
\chapter{End-wall constrained turbine fences}\label{Chapter3}

This chapter presents a numerical analysis of a four turbine long fence, examining the effect of reduced inter-turbine spacing on performance variation along the fence. It presents a potential solution to reduce this variation through a constraint to turbine fence end-losses by using a fence end-wall to reduce bypass flow and improve efficiency. 

Section \ref{CH3:Method1} explains the numerical model used to analyse the fence and design the end-wall, Section \ref{sc3:unducted_perf} examines the effect that altering the inter-turbine spacing has on a turbine fence of four turbines and Section \ref{sc3:sidewall_deisgn} presents an end-wall design methodology and characterises fence performance with an end-wall. A summary of the results is outlined in Section \ref{Ch3:conclusions}. The work presented in this chapter is closely linked with the experimental results presented in Chapter \ref{Chapter4}.


\section{Introduction}

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}, deviation from the theoretical framework arises at the end of the turbine fences due to an around-fence flow diversion, decreasing outboard turbine performance. This effect is not captured in \cite{Nishino2012} due to the assumption of a \enquote{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 two-scale approach to model shorter fences. However, this work did not include consideration for fence end-effects. 

While long fences may be desirable theoretically, shorter fences are more likely to be designed as technology develops. Within shorter fences, the losses resulting from flow diversion around the outer turbines and insufficient flow confinement 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 this \enquote{end-effect} by investigating flow diversion at varied inter-turbine spacing for a short fence of four turbines. Following this, a turbine fence end-wall is designed and characterised numerically as a possible mitigation strategy for end-losses. The end-wall is investigated further experimentally in Chapter \ref{Chapter4}.

Floating turbine structures require sufficient buoyancy to support the turbines, nacelles and connecting structure, this may require a buoyancy aid with relatively large displacement. There is an opportunity that this support structure in floating tidal fences can be designed to re-align flow normal to the turbine and potentially improve 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. 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 circumference improved the power density through the rotors; however, this improvement was not as effective as increasing rotor 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 buoyancy aids that are necessary 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 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 at floWave by \cite{McNaughton2019}. Towing tank testing of the end-wall at the SSPA facility is discussed in Chapter \ref{Chapter4}.

% 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. 




% 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.


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

A four turbine fence was analysed at varied inter-turbine spacings 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.

The turbine modelled in this study was that designed by \cite{Cao2018}, which uses a FX-84-W-140 aerofoil profile with a thickened trailing edge (defined as per \cite{Xu2014}) across the entire blade span. The aerofoil two-dimensional lift and drag coefficients were supplied by Cao from CFD simulation and used in the present study to model the turbines through a RANS embedded BE actuator disk (AD) method. The aerofoil data compared favourably to lift and drag measurements at $Re=1.6 \times 10^6$ from \cite{Bertagnolio2001} in \cite{Cao2018} and achieved a peak lift to drag ratio at $6^\circ$. The three-bladed rotor of \cite{Cao2018} achieves a $C_p=0.580$ at a design $\lambda=7$ for a blockage of 11.8\%.

\subsection{Numerical model}\label{Ch3:numerical}
The incompressible RANS equations were solved numerically using the computational fluid dynamics (CFD) software ANSYS Fluent 15.0 \citep{ANSYSInc.2013}. The $k-\omega$ shear stress transport (SST) turbulence model of \cite{Menter1994}, with updated coefficients from \cite{Menter2003}, was implemented. This modelling approach blends the $k-\omega$ model, which performs well in the near-wall region, with the $k-\epsilon$ model 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 flows through tidal rotors \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 steady analysis method utilises the zero thickness fan internal boundary condition available in Fluent, with the pressure drop and downstream swirl velocity determined by solving the blade element equations. The subroutine samples 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 local pressure drop and swirl velocity. The \cite{Glauert} tip correction model was applied to account for the required blade tip flow losses and reduce the loading on the outboard sections of the blades.

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 (Chapter \ref{Chapter4}); 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. The turbine centres are located 1 m below the simulated free surface which is represented by a symmetry plane. 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 conditions were applied to the upper, lower and side boundaries. By placing the turbines towards a side boundary using a symmetry condition we enable a simulation of a fence of four turbines placed centrally within a 20 m (16.67$D$) wide channel, thus halving the computational cost of the full simulation (the same approach was used for the physical experiments in Chapter \ref{Chapter4}). The inter-turbine spacing $s$ was one of the design variables in the study and so varied; the turbine-symmetry plane spacing was set to $s/2$ so that the four turbine fence was uniformly spaced. The global blockage of the fence was 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, $D=1.2$ m is the turbine diameter and $s$ is the inter-turbine spacing. Inner (red) and outer (blue) turbines are represented by disks, with the streamlined fence end wall shown beyond the outer blue disk.}
    \label{fig3:Domain}
\end{figure}

\subsection{Mesh convergence}

% - Mesh Convergence study
The disk meshing strategy followed that of \cite{Cao2019} which was developed for the same turbine model. The disk mesh used an identically parameterised structured mesh constructed using a total of 15 million hexahedral elements (27 million elements when the end-wall was included) with the rotor disk discretised using 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, reported in Table \ref{Tab_MCS}, was investigated in the zone between the turbines, with the element size growth rate fixed at 1.05 and the maximum element size varied. Table \ref{Tab_MCS} reports the integrated power and thrust coefficients for three inter-turbine mesh resolutions. Results illustrate close agreement between all three resolutions with the finest element sizes of 0.0083$D$ and 0.0125$D$ exhibiting less than 0.163\% relative difference in simulated fence-averaged $C_p$. A smaller relative error of less than 0.051\% was observed for the fence-averaged turbine thrust coefficient, $C_t$. Due to the close agreement in performance coefficients, 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}[]

\centering
\caption{Fence-averaged $C_p$ and $C_t$ for different inter-turbine meshes (two turbines, $D=1.2$ m, $s/D=0.25$). *Highest resolution model was used as the baseline for error calculations.}

\begin{threeparttable}
\label{Tab_MCS}


\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 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 is capable of implementing a log-law wall function to simulate the boundary layer where there is insufficient detail to resolve the viscous sub-layer, requiring $y^+$ as an input. Further discussion of this model can be found in Section \ref{Ch2:turbulence} and in \cite{Menter1994}.

Convergence was determined based on the wall's two-dimensional (pressure based) 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 end-wall, $u_{\infty}$ is the free-stream velocity and $L_{c}$ is its 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 log layer; \cite{ANSYSInc.2013} recommends that $y^+$ be no less than 30 for this reason. Accordingly, a mesh with $L_{el}/L_c$ of 0.056 that achieves a maximum $y^+$ of 100 at 50\% chord was adopted for the end-wall mesh in the present study.


\begin{figure}[ht]
\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{End-wall model mesh convergence of $C_l$ as defined by Equation \ref{CL_int} (a) with non-dimensional wall normal distance $y^+$ at 50\% chord, fixed $L_{el}/L_c=0.056$, and (b) with minimum element length to chord length ratio $L_{el}/L_c$, and fixed first wall normal dimension (achieves maximum $y^+=100$ at 50\% chord).}
\label{Fig3:Mesh_converges}
\end{figure}


\subsection{Root bending moment extraction}\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} (Equation \ref{EqPressureDrop} and \ref{EqSwirlVel}) 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_{\theta} = 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 per unit span 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_{\theta} 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. 

\subsection{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 average $C_p$ ($\lambda^*$) 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 \citep{ARC2022ARCUS}.  


\section{Effect of inter-turbine spacing on tidal fence performance}\label{sc3:unducted_perf}


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


\subsection{Turbine performance}

$\lambda^*$ and the corresponding turbine $C_p$ and $C_t$  were found for six inter-turbine spacings, $s$, ranging between 0.25$D$ to 2.333$D$. The tip-speed-ratio was specified to be the same for all turbines within the fence. Fence-averaged results from these simulations are presented in Figure \ref{Fig3:Unducted_perf}. As the fence spacing is decreased from $s=2.333D$ ($B_l = 5.7\%$) to $s=0.25D$ ($B_l = 15.1\%$), there is an increase in fence-averaged $C_p$ of 5.6\%. To achieve this improvement $C_t$ and $\lambda^{*}$ increased by 6.0\% and 4.5\%, respectively. This concurs with \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 gradient (thrust) that can be sustained across the rotor plane as turbine spacing is reduced and downstream flow expansion is constrained by neighbouring wakes and the free surface.



% $\lambda^*$ 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}[ht]
\centering
\includegraphics[width=0.7\textwidth]{Figures/Chapter_3/numerical/b_Unducted_Optimal_PowerCoefficent_TSR_curve.png}
\includegraphics[width=0.7\textwidth]{Figures/Chapter_3/numerical/b_Unducted_Optimal_ThrustCoefficent_TSR_curve.png}
\caption{Variation of (a) average fence power coefficient, $C_p$, and (b) average fence thrust coefficient, $C_t$, with tip-speed-ratio, $\lambda$, as a function of inter-turbine spacing, $s/D$. The optimal operating point, $\lambda^*$, is marked on each curve with a black cross and the global blockage is constant at 4.5\% for all cases.}
\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 are presented in Table \ref{Table:Inter_turbine_spacing}. This table highlights that as spacing is reduced, $C_p$ and $C_t$ increase, so too does the disparity between inner and outer turbines (Figure \ref{fig3:unducted}). The inner turbines can achieve a slightly higher $C_p$ (and corresponding $C_t$) than the outer turbines for all $s/D$ due to the greater flow confinement  effects experienced towards the centre of the fence; see Figure \ref{fig3:unducted}a and \ref{fig3:unducted}b. Wake expansion downstream of the inner turbines is laterally constrained by the presence of the wakes of the adjacent turbines, allowing larger stream-wise pressure gradients to be maintained through the wake, which in turn enables a greater pressure drop across each turbine improving their performance. Such lateral flow constraint is only present for the outer turbines as they pass the inner turbines, and so the reduced flow constraint for the outboard turbines presents as a fence end-loss. The resulting asymmetry can be quantified by a symmetry factor between the rotors, which we defined as:

\begin{equation}
    \label{Symm_fac}
    \begin{aligned} 
    \Phi_X = \frac{X_{inner}-X_{outer}}{X_{inner}}
    \end{aligned}
\end{equation}

\noindent where $\Phi$ is the symmetry factor for the parameter $X$, with positive values indicating 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 trends in the symmetry factor for both $C_p$ and $C_t$ are observed, with $\Phi C_p$ being more than twice that of $\Phi C_t$. The smallest asymmetry between the turbines' performance was observed when they were most separated at $s=2.33D$, although there still remained a 1.2\% bias towards the inner turbine's performance due to the slight but differing local blockage of the inner and outer turbines even at this wide spacing. 


\begin{figure}[ht]
\centering
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_TSR_Spacing.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_Symmetry_Factor.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_PowerCoefficent_Spacing.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_ThrustCoefficent_Spacing.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_alpha_Spacing.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/numerical/Unducted_Optimal_FlowRate_Spacing.png}

\caption{(a) $\lambda^*$, (b) $\Phi$, (c) $C_p$, (d) $C_t$, (e) $\alpha$, span-wise averaged between 0.3R-0.85R, and (f) turbine flow rate, $Q$, variation with inter-turbine spacing, $s/D$, at $\lambda^*$ for maximum power at each spacing. $Q_{ref}$ is the flow rate through each turbine with no flow resistance.}
\label{fig3:unducted}
\end{figure}

A maximum was observed for $\Phi$ at $s=0.75D$, with the inner turbine producing 2.9\% more power and 1.1\% more thrust. As turbine spacing decreases from $s=0.75D$, the difference in power and thrust along the fence is slightly reduced. Examining the mass flow rate, $Q$, in Figure \ref{fig3:unducted}f provides some evidence of the interference effects experienced along the fence. Moving the turbines closer together results in a small increase in mass flux for the inner turbine due to the increased local blockage. The picture is more complex for the outer turbine with close spacing permitting increased mass flux, and competition between end-losses and constructive interference constraints resulting in a minimum flow rate at $s=0.75D$. Examining the flow yaw angle in Figure \ref{fig3:yaw_plots} supports these results, with reducing yaw angles on the inner turbine as they are positioned closer together. Interplay between the constructive interaction of turbines and fence end-losses are responsible for the varied response of flux at decreased spacing for the outer turbine (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 $s=2.33D$ to $s=0.75D$, 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 $s<0.75D$.


The changes in flow-rate with inter-turbine spacing are small, remembering that $\lambda$ is re-tuned for each spacing. The increases in turbine thrust (and thus power), Figure \ref{fig3:unducted}d and \ref{fig3:unducted}c, are more substantial. The effect of turbine interference is to constrain the flow diversion around each turbine which helps to maintain a high flow rate. This enables a greater pressure drop to be achieved across each turbine than would be possible in unconstrained flow. As the turbines are brought closer together, at a given operating point $\lambda$, their flow rate at first increases, which then allows an increase in the resistive force to be applied whilst still increasing power. The increase in $\lambda$ to $\lambda^*$ (Figure \ref{Fig3:Unducted_perf}) for each $s/D$ to achieve this greater resistive force (Figure \ref{fig3:unducted}c) results in the fence achieving a total flow rate that is broadly independent of $s/D$ (Figure \ref{fig3:unducted}f).

It should be noted that while adjusting the operating point (tip-speed-ratio) as spacing is reduced, can help to bring the angle-of-attack distribution back towards the design point, changing angular speed does not change angle-of-attack uniformly along the blade and hence the resulting spanwise variation in angle-of-attack can be sub-optimal. Further performance enhancement is therefore achievable by redesign of the rotors to achieve optimum angle-of-attack along the entire length of the blade for the spacing in which they are to be deployed, see \cite{Vogel2019}. 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-0.85R), where span-wise flow terms are small and the 2-dimensional flow assumption is valid. The optimal angle of attack for the single aerofoil section used in this study was 6$^\circ$, close to the average $\alpha$ at the maximum spacing of 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. The distance from $\alpha=6^\circ$ highlights a potential for further performance improvements with altered designs at closer spacings.

% 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 $\alpha$ due to feedback between thrust and induction factors 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-0.85R). The optimal angle of attack for the single aerofoil section used in this study was 6$^\circ$, close to the average $\alpha$ at the maximum spacing of 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. The distance from $\alpha=6^\circ$ 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}[ht]
\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 $s=2.33D$ (top left), $s=0.75D$ (top right), $s=0.50D$ (bottom left) and $s=0.25D$ (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).



\subsection{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), the flow is not axisymmetric about the nacelle leading to a positional variation in blade root bending moments through a rotation, this can be extracted from the RANS-BE model following the method from Section \ref{S3_BERBM}. 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. The plot demonstrates that decreasing the inter-turbine spacing increases the root bending moments at all azimuthal angles and in both the edge-wise and flap-wise directions, with the inner turbine having greater bending loads than the outer turbine. 


\begin{figure}[ht]
\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 (left) and edge-wise (right) root bending moment coefficients with inter-turbine spacing for the inner (top) and outer (bottom) 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 (bold lines), with blue and red representing the outer and inner turbines respectively. Turbines are viewed from upstream so that flow is into the page and turbines rotate anti-clockwise. }
\label{fig3:Flap_bending_spacings}
\end{figure}


For the inner turbine, the edge-wise and flap-wise moments are typically highest in the upper right quadrant of rotation just after the blade makes its nearest pass to the centre of the fence at $270^\circ$. As the blades pass through this region they experience an increased bypass flow speed, and hence higher loading, due to constructive interference with the counter-rotating turbine on the other side of the symmetry plane. A similar observation can be made for the flap-wise moments of the outer turbine as it rotates through the same quadrant; although the loading is due to constructive interference from the co-rotating inner turbine. 

Both turbines experience elevated bending moments as blades rotate past the upper symmetry boundary. Although the distance to this surface is not altered, these elevated loads increase as the inter-turbine spacing is reduced and the flow between the turbines is diverted into alternate flow passages causing accelerated bypass flow above the turbines.

For the outer turbine, there is a reduction (relative to the average root bending moments 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 due to the relief on flow constraint as the flow diverts around (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}[ht]
\centering
\includegraphics[width=0.99\textwidth]{Figures/Chapter_3/numerical/047_Slices_pitch_angle.png}
\caption{Inclination of flow at the turbine plane for $s/D$=0.25 with turbines operating at $\lambda^*$. Positive is up.}
\label{fig3:Unduc_pitch}
\end{figure}


Using the results from the $s=0.25D$ 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. 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}[htp]
\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 $s=0.25D$ 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}



\section{Numerical design of turbine fence end-wall}\label{sc3:sidewall_deisgn}

Based on the results presented in Section \ref{sc3:unducted_perf}, the minimum inter-turbine spacing of $s=0.25D$ was selected for analysis with a end-wall. This spacing was chosen as it produced the highest power output from the spacings tested and a noticeable decrease in outer turbine performance relative to inner turbine performance was observed.

A parametric study of end-wall shape was carried out to determine whether an end-wall could be used to provide lateral flow constraint to improve flow alignment and reduce the loss in outer turbine performance at close spacing. Several end-wall cambers 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) so that inner and outer turbines saw similar geometric constraints. The length of the end-wall was constrained by the maximum size that manufacturing allowed for the physical testing discussed in Chapter \ref{Chapter4}.  The ellipse is centred on the plane of the rotor fence. 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, with the depth set to be symmetric above and below the turbine's axis of rotation. 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}. Positive camber implies that the end-wall's inner surface is more convex, and is achieved by displacing the leading and trailing edges of the cambered ellipse further outboard of the fence, so that the tip-to-wall spacing of $s/D=0.125$ is maintained at mid-length of the ellipse.

\begin{figure}[ht]
\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}


\subsection{Effect of end constraint on turbine performance}

Four metrics were used to quantify 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 $\eta$ following Equation \ref{Eq:Basin}. The performance of the fence with respect to these metrics is examined in this section. 

\subsection{Fence power}

Fence power at $\lambda^*$ for each end-wall profile, scaled by the optimal power coefficient achieved without a end-wall for the same spacing, is presented in Figure \ref{fig3:ducted_power}. Values greater than one 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. Improvement in the fence performance can be credited to the increase in global blockage from the thicker end-wall and 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 of the innermost turbine's flow alignment.


\begin{figure}[ht]
\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}[ht]
    \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 $s=0.25D$ 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}


\subsection{Fence thrust}
\label{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}[ht]
\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 $s=0.25D$.}
\label{fig3:ducted_thrust}
\end{figure}


\subsection{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}[ht]
\centering
\includegraphics[width=0.95\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}[ht]
\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 increased drag; 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, more closely 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}[htp]
\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}



\subsection{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. 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}. This plot shows 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{McIntosh2011,Schluntz2015,Cao2019} can also be applied to shorter fences of turbines.


\begin{figure}[ht]
\centering
\includegraphics[width=0.95\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 $s=0.25D$.}
\label{fig3:symfac_cp_wall}
\end{figure}


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

\begin{table}[h]
\centering
\caption{$C_p$ and $C_t$ at $\lambda^*$ for a four turbine fence at a spacing of $s=0.25D$ 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}





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 significantly improve performance due to the increased stream-wise load they experience (decreasing $\eta$) and shifting of the power bias towards the outboard turbine (negative $\Phi C_t$ and $\Phi C_p$). 



\subsection{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}[htp]
\centering
\includegraphics[width=0.80\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 $s=0.25D$ 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 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}[htp]
    \centering
    \includegraphics[width=0.65\textwidth]{Figures/Chapter_3/numerical/turbs_x_velocityDifference_plotsBase.png}
    \includegraphics[width=0.65\textwidth]{Figures/Chapter_3/numerical/turbs_x_velocityDifference_plotsDucted.png}
    \includegraphics[width=0.65\textwidth]{Figures/Chapter_3/numerical/turbs_x_velocityDifference_plotsDifference.png}
    \caption{Stream-wise flow velocity through the turbine plane for a fence with $s=0.25D$ 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). The 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}[ht]
\centering
\includegraphics[width=0.6\textwidth]{Figures/Chapter_3/numerical/RBM_BE_FW_wall.png}
\includegraphics[width=0.6\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 $s=0.25D$. 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}





\section{Conclusions}\label{Ch3: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 $s=2.33D$ to $s=0.25D$ 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 $s=0.75D$. 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 pressure drop over the turbine increased due to improved local blockage. This result suggests that there are spacings 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.  It was hypothesised that the fence end-loss could be reduced by introducing a fence end-wall, which could correct the skewed flow towards the ends of a fence, mimicking the effect of additional turbines in a longer 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.

% 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.


A parametric study of the effect of thickness and camber of an elliptical cross-sectioned end-wall of length (1$D$) was undertaken for an inter-turbine spacing of 0.25$D$. The end-wall was able to improve the power of all turbines in the fence, with fence $C_p$ improving by 3.95\%. This was achieved by better aligning the approach flow to the fence, such that turbines operated in closer to uniform flow. This 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 the overall basin efficiency (which included the parasitic power removed from the flow by the end-wall) was small, with reductions between 0.07\% 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. While 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. The 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.



%% Put a chapter split here

\chapter{Experimental testing of an end-wall constrained turbine fence}\label{Chapter4}

This chapter expands on the numerical analysis of the end-wall from Chapter \ref{Chapter3} through an experimental study of a candidate geometry. This experiment is carried out with two closely spaced turbines between an end-wall and the sidewall of a towing tank (representative of a symmetry plane), facilitating a comparison to the numerical results of the four turbine fence. 

Section \ref{C43:experimental} explains the experimental setup and manufacture of the end-wall, Section \ref{sc3:exper_turb_nowall} and Section \ref{sc3:expr_wall} present the results from the towing tank experiments and compares them to the numerical results from Chapter \ref{Chapter3} and Section \ref{CH3:model_limits} discusses potential sources of error in both the numerical and experimental model. 


\section{Introduction}

Details considering local blockage and the reasoning for installing a turbine fence end-wall have previously been discussed in Chapter \ref{Chapter3}. This section is limited to introducing the experimental testing and the comparison to numerical results.

Physical testing of tidal turbines is a necessary step in the development and eventual implementation of tidal turbine technology. Through physical testing a comparison can be made to numerical results to validate their accuracy and underlying assumptions. This work seeks to compare the RANS-BE results from Chapter \ref{Chapter3} with towing tank tests conducted at the SSPA test facility in Gothenburg, Sweden. Analysis of RANS-BE and experimental results has previously been conducted by \cite{Olczak2016} who examined the thrust and wake characteristics over a single turbine and multiple arrays configurations, including a fence. The comparison showed that for a turbine fence, the RANS-BE model over predicted thrust loading by between 3-12\% depending on location. In addition to investigating the end-wall improvement, this works looks to extend the comparison which was made by \cite{Olczak2016} through additional comparison data with experiments also conducted at larger geometric scale with lower $Re$ dependence. In addition to thrust measurements, the turbines were instrumented to measure torque, speed and angular positions, with each turbine blade instrumented to collect edge-wise and flap-wise blade root bending moments (r/R=0.175). 

The end-wall was instrumented to collect stream-wise and cross-stream loads; however, due to a strain gauge failure in the stream-wise direction, only cross-stream loads were measured. Limitations on the number of available test slots over the testing program also restricted the analysis to only consider one end-wall geometry (length $D$, 15\% thickness and 4\% camber) and inter-turbine spacing ($s=0.25D$). 

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{McNaughton2021} to firstly quantify the fence end-losses before new experimental results are presented for the end-wall constrained fence. The results in this chapter were collected alongside \cite{McNaughton2021} and are the author's own original work. 


\section{Experimental methodology}\label{C43:experimental}

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 \cite{McNaughton2021}. A total of ten tests were conducted across seven $\lambda$ ranging between 5.7 and 7.1 for the turbine fence with end-wall. Duplicate testing at three $\lambda$ close to $\lambda^*$ were run to confirm the results and quantify the uncertainty.

\subsection{Experimental facility}
\label{Expr_fac}
Experimental tests were conducted at the SSPA towing tank facility in Gothenburg, Sweden. This testing was part of the MaRINET2-funded ATHENA project, results on the impact of inter-turbine spacing on fence performance have already been reported in \cite{McNaughton2021}. The towing tank cross section matched the computational cross section (discussed in Section \ref{CH3:Method1}), measuring 10 m wide, 5 m deep and 240 m long. 

Two turbines of 1.2 m ($D$) diameter were deployed in a co-planar arrangement as per Figure \ref{fig3:exp_domain} with a tip-to-tip spacing of 0.25$D$ and a tip-to-wall spacing of 0.125$D$, matching the dimensions analysed in the numerical component of this study. Each turbine was supported from above by a tower located 0.33$D$ behind the turbine plane. The tank wall was used as a symmetry plane, allowing only two turbines to be tested while emulating a four turbine fence. The end-wall that was tested had a thickness of 0.15$D$ and a camber of 0.04$D$. 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.

Torque and thrust data were collected from a transducer on the drive shaft of each turbine. The root of each turbine blade was strain gauged to monitor the edge-wise and flap-wise root bending moments. Additional strain gauges were attached to the end-wall supports to measure the cross-stream forces acting on the wall.

Following some initial checks for Reynolds independence of the turbine performance, the testing speed was increased from 0.8 m/s (used in the RANS-BE model) to 1 m/s \citep{McNaughton2021}. Reynolds independence was checked through a comparison of $C_p-\lambda$ curves at 0.8, 0.9 and 1 m/s.   The towing carriage was accelerated 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, with tests running approximately every 20 minutes.

\begin{figure}[ht]
\centerline{
\includegraphics[width=0.7\textwidth]{Figures/Chapter_3/experiment/Domain_experiment_001.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}


\subsection{Prototype manufacture}


Loading on the end-wall was estimated from the RANS-BE model to be 300 N in the cross-stream direction and 10 N in the stream-wise direction at $\lambda^*$ (Section \ref{Fence_thrust}); 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 end-wall was constructed using a Computer Numerical Controlled router (CNC) to shape it into four segments. The segments were formed by layering sheets of marine grade plywood and cutting away the excess material to the appropriate size. Hollow sections on the internal faces of the wall were added to allow a steel frame to be inserted as well as lead weights. 80 kg of lead weights were added to offset the buoyancy of the structure so that it was approximately neutrally-buoyant when assembled. The wooden surface was coated in a marine grade epoxy and boltholes, which held the wood to the steel frame, were filled with sealant. Whilst every effort was made to ensure a smooth finish, the nature of the wooden material and epoxy resulted in some surface roughness on the exterior of the end-wall which will have increased the skin friction drag on the structure. The steel frame was constructed from 3 mm thick 50 mm square hollow sections. Strain gauges were attached to the top of the supports, allowing measurements of the force acting in the cross-stream direction to be calculated. An image of the partially assembled end-wall is presented in Figure \ref{fig3:sidewall_install}.


\begin{figure}[htp]
\centering
\includegraphics[width=0.7\textwidth]{Figures/Chapter_3/experiment/sidewall_manfactured.png}
\includegraphics[width=0.7\textwidth]{Figures/Chapter_3/experiment/Sidewall_side_view.png}
\includegraphics[width=0.7\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 front view of the installed end-wall in the turbine fence (bottom).}
\label{fig3:sidewall_install}
\end{figure}

%% HERE
\subsection{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. This method was able to distil the time signal on the 90 second data recording into a single azimuthal variation in root bending moments through averaging the experimental data into 1$^\circ$ bins. Triplicate data was used when possible as the three identical blades on each turbine were instrumented. Integrated results were compared to the thrust and torque transducer data and scaled to have an internally consistent results set. The thrust and torque transducer data is preferenced when comparing the signals due to the external calibration of this piece of equipment. The processing method is outlined below.

\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. This was to account for any changes in the sensors' baseline due to variations in temperature or 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$^\circ$ 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 the average measurement from the thrust and torque transducer matched the resultant sum from the blade root bending moment sensors. This is such that 
\begin{equation}
    SG_{new} = SG_{raw} C
\end{equation}
\noindent where $SG_{new}$ is the scaled strain gauge value, $SG_{raw}$ is the phase averaged strain gauge data, and $C$ is a factor to match the mean of the strain gauge data to the measurement from 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 sensors. 


\noindent \underline{Blade Averaging} 

Finally, the root bending moment is averaged over the phase-offset blade loads to arrive at a blade and azimuthally averaged root bending moment following
\begin{equation}
    M_{j} = \sum_{i=1}^{N_b} \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 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}[ht]
% \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}[ht]
% \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}



\section{Performance without end constraints}\label{sc3:exper_turb_nowall}

This section presents the experimental results of \cite{McNaughton2021} to provide context and as a reference case from which the effect of an end-wall constraint can be assessed. All results presented in this section were obtained with a constant tow speed of 0.8 m/s.

\subsection{Performance parameters}

Fence averaged $C_p$ and $C_t$ from the experimental work of \cite{McNaughton2021} are presented in Figure \ref{fig3:exper_perf}. These results confirm the observation of \cite{Nishino2013} 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 of inboard turbine performance; this is confirmed by $\Phi C_p$ and $\Phi C_t$ with bias being shifted towards the inboard turbine with decreased spacing. 

At larger turbine spacings ($s=0.72D$ and $s=D$), 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 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}[htp]
\centering
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Inboard_Turbine_C_p.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Inboard_Turbine_C_t.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Outboard_Turbine_C_p.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Outboard_Turbine_C_t.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Array_Averaged_C_p.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Array_Averaged_C_t.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Symmetry_Factor_C_p.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/experiment/perf_curve_varied_spacing_Symmetry_Factor_C_t.png}
\caption{$C_p$ and $C_t$ performance curves from the experimental testing of \cite{McNaughton2021} for a tidal turbine fence at varied inter-turbine spacings of $s=D$ (green) $s=0.72D$ (dark blue) and $s=0.25D$ (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. A second order polynomial has been fit to the data to improve readability.}
\label{fig3:exper_perf}
\end{figure}


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

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

Figure \ref{fig3:exper_rbm_spacing} demonstrates the effect of both local blockage and end-losses on the root bending moment. 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 increased performance of this turbine is due to various factors associated with local blockage (improved flow speed, wake constraint, inflow angle) and is examined in further detail by \cite{McNaughton2021}. Contrasting this result is the outboard turbine $C_{EW}$ signal, where a noticeable improvement is observed as it passes the inboard turbine ($\theta=270^\circ$). At the outboard location ($\theta=90^\circ$), the root bending moment appears to be unaffected by the change in spacing. Outer turbine $C_{FW}$ is also impacted by the varied spacing and has a minimum just prior to passing the exposed edge of the fence ($\theta=60^\circ$). This minimum decreases with inter-turbine spacing demonstrating a growing end-loss. These observations 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}[htp]
\centering
\includegraphics[width=0.85\textwidth]{Figures/Chapter_3/experiment/rbm_varied_spacing_C_{FW}_.png}
\includegraphics[width=0.85\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 $s=D$ (green) $s=0.72D$ (dark blue) and $s=0.25D$ (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. Data supplied by \cite{McNaughton2021}}
\label{fig3:exper_rbm_spacing}
\end{figure}



\section{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). As discussed in Section \ref{Expr_fac}, higher tow speeds were used to ensure Reynolds independence of the results. Consequentially there is a slight difference between the without end-wall results presented in this section at $s=0.25D$ and those from Section \ref{sc3:exper_turb_nowall}. 


\subsection{Performance parameters}

Table \ref{tab3:comp_perf} presents a comparison between performance metrics from the RANS-BE model and experimental study, highlighting that differences of up to 21.3\% in $C_p$ and 6.1\% in $C_t$ were observed. The causes for these disparities are primarily credited to the limitations of the RANS-BE model and experimental error (discussed further in Section \ref{CH3:model_limits}). However, 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 four 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} confirms the numerical model's observation that the outer turbine's performance ($C_p$ and $C_t$) is improved by the end-wall, albeit to a smaller degree than predicted in the numerical model. The experimental results show a 1.5\% increase in outer turbine power at $\lambda^*$, and a 0.8\% decrease for the inner turbine, to achieve an overall improvement of 0.5\% when the end-wall is in place. This is achieved at approximately the same $\lambda$ and fence thrust as the case without the end-wall. At $\lambda^*$, the end-wall caused the outer turbine thrust to increase and the inner turbine thrust to decrease, bringing both values closer to the mean fence thrust and reducing $\Phi C_t$.

Inner turbine $C_p$ was observed to decrease when the end-wall was installed, this goes against the findings of the numerical model which showed an increase in power alongside the outer turbine. This finding was consistent over the range of $\lambda$ tested.


% 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}[htp]
\centering
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Inboard_Turbine_C_pC_p_star.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Inboard_Turbine_C_tC_t_star.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Outboard_Turbine_C_pC_p_star.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Outboard_Turbine_C_tC_t_star.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Fence_Averaged_C_pC_p_star.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Fence_Averaged_C_tC_t_star.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/compare/perf_curve_compare_Symmetry_Factor_C_p.png}
\includegraphics[width=0.45\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 cross markers) and without \citep{McNaughton2021} (solid line with circle markers) an end-wall. The results for the inner turbine are presented on the top row, the outer turbine in the second row and the fence-averaged in the third 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 $C_p^*$ ($C_t^*$). A second order polynomial has been fit to the data to improve readability.}
\label{fig3:comp_wall_perf}
\end{figure}



% Paragraph 3 reason for this result
The experimental results 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 of the inboard turbine's performance may not be solely connected to an improvement mass flux and could also be a function of improved angle of attack on 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}. Given the lack of $Re$ independence observed for the \mbox{0.8 m/s} towing experiments \citep{McNaughton2021}, there is more confidence in the results from the RANS-BE model at higher $\lambda$. This is due to the higher $\lambda$ contributing to a higher relative velocity and $Re$.

Duplicate towing tank test results, shown in Figure \ref{fig3:comp_wall_perf}, demonstrated some variability in the experiment, around the design point. While small, this impacted the accuracy to which the optimal operating point is defined. It was hypothesised that this difference was a function of the build up of turbulence in the towing tank which was observed to not fully decay between tests.

% 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}[ht]
% \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}






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

% Paragraph 1 - Experimental observations
Phase-averaged blade root bending moments in the flap-wise ($C_{FW}$) and edge-wise ($C_{EW}$) directions at $\lambda^*$ are compared between the experimental results and the RANS-BE model  in Figure \ref{fig3:comp_rbm_wall}. Blade self weight effects in the edge-wise direction have been controlled for in the experimental data set and a 5$^\circ$ wide moving average has been applied to the experimental results to improve the clarity in this figure. Figure \ref{fig3:comp_rbm_wall} demonstrates the end-wall's influence on the blades of the outer turbine is localised around the $90^\circ$ position, where they are closest to the end-wall.

The end-wall introduces a second local maximum into the loading signal of the outer turbine where the minimum loading was previously experienced. This reduces the loading amplitude and brings the azimuthal variation closer to that experienced by the inner turbine. Comparing the outer turbine with and without end-wall $C_{EW}$ curves, the edge-wise root bending moment is increased in the 180$^\circ$ portion ($0^\circ$ to 180$^\circ$) closest to the end-wall, and the remaining portion is relatively unchanged. The total improvement results in a higher overall turbine torque and thus power. 

More subtle changes are observed for the inner turbine, with a marginal reduction in $C_{FW}$ just after the blade passes the outer turbine at 135$^\circ$ and negligible changes in $C_{EW}$.


\begin{figure}[ht]
\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{Phase-averaged flap-wise (top) and edge-wise (bottom) root bending moments comparison of RANS-BE (blue) and experimental (orange) results at $\lambda^*$ for outer (left) and inner (right) turbines. Solid lines represent the without end-wall case and dashed lines the with end-wall case. The illustrations at the top of each plot show the location of the blade (in bold) in the domain, with blue and red representing the outer and inner turbines.}
\label{fig3:comp_rbm_wall}
\end{figure}


Both the numerical and experimental results show a peak in $C_{FW}$ for the inner and outer turbines. The maximum flap-wise root bending moment was recorded at 260$^\circ$ in the experiment, whereas the simulations predicted this to be at 300$^\circ$. In both cases, the maximum coincides with passing an adjacent turbine. A secondary local maximum in the $C_{FW}$ signal was observed for the inner turbine as it passed the outer turbine in both data sets. Like the larger peak, a phase offset of $\approx40^\circ$ in blade position was observed between the data sets. The phase offset may be due to the RANS-BE model not explicitly modelling the rotation of the blades, and consequently phenomenon such as blade advancing and retreating in yawed flow are not resolved. 



% 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 difference 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. Unlike the numerical results the impact from the end-wall is limited to the outer turbine, with no clear offset of the inner turbine $C_{EW}$ or $C_{FW}$. A factor contributing to this is the different $\lambda^*$ the RANS-BE model predicted when the end-wall was introduced. Optimal fence $\lambda^*$ did not change in the experiments when the end-wall was introduced ($\lambda^*=6.44$) whereas in the numerical model $\lambda^*$ dropped from 6.59 to 6.37 with the end-wall. Consequently, the turbines are operating at different speeds between the experimental and numerical cases. 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. When the turbines are operated at the same $\lambda$ in both the numerical and experimental models, closer agreement in blade loading is observed. Despite the offset to the mean root bending moment, the RANS-BE model captures the approximate amplitude of the variations in root bending moment through a rotation, albeit with differences in phasing (especially for the outer turbine).


\begin{figure}[ht]
\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{Phase-averaged flap-wise (top) and edge-wise (bottom) root bending moments comparison of RANS-BE (blue) and experimental (orange) results \textbf{at the same $\lambda$=6.44} for outer (left) and inner (right) turbines. Solid lines represent the without end-wall case and dashed lines the with end-wall case. The illustrations at the top of each plot show the location of the blade (in bold) in the domain, with blue and red representing the outer and inner turbines.}
\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 four turbines operating at $\lambda^*$ and $s=0.25D$ 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}[ht]
% \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}






\subsection{End-wall loading}

Due to a strain gauge's failure during the end-wall installation,  data were available from only one strain gauge 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 the measured bending moment and cross-stream loading. A factor to convert bending moment to a cross-stream force 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 cross-stream loading on the end-wall, presented in Figure \ref{fig3:comp_sidewall_load}, corresponds to the confinement force on the bypass flow. Cross-stream loading was observed to increase approximately linearly with rotor $\lambda$, as higher turbine thrust tends to produce greater acceleration of the bypass flow. The loading on the wall has been non-dimensionalised using the same representative area as the turbines ($A_t$) to allow direct comparison with turbine $C_t$. At $\lambda^*$, the cross-stream force on the end-wall is close to half the turbine thrust.

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 blade passing and vortex shedding (not captured in the RANS-BE model), experimental uncertainty of the turbulence conditions in the tank and the fixity of the end-wall with the towing carriage.


\begin{figure}[ht]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/compare/Compare_endwall_cross_stream_force.png}
\caption{Cross-stream loading on the end-wall as a function of turbine tip-speed-ratio ($\lambda$) from the experimental (orange) and numerical (blue) studies. The 5th and 95th percentiles of the time series of the experimental data are indicated, and the marker is the median load. Results are non-dimensionalised using the turbine swept area.}
\label{fig3:comp_sidewall_load}
\end{figure}





% \begin{figure}[ht]
% \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}


\subsection{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{Payne2018} 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. The tower was located a distance of 0.33$D$ behind the turbine plane and its position corresponded to the $0^\circ$ blade location.

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.


\begin{figure*}[htp]
\centering
\includegraphics[width=0.875\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 spectrum's 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*}


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 result from an external low frequency source 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. 






\section{Numerical and experimental limitations} \label{CH3:model_limits}\label{Ch3:compare}
%SDE HERE 230222
% Talk about the blade advancing and retreating effect!

% 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}.

The limitations of the blade element method are well documented \citep{Vermeer2003,Snel2003,Malki2013} and are essential for the computationally efficient simulation of turbine rotors. Within the RANS-BE model used in the study the assumptions of azimuthal flow symmetry through the rotor and no span-wise flow were not strictly valid, especially as the turbines were positioned closely together. This is due to the closely spaced turbines introducing a shear profile to the incoming flow. Models such as the actuator line method (Section \ref{AL_model}) and unsteady blade resolved RANS \ref{CH2:BE_model}) are more capable of resolving these conditions, however come at a far higher computational cost. Regardless of this, the RANS-BE model was able to demonstrate the leading order physics at play and qualitatively match the experimental observations.

Furthermore, the RANS-BE method is a steady approximation to an unsteady process. As a result, the blade element method imposes resistance over the entire swept area throughout the simulation. Consequently, localised flow effects are smoothed over the rotor swept area as the model cannot resolve 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 that the change in flap-wise root bending moment is localised to the end-wall side in the experimental data, whereas in the RANS-BE model an offset over the entire azimuth is present in addition to the localised change. This is examined further in Figure \ref{fig3:RANS-BE_parms}, which plots the difference in pressure drop across 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 RANS-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\% (when operating at the same $\lambda$). This is in line with the work of \cite{Olczak2016} who observed a 3-12\% over-prediction of the RANS-BE model to experimental results.



% 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}[ht]
    \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 an 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}[ht]
\centering
\includegraphics[width=0.8\textwidth]{Figures/Chapter_3/compare/Profiles_047_294__velocity_magnitude.png}
\caption{Variation in tank sidewall shear stress at nacelle depth in the stream-wise direction (top) and profiles of velocity magnitude at nacelle depth for the inner turbine adjacent to the symmetry plane (orange) and wall (blue) boundaries. The boundary is located at the cross-stream position, 0.625$D$, and the dotted grey lines represent the free-stream flow condition (0.8m/s). Flow is from left to right.}
\label{fig3:BC_Compare}
\end{figure}

In the experiment, the presence of a boundary layer on the tank wall may have caused a change in loading on the inner turbine. The symmetry assumption for the tank wall in the experiment is only completely valid when the fluid and boundary move at the same speed. However, 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 numerically 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 tank wall. Figure \ref{fig3:BC_Compare} plots the evolution of the velocity profile at stream-wise positions relative to the turbine plane. 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 momentum flux and the third to energy flux) 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 momentum flux and a 3.5\% drop in energy 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}[htp]
\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}



\section{Conclusions}\label{Ch4:conclusions}


Towing tank tests were performed to establish an end-wall's potential to improve performance over a short turbine fence and re-distribute thrust and power along the fence. While only a marginal improvement in overall fence performance was observed, outer turbine power increased by 1.5\%, and the end-wall was observed to balance the thrust loading between the rotors and reduce the difference in the loading cycle magnitude between the inner and outer turbine. 

The experimental results were compared with the numerical results from Chapter \ref{Chapter3}. This comparison showed the improvement from installing an end-wall was significantly over-predicted by the RANS-BE model. The discrepancy was determined to be a factor of the model exaggerating 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 which was not resolved in the model. The study also highlighted some of the difficulties 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. 

This study highlights the potential for developers to enhance turbine performance by using elements of floating tidal platforms required to provide buoyancy to also align the flow approaching the turbines. Utilising end-walls may also be helpful as they allow for a reduced optimal tip-speed-ratio, reducing blade loading. Furthermore, the root bending moments between the inner and outer turbines were closer for both the averaged and cyclic loading components, equalising the accumulation of blade fatigue across the rotors in the fence. Integration of the end-wall into the turbine blade design methodology may enable further performance gains to be made by optimising the blade geometry for the flow conditions experienced by the turbines.





% \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}[ht]
% \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}[ht]
% \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}[ht]
% \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}[ht]
% \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



