\chapter{Yaw control of turbine fences}\label{Chapter6}


\noindent This chapter analyses the performance of a four turbine long fence by making adjustments to the yaw of the outermost turbines to constrain the flow and improve flow alignment. Building on the methods used in Chapter \ref{Chapter3}, where a turbine fence endwall was used to constrain the flow, this chapter examines a correction for the cross-stream flow component at the ends of a fence which cause a loss in end turbine performance. The correction is made by yawing the outer turbines within the fence to align them with the primary flow direction. Alternative yaw angles are also considered to investigate lateral confinement from the outer turbine on the fence. Yaw control is a practical method of providing lateral confinement for larger turbines (such as that designed in Chapter \ref{Chapter5}) as no additional structures are required were they to rotate about their existing tower connection. For this study, to avoid confusing the impact of yaw with changing local blockage (pivoting about the tower changes the inter-turbine spacing), a constant tip-to-tip spacing is maintained by yawing the outer turbine around the inner-most point of the turbine swept area.


\section{Yaw of turbines}

\subsection{Single turbines}

Misalignment of flow with the rotor plane is common in both wind and tidal turbines due to the continually varying flow direction. In the horizontal direction misalignment from normal to the turbine plane is referred to as yaw ($\Psi$). Yawed inflow to turbines is typically considered undesirable for a single turbine due to reduced volume flux through the rotor and inefficiencies from blade advancing and retreating effects; this effects is well understood for wind \citep{ Galloway2014,Bastankhah2016,Micallef2016AAerodynamics,Dou2019} and tidal \citep{Musa2020} turbines. Over one half of a rotation, a yawed turbine blade advances into the flow, increasing the relative velocity between the fluid and the blade and decreasing the blade angle of attack. Over the other half of the rotation, the blade and fluid are moving in the same direction, decreasing their relative speed and increasing the angle of attack. This is of concern for a tidal turbine as the variation in the angle of attack and relative flow speed results in increased unsteady loading, which can be detrimental to turbine life span. 

\cite{Bahaj2007} quantified some of the effects of yawing a turbine in their experimental study of a single turbine at various yaw angles within a cavitation tunnel. This work provided a comprehensive data set of power and thrust measurements for a range of yaw angles ($\Psi$), tow speeds and tip-speed-ratios ($\lambda$). This data set is commonly used when evaluating numerical model accuracy for tidal turbines in yawed flow, see e.g. \citep{Park2016,Baratchi2017}. \cite{Shahsavarifard2020} also performed experiments on a yawed rotor, noting that the power loss was negligible for yaw angles smaller than 10${}^\circ$ due to the strong relationship between the power output and the cosine of the yaw angle. Beyond this angle, there was a rapid decrease in performance. While this result highlighted the importance of correctly aligning rotors with the oncoming flow direction, it also suggests that there is scope for small yaw adjustments of turbines.

Numerically, several methods have been used to model yawed turbines. \cite{Galloway2014} accounted for the effects of yaw with a geometric correction to the inflow in an unsteady Blade Element Momentum (BEM) model. While the results captured the variability in blade loading, it was inaccurate when combined with additional flow phenomena, such as waves, under-predicting the peaks in blade root bending moment. \cite{Baratchi2017} compared both an actuator disk and actuator line model of a yawed turbine to the results of \cite{Bahaj2007}. The work showed that superior performance could be achieved from an actuator line than a disk model due to its ability to capture wake unsteadiness and the root and tip vortices; however, both models tended to over-predict power and thrust. Superior model accuracy was observed in \cite{Park2016} using a blade resolved sliding mesh model which was also compared to the results of \cite{Bahaj2007}. This model accurately captured the variation in power and thrust with yaw, however, this method is known to be computationally intensive as it resolved the blade geometry and motions explicitly.



\subsection{Turbine fences}

The previous section referenced studies which have been conducted for solitary turbines in yawed flow, however due to the high installation cost and capacity to share infrastructure between turbines, it is likely that tidal turbines may operate in close proximity to each other as an array. As discussed in Section \ref{Ch2_Blockage}, closely-spaced turbines offer an additional benefit of producing higher power output through the local blockage effect. 

%however it may not be feasible to have a turbine fence which fully spans the cross-section of a tidal channel. \cite{Nishino2012} extended the work of \cite{GARRETT2007} to consider a fence of turbines which only partially span the width of a wide channel. It was shown that adjacent turbines can impose constraints on the flow through neighbouring turbines by controlling inter-turbine spacing and, in turn, the local blockage, $B_l$ (ratio of device swept area, $A_t$, to the local passage cross-sectional area, $A_l$, $B_l=A_t/A_l$). Through the optimisation of $B_l$, \cite{Nishino2012} showed a new limit for power extraction from a fence partially spanning an infinitely wide channel of $C_p=0.798$ existed when increasing $B_l$ to $\sim$0.4, a 34.5\% improvement on the Betz limit. 


\begin{figure}[ht]
    \centering
    \includegraphics[width=1\textwidth]{Figures/Chapter_6/Approach_flow_With_Y_velocity_annotated.png}
    \caption{Streamlines of flow approaching a turbine fence, highlighting the greater flow diversion around the end turbines over the inner turbines. The contours are the absolute value of the cross-stream flow velocity.}
    \label{fig6:Approach_Flow}
\end{figure}


% - Turbine end-loss and flow skewing
While there is typically an improvement for all turbines within a fence when spacing is reduced, the performance of the outermost turbines lag behind as they do not experience the same onset flow as the inner turbines. End turbines do not have an adjacent turbine on both sides to constrain the flow, consequently, the approach flow towards the ends of the fence has a cross-stream flow component. This results in the outboard turbines encountering flow with a yaw component. Figure \ref{fig6:Approach_Flow} demonstrates this effect for a fence of four turbines. A greater cross-stream flow component is present over a larger area at either end of the fence than towards the middle. This so-called \enquote{end effect} has been investigated experimentally by \cite{Cooke2015} who used porous discs to examine the effects of inter-turbine spacing in a flume. \cite{Cooke2015} observed a reduction in performance of the outboard turbines, this result was supported numerically by \cite{Vogel2017} and \cite{Vogel2019} who observed a similar reduction in end turbine performance along a four turbine long fence using a RANS-BE model. More recently model turbines have been used by \cite{McNaughton2021} to perform towing tank tests of two turbines, specifically designed for high blockage conditions, adjacent the tank side-wall (which emulated a symmetry plane) and a full scale floating fence installation in Nova Scotia was analysed by \cite{Jeffcoate2021} (turbines were not design for blocked flow conditions). Both of these experiments showed a slight performance improvement for the outer turbines within the fence. The results of \cite{McNaughton2021} and \cite{Jeffcoate2021} are still the subject of further investigation to understand why the outer turbine have higher power. Nevertheless, consistent across all this work is the observation of a performance variation along the fence. 




To compensate for the diversion of the approach flow at the ends of a turbine fence, this chapter investigates outer turbine yaw to align the rotor normal to the local flow direction. Misalignment of the approach flow contributes to azimuthal variations in blade loading as the incident flow vector varies with the blade passing inboard and outboard of the fence. This is of relevance to turbine fence operators as the fluctuating blade loading may contribute to turbine fatigue. A secondary objective of yawing the outermost turbines within the fence is to also reduce this load variation. This is analysed using a blade resolved sliding mesh model of the fence, purpose-built to simulate a tidal fence of 4 turbines. 


\section{Methodology}

\subsection{Simulated turbine}

A four turbine fence of 8 m diameter ($D$), three-bladed turbines spaced at $s/D$=0.25 was used in this study. The turbine was designed using the Reynolds-Averaged Navier-Stokes (RANS) embedded Blade Element (BE) method outlined in Chapter \ref{Chapter5}, this method accounts for the effects of local blockage through implementing \cite{McIntosh2011}. Some alterations to the design were made towards the tip and root of the rotor to avoid rapid transitions in blade twist and intersecting blade geometries. A NACA-63-415 aerofoil profile was used along the length of the blade; the 2D aerofoil data input to the RANS-BE model was generated using ANSYS Fluent 19.2 \cite{ANSYSInc.2013}. The turbine was designed to operate at a tip-speed-ratio ($\lambda$) of 4.5 in order to limit cavitation at its operational depth (10 m) based on an approximation following \cite{Wimshurst2018}. The tip-speed-ratio was defined as $\lambda={\omega R}/{u_{\infty}}$ for this study, where $\omega$ is the rotational speed of the turbine, and $R$ is the turbine radius. Within this study, $\lambda$ has been scaled by the free-stream flow speed ($u_{\infty}$) and not the free-stream flow normal to the rotor ($u_{x'}$).


\subsection{Computational model}

Analogous to the turbine fence modelled in Chapter \ref{Chapter3}, a fence of four turbines (two outer and two inner rotors) was analysed in a planar fence configuration with an inter-turbine spacing of $s=0.25D$. The turbine fence was modelled in ANSYS Fluent 19.2 \citep{ANSYSInc.2013} using a transient sliding mesh method. This method solves the unsteady RANS equations over multiple domains; an inner domain which contains the turbine geometry and outer domain which prescribes the channel boundaries, an interface surface connects the domains. The turbine movement is directly simulated though a rotational transformation of the inner domain. The four turbine long fence, was positioned at the centre of a 550 m wide and 30 m deep channel ($B_g=1.2\%$) matching the channel dimensions of Strangford Lough in Northern Ireland. The fence was submerged at a nacelle depth of 10 m. To exploit the system's symmetry and reduce the computational cost, the rotors were designed to be counter-rotating about the fence's centre and positioned around the channel's centerline. This halved the computational load as only two turbines needed to be modelled by exploiting model symmetry. Figure \ref{fig6:Computational_domain} shows the computational domain highlighting the boundary conditions applied to the model. A uniform inflow velocity of 2.5 m/s was applied at the inlet. 

\begin{figure}
    \centering
    \includegraphics[width=1\textwidth]{Figures/Chapter_6/comp_domain_annotated.png}
    \caption{Computational domain for yawed turbine simulations with representative channel geometry based on Strangford Lough.}
    \label{fig6:Computational_domain}
\end{figure}


This simulation's computational mesh took the unstructured rotor mesh from Section \ref{CH5:sliding} and filled the surrounding domain with unstructured tetrahedral elements. In total, the mesh contained 23 million elements for the two rotor domain. This element count was kept relatively small due to the high $Re$ of the simulation permitting the use of the $k-\omega$ SST turbulence model. The unsteady Reynolds-Averaged Navier-Stokes equations were solved over a total of 12 complete turbine rotations.



\subsection{Yawed turbine simulations}

% Yawed sweep (constant TSR)
Seven yaw angles ($\Psi= [-10^\circ, -5^\circ, -2.5^\circ, 0^\circ, +2.5^\circ, +5^\circ, +10^\circ]$) were investigated as part of this study. The yaw angle was defined by a rotation of the outer turbine about the innermost position of the blade tip as shown in Figure \ref{fig6:Yaw_define}. Two other reasonable alternative yaw definitions are also presented in Figure \ref{fig6:Yaw_define}, yawing about a support tower and yawing about the centre of the rotor plane. While these alternative definitions are more practical for the physical implementation, the chosen method preserves the inter-turbine spacing between the inner and outer turbines, maintaining the local blockage of the inner turbine. Yawing the turbine about the tower would alter both the inter-turbine spacing and the stream-wise offset whereas yawing about the centre of the rotor plane would decrease the inter-turbine spacing with non-zero yaw. Both these methods make it difficult to isolate the effect of yaw in isolation. All turbines within the fence were operated at the same angular speed ($\omega = 2.81 \:\text{rad/s}$). This speed corresponded to the design $\lambda=4.5$ for a single turbine with a $u_\infty$ of \mbox{2.5 m/s} in the same global blockage conditions.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{Figures/Chapter_6/Alt_yaw_definePNG.PNG}
    \caption{Plan view of yaw applied to the outer turbine with the inner turbines remaining unchanged. (a) is yawed about the innermost region of swept area, (b) is yawed about a support tower and (c) is yawed about the center of the rotor. Yaw angles pointing towards the centre of the fence are positive in this study. }
    \label{fig6:Yaw_define}
\end{figure}


\subsection{Stream-wise adjustment}

% Streamwise displacement 
Stream-wise displacement of the outer turbine was also investigated for two yaw angles, $\Psi=0^\circ$ and $\Psi=+2.5^\circ$. Stream-wise offsets between -0.25$D$ (-2 m) and 0.25$D$ (2 m) were examined in this study to test the observations of \cite{Draper2014} and \cite{Hunter2015} around optimal stream-wise offsets. The study of \cite{Draper2014} examined two row's of turbines with the rows either staggered or aligned. \cite{Draper2014} suggested that the optimal layout for a fixed number of turbines is in a single plane, however, owing to the limitation of the LMADT model used in this study, only staggered turbines outside the near-wake region were considered. \cite{Hunter2015} was able to examine staggered fences in the near-wake region using a RANS actuator disk model and showed that a small performance improvement is possible when the disk's were differentially tuned. This aspect of the study aims to provide further support for \cite{Hunter2015} results by explicitly resolving turbine geometry in the model.

\subsection{Differential operation}
% Differential operation
The $\Psi=0^\circ$ and the $\Psi=5^\circ$ cases were further optimised by individually adjusting the rotational speeds of the turbines. To determine this operating point, the fence was first optimised for synchronised $\lambda$, then the outer and inner turbine $\lambda$ were adjusted individually until the total fence power was at a maximum. This process converged within a few iterations.  


\section{Results}


Excluding the differential speed controls analysis, the results presented in this parameter study are for a fixed turbine rotational speed ($\omega$) of 2.81 rad/s corresponding to a $\lambda$ of 4.5. Adjusting $\lambda$ had limited influence on the turbine power coefficients and the relative difference between the inner and outer turbines, as shown in Figure \ref{fig6:turbine_lambda_opt}. Increasing the tip speed ratio to $\lambda=5.4$ increased fence $C_p$ by 1.8\% to $C_p=0.542$ and fence $C_t$ by 2.4\% to 0.971 due to a higher turbine resistance being achievable through the local blockage created by the fence. Qualitatively the relationship of power and thrust between the inner and outer turbines was the same when operating at the optimal $\lambda$=5.4 condition and the sub-optimal $\lambda$=4.5.


\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{Figures/Chapter_6/Cp_ct_opti.png}

\caption{Inner (red), outer (blue) and fence-averaged (black) turbine thrust coefficient $C_t$ (left) and power coefficient $C_p$ (right) over a range of $\lambda$ for 0$^\circ$ (solid) and +5$^\circ$ (dashed) outboard turbine yaw angles. The maximum turbine $C_p$ is marked with crosses.}
\label{fig6:turbine_lambda_opt}
\end{figure}



\subsection{Non-yawed turbine flow}\label{non_yawed_turbine_flow}

The performance of the four turbine fence was initially characterised for the no outer turbine yaw condition. Outer turbine $C_p=0.529$ and $C_t=0.909$ were both observed to be lower than the equivalent inner turbine parameters ($C_p$ = 0.536, $C_t$ = 0.916). The difference between them is described by the symmetry factor, $\Phi$. $\Phi$ quantifies the balance between a parameter for the inner and outer turbines with a positive value indicating a bias towards the inner turbine. This is defined as

\begin{equation}
    \Phi_{X} = \frac{X_{inner} - X_{outer}}{X_{inner}}
\end{equation}

\noindent where $X$ is a turbine metric (e.g., $C_p$ or $C_t$). For the no yaw condition there was a 1.3\% bias in power production towards the inner turbine ($\Phi$ $C_p$ = 0.013) and a 0.8\% bias in thrust ($\Phi$ $C_t$ = 0.008).

% load variation on the blades
% 
\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{Figures/Chapter_6/RBM_No_yaw.png}
\caption{$C_{FW}$ (bold) and $C_{EW}$ (normal) azimuthal variation for inner (red) and outer (blue) turbines. }
\label{fig6:RBM_no_yaw}
\end{figure}


While the differences in integrated quantities of $C_p$ and $C_t$ were relatively small between the inner and outer turbines, a greater disparity was observed in the blade loading. Blade root bending moments were used to quantify the loading as the blade moved around the azimuth. This loading was split into two components, $C_{FW}$ which is the root bending moment coefficient normal to the rotor plane (flapwise direction), and $C_{EW}$, which is the root bending moment coefficient in the plane of rotation of the turbine (edgewise direction). These coefficients were defined as 
\begin{equation}
    C_{FW} =  \frac{M_{FW,i}}{\tfrac{1}{2} \rho u_{\infty}^2 A_t R} \quad
    C_{EW} =  \frac{M_{EW,i}}{\tfrac{1}{2} \rho u_{\infty}^2 A_t R}
\end{equation}


\noindent where $M_{FW,i}$ is the root bending moment of blade $i$ in the flapwise direction and $M_{EW,i}$ is the root bending moment of blade $i$ in the edgewise direction. Figure \ref{fig6:RBM_no_yaw} plots the azimuthal variation in $C_{FW}$ and $C_{EW}$ for the outer and inner turbines. This plot clearly demonstrates that the amplitude of loading experienced by the outer turbine (amplitude $C_{FW}$=0.015) is 48.0\% greater than the inner turbine (amplitude $C_{FW}$=0.010). The cause of this is explored in Figure \ref{fig6:turbine_yaw_appoch_angle} which shows that the flow angle at hub height through the turbine plane is significantly larger for the outboard side than the inboard side. This flow mis-alignment, in conjunction with the reduced mass flux through the rotor, results in a drop off in load as the blade passes the end of the fence. 

The load signal also demonstrates a once-per-revolution peak in load fluctuations for the outer turbine and a twice-per-revolution signal for the inner turbine, these peaks correspond to the blade passing an adjacent turbine. Due to the symmetry assumption, the peak for the inner turbine's root bending moment at 90$^\circ$ corresponds to the blade passing a counter rotating turbine, qualitatively, this is similar to the peak at 270$^\circ$ where the blade passes a co-rotating turbine. Given this observation, results presented here can be considered indicative of a fence of four turbines all spinning in the same direction.



% Difference in inflow angle
\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{Figures/Chapter_6/Approach_angle.png}
\caption{Flow approach angle relative to the turbine plane at hub height along the fence, 0$^\circ$ is normal to the turbine plane. The grey area marks the portion of the transect where the turbine blade sweeps through and the black area is the location of the nacelle. }
\label{fig6:turbine_yaw_appoch_angle}
\end{figure}






\subsection{Impact of outboard turbine yaw on performance}\label{ch6:int_perf}

The $C_p$, $C_t$ and volume flux ($Q$) (averaged over a complete rotation) are presented in Figure \ref{fig6:CP_yaw} for varied outer turbine yaw angles. The fence average for all three quantities are maximised when the outer turbine yaw angle is 0$^\circ$. The outer turbine produces maximum power when it is yawed inwards by +2.5$^\circ$, but this coincides with a decrease in inner turbine performance, resulting in an overall reduction in fence power. An approximately linear decrease in the inner turbine performance was observed as the outer turbine yaw angle increased. This result highlights an interaction between the turbine performance and the constraint on the bypass flow between inner and outer turbines. Figure \ref{fig6:yaw_streamlines} plots the velocity contours and streamlines at the hub height over half the turbine fence and Figure \ref{fig6:Approach_flow} plots the inflow angle and velocity normal to the turbines at hub height. At negative outer turbine yaw angles a subtle increase in velocity across the entire inner rotor is observed, leading to a increase in power. The increased velocity is sustained by the outer turbine imposing greater confinement on near wake expansion at negative yaw angles. 

\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{Figures/Chapter_6/Cp_ct_q.png}
\caption{Turbine thrust coefficient $C_t$ (left), power coefficient $C_p$ (middle) and normalised volume flux $\overline{Q/Q_\infty}$ (right), for inner (red) and outer (blue) turbines at varied yaw angles. The fence average is plotted in black.}
\label{fig6:CP_yaw}
\end{figure}


\begin{figure}[ht]
\centering
\includegraphics[width=0.32\textwidth]{Figures/Chapter_6/yaw-10_360_deg_zommed.png}
\includegraphics[width=0.32\textwidth]{Figures/Chapter_6/yaw0_360_deg_zommed.png}
\includegraphics[width=0.32\textwidth]{Figures/Chapter_6/yaw10_360_deg_zommed.png}
\caption{Velocity streamlines at hub-height for half of a four turbine fence. The inner turbine is on the left and the outer turbine is on the right. The outer turbine yaw alignment has been adjusted by -10$^\circ$ (left), 0$^\circ$ (middle) and +10$^\circ$ (right). }
\label{fig6:yaw_streamlines}
\end{figure}


\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{Figures/Chapter_6/Approach_angle_flow_Yaw (1).png}
\caption{Transect of turbine inflow angle (top) and inflow velocity (bottom) at rotor hub height for the inner (left) and outer (right) turbines. Inflow angle and velocity are determine relative to the turbine plane as it yaws. Data has been averaged over a complete turbine rotation.}
\label{fig6:Approach_flow}
\end{figure}


For the outer turbine, the volume flux through the turbine appears to be proportional to the cosine of the yaw angle (as is the swept area normal to the free stream flow). A slight angular offset to this relation is observed with outboard turbine performance optimised at a yaw angle of +2.5$^\circ$. The inflow alignment correction can be observed in Figure \ref{fig6:yaw_streamlines} on the outer turbine for the +10$^\circ$ yaw case. In this plot, the inflow angle passing the outer portion of the turbine is close to orthogonal to the blade; correcting for the misalignment observed in the 0$^\circ$ yaw case. Despite the correction over the outer portion of the rotation, the inner region is made worse at higher yaw angles, so too is the velocity normal to the rotor. Figure \ref{fig6:Approach_flow} plots the approach angle (averaged at hub height over a rotation) alongside the normal component of velocity to the turbine. A decrease in inflow angle magnitude is observed as the yaw is increased, with the largest yaw angle (+10$^\circ$) producing between 5$^\circ$ to -10$^\circ$ misalignment over the outer portion of the blade relative to the no yaw case which had between 2$^\circ$ and -22$^\circ$.

The integrated quantities presented in Figure \ref{fig6:CP_yaw} arise due to the combined effect of all the blades through a rotation, however due to the asymmetric flow an individual blade's contribution varies with azimuthal position. Figure \ref{fig6:CP-contours} plots this variation with different outboard turbine yaw angles. Also important to the blade loading of yawed turbines is the direction of rotation. Changing the spin direction causes a 180$^\circ$ phase shift of the loading from the blade advancing and retreating effect. To decouple this effect Figure \ref{fig6:CP-contours} includes the azimuthal variation of individual blade $C_p$ for co-rotating, counter-rotating (outer turbine spin direction reversed), and the average over both directions. This plot demonstrates that much of the azimuthal variation in blade performance results from the turbine blade advancing and retreating. As the rotation is reversed the blade advancing and retreating load is offset by 180$^\circ$; the super position of this loading with the confinement loads (passing the adjacent turbine's blades) results in a 90$^\circ$ shift in the peak load when the turbine spin direction is reversed. Averaging over the co-rotating and counter-rotating outer turbine results demonstrates two regions where flow properties are, on average, enhanced. The first is with a yaw angle of -5$^\circ$ as the outer turbine blade passes the inner turbine ($\theta$=90$^\circ$). At this azimuthal position, the constructive interference between the turbines is at a maximum due to the distance between adjacent blades being a minimum and the negative yaw angle imposes an additional lateral constraint on the inner turbine. The secondary peak observed in the plot occurs at the +10$^\circ$ yaw angle over the outermost portion of the fence ($\theta$=270$^\circ$). This localised improvement is primarily associated with flow realignment around the fence, as mentioned above and observed in Figure \ref{fig6:yaw_streamlines}. 



\begin{figure}[ht]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Chapter_6/Cp_all_Outboard_turbine_rotated_averaged (1).png}
\caption{Inner (left column) and outer (right column) turbine $C_p$ variation with angular position for varied turbine yaw angles. The top row corresponds to co-rotating inner and outer turbines, the middle to counter-rotating inner and outer turbines, and the bottom is the average of the rotation directions. The inner turbine always rotates in the anticlockwise direction.}
\label{fig6:CP-contours}
\end{figure}

The two regions of improvement highlight the balance between optimising the flow alignment relative to the outer turbine and increasing flow confinement, these effects are optimised at different yaw angles and azimuthal positions for the individual blade. Given that multiple blades experience this loading (albeit with a phase offset), the maximum rotor power is achieved at an outer turbine yaw angle somewhere between the positions which optimise the two regions of enhanced performance ($\Psi$=+2.5$^\circ$). This optimum yaw angle also corresponds to minimising the variation in the power production over a rotation. 

The differences between co-rotating and counter-rotating turbines had little influence on the integrated performance of the fence. Furthermore no significant impact on the inner turbine blade loading was observed when the direction of rotation of the outer turbine was changed. This supports the assertion presented in Section \ref{non_yawed_turbine_flow} that results from modelling two turbines with a symmetry plane are indicative of a fence of four turbines all spinning in the same direction.


\subsection{Impact of outboard turbine yaw on loading}\label{sec6:Blade_load_yaw}

Figure \ref{fig6:Blade_forces_dist} shows the flapwise ($C_{FW}$) and edgewise ($ C_{EW}$) root bending moments coefficients. The $C_{EW}$ plot supports the observation made in Section \ref{ch6:int_perf} that the variation in power production throughout a rotation is minimised at $\Psi=+2.5^\circ$ (given $P \propto \sum_{i=1}^3{M_{EW,i}*\omega}$). A similar trend is also observed in the flapwise direction for the same yaw position. The outboard turbine loading in both flapwise and edgewise directions is smaller in amplitude than the inboard turbine for $\Psi=+2.5^\circ$, in addition, at this yaw angle the outer turbine bending moment amplitude is smaller than both inner and outer turbines for the $\Psi=0^\circ$ case. This is of practical relevance to turbine manufacturers as it influences blade fatigue estimates. At the $\Psi=+2.5^\circ$ position, the outer turbine loading amplitude in the flapwise direction is 0.008 (47.4\% reduction from $\Psi=0^\circ$), this is similar to the inner turbine (0.010) and corresponds to the point that minimises the variance due to the combined effects of blade advancing and retreating and flow alignment. 

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.99\textwidth]{Figures/Chapter_6/box_whisker_Standard.png}
    \caption{Box and whisker plots of $C_{EW}$ (left) and $C_{FW}$ (right) for inner (red) and outer (blue) turbines for varied outboard turbine yaw angles. 
    }
    \label{fig6:Blade_forces_dist}
\end{figure}

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.99\textwidth]{Figures/Chapter_6/Line_RBM.png}
    \caption{Azimuthal variation in $C_{EW}$ (left) and $C_{FW}$ (right) for co-rotating (top), counter-rotating (bottom). Inner and outer turbines are plotted as if viewing from front on.}
    \label{fig6:Azi_lod_var}
\end{figure}

The azimuthal variation in $C_{EW}$ and $C_{FW}$ is plotted in Figure \ref{fig6:Azi_lod_var}. Concentric profiles of both $C_{EW}$ and $C_{FW}$ for the inner turbine indicate a uniform increase in loading around the azimuth, consistent with increasing the mass flux over the entire turbine. The outer turbine experiences more significant changes in loading from being yawed due to blade advancing and retreating effects. This is demonstrated by the $\sim$180$^\circ$ phase shift in $C_{EW}$ and $C_{FW}$ peaks between the co-rotating and counter-rotating turbines as the higher relative blade speed shift peaks from inboard to outboard of the fence. 


Figure \ref{fig6:Force_pres} plots the axial and tangential force distribution per unit span along a blade from the outer turbine as it passes the inner-outer turbine bypass ($\theta$=90$^\circ$) and the fence bypass ($\theta$=270$^\circ$). This plot shows an impact on both the axial and tangential force distribution ($F_x$ and $F_{\theta}$), with the highest impact of yaw being localised around the final quarter of the blade. The pressure distribution at the 85\% radius position ($r=0.85R$), also presented in Figure \ref{fig6:Force_pres}, shows that the change in blade force is associated with an uplift in static pressure around the entire section. Most notably, there is an increase in suction on the suction surface towards the peak (near the leading edge) however, there is also a small pressure rise over the pressure surface.

Figure \ref{fig6:Pressure_azi} plots the suction surface towards the tip of a blade on the outboard turbine. This plot demonstrates that the greatest improvement in suction peak is achieved for the outboard passage ($\theta$=270$^\circ$) for $\Psi=+10^\circ$, whereas there is a decrease in the suction peak when $\Psi=-10^\circ$ over the same region. This is intuitive when comparing to the inflow and alignment plot (Figure \ref{fig6:Approach_flow}) as there is a higher flow speed and improved alignment over the outboard region when the turbine is yawed by $\Psi=+10^\circ$. The azimuthal position of the higher suction also correspond to the regions of greater torque observed in Figure \ref{fig6:Azi_lod_var}. At locations around $\theta$=90$^\circ$ and $\theta$=270$^\circ$ the blade advancing and retreating effect is small due to the relative motion of the blade transitioning from advancing to retreating. This leads to a change in pressure over the blade surface as a result of the change in velocity, the magnitude of this effect is reversed between positive and negative yaw cases. 


\begin{figure}[ht]
    \centering
    \includegraphics[width=1\textwidth]{Figures/Chapter_6/pressure_dist.png}
    \caption{Turbine normal, $F_x$, (red) and turbine plane ,$F_{\theta}$, (left) force distribution and blade pressure distributions at r/R=0.85 (right) for a blade on the outer turbine passing inboard (top) and outboard (bottom) of the fence. Results are presented for the co-rotating turbines.}
    \label{fig6:Force_pres}
\end{figure}
\begin{figure}[ht]
    \centering
    \includegraphics[width=1\textwidth]{Figures/Chapter_6/Suction_surface_radial_rR0pnt85to1.png}
    \caption{Static pressure coefficient on the blade suction surface over the tip region (0.85R-1R) every 45$^\circ$ indicated by the blade plot angle for $\Psi=-10^\circ$ (left), $\Psi=0^\circ$ (middle) and $\Psi=+10^\circ$ (right). The $\Psi=-10^\circ$ and $\Psi=+10^\circ$ are plotted as the difference from the $\Psi=0^\circ$ case. }
    \label{fig6:Pressure_azi}
\end{figure}








\subsection{Differential operation of yawed turbines}

Allowing variable speed operation of the turbines permits additional power extraction from the fence as each turbine can be optimised for the specific flow conditions and constructive turbine-turbine interaction effects. Figure \ref{fig6:Differential_control_curves} shows that individually, the outer turbine achieves more power when the rotational speed is reduced from $\lambda=5.41$ (which is the optimal $\lambda$ for synchronised controls) to $\lambda=5.25$ due to the lower inflow velocity resulting from the fence end-losses. However, the speed reduction of the outer turbine required to maximise overall fence power is less due to the lateral flow confinement it provides to the inner turbine. The optimal fence power is achieved when the outer turbine operates at $\lambda=5.36$. This result is within 0.2\% for both the yaw cases tested ($\Psi=0^\circ$ and $\Psi=+5^\circ$). The figure also shows that spinning the outer turbine faster enhances the inner turbine $C_p$ as a linear function of the outer turbine $\lambda$ and reduces the outer turbine $C_p$ proportional to $\lambda^2$ over the range tested. 


Comparing between the $\Psi=0^\circ$ and $\Psi=+5^\circ$ simulations, the impact of yawing the outer turbine reduces performance, especially for the inner turbine. The physical intuition for this phenomenon is similar to that presented in the yaw adjustment section, where increased yaw removed the lateral constraint on the inner turbine. In this case, yaw angles are fixed, and a lateral constraint is imposed by increasing the outer turbine $\lambda$. This action is dampened when the turbine is in a yawed position leading to slightly less power in those cases. 

\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{Figures/Chapter_6/Cp_ct_differ (1).png}
\caption{Inner (red), outer (blue) and fence averaged (black) turbine $C_t$ (left) and $C_p$ (right) over a range of outboard turbine $\lambda$ with a fixed inboard turbine $\lambda$ of 5.41 for $\Psi$=0$^\circ$ (solid) and $\Psi$=5$^\circ$ (dashed). The maximum turbine $C_p$ is marked with crosses.}
\label{fig6:Differential_control_curves}
\end{figure}


\begin{figure}[ht]
    \centering
    \includegraphics[width=1\textwidth]{Figures/Chapter_6/differential_control_optimisation.drawio (2).png}
    \caption{Strategy to match thrust and blade load amplitude on a 4 turbine fence.}
    \label{fig6:differential_process}
\end{figure}


% Analysis of the combined optimisation
The surfaces in Figure \ref{fig6:Differential_optimised} are the output of the differential speed operation optimisation process. Average fence $C_p$ and $C_t$ have been scaled to show the percentage improvement relative to the optimal synchronised operation case of the $\Psi=0^\circ$ fence, $C_p^*$ and $C_t^*$.

Two differential speed control methods were investigated in this study. The first (strategy 1) was focused on optimising the total turbine power and was determined through a progressive iteration of inner and outer turbine $\lambda$. The second strategy (strategy 2) is presented in Figure \ref{fig6:differential_process}, focusing on the trade off between total power generation and turbine loading. Through this process average turbine $C_p$ and $C_t$ were matched between the inner and outer turbines. An additional degree of freedom is introduced by allowing the outer turbine to yaw. This makes it possible to also balance the amplitude of the blade root bending moments between the inner and outer rotors, ensuring that turbines experienced closely matched loading conditions.


The black cross (strategy 1) in Figure \ref{fig6:Differential_optimised} shows that there is a combination of turbine $\lambda$ for which fence performance is marginally improved ($<$0.25\%) above the synchronised control case. The green star (strategy 2) shows a combination of inner and outer turbine $\lambda$ which reduces the overall azimuthally-averaged thrust on the system and balances the azimuthally-averaged loading between the inner and outer turbine.

By operating the outer turbine at $\lambda=5.21$ and the inner turbine at $\lambda=5.07$, the two rotors have the same azimuthally-averaged thrust coefficient and the added benefit of a reduction in the total thrust on the system of 1.5\%. There is still a 1.5\% bias in power production towards the inner turbine. From a blade loading perspective this is of less concern as the edgewise bending moments are significantly smaller than the flapwise bending moments. The amplitude of the flapwise root bending moment for the outer turbine is close to unchanged under this operation strategy, being close to double that of the inboard turbine ($\Phi C_{FW} Amplitude=0.5$).

\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{Figures/Chapter_6/Contours_differential_alt.png}
\caption{Averaged turbine fence $C_p$ (left), $C_t$ (centre-left), $\Phi C_p$ (centre-right) and $\Phi C_{FW}$ Amplitude (right) with differential operation for $\Psi=0^\circ$ (top) and $\Psi=+5^\circ$ (bottom). $C_p$ and $C_t$ have been scaled by their respective values for the optimal operating condition with synchronised controls for $\Psi=0^\circ$ ($C_p^*$ and $C_t^*$). Red circles indicate the data points the contours were interpolated from and the black cross is the operating condition which produces the maximum power. The green star is the operating point for the operating strategy presented in Figure \ref{fig6:differential_process}}
\label{fig6:Differential_optimised}
\end{figure}


\begin{table}[]
\centering
\caption{Comparison between differential operation strategies with a 5\% outer turbine yaw}
\label{Tab:stat_compare}

\begin{tabular}{l|lll|lll}
                & \multicolumn{3}{c|}{Inner Turbine}   & \multicolumn{3}{c}{Outer Turbine}    \\
                & $C_p$ & $C_t$ & $C_{FW}\; Amplitude$ & $C_p$ & $C_t$ & $C_{FW}\; Amplitude$ \\ \hline
Strategy 1      & 0.543 & 0.972 & 2.251E-03            & 0.538 & 0.964 & 1.586E-03            \\
Strategy 2      & 0.543 & 0.953 & 2.221E-03            & 0.537 & 0.953 & 1.637E-03            \\ \hline
Difference (\%) & -0.2\% & -2.0\% & -1.3\%                & -0.1\% & -1.1\% & 3.2\%              
\end{tabular}
\end{table}


Using the same operating point but yawing the outboard turbine by 5$^\circ$ reduces the amplitude of $C_{FW}$ for the outer turbine by 49\%, less than inboard turbine (positive $\Phi C_{FW} $ amplitude) in Figure \ref{fig6:Differential_optimised}). This comes at the cost of a $<0.4\%$ reduction in fence power but it is advantageous as both the load magnitude and amplitude are closer to uniform along the fence. Further optimisation of yaw over the range of $0^\circ<\Psi<+5^\circ$ is expected to equalise both the azimuthally averaged thrust and the flapwise blade root bending moment amplitude. This effectively ensures that the same loading conditions are experienced by both turbines.

A comparison between the two strategies operating with a 5\% outer turbine yaw is made in Table \ref{Tab:stat_compare}. This table demonstrates how strategy 2 makes a small sacrifice in turbine power (reduction of less than 0.2\%) to reduce the thrust experienced by the turbines (reduction of more than 1.1\%). A slight increase is observed in the outer turbine $C_{FW}\; Amplitude$ relative to strategy 1, however outer turbine $C_{FW}\; Amplitude$ is less than inner turbine $C_{FW}\; Amplitude$ for both strategies. Further iterations of strategy 2 would result in the yaw angle decreasing and outer turbine $C_{FW}\; Amplitude$ increasing to match that of the inner turbine.


\subsection{Wake deflection}
Deflection of the outer turbines wake as a result of turbine yaw was observed in this study. As the turbine is yawed towards the centre of the fence, so too was the wake. Figure \ref{fig6:Velocity countours wake} shows the difference in the wake shape between the $\Psi$=0$^\circ$ and $\Psi$=+10$^\circ$ cases 4$D$ downstream of the turbine plane. This figure highlights a smaller wake width as flow progresses away from the rotor plane. 

Wake deflection may be a useful tool in larger scale tidal turbines farms to optimise total farm energy output. Wake steering has been shown to improve energy production in wind farms by diverting wakes away from downstream turbines \citep{Archer2019}. Similar methods using these controls may also provide benefit to tidal farms. The tidal application of these controls is slightly different from wind, which employs dynamic yaw controls to steer wakes from downstream turbines. The process needs to be dynamic in wind to adjust for the varying flow directions. For the tidal application, the predominate flow direction is known due to the periodic nature of the tides. In this case, yaw controls to reduced the wake width behind short turbine fences as shown in Figure \ref{fig6:Velocity countours wake} may allow a higher packing density of tidal turbines.

% \begin{figure}[]
%     \centering
%     \includegraphics[width=0.95\textwidth]{Figures/Chapter_6/yaw0_360_deg_1.png}
%     \includegraphics[width=0.95\textwidth]{Figures/Chapter_6/yaw10_360_deg_1.png}
%     \caption{Velocity magnitude contours and streamlines at mid-turbine depth of a four turbine fence with 0$^\circ$ outboard turbine yaw (top) and +10$^\circ$ outboard turbine yaw (bottom).}
%     \label{fig6:Velocity Sheets}
% \end{figure}

\begin{figure}[]
    \centering
    \includegraphics[width=0.95\textwidth]{Figures/Chapter_6/4D_ds_yaw.png}
    \caption{Velocity magnitude contours 4$D$ downstream of the turbine fence for 0$^\circ$ outboard turbine yaw (thin lines) and +10$^\circ$ outboard turbine yaw (thick lines).}
    \label{fig6:Velocity countours wake}
\end{figure}

\subsection{Stream-wise offset}

$C_p$, $C_t$ and $Q$ averaged over a complete rotation are plotted in Figure \ref{fig6:performance_streamwise}. This plot demonstrates that for a positive stream-wise offset (outer turbine advances into the flow), there is a decrease in outer turbine performance. This corresponded with an approximately equal (over the range of offsets tested) increase in inner turbine performance, having an overall neutral impact on fence power production. Examining the outer-inner turbine bypass region in Figure \ref{fig6:plan_offset} provides an insight to the physical process responsible for this observation. As the outer turbine advances into the flow, the lateral constraint on the inboard side is reduced. This leads to a reduction in mass flux through the turbine (Figure \ref{fig6:performance_streamwise}) and an increased bypass flow which is diverted towards the adjacent turbine. Because of this, mass flux through the system is close to constant. This phenomenon is an artefact of the turbine stagger being smaller than the length required for pressure recovery. Were the stream-wise offset much larger, performance of the outer turbine is expected to decay to that of a single turbine the inner turbines to that of a two turbine fence. This result expands on those of \cite{Draper2014} and \cite{Hunter2015}, suggesting that some small stagger is acceptable within a turbine farm when targeting optimal power. 

\begin{figure}
    \centering
    \includegraphics[width=1\textwidth]{Figures/Chapter_6/Cp_ct_q_offset.png}
\caption{Turbine thrust coefficient $C_t$ (left), power coefficient $C_p$ (middle) and normalised volume flux $\overline{Q/Q_\infty}$ (right), for inner (red) and outer (blue) turbines at varied outer turbine stream-wise offsets. The fence average is plotted in black, the no turbine yaw case is plotted with dashed lines and 2.5$^\circ$ yaw case is with a dotted line.}
    \label{fig6:performance_streamwise}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[width=0.32\textwidth]{Figures/Chapter_6/Yaw_0pnt0_offset_ 2pnt0_TSRfac_1pnt0_360_deg_1.png}
\includegraphics[width=0.32\textwidth]{Figures/Chapter_6/Yaw_0pnt0_offset_0pnt0_TSRfac_1pnt0_360_deg_1.png}
\includegraphics[width=0.32\textwidth]{Figures/Chapter_6/Yaw_0pnt0_offset_2pnt0_TSRfac_1pnt0_360_deg_1.png}
\caption{Velocity streamlines at hub-height for half of a four turbine fence. The inner turbine is on the left and the outer turbine is on the right. The outer turbine yaw offset has been adjusted by -2 m (left), 0 m (middle) and +2 m (right). }
\label{fig6:plan_offset}
\end{figure}

Figure \ref{fig6:contours_offset} plots the $C_p$ contribution of a blade around the azimuth of a turbine with varied outer turbine stream-wise offsets. This plot shows how the change in flow, observed in Figure \ref{fig6:plan_offset}, translates to a change in power. The outboard region of the inner turbine ($\theta$=270$^\circ$) and the inner region of the outer turbine ($\theta$=90$^\circ$) experience opposite trends with changing stream-wise offsets; the outer turbine increasing as the inner turbine decreases. This results in approximately constant total fence power. Given this result, it may appear that a slight negative offset is beneficial to match the $C_t$ or $C_p$ of the outer and inner turbines without a sacrifice on turbine power. However this adjustment impacts flow conditions primarily around the inner-outer turbine bypass (as shown by the outer turbine power production in Figure \ref{fig6:contours_offset}) and increases the unsteady loading on the system. 

% The contour plot in Figure \ref{fig6:contours_offset} helps to further the understanding of this problem. For the outer turbine, a single peak in power production is observed as the blade passes the adjacent turbine at a negative stream-wise offset. The peak softens as the turbine advances into the flow and flow conditions become closer to those of a single turbine. The inner turbine experiences two loading peaks at the 2 m stream-wise offset position for both 90$^\circ$ and 270$^\circ$ representing passing the inner and outer region of the fence respectively. The peak at 270$^\circ$ rapidly decreases at the lateral constraint is reduced, shifting the peak power producing zone of the rotation to the 90$^\circ$ position after the stream-wise offset is reduced below 1.3 m. 


% Blade loading

\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{Figures/Chapter_6/Cp_all_offsOutboard_turbine_rotated_averaged (2).png}
\caption{Inner (left) and outer (right) turbine $C_p$ variation with angular position for varied outer turbine stream-wise offsets.}
\label{fig6:contours_offset}
\end{figure}







% Outboard turbine yaw realignment towards the centre of the fence decreased the mass flux through the turbine when the turbines were constrained to operate at the same speed. This is a counter-intuitive result if the angled ends of the fence are considered to prevent flow from escaping around the fence. While, to some extent, this was shown to be true (more power was produced at the very ends of the fence), the interaction between turbines and redistribution of thrust along the fence is responsible for a drop in mass flux. To understand the cause for this, it is helpful to recall how performance is improved with local blockage. As local blockage increases, the turbine wake is more tightly constrained and a greater pressure drop can be sustained over a rotor. Adjusting the outboard turbine yaw in the way described in this study imposes a lateral thrust component upstream of the wake and removes the lateral constraint from the downstream side. Consequently, this has the effect of reducing the pressure drop and power for the inboard turbine. Therefore, it is expected that yawing the outboard turbines in the other direction will impose the lateral constraint on the wake and improve inboard turbine performance. 

%% ADD in additional plots


% greater area for flow to squeeze past reducing performance improvement


% Turbine blade loading magnitude and distribution were shown to be parameters augmented by turbine yaw in this study. It was shown that when the outboard turbine was at a yaw angle of 2.5 degrees, the loading was reduced from the without yaw case and the variation in blade loading was approximately equal to the inboard turbine. This loading was only slightly higher for the 5-degree case where the outboard turbine power was highest. This demonstrates that it is possible to get very close to balancing turbine thrust, power and loading cycles between the inboard and outboard turbines by yawing the outboard turbine. While this cases slightly reduces total fence power, the benefit of balancing these parameters is that the loading is the same along the fence and a designer can treat them as equals for maintenance scheduling purposes. 

\section{Model limitations}

This study employed an unsteady RANS model with a sliding mesh to resolve the rotation of a turbine within a fixed domain. The model has been useful to characterise the flow physics associated with the yawing of an outboard turbine within a fence; however, some key physics have been removed to facilitate a quick simulation. Notably, flow conditions have been simplified to allow the impact of outer turbine yaw to be isolated. Surface deformation, flow profiles and waves all impact the turbine loading and power production which may be influenced by yaw. Non-uniform flow in each axis will impact the individual turbine loading characteristics and the variation of the flow conditions along the length of the fence with interfere with how the turbines interact. Furthermore, the turbine towers have not been included in this study; the tower has been observed in experiments to reduce blade loading as it passes the tower \cite{Bahaj2007}. This effect is typically small; however, the physics of the problem changes when the turbine is yawed, and the downstream projection of the turbine on the tower is reduced.


The choice to analyse the outer turbine yawing about the inner-most point of the turbine swept area was an effective means to isolate the impact of yaw on the turbine fence separate to local blockage and stream-wise displacement. However, the practical implementation would likely require the turbine to rotate around the tower, located at an offset downstream of the turbine plane. A preliminary set of simulations with the turbine yawing around a tower has produced qualitatively similar results. Further simulation is required to determine the combined effect of varying the inter-turbine spacing and outer turbine yaw angle when the outer turbine is yawed about a fixed tower.

% An ensemble of simulations were run considering this effect with the outer turbine parameterised to yaw about a tower offset 0.33$D$ from the rotor plane. These results are plotted alongside the results using the primary yaw definition from this study in Figure \ref{fig6:yaw_alter_perf}. Qualitatively these definitions of yaw produce similar results, however there were some quantitative differences as observed in Figure \ref{fig6:yaw_alter_perf}. Notably a slight increase in power for the outer turbine in the tower yaw case. }

% \begin{figure}[ht]
% \centering
% \includegraphics[width=0.8\textwidth]{Figures/Chapter_6/Ct_compare_yaw.png}
% \includegraphics[width=0.8\textwidth]{Figures/Chapter_6/Cp_compare_yaw.png}

% \caption{Inner (red) and outer (blue) turbine thrust coefficient $C_t$ (top) and power coefficient $C_p$ (bottom) for yawing the outer turbine about the innermost portion of the turbine swept area (dashed lines) and yawing about a tower $0.33D$ downstream of the turbine plane.}
% \label{fig6:yaw_alter_perf}
% \end{figure}

Due to computational constraints, only a single yaw angle was investigated for differential operation; defining the optima requires significant iterations and model runs, making it impractical to consider a range of outer turbine yaw angles. For the $\Psi=5^\circ$ case, which was optimised for differential controls, the resulting operating condition was similar to that of the $\Psi=0^\circ$ suggesting only a small variation in yaw angle is required to tune the differential controls; further simulation is required to verify this hypothesis. This physical reason is that small yaw angles do not significantly impact the inflow (inflow is proportional to the cosine of the yaw), and the leading order effect from the yaw is blade advancing and retreating which quickly increases when outside of the optimal alignment conditions. 

%Expand on the types of (more) realistic flow conditions that would need to be considered:

%e.g. non-uniform in each axis, extent of variation expected in undisturbed flow along length of fence. 



\section{Conclusions}

This study has analysed the impact of yaw controls on the outermost turbines of a four turbine fence operating in blockage conditions representative of Strangford Lough. It has been demonstrated that a small adjustment in the outboard turbine yaw angle towards the fence centre results in a performance improvement for that turbine. However, this comes at the cost of reduced flow confinement around, and hence performance of, the inboard turbines.

Both co-rotating and counter-rotating turbines operating with synchronised controls were analysed in this study. While the direction of rotation was not observed to influence power output, it did significantly influence the azimuthal variation in outer turbine blade loading due to the advancing and retreating blade effect. Averaging over co-rotating and counter-rotating simulations allowed the blade advancing and retreating effect to be controlled for and highlighted the two factors influencing performance within the fence with the outer turbine yawed. These were confinement of the bypass flow between the inner and outer turbines and improved alignment of the incident flow for the outer turbine. The former was optimised for the inner turbine at $\Psi>-10^\circ$ and the latter was optimised for the outer turbine at $\Psi=+2.5^\circ$. Given the balance of the two effects, the optimal arrangement for power generation when the rotors are performing with synchronised controls was $\Psi=0^\circ$. Despite this result, yaw was effective in reducing the amplitude of loading of the outer turbine, with a yaw angle of +2.5$^\circ$ reducing the $C_{FW}$ amplitude by 47.5\% (from 48\% greater than the inner turbines amplitude to 22\% less). At this yaw angle, the influence of the blade advancing and retreating effect was minimised; this is advantageous for fatigue and maintenance planning as load cycles on the inner and outer turbines can be treated as equivalent. 

Differential turbine speed operation was also investigated in this study. While it was found that no significant improvements in power were realised by operating the turbines at different speeds, an operating point was identified for the $\Psi=0^\circ$ case which produced the same power as the optimal synchronised turbine operation case, reduced fence $C_t$ by 1.5\% and balanced the thrust between the inboard and outboard turbines. A similar operating condition was found for the $\Psi=+5^\circ$ case; however, this configuration produced 0.4\% less power than the $\Psi=0^\circ$ layout.  Despite this, the configuration may be preferable as the variation in outboard turbine loading was halved.






% An outboard turbine yaw control strategy for a turbine fence consisting of four 8 m diameter turbines was investigated. A range of turbine yaw angles between 0 and 15 degrees were modelled using a RANS sliding mesh model in oceanographic conditions representative of Strangford Lough. It was found that yaw adjustments made within the range considered in this study acted to reduce the volumetric flow rate through both inboard and outboard turbines due to reduced frontal projection of the outboard turbine and reduced pressure drop for the inboard turbine. Despite this, a marginal performance improvement was observed for the outboard turbine when it was yawed by 5 degrees. This acted to correct for flow misalignment due to diversion around the fence, permitting the turbine to extract slightly more power. This power was extracted at a lower thrust and with reduced variation in blade loading. While the percentage improvement was marginal for the integrated $C_t$ and $C_p$ parameters, significantly less variation in the loading over a blade rotation was observed with the yaw adjustment. This resulted in the amplitude of the loading on the outboard being close to equal to the inboard turbine, with the inboard turbine mean loading being slightly higher. 

% The wake was observed to reduce in width downstream of the fence as the outboard turbines were yawed, steering the outboard turbine wakes into the fence's centre-line. This can improve turbine farm layout by decreasing downstream separation between turbines and improving turbine density. 

% Yaw adjustments of outboard turbines present an exciting opportunity within turbine fences to improve the load distribution along a fence with minimal alteration of the turbines, relying on the existing yaw mechanisms. More work still has to be undertaken in this space to analyse the impact of a stream-wise offset to enhance the turbine performance and investigate if inboard turbine performance can be improved alongside the outboard turbine. 




% \subsection{Integrated performance parameters}\label{ch6:int_perf}

% Figure \ref{fig6:Volume_flux} plots the volume flux averaged over a complete rotation of a turbine and the flux variation over a rotation. It was anticipated that yawing the outboard turbine would correct the flow that is diverted around the fence, improving the mass flux through the turbine. However, examining this figure shows a monotonic decrease in the flux of both inner and outer turbines with yaw angle. The variation in flux with rotation shows the drop in flux is close to uniform at all azimuthal angles when yaw is adjusted. This implies the critical driver for the change impacts the entire swept area of the turbine instead of localised changes within the swept area. 

% \begin{figure}[]
%     \centering
%     \includegraphics[width=\textwidth]{Figures/Chapter_6/Average_flux.png}
%     \includegraphics[width=\textwidth]{Figures/Chapter_6/Flux_out.png}
%     \caption{Nondimensionalised flux through the turbine swept area averaged over a rotation (top) and variation in nondimensionalised flux over a turbine rotation (bottom) for varied yaw angles.}
%     \label{fig6:Volume_flux}
% \end{figure}

% Also of note, both inboard and outboard turbines have a 3 peaks-per-revolution signal. This signal is out of phase between the inboard and the outboard turbines and highlights the transient nature of the bypass flow from tip vortices interacting between the turbines. The velocity contour plot in Figure \ref{fig6:Velocity Sheets} shows this interaction. 

% % \begin{figure}[]
% %     \centering
% %     \includegraphics[width=\textwidth]{Figures/Chapter_6/yaw0_360_deg_1.png}
% %     \caption{Velocity contours at mid-turbine depth of a 4 turbine fence highlighting the out of phase tip vortices in the bypass flow between inboard and outboard turbines. }
% %     \label{fig6:velocity_contoures}
% % \end{figure}



% Figure \ref{fig6:Perf_Yaw} plots the variation in $C_p$ and $C_t$ with yaw angle. A similar trend to the mass flux, presented in Figure \ref{fig6:Volume_flux}, is observed in these two parameters with a linear decay of inboard turbine performance with yaw angle. This is unsurprising given the relationship of these parameters to velocity ($C_t \propto u^2$ and $C_p \propto u^3$). It is worth noting that the outboard turbine's performance is initially improved until the turbine reaches a yaw angle of 5 degrees ($C_p = 0.533$), where outboard turbine performance is optimised for yaw angle. At this configuration, the difference in thrust between inboard and outboard turbines is also at a minimum ($C_{t,inner} = 0.916$, $C_{t,outer} = 0.911$). This configuration does not optimise power over the entire fence due to the more significant decline of inboard turbine performance; instead, fence power is highest at 0 degrees yaw for the parameter space investigated ($\overline{C_p}=0.536$). There also appears to be a rapid decay in outboard turbine $C_p$ and $C_t$ beyond 10 degrees; this observation is consistent with those made by \cite{Shahsavarifard2020} in their study of a single turbine in yawed flow, crediting this decay to the reduction in flow through the turbine which is proportional to $\cos{\Psi}$. Nevertheless, the results indicate that some flow correction is achieved; the specific region on the rotation where this is achieved is presented in Figure \ref{fig6:Perf_theta}. 

% \begin{figure}[]
%     \centering
%     \includegraphics[width=\textwidth]{Figures/Chapter_6/Cp_Yaw.png}
%     \includegraphics[width=\textwidth]{Figures/Chapter_6/Ct_Yaw.png}
%     \caption{$C_p$ (top) and $C_t$ (bottom) averaged over a complete rotation of a turbine for varied yaw angles.}
%     \label{fig6:Perf_Yaw}
% \end{figure}

% % correct the legend
% \begin{figure}[]
%     \centering
%     \includegraphics[width=0.49\textwidth]{Figures/Chapter_6/Cp_inner_yaw.png}
%     \includegraphics[width=0.49\textwidth]{Figures/Chapter_6/Cp_outer_yaw.png}
%     \includegraphics[width=0.49\textwidth]{Figures/Chapter_6/Ct_inner_yaw.png}
%     \includegraphics[width=0.49\textwidth]{Figures/Chapter_6/Ct_outer_yaw.png}
%     \caption{Variation in blade $C_p$ (top) and $C_t$ (bottom) for inboard (left) and outboard (right) turbines. The subplots above the indicate the blade position for the outboard (blue) and inboard (red) turbines. The thicker blade in the subplot indicates the blade angular position.}
%     \label{fig6:Perf_theta}
% \end{figure}

% % Explain the physics in this paragraph
% Figure \ref{fig6:Perf_theta} plot the variation in blade $C_p$ and $C_t$ with the azimuthal position. The amplitude of the $C_p$ and $C_t$ signals varies significantly for the outboard turbine due to the yaw adjustment. For the outboard turbine at 0-degree yaw, there is a peak in blade $C_p$ at the 90-degree position (as it passes the inner turbine) and a minimum loading around the 300-degree position (at the end of the fence). The trend reverses at higher yaw angles, with maximum loading occurring around the 300-degree position and a new minimum loading occurring as the blade passes underneath the turbine. The result suggests an outboard turbine yaw angle of 10 degrees optimises power production around the end of the fence; however, there is an overcorrection at the inboard section of the rotor at this angle, leading to an overall loss in performance. The turbine yaw of 2.5 degrees appears to reduce variation in the $C_p$ signal by producing close to constant power over the turbine rotation. 

% Examining the inboard turbine's performance, we see a close to uniform offset with outboard turbine yaw due to the reduced flow rate discussed previously in this section. In addition to this, the peak at 270 degrees appears to soften with increased outboard turbine yaw, suggesting that constructive interference between the turbines is reduced with yaw. This is expected given the offset between adjacent turbine tips in the stream-wise direction, increasing the bypass flow size. 


% The loading envelope along the turbine blade length for inner and outer turbine blades in the stream-wise ($F_x$) and torque direction ($F_\theta$) is presented in Figure \ref{fig6:blade_load}. The loading envelope has been calculated by sampling the output at 15-degree intervals and tracking the min, mean and maximum at each radial station. As shown in Figure \ref{fig6:blade_load}, a greater change in loading is observed in the outboard turbine due to the rotor's yaw. An increase in load amplitude and a decrease in load magnitude is observed for high yaw angles ($\Psi > 10$ degrees) in both directions. At lower yaw angles, the average loading along the outboard blades is approximately the same; this is consistent with the observations made for the integrated $C_t$ coefficient presented in Figure \ref{fig6:Perf_Yaw}, where $C_t$ was close to constant for $\Psi < 7.5$ degrees. The loading amplitude relationship with smaller outboard turbine yaw angles is less readily observed in Figure \ref{fig6:blade_load}. Removing the annular averaged component from the force at each radial station allows a direct comparison of the deviatoric loading component to be made in Figure \ref{fig6:demean_blade_load}. This plot shows a non-linear relationship between the loading amplitude and the yaw angle for the outboard turbine. Moving from a yaw angle of 0 degrees to 5 degrees leads to a reduction in loading range, yaw angles beyond this point increase loading range. It is interesting to note that at the 2.5-degree and 5-degree yaw angles, the loading amplitude was of a similar range to the outboard turbine for both $F_x$ and $F_\theta$. This suggests that the yaw adjustment has effectively corrected the inflow angle and balanced the variation in loading between the inboard and outboard rotors. 

% Change in outboard turbine yaw angle appears to have a significantly smaller impact on inboard turbine loading. Figure \ref{fig6:blade_load} shows a slight decrease in loading in the $F_\theta$ direction across the entire blade length. In the stream-wise direction, the blade is most impacted over the range $0.4<r/R<0.9$. These two results support the observation made previously that $C_p$ and $C_t$ gradually decrease with outboard turbine yaw angle. Examining the deviatoric component in Figure \ref{fig6:demean_blade_load}, a marginal increase in loading envelope size with yaw angle can be observed. While this is small, it demonstrates the impact of reduced flow interaction between the turbines. It was noted in Section \ref{ch6:int_perf} that there was an increase in $C_p$ at the 270-degree position where the inboard and outboard turbines were closest together; as the yaw angle increased, this effect was reduced. This is the primary cause of the change in deviatoric loading on the inner turbine. 


% \begin{figure}[h]
%     \centering
%     \includegraphics[width=\textwidth]{Figures/Chapter_6/Blade_forces_Normal.png}
%     \caption{Blade loading variation range for different outboard turbine yaw angles in the stream-wise (top) and blade torque direction (bottom) for the inboard (left) and outboard (right) turbines.}
%     \label{fig6:blade_load}
% \end{figure}


% \begin{figure}[h]
%     \centering
%     \includegraphics[width=\textwidth]{Figures/Chapter_6/Blade_forces_Demeaned.png}
%     \caption{Blade loading variation with the annular average removed for different outboard turbine yaw angles in the stream-wise (top) and blade torque direction (bottom) for the inboard (left) and outboard (right) turbines.}
%     \label{fig6:demean_blade_load}
% \end{figure}


