
\chapter{Sidewall constrained turbines}\label{Chapter3}

This chapter presents the numerical and experimental analysis of a sidewall constrained fence of 4 turbines. Section \ref{Ch3:numerical} provides details on the Reynolds-Averaged Navier-Stokes (RANS) Blade Element (BE) numerical modelling approach and makes use of this model to characterise turbine performance with and without the inclusion of a sidewall. The design methodology and justification of the chosen design is also provided in this section. 

Section \ref{Ch3:experimental} provides details of the testing methodology including the manufacture of the prototype design and the experimental test facility. Experimental results with and without the inclusion of the end wall are also provided in this section. A comparison is made between the numerical and experimental models in Section \ref{Ch3:compare}. This section compares the performance of the turbines with the inclusion of the sidewall and provides a critical discussion of the limitations of both methodologies. Finally, a summary of the results is provided in Section \ref{Ch3:conclusions}.


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

A complete overview for the numerical modelling and design of a sidewall for an array of four turbines is presented in this section. An outline of the numerical model is provided prior to the presentation of results allowing the reader to appreciate the strengths and limitations of the selected modelling methodology.

\subsection{RANS-BE model}

This section provides an overview of the RANS-BE numerical modelling methodology citing relevant literature with a specific focus on the application to tidal turbines. The RANS-BE model was selected for this application due to the speed with which it provides a reasonably accurate result, which facilitates the investigate a large parameter space for the design of the sidewall. The trade offs between speed and accuracy are discussed later in this section.

\subsubsection{Reynold-Averaged Navier-Stokes equations}

The RANS equations are a fundamental component of modern computational fluid dynamics (CFD). These equations are a time-averaged version of the Navier-Stokes equations which themselves are a per-volume expression of Newtons second law:

\begin{equation}
    \frac{ma}{V} = \frac{\sum F}{V}
\end{equation}

\noindent where $F$ is the forces acting on a control volume of fluid, $a$ is the acceleration of the fluid, $V$ is the volume of the fluid and $m$ is the fluid mass. The fluid density, $\rho$ can be used in place of the $\frac{m}{V}$ fraction. Considering the shear and normal stresses drawn on the infinitesimally small control volume in Figure \ref{fig:3-Stresses} and making the assumption of an isotropic incompressible fluid, an expression for fluid motion can be derived:


\begin{equation}
    \label{EQ:NS}
    \rho \frac{\mathrm{D} u }{\mathrm{D} t} = \rho g -\bigtriangledown T
\end{equation}

\noindent where $u$ is the velocity, $g$ is the acceleration due to gravity, $\bigtriangledown$ is the grad operator and $T$ is the Cauchy stress tensor. In 3D $T$ is defined as

\begin{equation}
    T = \begin{bmatrix}
        \sigma_{x} & \tau_{xy} & \tau_{xz} \\
        \tau_{yx} & \sigma_{y} & \tau_{yz} \\
        \tau_{zx} & \tau_{zy} & \sigma_{z} \\
    \end{bmatrix}
\end{equation} 

\noindent where $\sigma$ are normal forces and $\tau$ are shear forces. The expression for motion prescribed by Equation \ref{EQ:NS} is useful as it describes how fluid will behave when stresses act upon it. However, knowing what these stresses are requires the formation of a constitutive relation between the stress and strain rate of the fluid (transverse velocity) through the viscosity ($\mu$). Details of how this constitutive law is applied can be found in  \cite{Schlichting2000Boundary-LayerEdition}. This leads to the more common expression for the Naiver-Stokes equation as:

\begin{equation}
    \rho \frac{\mathrm{D} u }{\mathrm{D} t} = \rho g -\bigtriangledown p +\mu \bigtriangleup u
\end{equation}

\noindent where $p$ is the fluid pressure.

\begin{figure}
    \centering
    \includegraphics[width=0.75\textwidth]{Figures/Chapter_3/numerical/BEM.png}
    \caption{Stresses acting on an infinitesimally small control volume in 2D}
    \label{fig:3-Stresses}
\end{figure}

When appropriately discretised, the equation presented above is capable of describing fluid flow across all length and time scales with a simulation methodology know as direct numerical simulation (DNS). This is an exceptionally computationally intense model both in terms of memory and computational requirements. For this reason additional assumptions must be made to reduce the complexity of the problem. The approach adopted by the RANS equations is to decompose the velocity into a time-averaged and a fluctuating components such that

\begin{equation}
    u(\mathbf{x},t) = \overline{u}(\mathbf{x}) + u^{\prime}(\mathbf{x},t),\;   \overline{u^{\prime}} = 0
\end{equation}

\noindent where u is the velocity in the $\mathbf{x}$ direction, primes represent fluctuating components and the overbar indicates a time average. 

Substituting this expression into Equation \ref{EQ:NS} yields

\begin{equation}
    \rho \frac{\mathrm{D} \overline{u} }{\mathrm{D} t} = \rho g - \frac{\partial p}{\partial x_{i}} +  \frac{\partial}{\partial x_{j}}(\mu \frac{\partial u_i}{\partial x_j} - \rho \overline{u^{\prime}_i u^{\prime}_j})
\end{equation}

\noindent the $\rho  \overline{u^{\prime}_i u^{\prime}_j}$ is commonly referred to as the Reynolds stress tensor. Similar to $T$ this expression describes how fluid moves when stresses are applied, in this case the stresses arrive from fluctuations in the flow. However, unlike $T$ there is not a constituent law relating the stresses, this leads to a closure problem and the need for additional models to model the turbulence.

For the work presented in this section, the $k-\omega$ shear stress transport (SST) turbulence closure scheme of \cite{Menter1994} is implemented. This modelling approach blends the $k-\omega$ model, which performs well in the near-wall region, with the $k-\epsilon$ in the far field \citep{ANSYSInc.2013}. This approach has previously been used to model flow over an airfoil profile with a favourable comparison to observations \citep{Menter1994} and to directly simulate flow through a tidal rotor \citep{Cao2019}. Whilst the simulations presented in this section do not directly model a rotor, this turbulence model is useful to describe the formation of a boundary layer around the sidewall without directly direct resolving it allowing for fewer computational elements to be used. 

\subsection{Blade element equations}\label{S3_BE}

As discussed in the previous section the direct simulation of turbine blades is a time consuming and computationally expensive process. For this reason alternative methods to attempt to accurately capture the dynamics of turbine have been developed including blade element momentum, porous disc models, actuator line and blade element models. The later will be the focus of this section.

The blade element (BE) method is a widely used \citep{Olczak2016ComparisonThrust,Malki2013,Belloni2017,Cao2019} and useful tool to simulate the performance of tidal turbines. This method is similar to the blade element momentum method with the key exception being that the momentum terms are calculated by an external model, commonly a RANS solver. The method uses 2D lift and drag data from hydrofoil profiles along the radial length to approximate a pressure drop and swirl velocity. The use of 2D data requires that there is no flow along the radial span of the rotor and the use of annular averaged values imposes a force on the flow at all point within the swept area of the turbine at all times. 

\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{Figures/Chapter_3/numerical/BEM.png}
\caption{(a): Rotor discritisation; (b): resolution of lift and drag forces.}
\label{Fig3:BEM}
\end{figure}

The BE equations can be derived through a considering the velocity triangle at a point along the blade and equating the relevant forces as described by Figure \ref{Fig3:BEM}. This yields the following expressions for pressure drop and swirl velocity on the swept area: 

\begin{equation} 
\label{EqPressureDrop}
\begin{aligned}
\Delta p &=\frac{1}{2} \rho W^{2} \sigma_{r}\left(C_{l} \cos \phi+C_{d} \sin \phi\right) ,
\end{aligned}
\end{equation}

\begin{equation}
\label{EqSwirlVel}
\begin{aligned} 
u_{\theta} &= \frac{W^{2} \sigma_{r}\left(C_{l} \sin \phi-C_{d} \cos \phi\right)}{4u_{x}}, \end{aligned}
\end{equation}

\noindent where $\Delta p$ is the pressure drop, $W$ is the relative speed of the flow to the blade, $\phi$ is the flow angle between the approach flow and rotor plane, $u_x$ is the stream-wise velocity component at the rotor plane and $\sigma_{r}$ the blade solidity defined as 

\begin{equation}
\label{EqSolidity}
\begin{aligned} 
\sigma_{r} = \frac{N_b c(r)}{2 \pi r}
\end{aligned}
\end{equation}

\noindent where $N_b$ is the number of blades and $c(r)$ is the chord length at radial position $r$. The relative flow speed can be determined by analysing the components of the flow and the rotation of the blades to arrive at the expression,

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

\noindent where $\Omega$ is the rotational speed of the turbine.


The derivation of the BE model constrains fluxes through each concentric streamtube to the stream-wise and azimuthal directions, not permitting an exchange of radial momentum. In reality a turbine rotor introduces a pressure gradient between the pressure and suction surface of the hydrofoil at the blade tip \citep{Masters2011ACorrections} driving a spanwise flow. Various tip correction models attempt to account for the reduction in power resulting from this flow, an overview of the performance of some common implementation can be found in \cite{Koh2016EffectsExperiments}. For this study the commonly used tip correction model of \cite{Glauert} was included in the BE model; this factor reduces the loading on the outboard blade span following:

\begin{equation}\label{eq3:TCF}
    \begin{aligned}
        F = \frac{2}{\pi} \cos^{-1}\left[\exp{\frac{N_b\left(1-\tfrac{R}{r}\right)}{2 \sin{\phi}}}\right] 
    \end{aligned}
\end{equation}

\noindent where $R$ is the turbine radius. 

Careful consideration of the limitations to this model have to be made prior to the implementation. As discussed previously in this section the set up of the model restricts radial flow and imposes a permanently present resistance at all radial locations in the rotors swept area. The later restriction limits the use of the model to steady conditions, however some unsteady implementations of BEM models have been shown to have favourable results to experimental data \citep{Galloway2014}, suggesting a similar implementation may be possible in a RANS-BE model. Nevertheless, the restrictions to steady conditions limits the ability to simulate turbines under wave motion or rapidly varying flow conditions. Furthermore the restriction limiting the radial exchange of momentum along the blades also has implications for the flow conditions the RANS-BE model can be used in. Conditions where the flow is not orthogonal to the rotor, such as yawed inflow, induce a radial component to the flow limiting the accuracy of the model. 



\subsection{Numerical implementation}

The numerical analysis of the RANS-BE tidal rotor was completed using the steady RANS solver in ANSYS Fluent 15.0 with a user defined function (UDF) to interface with a BE model. A structured computational mesh was developed using the commercial CDF software ANSYS ICEM \citep{Inc.2012}. The mesh is a rectangular domain matching the SSPA towing tank facility where the 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 12 m downstream of a velocity inlet boundary and 24 m upstream of a pressure outlet boundary. The global blockage (ratio of total turbine cross sectional area to channel cross-sectional area) is 4.5\%. The velocity inlet specifies a 0.8 m/s inflow speed with a uniform velocity profile and an approximate turbulence intensity of 5\% at the rotor plane. A symmetry boundary condition is applied to the side boundaries to simulate the effect of a fence of 4 turbines in a 20 m wide channel, halving the computational cost. The turbine design was that of \cite{Cao2019} which used a Wortmann FX-84-W-140 profile with a curtailed trailing edge to achieve improved performance with Reynolds number $O(10^5)$.


\begin{figure}[ht]
    \centering
    \includegraphics[width=0.8\textwidth]{Figures/Chapter_3/numerical/m.png}
    \caption{Computation domain and boundary conditions for the open channel model, D is the turbine diameter (1.2 m) and s is the inter-turbine spacing.}
    \label{fig3:Domain}
\end{figure}

\subsubsection{Mesh convergence}

A mesh convergence study was completed to ensure accuracy of the numerical results. Following the work of \cite{Cao2019} an identically parameterised mesh was initially used as the starting point for the study. The final structured mesh contained 15 million hexahedral elements and 27 million when the sidewall 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 used in the vicinity of the rotors and was relaxed towards the boundaries. Changes to the mesh resolution were made to investigate the refinement in the zone between turbines as this spacing was varied in the initial phase of the study. Using a fixed element size growth rate of 1.05, the maximum element size in this region was altered to determine the results dependency on the mesh. 

The results of this convergence study, presented in Table \ref{Tab_MCS}, illustrate close agreement between all three resolutions with the element size of 0.010 and 0.015 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 the turbine thrust coefficient, $C_t$. Therefore an element size of 0.015 was selected as the maximum inter-turbine spacing element size.

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

\begin{threeparttable}



\begin{tabular}{ccccc}
\hline
\multicolumn{1}{l}{Max element size} & \multicolumn{1}{l}{$C_p$} & \multicolumn{1}{l}{$C_t$} & \multicolumn{1}{l}{\% $C_p$ Error $^*$} & \multicolumn{1}{l}{\% $C_t$ Error $^*$} \\  \hline
0.010                                & 0.580                     & 1.199                     & -                                  & -                                  \\
0.015                                & 0.579                     & 1.199                     & -0.163\%                           & -0.051\%                           \\
0.020                                & 0.576                     & 1.196                     & -0.691\%                           & -0.245\%                           \\ \hline

\end{tabular}

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

The mesh independence of the sidewall was also investigated to assess the side wall mesh resolution. Both the non-dimensional wall distance, $y^+$, and the maximum stream wise element length, $L_{el}$, were varied. $y^+$ is defined by

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

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

\noindent where $\tau_w$ is the wall shear stress. $y+$ is an important criteria for resolving the loading on the sidewall due to the $k-\omega$ SST model using a wall model to simulate the boundary layer which requires the $y^+$ as an input.

The convergence of a 2D lift coefficient for a cross section of the sidewall at the rotor hub height was used to assess mesh performance. Lift was selected as the key criteria to achieve convergence due to this force imposing a confinement on the flow through the turbines. The cross-section lift coefficient was determined 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}

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

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

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

Individual blade root bending moments were calculated from the RANS-BE model by applying the blade element equations to the converged velocity field form the RANS-BE model. 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, azimuthal and radial components following,

\begin{equation}
\begin{bmatrix} u_{norm}\\u_{r}\\u_{swirl}\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 $u_{swirl}$ are the transformed velocities from the RANS-BE model normal to the turbine, in the radial direction and in the azimuth direction. Next the angle between the approach flow and rotor plane ($\phi$) and the blade velocity ($u_{mag}$) was determined following,
\begin{equation}
\begin{aligned}
        \alpha = \tan^{-1} \frac{ (u_{\infty} (1- (1- \frac{u_{norm}}{u_{\infty}})))/F }{\Omega r - \frac{u_{swirl}}{F}}, \\ \\
    u_{mag} = \sqrt{ \left(\frac{u_{\infty} (1- (1- \frac{u_{norm}}{u_{\infty}}))}{F}\right ) ^2 +  \left(\Omega r - \frac{u_{swirl}}{F} \right )^2}
\end{aligned}
\end{equation}
\noindent where $r$ is the radial position of the node and $F$ is the tip correction factor defined in Equation \ref{eq3:TCF}. The angle of attack ($\alpha$) on the blade was then calculated using $\phi = \alpha +\beta$, where $\beta$ is the blade twist. Lift and drag coefficients, $C_l$ and $C_d$, were interpolated from the 2D hydrofoil data for the specified $\alpha$ and transformed into streamwise and rotational thrust coefficients, $C_x$ and $C_t$,

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

\noindent Forces in the streamwise and rotational directions on each element were calculated using
\begin{equation}
    \begin{aligned}
    F_x = \frac{1}{2} \rho u_{mag}^2 C_x c(r), \\
    F_t = \frac{1}{2} \rho u_{mag}^2 C_t c(r).
    \end{aligned}
\end{equation}

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

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

\noindent This process was repeated for each annular segment in the turbine swept area to extract the variation in root bending moment with azimuthal position. Finally, the 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} \lambda}{\tfrac{1}{2} \rho u_{\infty}^2 A_t R}
\end{equation}

\noindent where $R$ is the turbine radius. 

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

Pertinent to understanding the effects on performance resulting from placing a sidewall adjacent to a turbine fence is being able to find the tip-speed-ratio ($\lambda$) at which the array configuration power output is optimal. Due to shifting inter-turbine spacing and varied sidewall designs, the optimal tip-speed ratio ($\lambda^*$) is expected to change across the simulations. Efficiently locating this value reduces the overall computational cost of investigating this parameter space. To limit the scope of the analysis, inboard and outboard turbines were operated at the same $\lambda$; differential performance is investigated in Section \ref{Chapter4}.

$\lambda^*$ was determined by first running a set of three simulations at different $\lambda$ close to what was expected to be optimal based on previous studies by \cite{Cao2019}. A quadratic curve was then fitted to the average turbine $C_p$ output from the model and $\lambda$. The $\lambda$ corresponding to the maximum $C_p$ from this curve was then used for the next simulation. The results from this iteration were then 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 $\lambda^*$. When this criteria was not met the results from the new iteration were used to fit a curve to the three largest values of $C_p$ and the search process was repeated. This methodology was useful in determining the optimal operating point in the 100+ simulations examined as part of this study, as it enabled exploitation of the parallel computing resources available on the ARCUS computing system. This allowed the initial three simulations to run con-currently; in general only one additional simulation was required to meet the specified criteria for defining $\lambda^*$.


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

Prior to the design and testing of the sidewall, an appropriate inter-turbine spacing was investigated. The spacing selected for further analysis had to balance the performance improvement due to increased local blockage \citep{Nishino2012} with the practicality constraints of placing turbines close together. The performance improvement due to decreased turbines spacing (increased local blockage) has previously been demonstrated numerically \citep{Schluntz2015,Vogel2017} and experimentally \citep{McNaughton2019,Cooke2016} however a noticeable deviation from the theoretical framework of \cite{Nishino2012} arises at the end of the turbine arrays due to diversion of the flow, decreasing overall performance. This effect is not captured in \cite{Nishino2012} due to assumptions which are made in the analysis of a long array but are observed in the experiments of \cite{Cooke2016} and the numerical analysis of \cite{Vogel2017}. Some improvements to the theory based modelling for shorter turbines fences were made by \cite{Nishino2013} who considered a 2-scale approach to model shorter fences however this work did not include a consideration for array end effects. The aim of the work presented in this section is to further the understanding of local blockage by investigating the effects of varied inter-turbine spacing on a short array of 4 turbines. Whilst long fences may be desirable commercially, shorter fences are more likely to be designed initially as technology develops. Within shorter fences the losses resulting from flow diversion around the outer turbines represent a larger percentage of the total array power and therefore cannot be ignored when optimising overall performance.

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

Using the methodology outlined in Section \ref{CH3:iteration_method}, the optimal operating point $\lambda^*$ and the corresponding turbine $C_p$ and $C_t$ was determined at a total of 7 inter-turbine spacing (s) ranging between 0.125D and 2.333D. The array averaged results from these simulations are presented in Figure \ref{Fig3:Unducted_perf}. The results show that as the array spacing is decreased from 2.333D ($B_l = 5.7\%$) to 0.125D ($B_l = 16.8\%)$ there is a monotonic increase in power production and thrust when $\lambda$ is held constant. This complements \cite{Nishino2012} work, demonstrating for a fixed global blockage ($B_G$), an increase in $B_L$ will improve the performance of the array. An improvement in performance at the optimal operating point of 6.02\% was achieved by increasing the $B_L$ from 0.057 to 0.168; the corresponding increase in thrust and $\lambda^{*}$ was 6.68\% and 5.13\% respectively. These results suggest that higher turbine thrusts can be sustained when the rotors are spaced closer together, increasing $\lambda^*$, allowing a greater pressure gradient to be maintained across the rotor plane resulting in higher performance. 



\begin{figure}[]
\centering
\includegraphics[width=\textwidth]{Figures/Chapter_3/b_Unducted_Optimal_PowerCoefficent_TSR_curve.png}
\includegraphics[width=\textwidth]{Figures/Chapter_3/b_Unducted_Optimal_ThrustCoefficent_TSR_curve.png}
\caption{Variation in array averaged $C_p$ (a) and $C_t$ (b) with $\lambda$ over a range of inter-turbine spacing's. The optimal operating point $\lambda^*$ is marked on each curve with a black cross.}
\label{Fig3:Unducted_perf}
\end{figure}


\begin{table}[h]
\centering
\caption{Turbine $C_p$ and $C_t$ at $\lambda^*$ for different 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 the inner, outer and averaged turbines over the range of inter-turbine spacing's are presented in Table \ref{Table:Inter_turbine_spacing}. This table highlights an increase in $C_p$ and $C_t$ with reduced spacing as well as a growing disparity between the performance of the inner and outer turbines. Across the entire parameters set tested, it was observed that the inner turbine was able to sustain a larger $C_p$ and $C_t$. This result is expected when considering the local confinement of the inner and outer turbines. The wake expansion of the inner turbines is laterally constrained by the presence of adjacent turbines, allowing larger pressure gradients to be maintained resulting in improved power. On the outermost turbines this lateral constraint is only present for the inboard region, allowing flow to bypass the outboard section of the rotor and reducing performance. This effect can be quantified by considering a symmetry factor between the rotors defined as:

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

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

Figure \ref{fig3:unducted} plots the variation in turbine symmetry factor alongside other key metrics. As can be observed in the plot, similar treads in the symmetry factor for both $C_p$ and $C_t$ are present with $SF_{C_p}$ being approximately 2.6 times larger than $SF_{C_t}$. As expected the smallest values for $SF$ were observed when the turbines were at the largest spacing of s/D=2.33. Despite the greater separation there was still a 1.5\% bias in the performance of the inner turbine due to the differing local blockage of the inboard and outboard turbines. A maximum was also observed for $SF$ at s/D=0.75, with the inner turbine producing 2.9\% more power and 1.1\% more thrust. As turbine spacing decreases from s/D=0.75 there is an improvement in the balance of power and thrust along the array. From a turbine fatigue perspective this result suggests that there are spacing's which give greater imbalances between inboard and outboard turbine loading which may not necessarily be at the closest spacing. These spacing's should be avoided as they give the greatest imbalance in loading between the rotors and do not optimise the power production. The cause of this may be explained by examining the flow rate through the rotor swept area (Figure \ref{fig3:unducted}f). 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 array, 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 array (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 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).


\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/a_Unducted_Optimal_TSR_Spacing.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/b_Unducted_Optimal_Symmetry_Factor.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/c_Unducted_Optimal_PowerCoefficent_Spacing.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/d_Unducted_Optimal_ThrustCoefficent_Spacing.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/e_Unducted_Optimal_alpha_Spacing.png}
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/f_Unducted_Optimal_FlowRate_Spacing.png}

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


\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/yaw/052_Slices_yaw_angle.png} %2.33D
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/yaw/049_Slices_yaw_angle.png} %0.75D
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/yaw/047_Slices_yaw_angle.png} %0.25D
\includegraphics[width=0.45\textwidth]{Figures/Chapter_3/yaw/046_Slices_yaw_angle.png} %0.125D

\caption{Yaw angle at the turbine plane for s/D=2.33 (top left), s/D=0.75 (top right), s/D=0.25 (bottom left) and s/D=0.125 (bottom right).}
\label{fig3:yaw_plots}
\end{figure}


Based off the results presented in Figure \ref{fig3:unducted}, it is clear that the minimum inter-turbine spacing of s/D=0.125 is optimal with respect to increasing the power output of the array. However, the practicalities of conducting experiments and being able to take measurement between arrays at this spacing may be impractical due to clearance requirements for the blades. As a result a inter-turbine spacing of s/D=0.25 was chosen to investigate in further detail as it demonstrated a performance improvement and allowed sufficient clearance for experimental measurements. 


The performance improvements observed here are based solely on the operation and inter-turbine spacing, further enhancement of the power output could be made 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. It is expected that the optimal shape of the turbine blades would change as the global blockage is reduced by a factor of 3. Furthermore, the operating conditions examined in this study restricted $\lambda$ to be the same for all turbines along the array. It is expected that, given the different flow conditions between inboard and outboard rotors, altering $\lambda$ along the array would result in improved performance. This is discussed further in Chapter \ref{Chapter4}. 



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

Due to the non-uniform confinement between the inboard and outboard turbines and the distorted aspect ratio of the local flow area (nacelle of the turbine at 0.83D depth and 3.33D from bottom of the domain) variation in the blade root bending moments was observed in the RANS-BE model. Figure \ref{fig3:Flap_bending_spacings} plots the azimuthal variation in flap-wise root bending moment with inter-turbine spacing using the method described in Section \ref{S3_BE}. As can be observed from this plot, decreasing the inter-turbine spacing increases the root bending moment with the inner turbine having greater bending loads than the outer turbine. Decreased spacing for the inboard turbine increases the flap-wise bending moment most rapidly in the region just prior to the blade passing the outboard turbine. The cause of this is the increased diversion of the flow under the array (Figure \ref{fig3:Unduc_pitch}) and constructive interference with the adjacent turbine. A similar observation can be made for the outboard turbine at the same angular position, however this loading is primarily associated flow diversion around the array. As a result there is a larger increase in the amplitude for the bending moment (max bending moment - min bending moment) for the inboard turbine as they are moved closer together. In summary, as turbines are spaced closer together there is a growing disparity between the blade loading conditions of the inboard and outboard rotors. 


\begin{figure}[h]
\centering
\includegraphics[width=0.72\textwidth]{Figures/Chapter_3/turb_2_Flap.png}
\includegraphics[width=0.72\textwidth]{Figures/Chapter_3/turb_1_Flap.png}

\caption{Azimuthal variation in flap-wise root bending moment with inter-turbine spacing for the inboard (top) and outboard (bottom) turbines. 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:Flap_bending_spacings}
\end{figure}

Using the results from the s/D=0.25 simulation at $\lambda^*$, the stream-wise velocity ($u_x$) through the rotor plane was extracted and decomposed into an annular averaged ($\left<{u_{x}}\right>$) and a deviatoric component ($u_{x}'$) such that $u_x= \left<{u_{x}}\right>+u_{x}'$. This metric was used as it indicated the angular and radial position where load is changed where as root bending moments only illustrate the angular position of load increase. 

The results of the velocity decomposition ($u_x$, $\left<{u_{x}}\right>$ and $u_{x}'$) are presented in Figure \ref{fig3:UnducDeviatoric_analysis} for the inboard and outboard rotors. This figure clearly highlights that there is a significantly reduced flow rate through the outer turbine with the $\left<{u_{x}}\right>$ at all radial locations being lower than the inner turbine; implying a reduced power output from this turbine. Examining $u_{x}'$ demonstrates the cause for the lower flow rate at the outboard turbine being due to lower $u_{x}'$ at the outer most section of the outboard turbine. Higher $u_{x}'$ for the left and right regions of the inner turbine and the inboard region of the outboard turbine highlight the constructive interference effects between the turbines. At the outboard section of the outboard turbine, bypass flow around the rotor interacts with the flow causing it to yaw, reducing the flow rate in the $u_x$ direction (this can be observed in Figure \ref{fig3:yaw_plots}). 

\begin{figure}[]
\centering
\includegraphics[width=0.8\textwidth]{Figures/Chapter_3/047_Slices_x_velocityturbines.png}
\caption{Total (top), azimuthally averaged (middle) and deviatoric (bottom) components of the stream-wise velocity at the inner (left) and outer (middle) turbines as well as the difference, $\Delta$, of the inner turbine's velocity component minus the outer turbine's velocity components turbine (right) for s/D=0.25 spaced rotors operating at $\lambda^*$. 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 of 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 prior to the blades passing the outermost portion of the rotation at 240$^\circ$ and the minimum loading 

%%%
is a the blade passes the top of the tank. Figure \ref{fig3:Unduc_pitch} demonstrates that there is a higher pitch angle in the region of flow bypassing underneath the array, this is analogous to what was observed in the yaw angle at the end of the array in Figure \ref{fig3:yaw_plots}. The distorted aspect ratio of the local flow area results in non-uniform flow confinement causing enhanced bypass flow in regions of lower flow constraints analogous to the end loss from a turbine array. 


\begin{figure}[]
\centering
\includegraphics[width=0.72\textwidth]{Figures/Chapter_3/RBM_BE_FW.png}
\includegraphics[width=0.72\textwidth]{Figures/Chapter_3/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}


\begin{figure}[]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/047_Slices_pitch_angle.png}
\caption{Pitch angle at the turbine plane for s/D=0.25.}
\label{fig3:Unduc_pitch}
\end{figure}


\subsubsection{Conclusions}

%In summary, as turbines are spaced closer together there is a growing disparity between the blade loading conditions of the inboard and outboard rotors. 

Analysis of the performance variation with inter-turbine spacing for an array of four 1.2 m diameter rotors was undertaken. It was determined that decreasing the spacing between the turbines from s/D=2.33 to s/D=0.125 increased the performance of the array by 6.02\%. A smaller improvement in performance of the outboard turbine (relative to the inboard turbine) was observed as the turbines were placed closer together with the greatest imbalance occurring at s/D=0.75. This spacing produced the greatest relative difference in loading between the inboard and outboard turbines due to a negative interaction from the inboard turbine on the outboard turbine leading to a drop in flow rate for the outboard turbine. As the spacing decreased further this interaction became positive and the flow rate through the turbine was increased. This result suggests that there are spacing’s which give greater imbalances between inboard and outboard turbine loading that may not necessarily be at the closest spacing; indicating that inter-turbine spacing should be analysed if a designer is trying to balance fatigue along an array.

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 flapwise and edgewise root bending moments were greater for the inboard turbine except in the inboard region of the outboard turbine where constructive interference effects resulted in comparable loading for the two turbines. Maximum loading of the blade was achieved as the blade passed the bottom of the computational domain. This loading was credited to the greater bypass flow in this region due to lower confinement in this portion of the asymmetrical local flow cross-section, this is analogous to the end losses observed in the yaw direction for the array. 



\subsection{Numerical design of turbine fence side wall}

Following the analysis of the 4 turbine array, a parametric study of sidewall shape was carried out to determine if a sidewall could be used to improve flow alignment and reduce end losses of the array. Using a turbine spacing of s/D=0.25, the effect of sidewall camber and thickness was investigated for a 1D long elliptical cross-section placed adjacent to the outboard turbine at half the inter-turbine spacing (s/D=0.125 between the tip of the outboard turbine and the interior face of the wall). A symmetric (upstream to downstream) parabolic distribution was applied to the major axis of the ellipse for the camber and a semi-circular section was used to round the bottom of the 2 m (1.67D) deep sidewall. For each of the sidewall designs considered, $\lambda^*$ was determined following the method outlined in Section \ref{CH3:iteration_method}. 3 sidewall thicknesses (0.05D, 0.10D and 0.15D) and 5 cambers (-4\%, -2\%, 0\%, 2\%, 4\%) were considered in the design of the sidewall, these cross-section profiles are presented in Figure \ref{fig3:wall_sections}.

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


\subsubsection{Effect of end constraint on turbine performance}

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

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

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

\subsubsection{Array power}

Average array power at $\lambda^*$ for each of the sidewall profiles, scaled by the optimal without sidewall array averaged power, is presented in Figure \ref{fig3:ducted_power}. Values on this plot greater than 1 represent regions where there has been a improvement in array power output. As can be observed from the plot it appears that high positive camber (curvature of the sidewall nearest to the turbines is convex) and greater thickness of the sidewall produces more power. An increase in power of 3.95\% relative to the without sidewall case was observed for the highest camber and thickness case (4\% camber and 15\% thickness). The outer turbine benefited the most from the inclusion of the sidewall with a 5.59\% increase in performance an increase in power production for the inner turbine of 2.36\% was also observed. Whilst an increase in fence performance may be expected due the higher global blockage from the thicker sidewall, the additional improvement can also be credited to improved alignment of the flow due to the presence of the sidewall which increases the mass flow rate through the turbine and the angle of attack on the turbine blades. Figure \ref{fig3:Yaw_sidewall} plots the impact on the flow yaw due to installing a sidewall adjacent to the end turbine. A significant reduction in end losses of the turbine array can be observed adjacent to the interior face of the wall. Alongside the reduced end loss, a minor improvement in the flow alignment of the innermost turbine is also observed, this is likely a result of the improved alignment of flow at the outboard turbine.

\begin{figure}[]
\centering
\includegraphics[width=0.8\textwidth]{Figures/Chapter_3/PowerCoefficent_Geom.png}
\caption{Array averaged power with a sidewall at $\lambda^*$ scaled by the without sidewall array averaged power for s/D=0.25. The dashed line plots the region where the with and without sidewall power is the same.}
\label{fig3:ducted_power}
\end{figure}


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


\subsubsection{Array thrust}

Across the entire parameter space investigated for the sidewall geometry the array averaged thrust was found to decrease (Figure \ref{fig3:ducted_thrust}) as a result of the lower $\lambda^*$ arrays with sidewalls operate at. Figure \ref{fig3:ducted_thrust} demonstrates how both camber and thickness of the sidewall can influence the thrust on the turbines with both increased thickness and camber leading to an higher turbine thrust. The array with the sidewall which achieved the highest $C_p$ (4\% camber and 15\% thickness) also had the highest thrust. For this array an averaged $C_t$ of 1.108 was observed, this is a decrease of 0.67\% relative to the without sidewall case. 
 

\begin{figure}[]
\centering
\includegraphics[width=0.8\textwidth]{Figures/Chapter_3/ThrustCoefficent_Geom.png}
\caption{Array averaged thrust with a sidewall at $\lambda^*$ scaled by the without sidewall array averaged power for s/D=0.25.}
\label{fig3:ducted_thrust}
\end{figure}

\begin{figure}[]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/Sidewall_force.png}
\caption{Resultant forces (pressure and viscous), normalised by the free-stream momentum flux through a turbine, acting on the sidewall in the stream-wise (a) and cross-stream (b) directions for a variety of wall cambers and thicknesses, turbines are operating at $\lambda^*$. Positive stream-wise thrust is in the free-stream flow direction and positive cross-stream thrust is away from the array.}
\label{fig3:wall_loads}
\end{figure}

An important consideration for the installation of the sidewall is not just the influence it has on the loading of the turbines but the loading experienced by the sidewall itself. Figure \ref{fig3:wall_loads} plots the stream-wise and cross-stream loading on the sidewalls considered in this study. An increasing relation was observed for stream-wise thrust with increased thickness and camber. This is a result of the larger cross-section area projected in the flow direction as both camber and thickness increased. Separation was found to occurring on the more highly 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 sidewall in Figure \ref{fig3:wall_loads}. This highlights the importance of thickness in the sidewall 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. This plot clearly shows flow separation at the leading edge of the thinner sidewall. For the thicker wall flow appears to remain attached longer. A similar finding was made by \cite{Fleming2016} in their analysis of a variety of turbine ducting profiles. 


Also shown in Figure \ref{fig3:Ducting_Separation} is a improvement in flow symmetry about the centre-line of the turbine with the introduction of the sidewall at 4\% camber. A clear increase in bypass flow speed can be observed at the outboard section of the turbine when the sidewall is in place, matching the bypass flow field of the interior side. This improved bypass flow speed is a function of the lateral confinement and is observed to be smaller when the camber of the sidewall 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 profiles of matched camber (but varied thickness) suggesting that lateral confinement is not heavily influenced by sidewall thickness. This is confirmed by Figure \ref{fig3:wall_loads} which shows similar lateral force on the sidewall across profiles thicknesses. 


\begin{figure}[]
\centering
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/047_sidewall_velocity_magnitudelong_section.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/176_sidewall_velocity_magnitudelong_section.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/180_sidewall_velocity_magnitudelong_section.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/189_sidewall_velocity_magnitudelong_section.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/172_sidewall_velocity_magnitudelong_section.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/178_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 a sidewall (a), a sidewall of 4\% camber and 5\% thickness (b), a -4\% camber and 5\% thickness (c), a 0\% camber and 15\% thickness (d),  a 4\% camber and 15\% thickness sidewall (e) and a -4\% camber and 15\% thickness (d).}
\label{fig3:Ducting_Separation}
\end{figure}



\subsubsection{Array basin efficiency}

The basin efficiency brings together the two performance metrics (power and thrust) discussed previously, relating them together as the efficiency with which power is extracted from the flow. Figure \ref{fig3:basin_eff} plots the basin efficiency over the set of sidewall profiles investigated. Across all profile geometries $\eta$ was found to decrease from the without sidewall case due to the contribution of thrust in the stream-wise direction from the sidewall. Due to the relatively small thrust on the sidewall $\eta_{ducted}/\eta_{unducted}$ remained close 1 for all the cases examined. The maximum basin efficiency with sidewall was found at approximately 2\% camber and 10\% thickness, this efficiency was 99.93\% of the unducted basin efficiency.

\begin{figure}[]
\centering
\includegraphics[width=0.8\textwidth]{Figures/Chapter_3/BasinEfficiency_Geom.png}
\caption{Array basin efficiency with a sidewall at $\lambda^*$ scaled by the without sidewall array basin efficiency for s/D=0.25.}
\label{fig3:basin_eff}
\end{figure}


\subsubsection{Array symmetry factor}

An analysis of the distribution of thrust and power along the array was undertaken using the symmetry factor, this metric quantifies how unbalanced the loading (and therefore the fatigue) is between along the array. Positive values indicate higher loading of the inboard turbine and negative values are an indicator of higher loading on the outboard turbine; a value of zero implies that the thrust and power are evenly balanced along the array. As observed for the array without a sidewall in Section \ref{sc3:unducted_perf}, the $SF_{C_p}$ and $SF_{C_t}$ were offset by a constant value, for conciseness only $SF_{C_p}$ is plotted in Figure \ref{fig3:symfac_cp_wall}. As can be observed on this plot the majority of sidewall profiles continue to bias the inner turbine. This bias is slightly reduced from the without sidewall case (2.7\%) for the thicker and positively cambered profiles. A single sidewall profile was found to bias the performance of the outboard turbine at 4\% camber and 15\% thickness. This configuration resulted in 0.4\% more power production for the outboard turbine than the inboard turbine and also produced the most power in the array overall. The improved balance between inboard and outboard turbines may imply that methodologies which design turbines using a uniform blockage, such as that used in \cite{Cao2019,Schluntz2015}, is valid for shorter fences with sidewall constraints due to the flow conditions at the outer turbines being more representative of the uniformly confined case.


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


\subsubsection{Sidewall profile selection}

Considering the performance metrics above, a sidewall of stream-wise length D, 15\% thickness and positively cambered by 4\% was determined to be the best candidate geometry for the 4 turbine array of 0.25D tip-to-tip spacing. When the sidewall was placed at a 0.125D tip-to-wall spacing it provided sufficient lateral constraint to increase the power and decrease the thrust on the turbines, 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 result in improved performance due to the increased stream wise load they will experience (decreasing $\eta$) and increased bias of outboard turbine load and power production (negative $SF_{C_t}$ and $SF_{C_p}$). 


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



\subsubsection{Unsteady loading on turbine blades}

Following the methodology used in Section \ref{sc3:unsteady_unduct} the azimuthally averaged ($\left<{u_{x}}\right>$) and deviatoric components ($u_{x}'$) of the stream-wise velocity through the turbine is presented in Figure \ref{fig3:Deviatoric_analysis_sidewall}. This figure highlights the differences across all the components of the velocity between the inner and outer turbines. The difference in total velocity suggests that the close proximity of the outer turbine to the sidewall could introduce some unsteady loading to the turbine blades as they move past this region.  The increased flow velocity suggests that the sidewall may have slightly overcompensated for the bypass flow losses. Figure \ref{fig3:flow_sidewall} shows the effect on stream-wise velocity from introducing the sidewall; clearly demonstrating localised increases in flow velocity adjacent to the wall which results in greater unsteady loading on the rotor. This can be directly observed by comparing the $u_{x}'$ component of the flow through the outer turbine for the turbine with and without sidewall (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 root bending moment in the flap-wise direction was slightly reduced and the edge-wise bending was slightly decreased when the wall was included. This is consistent with the results of the integrated coefficients $C_p$ and $C_t$. The difference in magnitude between the root bending moment between outer and inner turbines is also reduced (Table \ref{tab3:RBM_loading}). Consequently, throughout the life cycle of the array similar fatigue damage would be expected on the turbines, simplifying the scheduling of maintenance and improving the design life of the structure 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 sidewall}
\label{tab3:RBM_loading}
\begin{tabular}{|c|c|rrr|rrr|}
\hline
\multirow{2}{*}{Bending Direction}                                                     & \multirow{2}{*}{Turbine} & \multicolumn{3}{c|}{Without sidewall}                                         & \multicolumn{3}{c|}{With Sidewall}                                            \\ \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@{}}Flapwise \\ root bending (Nm)\end{tabular}} & Inner                    & 48.90                   & 51.05                   & 49.59                     & 47.98                   & 50.68                   & 48.94                     \\
                                                                                       & Outer                    & 48.03                   & 50.05                   & 49.04                     & 47.63                   & 50.21                   & 48.90                     \\
                                                                                       & Difference               & 0.87                    & 1.00                    & 0.55                      & 0.35                    & 0.47                    & 0.04                     \\ \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Edgewise \\ root bending (Nm)\end{tabular}} & Inner                    & 5.96                    & 6.72                    & 6.25                      & 6.25                    & 7.16                   & 6.56                     \\
                                                                                       & Outer                    & 5.69                    & 6.48                    & 6.08                      & 6.07                    & 7.08                   & 6.55                     \\
                                                                                       & Difference               & 0.26                    & 0.24                    & 0.17                      & 0.18                    & 0.07                    & 0.01                      \\ \hline
\end{tabular}
\end{table}



\begin{figure}[]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/RBM_BE_radial_wall.png}
\caption{Flap-wise (left) and edge-wise (right) root bending moment for inboard (red) and outboard (blue) turbines operating at $\lambda^*$ at s/D=0.25. Solid lines represent the without sidewall case and the dotted line represent the with sidewall case. The 0, 90, 180, 270 degree positions correspond to the top of the tank, inboard of the array, bottom of the tank and outboard of the array respectively.}
\label{fig3:ducted_RBM}
\end{figure}


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


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


 \subsubsection{Conclusions}

A parametric study of a cambered elliptical cross-sectioned sidewall of length D, thickness (5\%, 10\% and 15\% length) and camber (-4\%, -2\%, 0\% 2\% and 4\% length) was undertaken for an inter-turbine spacing of s/D=0.25. The results of this study showed that in some cases the installation of a sidewall was able to improve the power generating capability of all turbines in the fence with an improvement in average fence $C_p$ of 3.95\%. This was achieved by better aligning the approach flow to the fence, so that turbine blades operated closer to the optimal angle of attack for which the turbine was designed for. This improvement in turbine performance coincided with a small reduction in thrust applied to the turbine blades owing, in part, to the reduction of $\lambda^*$. As a result the impact on the overall basin efficiency (which included the parasitic power removed from the flow by the side wall) was a minor reduction in of between 0.5\% and 3\% relative to the no sidewall case. 


The introduction of the sidewall was also observed to reduce the difference in root bending moments between inboard and outboard turbines bringing both the averaged and cyclic loading components closer together and balancing the turbine fatigue across the rotors in the array. Whilst this work is promising in terms of the improved performance, further investigation into integrating the sidewall into the BE turbine blade design methodology may further improve performance. 



\section{Experimental model}\label{Ch3:experimental}

Complementary to the numerical study presented in Section \ref{Ch3:numerical}, an experimental study of the performance of an array of 4 turbines constrained by a sidewall is presented in this section. Due to limitations on the number of tests which could be run over the testing program only one sidewall wall geometry (length D, 15\% thickness and 4\% camber) and inter-turbine spacing (s/D=0.25) was considered. 

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


\subsection{Experimental methodology}

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

\subsubsection{Experimental facility}

The towing tank testing was completed at the SSPA test facility in Gothenburg, Sweden. The facility has a 10 m wide, 5 m deep and 240 m long towing tank in which two turbines of 1.2 m ($D$) diameter were placed. Turbines were arranged as per Figure \ref{fig3:exp_domain}, spaced at 0.25$D$ (0.3 m) tip-to-tip with the turbine sidewall placed a distance of 0.125$D$ tip-to-wall from the outboard turbine. The inboard turbine was spaced at a distance of 0.125$D$ from the tank side wall to take advantage of symmetry, emulating a fence of four turbines. The turbines were instrumented with thrust and torque transducers and the individual blades of each turbine were instrumented with strain gauges to measure the flap-wise and edge-wise root bending moments. Additional strain gauges were fastened to the sidewall to record both stream-wise and cross-stream loading. However, due to instrument failure, only the cross-stream loading data is available. Tests were carried out at a speed of 1 m/s with the towing carriage accelerating at a rate of 0.05 $\mathrm{m}/\mathrm{s}^2$, providing a minimum of 90 seconds of usable data for each run. Between tests a minimum settling time of 10 minutes was observed, allowing the tank to return to still conditions.

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


\subsubsection{Prototype manufacture}

Like the sidewall examined in the numerical study, the sidewall used in this experimentally study measured 1.2 m (1$D$) in the stream-wise direction and 0.18 m (0.15$D$) in the cross-stream direction with a maximum camber of 4\% towards the rotor. The bottom of the sidewall was 2 m (1.67$D$) below the water surface and was rounded in order to reduce drag. The turbines were submerged to a hub depth of 1 m and thus were symmetrically placed about the sidewall mid-depth.

Loading on the sidewall was estimated from the RANS-BE model to be 300N in the cross-stream direction and 10N in the stream-wise direction for the design conditions. These loads were used to determine the structural design of the wall. A conservative assumption of the load acting at the unsupported tip of the sidewall was used to estimate bending at the connections from the sidewall to the support structure (Figure \ref{fig3:sidewall_install}). Checks for shear, axial loads and bending were also done at key locations in the frame. The critical loading case for this design was determined to occur during the installation if it were to be suspended by a single one of the two connecting members. In this scenario one of the connecting plates within the steel frame was at 91\% of its capacity. To avoid this the wall was carefully rigged to distribute the 3600 N self weight as it was placed into the water. The purple rigging equipment is visible in Figure \ref{fig3:sidewall_install}.

The wall was manufactured using a Computer Numerical Controlled router (CNC) to shape it into four segments. The segments were formed by layering 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 sidewall which 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 images of the partially assembled sidewall and the sidewall installed adjacent to the fence are presented in Figure \ref{fig3:sidewall_install}.


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

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

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

Blade root bending moment data were first processed by removing the zero offsets from the strain gauges in the root bending moment sensors and the thrust and torque transducer to account for change in the gauge zero from variations in temperature and other environmental factors over the experiment. Zero offsets were recorded daily before and after experimentation, linear interpolation was used to adjust the offset for each experiment. Next an azimuthal average was calculated for each blade and each bending direction by averaging the bending moment in 1 degree bins. A blade averaged value was also derived from the thrust and torque transducer data following
\begin{equation}
        M_{FW}^* = \frac{T}{N_b}r_T, \quad
        M_{EW}^* = \frac{Q}{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, $T$ is the thrust, $Q$ is the torque, $N_b$ is the number of blades and $r_T$ is the distance to the line of action in the thrust direction. $r_T$ was derived numerically from blade resolved simulations from \cite{Cao2019} to be 0.4 m. 

Next, an offset was applied to each strain gauge such that
\begin{equation}
    SG_{new} = SG_{raw} C
\end{equation}
\noindent where $SG_{new}$ is the new strain gauge value, $SG_{raw}$ is the phase averaged raw strain gauge data and $C$ is factor to match the mean of the strain gauge data to the thrust and torque transducer. The factor ($C$) was taken to be the ratio of the $M_{FW}^*$ or $M_{EW}^*$, dependant on the direction, and the mean strain gauge value over all three turbine blades. This process adjusts the loading such that the average load measured from the thrust and torque transducer are the same as the average for each blade. Finally the root bending moment is averaged over the phase corrected blades loads to arrive at a blade and azimuthally averaged root bending moment following
\begin{equation}
    M_{SG} = \sum_{N_b}^{i=1} \frac{SG_{new_i}(\theta_i)}{N_b} 
\end{equation}
\noindent where $M_{SG}$ is the blade and azimuthally averaged root bending moment, $SG_{new_i}$ is the azimuthally averaged strain gauge data for blade $i$ which has been phase corrected for the blade position ($\theta=0$ for blade 1, $\theta=120$ for blade 2 and $\theta=240$ for blade 3). An additionally quantity $M_{SG}^*$ was also derived to compare directly against $M_{FW}^*$ and $M_{EW}^*$ by using $\theta=0$ for all blades. 

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

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


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


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



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

This section presents the experimental results of \textbf{JAMES PAPER} under a shifted perspective which focuses on array end losses to provide context and justification for placing a sidewall adjacent to the end turbine. All results presented in this section were obtained with a constant tow speed of 0.8 m/s.

\subsubsection{Performance parameters}

Array averaged $C_p$ and $C_t$ from the experimental work of \textbf{JAMES PAPER} are presented in Figure \ref{fig3:exper_perf}. This figure shows that at the optimal operating condition the array with the closest spacing produces the highest array averaged power. The primary cause of the increased power for the array is due to an increase in the power output of the inboard turbine. Power production from the outboard turbine was observed to be relatively constant with respect to the inter-turbine spacing. A slightly higher array averaged thrust is observed for the s/D=0.25 spacing, like the power, this is primarily due to an increase of the inboard turbine thrust with the performance of the outboard turbine having a small variation over the range of spacing's tested. These results contrast the numerical model results presented in \ref{fig3:unducted} which demonstrate noticeable increases in both inboard and outboard turbine $C_p$ and $C_t$ with decreased spacing. 

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 array 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.

At the higher turbine spacing's of s/D=0.72 and s/D=1, the inboard turbine was observed to generate higher power as indicated by negative $SF_{C_p}$. This result is surprising given that the local blockage of the inboard turbine is higher than the outboard turbine. This cause of this may be due to the assumption of the tank sidewall as symmetry plane in this experiment. 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 inducing a relative motion between the fluid and the wall, in-turn forming a boundary layer which alters the inflow condition. This is considered in further detail in Section \ref{CH3:model_limits}.




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


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


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

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. 

Edgewise 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 edgewise 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 off attack of the turbine blade to a sub-optimal position, resulting in increased drag (higher flap-wise bending) and decreased lift (reduced edge-wise bending moment). 


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

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



\subsection{Performance of turbines with sidewall constraint}

The performance of the turbine array with the sidewall is examined in this section. Unlike the results presented in Section \ref{sc3:exper_turb_nowall}, the towing tank speed used for these tests was 1 m/s (instead of 0.8 m/s). This was done to ensure Reynolds independence of the results as discussed in \textbf{Ref to James paper}. Consequentially there is a small difference between the results presented in this section for the without wall case at s/D=0.25 and those from Section \ref{sc3:exper_turb_nowall}.  


\subsubsection{Performance parameters}

Figure \ref{fig3:exper_rbm_wall} plots $C_p$ and $C_t$ over a range of $\lambda$. It can be observed from this figure that the sidewall increases the power at $\lambda^*$ of the outboard turbine (+1.6\%) and decreases the power of the inboard turbine (-0.5\%) to achieve an overall improvement (+0.5\%) when the sidewall is in place. The improved performance is able to be achieved at approximately the same $\lambda$ and array averaged thrust. Installation of the sidewall did result in the outboard turbine thrust increasing and the inboard turbine thrust decreasing, bringing both values closer to the mean array thrust, therefore minimising $SF_{C_t}$.

Typically, power output from a turbine is increased through a combination of increasing the mass flow rate through the turbine and achieving a higher lift-to-drag ratio along the blades. It is anticipated that the mass flow rate through the outboard turbine was increased as the sidewall constrains the flow, preventing some it from bypassing the end turbine as easily. The change in flow speed also adjusts the angle of attack along the blades to locally maximise the lift-to-drag ratio and deliver optimum performance in the presence of a sidewall. For the inboard turbine this is not the case and power is reduced due to changed inflow conditions resulting from the sidewall and outboard turbine operation, this is explored further in Section \ref{sc3:exp_rbm_wall}.

The drop in power of the inboard turbine and increase from the outboard resulted in the outboard turbine producing more total power than the inboard turbine. This effect occurs for all $\lambda$ tested and can be identified by negative $SF_{C_p}$ values. Higher power from end turbines is not typically expected for a turbine array due to flow diversions around the end turbine. This effect demonstrates the sidewalls capacity to improve the performance of the end turbine, shifting the $SF_{C_p}$ down. In this case however, the sidewall used in this experiment was too effective in reducing the $SF_{C_p}$ and over corrected the bias to the outboard turbine such that the magnitude difference between the turbine performance was approximately the same (absolute value of $SF_{C_p}$ is unchanged). Instead, a wall placed sightly further away may have been more effective in balancing the power between the turbines. However, at this spacing the sidewall was effective in balancing the thrust between the turbines. This is perhaps a more critical consideration as thrust loading is typically higher than the torque loading making it a critical input to maintenance scheduling and fatigue estimations. By reducing the disparity in loading between the turbines (as achieved from the sidewall installation) turbine degradation is better balanced along the array, improving maintenance planning. 

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


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

Phase averaged root bending moments are presented in Figure \ref{fig3:exper_rbm_wall} following the methodology outlined in Section \ref{SC3:RBM_expr_method}. Modest changes to the phase averaged root bending moments are observed for the inboard turbine. In the flap-wise direction a slight reduction is observed at the 150$^\circ$ position, leading to a decrease in $C_t$ for that turbine. In the edgewise direction a reduction in the root bending moment at the 90$^\circ$ position causes a drop in $C_p$. The increased thrust of the outboard rotor (Figure \ref{fig3:exper_wall_perf}) can most likely be credited for the change in the inboard turbine behaviour given these observations coincide with the blades passing the inboard turbine. 

More significant changes are observed for the outboard turbine than the inboard turbine. The presence of the sidewall introduces a second peak to the outboard turbine at the 70$^\circ$ position. A similar response was observed at this position for the inboard turbine as the turbines were spaced closer together (Figure \ref{fig3:exper_rbm_spacing}). As with the reduced spacing case this spike is likely due to increased bypass flow speed resulting from the confinement imposed by the wall. A significant drop in edge-wise root bending at this location was also observed. It is hypothesised that the increased inflow speed has resulted in a sub-optimal angle of attack leading to a drop in edge-wise root bending moment over this portion of the rotation. Over the remaining rotation the edge-wise root bending moment is improved and the loading range is reduced. The loading range for the flap-wise root bending moment is reduced, however loading in a 180$^\circ$ portion of the rotation from 300$^\circ$ to 120$^\circ$ is increased. This increase more closely aligns the loading cycle of the outboard turbine with the inboard turbine. 

\begin{figure}[]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/experiment/rbm_wall_exper_Edgewise_Root_Bending_Moment_.png}
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/experiment/rbm_wall_exper_Flapwise_Root_Bending_Moment_.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 edgewise root bending moment signal. The three illustrations at the top of each plot show the location of the blade (in bold) in the domain, with blue and red representing the outboard and inboard turbines in these illustrations.}
\label{fig3:exper_rbm_wall}
\end{figure}




\subsubsection{Sidewall loading}

Due to failure of a strain gauge in the installation of the sidewall additional processing was required to determine the sidewall loading from the experimental results. The data available was from one of two strain gauges measuring bending moment in the cross-stream direction at the connection between the sidewall and the supporting beam. A linear elastic frame model was developed in SPACE GASS 12.85 \cite{SPACEGASSSPACEPage} to determine the relationship between bending moment at this member and cross-stream loading. A factor to convert the bending moment to a cross-stream force from this strain gauge was determined by applying a unit load in the cross-stream direction at the center of pressure (determined from the RANS-BE model) and analysing the bending moment in at that location. The results of the analysis are presented in Figure \ref{fig3:expr_sidewall_load}.

The sidewall cross-stream loading presented in Figure \ref{fig3:expr_sidewall_load} corresponds to the confinement being imposed to constrain the wake expansion and re-align the inflow with the free-stream. A strong linear relationship between rotor $\lambda$ and cross-stream loading was observed with greater $\lambda$ corresponding to higher sidewall loading. The increased loading is expected to be a result of the increase in both bypass flow speed and pressure drop across the turbine with $\lambda$. Whilst $\lambda$ clearly has an impact on the sidewall loading, individual blade passing effects did not appear to cause significant variation in loading, this is explored further with spectral analysis Section \ref{sc3:expr_spec}.

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

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


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

Fast Fourier Transforms (FFTs) of the blade root bending moment, integrated power and thrust coefficients and sidewall cross-stream force are presented in Figure \ref{fig3:fft_wall}. The blade root bending moments all exhibit a peak at the once per-rotation frequency ($f_0$) with distinctive peaks at 2$f_0$, 3$f_0$ ,4$f_0$, 5$f_0$ and 6$f_0$ in some cases. As observed by \cite{Payne2018VariationPosition} the once-per-revolution loading spikes can be credited to a tower passing effect where the blade passing the support tower causes a drop in thrust leading to a once-per-rotation frequency spike in the blade loads. Examining Figure \ref{fig3:exper_rbm_wall} it is observed that an increase in flap-wise bending occurs just prior to the rotors passing each other (90$^\circ$ and 270$^\circ$ for the inboard turbine and 270$^\circ$ for the outboard turbine). Whilst there are two peaks for the inboard turbine, flap-wise root bending moment is at a maximum when the blade is positioned at 250$^\circ$ (just before it passes the tank sidewall), this is the same location for the outboard turbine. Consequently, this loading contributes to the once-per-revolution spikes in the FFT. When the sidewall is introduced a second peak in the outboard turbine loading immediately prior to passing the sidewall is present at 70$^\circ$, this peak is offset 180$^\circ$ from the without sidewall 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 sidewall is installed with energy at $f_0$ remaining relatively unchanged.

With respect to the edge-wise root bending moment without a sidewall, a single peak is observed at 110$^\circ$ and 160$^\circ$ for the outer and inner turbines respectively. When the wall is installed the location of maximum loading for the outer turbine is shifted to 180$^\circ$ and becomes more distinct and a second peak is introduced around 15$^\circ$. This results in an increase in the energy at $f_0$ and the harmonics (up to the 6th). 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.


Analysing the power spectrum for the thrust and power coefficients, harmonics exist at 2$f_0$ and 3$f_0$. The 3$f_0$ peak is likely associated with blade passing effects. The even harmonics may be explained through the asymmetry of the tank and the turbine positions within the fence as well as yaw misalignment of the rotors which would result in 2$f_0$ spectral content. 

Finally, 3 significant peaks were observed in the cross-stream loading acting on the sidewall, the first two spikes corresponded to frequencies less than $f_0$ at 0.5$f_0$ and 0.67$f_0$, these varied slightly between the experiments as the $\lambda$ was changed. There is no clear signal in the root bending moment or the integrated quantities at these frequencies suggesting it may come from some external low frequency forcing such as vortex shedding from the sidewall itself. The final peak occurs at 2.2$f_0$, unlike the other frequencies this was observed to change by a significant amount between experiments at different $\lambda$, with the frequency being linearly related to the rotor $\lambda$, and was observed both with and without the sidewall present. 


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

\subsection{Conclusions}


Towing tank experimental results of a turbine fence with and without a sidewall have been analysed and a performance improvement in array average power has been observed when a sidewall is installed. Whilst only a marginal improvement in performance was observed in the experiment it is believed that improved design of a sidewall could result in further increases in power output as the inboard and outboard turbine performance is matched. Specifically the placement of the sidewall at 0.125D tip-to-wall spacing was determined to be too close to the end turbine as it caused a significant improvement in outboard turbine performance and a decrease in inboard turbine performance. This resulted in the outboard turbine producing more power than the inboard turbine. 

The installation of the sidewall was shown to have an analogous effect on the flap-wise root bending moment as reducing inter-turbine spacing on an interior turbine within an array. This resulted in an increased contribution at the 2$f_0$ frequency band for the outboard turbine. The similarity between inboard and outboard turbine loading cycles simplifies maintenance scheduling and improving design life.  



\section{Comparisons between numerical and experimental models}\label{Ch3:compare}

This section presents a comparison between the numerical results presented in 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, this is discussed in \textbf{JAMES PAPER}. As a result this section makes comparison solely on non-dimensionalised quantities.

\subsection{Comparison of performance metrics}

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 array 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.

A slight decrease in $C_p$ was observed for the inboard rotor when the sidewall was installed in the experiment whereas an increase was observed in the numerical model. The experimental result was not expected given the additional flow confinement from the wall and increased thrust from the adjacent turbine typically improves performance. However, a very small decrease in $C_t$ was observed between the experimental results with and without a sidewall and subsequently the inboard turbine power is reduced. Examining the thrust on the outboard turbine suggests 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 inboard turbine is the formation of a boundary layer from the side of the tank which reduced inflow speed and altered inflow angle, this is discussed further in Section \ref{CH3:model_limits}. 

The inclusion of a sidewall in the experiment leads to a modest increase in $C_p$ for the outboard turbine and a decrease in performance for the inboard turbine causing an increase in array averaged power of 0.5\%, in contrast the RANS-BE model saw an improvement in performance along the entire array leading to an increase in array averaged $C_p$ of 3.95\%. Whilst this result suggests that the RANS-BE model may not be able to capture the exact values of integrated parameters $C_p$ and $C_t$ when the sidewall is installed, the symmetry factors indicates that the model is able to capture the relative change along the array. 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 similar 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 range shown in the experimental data highlights the variability in the cross-stream force (which is also the confining force acting on the array). This fluctuation is due to transient effects such as blade passing and vortex shedding (Figure \ref{fig3:fft_wall}). This is not captured in the RANS-BE model however, the close agreement of the average loading from the experiment and the RANS-BE model supports the use of RANS-BE models in simulating the lateral confinement force of a sidewall. 

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





\begin{figure}[]
\centering
\includegraphics[width=0.49\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 a numerical 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 array 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 array averaged maximum power ($C_p^*$) and the $C_t$ curves have been scaled by the array 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}


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



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

\subsubsection{Without sidewall}
Figure \ref{fig3:comp_rbm_wall} plots the phase averaged root bending moment from the towing tank experiments and the RANS-BE results for flap-wise and edge-wise directions with and without a sidewall. Overall the numerical model performs well to capture the magnitude of the variation in the root bending moments however the phasing (especially for the inboard turbine) is not accurately modelled.

Examining the flap-wise root bending moment variation for the numerical model and experimental data plotted in Figure \ref{fig3:comp_rbm_wall}, it was observed that both data sets have a clear single peak for inboard and outboard turbines. However, the locations of the peaks between the data sets were observed to occur approximately out of phase with each other. For the experimental data the peak loading occurs at 260$\circ$ for inboard and outboard turbines, corresponding to passing the tank wall and passing the adjacent turbine respectively. In contrast the numerical model has loading peaks at 60$\circ$ for the inboard turbine and 110$\circ$ for the outboard turbine. For the outboard turbine this suggests that the peak in the loading is associated with diversion of flow around the array in the numerical model and with constructive interference between the rotors in the experiment. In contrast, the inboard rotor peak in the experimental data is associated with passing the tank wall 'symmetry plane' and is a product of constructive interference with the outboard turbine in the numerical model. 

Comparing the edgewise bending moment, a closer agreement in the phasing of the loading peaks is observed between the RANS-BE and experiment. However a significant difference in shape of the root bending moment variation is observed for the inboard turbine with a sharp peak in the numerical model and a wide peak in the experimental data. 

% 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 flapwise and edgewise 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 flapwise 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 flapwise thrust coefficient ($C_x$) and a decrease in local edgewise 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 flapwise and edgewise 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 flapwise and edgewise moments are in phase in the numerical model.

The above explains why the RANS-BE model has aligned peaks in the edgewise and flapwise bending moments but does not explain why the locations of the peaks in the flapwise direction do not match between the model and the experiment. As mentioned above the flapwise 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.


\subsubsection{With sidewall}

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 in root bending moment at both rotors and in both directions is present for all azimuthal locations. Where as the experimental data does not measure significant change in the inboard turbine root bending moment loading and only changes in the vicinity of the sidewall for the outboard turbine. As mentioned above, the cause of this discrepancy may be due to the RANS-BE model distributing the effect of the local confinement of the wall around the azimuth leading to an offset of the root bending moment. 

Despite these difference, both the experimental data and the RANS-BE model have a second peak in flapwise bending moment as it passes the sidewall. Two peak are also observed in both data sets for the outboard turbine in the edgewise direction, this is slightly more subtle for the RANS-BE model due to the large increase in bending moment as the rotor blade passes the sidewall. 


%%% Up to here


% \textcolor{red}{Considering the effect of the sidewall, experimental results indicate that flapwise root bending is increased on both inner and outer turbines. In contrast the experimental results showed that only the outer turbine flapwise root bending was increased, primarily as the blade was passing the wall due to the increased flow speed in this region. The experimental result also demonstrates the load distribution is now better matched between the inboard and outboard rotor when the sidewall is used.}


% mention the average loading ...
% generally speaking edgewise loading isnt as critical as flapwise


\begin{figure}[]
\centering
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Inboard_Flapwise_Root_Moment_.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Outboard_Flapwise_Root_Moment_.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Inboard_Edgewise_Root_Moment_.png}
\includegraphics[width=0.49\textwidth]{Figures/Chapter_3/compare/compare_rbm_varied_spacing_Outboard_Edgewise_Root_Moment_.png}

\caption{A comparison between experimental (orange) and numerical (blue) phase averaged flapwise (top) and edgewise (bottom) root bending moments for inboard (left) and outboard (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 array. Experimental results for the edgewise root bending moment have controlled for blade self weight and buoyancy. Blade root bending moments have been scaled by the mean root bending moment for each turbine and direction. 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}



\subsection{Limitations of modelling methodologies}\label{CH3:model_limits}

Both the experimental and numerical models used in this study faced limitations. Most noticeably in the numerical model was the use of the RANS-BE model methodology in the simulations of the rotors. The blade element method relies on the key assumption of radial independence of flow over the rotor, this permits the use of 2D aerofoil data to approximate lift and drag on each radial element. The limitations of assumption are well understood \cite{Vermeer2003,Snel2003,Malki2013} and essential for the computationally efficient simulation of turbine rotors. However, decreasing the inter-turbine spacing causes greater flow diversion (especially at the end turbine), which has a radial flow component suggesting that the use of the model may not be appropriate for highly blocked or isotropic blockage conditions. A less commonly discussed feature of the blade element approximation is the time independent nature of this solution which imposes resistance on the rotor plane at all locations. This limitation may have facilitated greater interaction between the rotors and by increasing the flow diversions around the rotor. In reality there is a time dependence of blade position which means the resistance is varied in the flow as the blade rotates and the velocity is varied with time. 

The use of the RANS-BE model in the design of the sidewall is a simplifying assumption which was found to artificially smoothed the performance improvement of the turbine over the entire turbine. This is partly due to the RANS-BE model imposing turbine resistance being over the complete swept area instead of the region directly effected by endwall action. Experimental results suggested that the majority of the interaction between the endwall and the fence was localised in the region where the outboard turbine blade passes. Despite this the RANS-BE model was able to accurately predict the loading on the sidewall which suggests it may be an appropriate tool for the design loading on the wall and a first iteration tool for the detailed design of a sidewall. 

In the experiment the formation of a boundary layer from the tank wall may have caused an increase in loading on the inboard turbine. They symmetry assumption for the tank wall in the experiment is only valid when the fluid and boundary move at the same speed. This is not the case, despite the wall and the water being 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 which changes the velocity distribution across the rotor. The impact of this assumption can be observed in Figure \ref{fig3:BC_Compare} where a slight decrease in velocity on the inboard section of the turbine is observed when the moving wall boundary is used. This reduction in velocity is responsible for a 2.7\% decrease in $C_p$ of the inboard turbine, a significant portion of the 15.6\% error in $C_p$ estimates of the inboard turbine from the RANS-BE model. Higher flow speeds used in the experiment are expected to further increase this discrepancy due to the increased thrust contribution of the rotor resulting in a larger velocity gradient between the flow and the tank wall. 


% location in addition to the constructive interference expected from an adjacent turbine through the symmetry plane. For reference, a small deviation is observed at 60$\circ$ in the data which was credited to constructive interference between adjacent turbines (Section \ref{sc3:rbm_exper_spacing}). A similar signal was expected in the experimental data as the blade passes the 'symmetry plane'. 

% The ability to accurately predict the thrust distribution along the array is a key component to being able to properly design a sidewall due to the interaction it has with the fluid and turbines. When comparing the model and experimental data it was discovered that the model consistently under predicted the observed stream-wise and cross-stream load contribution form the sidewall. 


\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_3/compare/Profiles_047_294__velocity_magnitude.png}
\caption{Profiles of velocity magnitude at the mid-depth of the inboard turbine adjacent to the symmetry plane (orange) and wall (blue) boundary condition. The boundary is located at 5 m in the cross-stream direction and the dotted grey lines represent the free-stream flow condition (0.8m/s) at that location.}
\label{fig3:BC_Compare}
\end{figure}




\section{Conclusions}\label{Ch3:conclusions}

Model is good at predicting the condining force from a sidewall - Typical RANS application 
Not great at predicting the change in thrust resulting from the 

To conclude an investigation into the performance of inter-turbine spacing has been undertaken for a fence of 4 turbines using a RANS-BE model. It was shown that as the inter-turbine spacing was decreased (local blockage increased) greater power output could be achieved by allowing the turbines to operate at a slightly higher $\lambda^*$. This improvement in overall fence performance was combined with an increase in end losses of the turbine fence due to a greater disparity between inner and outer turbine performance at close spacing. It was hypothesised that the array end loss could be reduced by introducing an array sidewall which could help to align the flow approaching the array; mimicking the effect of additional turbines in a longer fence. 

A parametric study of a cambered elliptical cross-sectioned sidewall of length (D), thickness (5\%, 10\% and 15\% length) and camber (-4\%, -2\%, 0\% 2\% and 4\% length) was undertaken for an inter-turbine spacing of 0.25D. The results of this study showed that in some cases the installation of a sidewall was able to improve the power generating capability of all turbines in the fence with an improvement in average fence $C_p$ of 3.95\%. This was achieved by better aligning the approach flow to the fence, so that turbine blades operated closer to the optimal angle of attack for which the turbine was designed for. This improvement in turbine performance coincided with a small reduction in thrust applied to the turbine blades owing, in part, to the reduction in $\lambda^*$ for the optimal operating point. As a result the impact on the overall basin efficiency (which included the parasitic power removed from the flow by the side wall) was a minor reduction in efficiency of between 0.5\% and 3\% relative to the no sidewall case. 

Comparisons to towing tank experiments demonstrated a sidewalls capability to adjust the thrust and power distribution along an array. Experimental results showed that the sidewall was able to increase the power of the outboard turbine by 1.8\%. However this increase was less than predicted by the RANS-BE model. This result was determined to be a factor of the model over predicting the response of the flow from the installation of the wall, limitations of the RANS-BE model and the formation of a small boundary layer in the towing tank experiment. This study also highlighted some of the difficulty in accurately modelling closely spaced tidal turbines especially in isotropic blockage conditions.

The implementation of a sidewall presents an exciting opportunity to tidal device manufactures to obtain a performance improvement to floating tidal fences by using the support structure to both provide buoyancy and align the flow to reduce end effects. The introduction of the sidewall also brings the root bending moments between inboard and outboard turbines closer together for both the averaged and cyclic loading components balancing the turbine fatigue across the rotors in the fence and simplifying maintenance scheduling.

% Additionally, the effect of ducting has only been investigated on the ends of the array, there is a potential to further improve the confinement of the array by introducing a bottom wall to this system. This would also provide additional support for the sidewalls in the most heavily loaded direction.
