to the higher thrust increasing bypass flow. The loading on the wall has been non-dimensionalised using the same representative area as the turbines ($A_t$), allowing direct comparison between $C_t$ from the turbines. At $\lambda^*$, the confining force on the fence is close to half the turbine thrust; this force is required to remove lateral momentum from the flow and align it normal to the rotor plane.

Good agreement between the experimental results and the RANS-BE model is observed for the end-wall cross-stream loading. The loading range shown in the experimental data highlights the variability in the cross-stream force, which can be up to 15\% of the turbine thrust. This fluctuation is likely a result of transient effects which include:
\begin{itemize}
    \item blade passing and vortex shedding (Section \ref{sc3:expr_spec}) which is not captured in the RANS-BE model
    \item experimental uncertainty of the turbulence conditions in the tank
    \item fixity of the end-wall with the towing carriage.
\end{itemize}
\noindent Despite the transient effects, close agreement between the average loading from the experiment and the RANS-BE model supports the use of the RANS-BE model for simulating the lateral confinement force of an end-wall. 

\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/compare/Compare_endwall_cross_stream_force.png}
\caption{Comparison of cross-stream loading on the end-wall between experimental (orange) and numerical (blue) studies. The range of the experimental data is the 5th and 95th percentile of the loading on the wall over the experiment, the marker is the median load. Results have been non-dimensionalised using the turbine area as reference.}
\label{fig3:comp_sidewall_load}
\end{figure}





% \begin{figure}
% \centering
% \includegraphics[width=0.9\textwidth]{Figures/Chapter_3/experiment/Endwall_cross_stream_force.png}

% \caption{Cross-stream loading on the sidewall over a range of $\lambda$. The range indicated is the 5th and 95th percentile of the loading on the wall over the experiment, the marker is the median load.}
% \label{fig3:expr_sidewall_load}
% \end{figure}


\subsubsection{Frequency analysis}\label{sc3:expr_spec}

Fast Fourier Transforms (FFTs) of the blade root bending moment, integrated power and thrust coefficients and end-wall cross-stream force are presented in Figure \ref{fig3:fft_wall}. The blade root bending moments all exhibit a peak at the once per-rotation frequency ($f_0$) with distinctive peaks at 2$f_0$, 3$f_0$, 4$f_0$, 5$f_0$ and 6$f_0$ in some cases. As observed by \cite{Payne2018VariationPosition} the once-per-revolution loading spikes can be credited to a tower passing effect where the blade passing the support tower causes a drop in thrust leading to a once-per-rotation frequency spike in blade loads. 

Examining Figure \ref{fig3:comp_rbm_wall}, an increase in flap-wise bending just before the rotors pass each other (90$^\circ$ and 270$^\circ$ for the inboard turbine and 270$^\circ$ for the outboard turbine) is observed. Whilst there are two peaks for the inboard turbine, the flap-wise root bending moment is maximum when the blade is at 250$^\circ$ (just before it passes the tank sidewall); this is the same location for the outboard turbine. When the end-wall is introduced, a second peak in the outboard turbine loading immediately before passing the end-wall is present at 70$^\circ$; this peak is offset 180$^\circ$ from the without end-wall maximum load and is slightly smaller in magnitude. As a result, an increase in the energy at 2$f_0$ and its harmonic (4$f_0$) is observed when the end-wall is installed with energy at $f_0$ remaining relatively unchanged.

For the edge-wise root bending moment without an end-wall, a single loading peak is observed at 270$^\circ$ for the outboard turbine and two peaks at 90$^\circ$ and 270$^\circ$ for the inboard turbine. When the wall is installed, a second peak is introduced around 90$^\circ$ in the outboard turbine loading. This results in an increase in the energy at $f_0$ and the harmonics (up to the 6$f_0$). For the inner turbine, little change is observed when the wall is in place, corresponding to no distinct difference in the flap-wise and edge-wise energy spectrum.


Harmonics exist at 2$f_0$ and 3$f_0$ in the power spectrum for the thrust and power coefficients. The 3$f_0$ peak is likely associated with blade passing effects. The even harmonics may be explained through the tank's aspect ratio (5 m depth and 10 m width) and potential yaw misalignment of the rotors or end-wall, which would result in 2$f_0$ spectral content. 

Finally, three significant peaks were observed in the cross-stream loading acting on the end-wall; the first two spikes corresponded to frequencies less than $f_0$ at 0.5$f_0$ and 0.67$f_0$, these vary slightly between the experiments as the $\lambda$ was changed. There is no clear signal in the root bending moment or the integrated quantities at these frequencies, suggesting it may come from some external low frequency forcing such as vortex shedding from the end-wall. The final peak occurs at 2.2$f_0$; however, its value was observed to vary linearly with $\lambda$ and was present with and without an end-wall suggesting it may be the tow carriage response to the turbine operation. 


\begin{figure*}[]
\centering
\includegraphics[width=0.95\textwidth]{Figures/Chapter_3/experiment/Blade_fft_adjusted_TSR5pnt74.png}
\caption{Typical power spectrum of turbine loading at $\lambda$=5.74. Flap-wise root bending moment is plotted on top with the edge-wise direction plotted below. The third row plots the power spectrum for the turbine power coefficient on the left and thrust coefficient on the right. The bottom level plots the power spectrum for the cross-stream loading on the end-wall. Red is the inboard turbine and blue is the outboard turbines, solid lines represent the case without the end-wall and dotted lines the case with the end-wall.}
\label{fig3:fft_wall}
\end{figure*}



\subsection{Conclusions}

The towing tank testing demonstrated that including an end-wall in a short turbine fence leads to increased power production. While only a marginal performance improvement was observed, it is anticipated that improved end-wall design could result in a further increase in power output and match inboard and outboard turbine performance.

The end-wall was also observed to balance the thrust loading between the rotors, reducing the difference in the loading cycle between turbines. This potentially simplifies the maintenance scheduling and improves the design life as all turbines can be designed for the same flow and loading conditions.



\section{Limitations of modelling methodologies}\label{CH3:model_limits}\label{Ch3:compare}

% This section presents a comparison between the numerical results from Section \ref{Ch3:numerical} and the experimental results from Section \ref{Ch3:experimental}. A direct comparison between these quantities was not possible due to Reynolds independence issues identified over the course of the towing tank testing. Higher tow velocities were required for Reynolds independence in the experiments leading to measurements being taken at 1 m/s; in contrast, the RANS-BE models simulated a 0.8 m/s free-stream flow rate. This is discussed in \textbf{JAMES PAPER}.

Both the experimental method and the RANS-BE model used in this study faced limitations. The BE turbine model (within the RANS-BE model) is derived as an azimuthally averaged representations of rotor behaviour; however, the model's specific implementation violated this assumption by using the local velocities incident on a persistently present rotor disk. This assumption was relaxed to investigate the parameter space as it was impractical to use more sophisticated and computationally intense methods such as blade resolved simulation due to the number of simulations in this study. The result of violating this assumption was a smoothed response to a change in velocity over the rotor plane, and an overestimate in turbine torque and thrust. Additionally, the blade element method relies on radial independence of flow over the rotor, permitting 2D aerofoil data to approximate lift and drag on each radial element. The limitations of these assumptions have been widely reported \citep{Vermeer2003,Snel2003,Malki2013} and are essential for the computationally efficient simulation of turbine rotors. However, decreasing the inter-turbine spacing causes greater flow diversion (especially at the end turbine), the flow diversion has a lateral component, which manifests in a radial component for some parts of the disc, suggesting that the use of the model may not be appropriate for highly blocked or anisotropic blockage conditions which induce flow components in the turbine plane. 

Furthermore, the RANS-BE method is a steady approximation for a transient process. As a result, the blade element method imposes resistance over the entire swept area. Consequently, localised flow effects are smoothed over the rotor swept area as the model cannot simulate the pressure relief on blades as flow accelerates between them. The effect of this is highlighted in Figure \ref{fig3:comp_rbm_wall_same_TSR}, which shows the change in flap-wise root bending moment is localised to the end-wall in the experimental data. The RANS-BE model has an offset over the entire azimuth in addition to the localised change. This is examined further in Figure \ref{fig3:RANS-BE_parms}, which plots the difference in pressure drop of the outboard turbine due to the end-wall. Whilst there is a noticeable increase in pressure drop for the outboard section of the rotor as it passes the end-wall, there is a significant increase over the remainder of the swept area; this is the smoothed response of the RAN-BE model, causing the offset in the root bending moment. Despite the RANS-BE model's shortcomings in evaluating the fence's performance, it accurately predicts the loading on the end-wall and approximates the turbine thrust to within 5\%. 

% Correction factor to account for the local blockage. ?

% Phasing between the RANS-BE model and experimental results was also observed to be off with the RANS-BE results shifted ~30$^\circ$ ahead. The cause of this may have been due to the use of the symmetry condition for the water surface, causing artificially higher bypass flows in the upper portion of the flow domain appearing like a phase offset in the results. This is a common methodology and has been adopted in previous numerical studies 


\begin{figure}
    \centering
    % \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/47__Outer_Turbine_delta_p_out_turbines.png}
    % \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/168__Outer_Turbine_delta_p_out_turbines.png}
    \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/Change_in_168__Outer_Turbine_delta_p_out_turbines.png}

    \caption{Change in pressure drop over the outboard turbine swept area from the RANS-BE model after the installation of a end-wall. The difference is calculated as the with end-wall case minus the without end-wall case for $\lambda$=6.44.}
    \label{fig3:RANS-BE_parms}
\end{figure}


\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{Figures/Chapter_3/compare/Profiles_047_294__velocity_magnitude.png}
\caption{Profiles of velocity magnitude at the mid-depth of the inboard turbine adjacent to the symmetry plane (orange) and wall (blue) boundary condition. The boundary is located at 5 m in the cross-stream direction and the dotted grey lines represent the free-stream flow condition (0.8m/s) at that location.}
\label{fig3:BC_Compare}
\end{figure}


In the experiment, a boundary layer on the tank wall may have caused an increase in loading on the inboard turbine. The symmetry assumption for the tank wall in the experiment is only valid when the fluid and boundary move at the same speed. This is not the case; while the wall and the water are still during the experimentation, the force exerted by the turbine on the flow causes relative motion between the fluid and wall, inducing a boundary layer that changes the velocity distribution across the rotor. The impact of this assumption was investigated by using a smooth moving wall boundary condition in place of the symmetry plane in the RANS-BE model. Additional mesh refinement was required between the side of the domain and rotors to achieve a $y^+ <$100 over the wall. Figure \ref{fig3:BC_Compare} plots the evolution of the boundary layer at sections along the tank. A slight decrease in velocity at the inboard section of the turbine is observed when the moving wall boundary is used; however, the greatest impact on velocity appears to be in the wake. Figure \ref{fig3:BC_Compare_velocity} compares the first three powers of the velocity normal to the turbine (the first power corresponds to flow rate, the second to kinetic energy flux and the third to power) and presents the area-weighted integral over the rotor plane. This boundary condition primarily affects the inboard section of the rotor resulting in a 1.8\% drop in flow rate, 2.9\% drop in kinetic energy flux and a 3.5\% drop in power flux. The analysis goes part of the way to explaining why the power of the inboard turbine did not increase as predicted by the RANS-BE model when the end-wall was installed. However, it does not explain why the power was reduced when the end-wall was installed. The reason for this remains uncertain, but it may be an effect of the change in thrust distribution (reduced inboard turbine thrust and increased outboard turbine thrust) skewing the inflow angle; evidence for this is seen in the flap-wise root bending moment as the inboard turbines blades pass the outboard turbine in Figure \ref{fig3:comp_rbm_wall}. 


\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/compare/Compare_Inner_BCs.png}
\caption{Powers of velocity normal to the turbine for the inboard turbine with a symmetry plane boundary condition (left) and a moving wall boundary condition (middle) and the difference between them (right). The location of the boundary is to the right of the turbine. The difference is calculated as the results using the moving wall boundary condition minus the symmetry plane boundary condition. The area weighted average is presented at the top of each plot.}
\label{fig3:BC_Compare_velocity}
\end{figure}



% SDE HERE %%% 14/04/21
\section{Conclusions}\label{Ch3:conclusions}

An investigation into turbine fence performance at varied inter-turbine spacing has been undertaken for a fence of four turbines using a RANS-BE model. It was shown that as the inter-turbine spacing decreased (local blockage increased), greater power output is achieved by allowing the turbines to operate at a slightly higher $\lambda^*$. This improvement in fence performance also saw an increase in the end losses from the outboard turbine due to a growing disparity between inner and outer turbine performance at close spacing. It was hypothesised that the fence end loss could be reduced by introducing a fence end-wall, which could correct the flow skewed around the ends, mimicking the effect of additional turbines in a longer fence. 

A parametric study of a cambered elliptical cross-sectioned end-wall of length ($D$), thickness (5\%, 10\% and 15\% length) and camber (-4\%, -2\%, 0\% 2\% and 4\% length) was undertaken for an inter-turbine spacing of $0.25D$ using a RANS-BE solver. The results of this study showed that an end-wall could improve the power generating capability of all turbines in a fence with an improvement in average fence $C_p$ of 3.95\%. This was achieved by improving the alignment of the approach flow, which allowed turbine blades to operate closer to the optimal angle of attack. The improvement in turbine performance coincided with a small reduction in thrust owing, in part, to the reduction in $\lambda^*$. As a result, the impact on overall basin efficiency (which included the parasitic power removed from the flow by the end-wall) was a minor loss of between 0.5\% and 3\% relative to the no end-wall case. 

Comparisons to towing tank experiments demonstrated an end-walls capability to adjust the thrust and power distribution along a fence. Experimental results showed that the end-wall increased the outboard turbine's power by 1.5\%, significantly less than the increase predicted by the RANS-BE model. This result was determined to be a factor of the model over predicting the flow's response from the end-wall, limitations of the RANS-BE model, and the formation of a small boundary layer in the towing tank experiment. The study also highlighted some of the difficulty in accurately modelling closely spaced tidal turbines, especially in anisotropic blockage conditions. Despite the RANS-BE model's poor performance in predicting power extraction, the average root bending moment in the flap-wise direction was conservatively estimated within 5\% of the experimentally determined value. The model was also able to accurately predict the mean loading on the end-wall in the cross-stream direction. 

Installing an end-wall presents an exciting opportunity for tidal device manufacturers to obtain a performance improvement to floating tidal fences by using the support structure to provide buoyancy and align the flow to reduce end effects. The introduction of the end-wall also reduces the disparity in loading between the inboard and outboard turbines for both the averaged and cyclic loading components, balancing the turbine fatigue across the rotors in the fence and simplifying the maintenance schedule.




% \subsection{Comparison of performance metrics}

% A comparison between performance metrics from the RANS-BE model and experimental study is presented in Table \ref{tab3:comp_perf}. As can be observed from this table there is a significant difference between the $C_p$ (up to 21.3\%) and $C_t$ (up to 6.1\%) between the models. The cause for this disparity is investigated in Section \ref{CH3:model_limits} and can be primarily credited to the limitations of the RANS-BE model. For comparison purposes scaled values of $C_p$ and $C_t$ are presented in this section. 

% Figure \ref{fig3:comp_wall_perf} compares performance parameters ($C_p$ and $C_t$) and symmetry factors between the RANS-BE model and experimental results of the turbine fence when a sidewall is in place. Examining the outboard turbine it was observed that both numerical and experimental data capture a performance improvement with the installation of a sidewall. However, the performance increase measured from the experimental model was significantly lower (approximately one third) than predicted from the numerical model. This can be explained in part by the lower thrust at outboard turbine for the experiment.

% \begin{table}[]
% \centering
% \caption{A comparison of turbine performance metrics $C_p$ and $C_t$ at $\lambda^*$ for a 4 turbine fence at a spacing of s/D=0.25 from a RANS-BE model and an experimental study.}
% \label{tab3:comp_perf}
% \begin{tabular}{cc|rrr|rrr|}
%                               &            & \multicolumn{3}{c|}{Without sidewall}                                                & \multicolumn{3}{c|}{With sidewall}                                                   \\ \cline{3-8} 
%                               & \textbf{}  & \multicolumn{1}{c}{RANS-BE} & \multicolumn{1}{c}{Exper} & \multicolumn{1}{c|}{Error} & \multicolumn{1}{c}{RANS-BE} & \multicolumn{1}{c}{Exper} & \multicolumn{1}{c|}{Error} \\ \hline
%                               & $\lambda*$ & 6.52                        & 6.44                      & 1.2\%                      & 6.37                        & 6.44                      & -1.1\%                     \\ \hline
% \multirow{2}{*}{Inner Turbine} & $C_p$      & 0.569                       & 0.484                     & 17.6\%                     & 0.582                       & 0.480                     & 21.3\%                     \\
%                               & $C_t$      & 1.119                       & 1.058                     & 5.8\%                      & 1.107                       & 1.053                     & 5.1\%                      \\ \hline
% \multirow{2}{*}{Outer Turbine} & $C_p$      & 0.553                       & 0.479                     & 15.6\%                     & 0.584                       & 0.486                     & 20.2\%                     \\
%                               & $C_t$      & 1.108                       & 1.044                     & 6.1\%                      & 1.108                       & 1.048                     & 5.7\%                     
% \end{tabular}
% \end{table}

% Contrasting results were observed in the performance of the inboard turbine with the experiment indicating a slight decrease in $C_p$ and the numerical model suggesting that $C_p$ would increase when a sidewall was installed. The experimental result was not expected given the additional flow confinement from the wall and increased thrust from the adjacent turbine was expected to improve performance. Instead a very small decrease in $C_t$ was observed between the experimental results with and without a sidewall for the inboard turbine. Examining the thrust on the outboard turbine suggests that the RANS-BE model over-predicted the change in thrust of the outboard turbine which resulted in greater confinement (and therefore power generating capacity) for the inboard turbine leading to an over prediction of inboard turbine power. The drop in thrust in the experimental data is a result of a small change in thrust just after the turbine blade passes the inner turbine, this is discussed in further detail in Section \ref{SC3:loading_compar}. A factor to be considered in the analysis of the experimental results for the inboard turbine is the formation of a boundary layer from the side of the tank which slightly reduces the inflow speed and alters the inflow angle, this is discussed further in Section \ref{CH3:model_limits}. 


% \begin{figure}
% \centering
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Inboard_Turbine_C_pC_p_star.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Inboard_Turbine_C_tC_t_star.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Outboard_Turbine_C_pC_p_star.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Outboard_Turbine_C_tC_t_star.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Array_Averaged_C_pC_p_star.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Array_Averaged_C_tC_t_star.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Symmetry_Factor_C_p.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Symmetry_Factor_C_t.png}

% \caption{$C_p$ (left) and $C_t$ (right) performance  curves  comparing between  experimental  testing (orange) and the RANS-BE model (blue) for  a  tidal turbine with (dashed line) and without (solid line) a sidewall. The results of the inboard turbine are presented on the top row, the outboard turbine in the top middle row and the fence averaged in the bottom middle row. The symmetry factors for $C_p$ and $C_t$ are plotted on the bottom row. $C_p$ curves have been scaled by the without sidewall fence averaged maximum power ($C_p^*$) and the $C_t$ curves have been scaled by the fence averaged thrust at $C_p^*$ ($C_t^*$). For the experimental results the wait time between tests is between 10 and 20 minutes and duplicate data has been averaged.}
% \label{fig3:comp_wall_perf}
% \end{figure}

% Over the fence, the inclusion of a sidewall in the experiment leads to a modest increase in $C_p$ of 0.5\% with the performance improvement of the outboard turbine compensating for the decline in performance of the inboard turbine. In contrast the RANS-BE model saw an improvement in performance along the entire fence leading to an increase in fence averaged $C_p$ of 3.95\%. This result suggests that the RANS-BE model may not be able to capture the exact change in integrated parameters $C_p$ and $C_t$ when the sidewall is installed. However it is a useful tool for investigating relative change in these parameters when a sidewall is installed. As can be observed in the symmetry factor plots in Figure \ref{fig3:comp_wall_perf} both the experimental data and RANS-BE model capture a downwards shift in bias from the inboard turbine to the outboard turbine when the sidewall is installed. 

% Good agreement between the experimental results and the RANS-BE model are observed for the cross-stream loading on the sidewall (Figure \ref{fig3:comp_sidewall_load}). The loading on the wall has been non-dimensionalised using the same representative area as the turbines ($A_t$) allowing direct comparison between $C_t$ from the turbines. At $\lambda^*$ the confining force on the fence is close to half the thrust of a turbine. This significant force is required take away lateral momentum from the flow and align it with the rotor plane. The loading range shown in the experimental data highlights the variability in the cross-stream force which can be up to 15\% of the thrust on the turbines. This fluctuation is likely a result of transient effects such as blade passing and vortex shedding (Figure \ref{fig3:fft_wall}) which are not captured in the RANS-BE model as well as experimental uncertainty in the turbulence conditions in the tank and fixity of the sidewall with the towing carriage. The close agreement of the average loading from the experiment and the RANS-BE model does however support the use of RANS-BE models in simulating the lateral confinement force of a sidewall. 

% % This plot shows that the RANS-BE model predicts optimal power for the fence to be achieved when the symmetry factor for $C_p$ approaches zero at the arrays optimal TSR, this is to say the power production is evenly distributed along the array at the operating point. It was anticipated that this balance could be achieved through augmenting the flow with a sidewall, applying lateral confinement to the flow to improve the power output of the outboard turbine. Whilst the model correctly predicted the direction of the shift in both thrust and power along the fence due to a sidewall, it was not able to accurately predict the magnitude due to the wall having a larger than expected change in thrust (especially at high TSR).
 

% \begin{figure}
% \centering
% \includegraphics[width=0.9\textwidth]{Figures/Chapter_3/compare/Compare_endwall_cross_stream_force.png}
% \caption{Comparison of cross-stream loading on the sidewall between experimental (orange) and numerical (blue) studies. The range of the experimental data is the 5th and 95th percentile of the loading on the wall over the experiment, the marker is the median load. Results have been non-dimensionalised using the turbine area as reference.}
% \label{fig3:comp_sidewall_load}
% \end{figure}



% \subsection{Root bending moment analysis}\label{SC3:loading_compar}

% Figure \ref{fig3:comp_rbm_wall} plots the phase averaged root bending moment from the towing tank experiments and the RANS-BE results in the flap-wise and edge-wise directions with and without a sidewall at $\lambda^*$. Overall the RANS-BE model appears to overestimate the loading on the turbines. This is quantified in table \ref{tab3:comp_rbm} which shows the error in the average root bending moments between the RANS-BE model and the experimental data. The error from the RANS-BE model was between 3.9\% and 4.4\%  in the flap-wise direction and 12.2\% and 14.1\% in the edge-wise direction. A factor which contributed to this poor agreement is the change in $\lambda^*$ in the RANS-BE model when the sidewall was introduced. The RANS-BE model predicted a reduction in $\lambda^*$ which in turn reduced the thrust on the fence and altered the power production, compounding the model error over these variable. A reduction in $\lambda^*$ was not observed in the experiment, perhaps due to the jagged shape of the performance curve around $C_p^*$ making it tough to correctly identify $\lambda^*$. Figure \ref{fig3:comp_rbm_wall_same_TSR} plots the azimuthal variation in root bending moment with and without a sidewall for rotors operating at a constant $\lambda$=6.44. As can be observed in the plot a closer agreement with the experimental data is achieved when simulating the rotors at the same $\lambda$ however there is still a significant difference between the root bending moments. The cause of this error is discussed further in Section \ref{CH3:model_limits}. Despite offset to the mean root bending moment, the RANS-BE model captures the approximate amplitude of the variations in root bending moment over a rotation, however the phasing (especially for the outboard turbine) is not accurately modelled.


% \begin{table}[]
% \centering
% \caption{A comparison of mean root bending moments over a rotation for a fence of 4 turbines operating at $\lambda^*$ and s/D=0.25 with and without a sidewall.}
% \label{tab3:comp_rbm}
% \begin{tabular}{cc|rrr|rrr|}
%                               &            & \multicolumn{3}{c|}{Without End-wall}                                                & \multicolumn{3}{c|}{With Sidewall}                                                   \\ \cline{3-8} 
%                               &            & \multicolumn{1}{c}{RANS-BE} & \multicolumn{1}{c}{Exper} & \multicolumn{1}{c|}{Error} & \multicolumn{1}{c}{RANS-BE} & \multicolumn{1}{c}{Exper} & \multicolumn{1}{c|}{Error} \\ \hline
%                               & $\lambda^*$ & 6.52                        & 6.44                      & 1.2\%                      & 6.37                        & 6.44                      & -1.1\%                     \\ \hline
% \multirow{2}{*}{Inner Turbine} & $\overline{C_{FW}}$   & 0.748                       & 0.720                     & 3.9\%                      & 0.737                       & 0.718                     & 2.6\%                      \\
%                               & $\overline{C_{EW}}$   & 0.552                       & 0.483                     & 14.1\%                     & 0.566                       & 0.481                     & 17.7\%                     \\ \hline
% \multirow{2}{*}{Outer Turbine} & $\overline{C_{FW}}$   & 0.741                       & 0.710                     & 4.4\%                      & 0.738                       & 0.712                     & 3.6\%                      \\
%                               & $\overline{C_{EW}}$   & 0.537                       & 0.479                     & 12.2\%                     & 0.567                       & 0.486                     & 16.6\%                    
% \end{tabular}
% \end{table}

% Comparing the without sidewall flap-wise root bending moment between the RANS-BE model and experimental data plotted in Figure \ref{fig3:comp_rbm_wall}, shows that both data sets have a clear peak for inboard and outboard turbines. The maximum root bending moment was recorded at 260$^\circ$ in the experiment whereas the RANS-BE predicted this to be at 300$^\circ$. In both cases this loading coincided with passing an adjacent turbine; passing the symmetry plane for the inboard turbine and passing the inboard turbine for the outboard turbine. A secondary local maximum in the flap-wise root bending moment signal was observed for the inboard turbine as it passed the outboard turbine in both the RANS-BE model and the experimental model. Like the larger peak, a phase offset of 40$^\circ$ between the data sets was observed.


% \begin{figure}
% \centering
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Outboard_C_{FW}_.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Inboard_C_{FW}_.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Outboard_C_{EW}_.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Inboard_C_{EW}_.png}

% \caption{A comparison between experimental (orange) and numerical (blue) phase averaged flap-wise (top) and edge-wise (bottom) root bending moments for outboard (left) and inboard (right) turbines. Open circles and dots are used to distinguish between the with sidewall and without sidewall cases in the experimental results whereas dashed lines (with sidewall) and solid lines (without sidewall) are used in the plots for the numerical results. All root bending moments presented are at $\lambda^*$ for the fence. Experimental results for the edge-wise root bending moment have controlled for blade self weight and buoyancy. The three illustrations at the top of each plot show the location of the blade (in bold) in the domain, with blue and red representing the outboard and inboard turbines in these illustrations.}
% \label{fig3:comp_rbm_wall}
% \end{figure}

% \begin{figure}
% \centering
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Outboard_C_{FW}_same_tsr.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Inboard_C_{FW}_same_tsr.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Outboard_C_{EW}_same_tsr.png}
% \includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Inboard_C_{EW}_same_tsr.png}

% \caption{A comparison between experimental (orange) and numerical (blue) phase averaged flap-wise (top) and edge-wise (bottom) root bending moments for outboard (left) and inboard (right) turbines. Open circles and dots are used to distinguish between the with sidewall and without sidewall cases in the experimental results whereas dashed lines (with sidewall) and solid lines (without sidewall) are used in the plots for the numerical results. All root bending moments presented are at $\lambda$=6.44 for the fence. Experimental results for the edge-wise root bending moment have controlled for blade self weight and buoyancy. The three illustrations at the top of each plot show the location of the blade (in bold) in the domain, with blue and red representing the outboard and inboard turbines in these illustrations.}
% \label{fig3:comp_rbm_wall_same_TSR}
% \end{figure}


% Comparing the without sidewall edge-wise bending moment, a noticeably worse agreement is observed between the RANS-BE and the experimental data. The 13\% higher root bending moment in the RANS-BE model demonstrates that the reason for higher $C_p$ than the experiments is due to higher torque at all azimuthal locations on the RANS-BE model. The experimental results show the maximum loading coinciding with the blade passing the inboard section of the fence at 270$^\circ$. This point is offset in the RANS-BE model by 90$^\circ$ for the outboard turbine and 60$^\circ$ for the inboard turbine. 

% The introduction of a sidewall appears to have a very different response between the RANS-BE model and the experimental data. In the RANS-BE model a noticeable positive offset at all azimuthal locations for edge-wise root bending moment and a negative offset in the flap-wise root bending moment was observed due to the decrease in $\lambda$. In addition to this offset an additional loading peak was observed in the outboard rotor. In contrast the experimental results do not significantly change for the inboard turbine root bending moment in either direction. However, like the RANS-BE model, the root bending moment distribution is altered as the blade passes the sidewall for the outboard turbine introducing a secondary peak similar to the loading on the inboard turbine. This secondary peak is observed in both the flap-wise and edge-wise directions for both the RANS-BE model and experimental model. Like the maxima observed in the without sidewall case, the RANS-BE model predicts this peak to occur approximately 40$^\circ$ out of phase in the flap-wise direction.

% % pressure drop is not azimuthally averaged - fix this text - thrust incorrect 
% The difference in the flap-wise and edge-wise root bending between the RANS-BE and the experimental data highlights some of the limitations of the RANS-BE model. Firstly, the peaks of the flap-wise and edge-wise loading are in phase for the the RANS-BE results and out of phase for the experimental results. The reason for the discrepancy can be explained by examining the variation in velocity and angle of attack of the rotor. The peaks in the flap-wise moment correspond to regions of higher velocity as was observed in the RANS-BE model, higher inflow velocity results in an increased angle of attack. If the turbine is operating at close to optimal conditions where the lift-to-drag ratio is close to optimal averaged around the rotor, the higher angle of attack will result in a decrease in the local lift-to-drag ratio on the blade. For the areofoil used in the experiment a higher angle of attack increases both $C_l$ and $C_d$, increasing $C_d$ by a larger amount due to the lift-to-drag ratio decreasing. This causes an increase in local flap-wise thrust coefficient ($C_x$) and a decrease in local edge-wise thrust coefficient ($C_e$) following

% \begin{equation}
%     C_x= C_l \cos(\phi) + C_d \sin(\phi), \quad C_e =C_l \cos(\phi) - C_d \sin(\phi).
% \end{equation}

% As a result of this the experimental model has variations in flap-wise and edge-wise loading which is out of phase. In the RANS-BE model the variation in the angle of attack is reduced due to annular averaging of flow properties, this means that changes in local thrust and lift is directly related to the local velocity, hence the flap-wise and edge-wise moments are in phase in the numerical model.

% The above explains why the RANS-BE model has aligned peaks in the edge-wise and flap-wise bending moments but does not explain why the locations of the peaks in the flap-wise direction do not match between the model and the experiment. As mentioned above the flap-wise bending moment is closely associated with the inflow velocity for the rotor, this suggests that the RANS-BE model does not capture the correct azimuthal variation in inflow velocity. There may be various reasons for this however the most probable may be the RANS-BE model simulating a permanently present disc over the turbine swept area. The resistance imposed by this disc is azimuthally averaged which means that for the outboard turbine the pressure drop across the rotor is equal at all azimuthal locations facilitating higher flow through the turbine at the outboard region. The RANS-BE model may therefore not be suitable in environments where there is non-uniform azimuthal confinement of turbine such as at the ends of arrays or in closely spaced turbine arrays where the local flow passage has a high aspect ratio.



% generally speaking edge-wise loading isnt as critical as flap-wise



