2.1. Staggered rows of actuator discs

A schematic of the theoretical model for the staggered case is shown in figure 2. Here we consider a straight local flow passage containing only one-half of an actuator disc due to the periodic and symmetric nature of the array as shown in figure 1. The cross-sectional area of the flow passage is $A/B$, where $A$ is the half-disc area and $B$ is the area blockage ratio. Although the figure is depicted in a two-dimensional manner, this is still a quasi-one-dimensional flow model as we do not consider any variation of flow quantities (velocity and pressure) over the cross-section of each streamtube in the inviscid zone. Assuming that the flow is incompressible, the average velocity over the cross-section of the entire flow passage, $u_{av}$, does not change in the streamwise direction.

Figure 2. Schematic of the quasi-one-dimensional theoretical model for a periodic staggered array of tidal turbines, with examples of how cross-sectional flow patterns may appear in a corresponding three-dimensional flow problem. The three vertical thin black lines in the top and middle drawings indicate the locations of stations 1, 4 and 5. In the middle drawing these thin black lines also indicate the (non-dimensional) cross-sectional average velocity, $\psi = u_{av}/u_{ref}$, for the superposed plot of $u/u_{ref}$ at each station, whereas the thick black lines show example profiles of $u/u_{ref}$ at the three stations (calculated for the case with $B=0.2$, $K=3$ and $m=0.7$). Reproduced from Nishino & Draper (Reference Nishino and Draper2019) with modifications.

Following the common notations used in the LMADT (Houlsby et al. Reference Houlsby, Draper and Oldfield2008) we define four stations within the inviscid zone: station 1 is at the inlet of the inviscid zone where the core-flow streamtube (that encompasses the actuator disc) starts to expand, stations 2 and 3 are immediately upstream and downstream of the disc, respectively, and station 4 is at the outlet of the inviscid zone where the pressure is equalised between the core- and bypass-flow streamtubes. In addition, we describe the outlet of the viscous zone as station 5 (which is station 1 for the next row of discs). The pressure at stations 1 to 5 is denoted by $p_1$ to $p_5$, respectively, whereas the velocity is described using the velocity coefficients $\alpha$ and $\beta$ with subscripts as in figure 2. Each velocity coefficient represents the ratio of the velocity there to a reference velocity, $u_{ref}$. In the following we take the core-flow velocity at station 1 as $u_{ref}$ (i.e. $\alpha _1 = 1$) for convenience.

We now consider the conservation of mass, momentum and energy in the inviscid flow zone in a similar manner to the work of Draper & Nishino (Reference Draper and Nishino2014). First, since the Bernoulli equation must be satisfied to conserve energy in each of the three bypass-flow streamtubes between stations 1 and 4, we obtain

(2.1)\begin{gather} p_1-p_4 = \tfrac{1}{2}\rho u_{ref}^{2} \left( \beta_{4a}^{2}-1 \right), \end{gather}

(2.2)\begin{gather}p_1-p_4 = \tfrac{1}{2}\rho u_{ref}^{2} \left( \beta_{4b}^{2}-\beta_5^{2} \right), \end{gather}

(2.3)\begin{gather}p_1-p_4 = \tfrac{1}{2}\rho u_{ref}^{2} \left( \alpha_8^{2}-\alpha_5^{2} \right), \end{gather}

where $\rho$ is the fluid density. Substituting (2.2) and (2.3), respectively, into (2.1) gives

(2.4)\begin{gather} \beta_{4b} = \left( \beta_{4a}^{2} + \beta_{5}^{2} - 1 \right)^{1/2}, \end{gather}

(2.5)\begin{gather}\alpha_{8} = \left( \beta_{4a}^{2} + \alpha_{5}^{2} - 1 \right)^{1/2}. \end{gather}

As the Bernoulli equation must also be satisfied in the upstream part (between stations 1 and 2) and downstream part (between stations 3 and 4) of the core-flow streamtube, we also obtain

(2.6)\begin{equation} p_2-p_3 = p_1-p_4 + \tfrac{1}{2}\rho u_{ref}^{2} \left( 1-\alpha_4^{2} \right), \end{equation}

which, together with (2.1), leads to an expression for the half-disc thrust, $T$, as

(2.7)\begin{equation} T = (p_2-p_3)A = \tfrac{1}{2}\rho u_{ref}^{2} A \left( \beta_{4a}^{2} - \alpha_4^{2} \right), \end{equation}

and, thus, an expression for the half-disc power, $P$, as

(2.8)\begin{equation} P = T\alpha_2 u_{ref} = \tfrac{1}{2}\rho u_{ref}^{3} A \alpha_2\left( \beta_{4a}^{2} - \alpha_4^{2} \right). \end{equation}

Hence, to obtain $P$ for a given $\alpha _2$, for example, we need to know how $\alpha _4$ and $\beta _{4a}$ depend on $\alpha _2$. By considering the conservation of mass in the ‘main’ bypass-flow streamtube between station 1 (where the velocity is $\beta _5 u_{ref}$) and station 4 (where the velocity is $\beta _{4b} u_{ref}$) we obtain

(2.9)\begin{equation} \beta_5 u_{ref} A \left(\frac{1}{B} - \frac{\alpha_2}{\alpha_4} - \gamma \right) = \beta_{4b} u_{ref} A \left(\frac{1}{B} - \frac{\alpha_2}{\alpha_4} - \frac{\gamma - \alpha_2}{\beta_{4a}} - \gamma \right), \end{equation}

which leads to

(2.10)\begin{equation} \alpha_4 = \frac{\alpha_2}{\dfrac{1}{B} - \dfrac{\beta_{4b}}{\beta_{4a}} \left(\dfrac{\gamma - \alpha_2}{\beta_{4b} - \beta_5} \right) - \gamma}, \end{equation}

where

(2.11)\begin{equation} \gamma = \frac{\alpha_2 \alpha_5}{\alpha_4 \alpha_8} \end{equation}

is the area ratio for the wake flow upstream of the disc as depicted in figure 2. Note that this wake originates from the disc located two rows upstream (and its velocity increases from $\alpha _5 u_{ref}$ to $\alpha _8 u_{ref}$ when it passes through the row immediately upstream) due to the staggered configuration. As for $\beta _{4a}$, we consider the conservation of momentum for the entire flow passage to obtain

(2.12)\begin{align} (p_1 - p_4)\frac{A}{B} - T &= \rho u_{ref}^{2} A \left[ \alpha_2 (\alpha_4 - 1) + (\gamma - \alpha_2)(\beta_{4a} - 1) + \gamma \alpha_8(\alpha_8 - \alpha_5) \vphantom{\left( \frac{1}{B} -\frac{\alpha_2}{\alpha_4} -\gamma \right)}\right.\nonumber\\ &\quad \left.+ \beta_5 \left( \frac{1}{B} - \frac{\alpha_2}{\alpha_4} -\gamma \right) \left( \beta_{4b} - \beta_5 \right)\right], \end{align}

which, together with (2.1) and (2.7), leads to

(2.13)\begin{align} & \left( 1-B \right)\beta_{4a}^{2} - 2B \left( \gamma-\alpha_2 \right) \beta_{4a} - 2B \left[ \alpha_2 \alpha_4 - \frac{1}{2}\alpha_4^{2} + \gamma \left( \alpha_8^{2} - \alpha_5 \alpha_8 - 1 \right) \right] \nonumber\\ &\quad -2\beta_5 \left( \beta_{4b} - \beta_5 \right) \left[ 1-B \left( \frac{\alpha_2}{\alpha_4} + \gamma \right) \right] - 1 = 0. \end{align}

The above set of equations for the inviscid zone is not closed as it includes $\alpha _5$ and $\beta _5$, which need to be modelled considering the effect of mixing in the viscous zone. There are several possible ways to model the effect of mixing but here we employ a very simple approach using a single non-dimensional input parameter called the mixing factor, $m$, which represents the completeness of mixing in the viscous zone. Specifically, the value of $m$ (between 0 and 1) describes how much the velocity of flow at each cross-sectional position returns to the cross-sectional average velocity of the entire flow passage, $u_{av}$, as the flow passes through the viscous zone. By applying this to the wake flow of the disc we obtain

(2.14)\begin{equation} \alpha_5 = m\psi + \left( 1-m \right) \alpha_4 , \end{equation}

where $\psi = u_{av}/u_{ref}$ is the non-dimensional cross-sectional average velocity and this can be calculated from the velocity profile at station 1, for example, as

(2.15)\begin{equation} \psi = B \left[ \gamma \left( \alpha_8 + 1 \right) + \beta_5 \left( \frac{1}{B} - \frac{\alpha_2}{\alpha_4} - \gamma \right) \right] . \end{equation}

For the bypass flow, however, the difficulty is that the number of flow passages at station 4 does not agree with that at station 1, since the actuator disc creates the additional narrow bypass streamtube that has $\beta _{4a}$ at station 4. To obtain a closed set of equations for this periodic flow problem within the framework of quasi-one-dimensional modelling, here we assume that the narrow bypass flow immediately outside of the wake is ‘merged’ or fully mixed with the main bypass flow regardless of the value of $m$. This assumption seems reasonable since the difference between $\beta _{4a}$ and $\beta _{4b}$ tends to be small as depicted in figure 2, and eventually leads to

(2.16)\begin{equation} \beta_5 = m\psi + \left( 1-m \right) \beta_{4m}, \end{equation}

where $\beta _{4m}$ is the ‘area-weighted’ average of $\beta _{4a}$ and $\beta _{4b}$, i.e.

(2.17)\begin{equation} \beta_{4m} = \frac{\left( \gamma - \alpha_2 \right) + \beta_5\left(\dfrac{1}{B} - \dfrac{\alpha_2}{\alpha_4} - \gamma \right)}{\dfrac{\left( \gamma - \alpha_2 \right)}{\beta_{4a}} + \dfrac{\beta_5}{\beta_{4b}} \left( \dfrac{1}{B} - \dfrac{\alpha_2}{\alpha_4} - \gamma \right)} . \end{equation}

It should be noted that $\alpha _9 = m\psi + ( 1-m ) \alpha _8$ is required together with (2.14) to (2.17) to conserve the total mass in the viscous zone, but this is automatically satisfied as we enforce $\alpha _9=\alpha _1=1$ in this analysis due to the periodicity of the flow.

Finally, the thrust and power coefficients of the disc are expressed (for a given cross-sectional average velocity of the entire flow, $u_{av}=\psi u_{ref}$) as

(2.18)\begin{gather} C_T = \frac{T}{\dfrac{1}{2}\rho \left( \psi u_{ref} \right)^{2} A} = \frac{\beta_{4a}^{2} - \alpha_4^{2}}{\psi^{2}}, \end{gather}

(2.19)\begin{gather}C_P = \frac{P}{\dfrac{1}{2}\rho \left( \psi u_{ref} \right)^{3} A} = \frac{\alpha_2\left( \beta_{4a}^{2} - \alpha_4^{2} \right)}{\psi^{3}}. \end{gather}

For convenience, we also define the resistance coefficient (or local thrust coefficient) of the disc as

(2.20)\begin{equation} K = \frac{T}{\dfrac{1}{2}\rho \left( \alpha_2 u_{ref} \right)^{2} A} , \end{equation}

from which and (2.7), we obtain

(2.21)\begin{equation} \alpha_2 = \left( \frac{\beta_{4a}^{2} - \alpha_4^{2}}{K} \right)^{1/2} . \end{equation}

In summary, the above theoretical model for an infinite number of staggered rows of identical ideal turbines consists of three non-dimensional input parameters ($B$, $K$, $m$), ten non-dimensional unknowns to be determined ($\alpha _2$, $\alpha _4$, $\alpha _5$, $\alpha _8$, $\beta _{4a}$, $\beta _{4b}$, $\beta _{4m}$, $\beta _5$, $\gamma$, $\psi$) and a set of ten equations to be solved numerically: (2.4), (2.5), (2.10), (2.11), (2.13), (2.14), (2.15), (2.16), (2.17) and (2.21).

2.2. Aligned rows of actuator discs

Compared with the staggered case, the theoretical model for the aligned case becomes simpler as only two bypass-flow streamtubes (instead of three) need to be considered in the inviscid zone. This is because the wake encounters the disc immediately downstream (instead of two rows downstream). In the following, we again consider the conservation of mass, momentum and energy in the inviscid zone, and the simplified mixing process in the viscous zone, to derive a similar but smaller set of equations for the aligned case.

By applying the Bernoulli equation to the two bypass-flow streamtubes and the core-flow streamtube in the same manner as in the staggered case, we obtain (2.1), (2.2) and (2.6), which lead to (2.4), (2.7) and (2.8) for $\beta _{4b}$, $T$ and $P$, respectively. Note that these equations are identical for both aligned and staggered cases, whereas (2.3) and (2.5) are only for the staggered case (since the third bypass-flow streamtube does not exist in the aligned case). Next, from the conservation of mass in the ‘main’ bypass-flow streamtube, we obtain

(2.22)\begin{equation} \beta_5 u_{ref} A \left(\frac{1}{B} - \frac{\alpha_2}{\alpha_4} \right) = \beta_{4b} u_{ref} A \left[\frac{1}{B} - \frac{\alpha_2}{\alpha_4} - \frac{1}{\beta_{4a}}\left( \frac{\alpha_2}{\alpha_4}-\alpha_2 \right) \right], \end{equation}

which leads to an expression for $\alpha _4$ as

(2.23)\begin{equation} \alpha_4 = \frac{\alpha_2 \left( 1 + \dfrac{1}{\beta_{4a}} - \dfrac{\beta_5}{\beta_{4b}} \right)}{\dfrac{1}{B}\left( 1 - \dfrac{\beta_5}{\beta_{4b}} \right) + \dfrac{\alpha_2}{\beta_{4a}} }. \end{equation}

For $\beta _{4a}$, we consider the conservation of momentum for the entire flow to obtain

(2.24)\begin{align} & (p_1 - p_4)\frac{A}{B} - T \nonumber\\ &\quad = \rho u_{ref}^{2} A \left[ \alpha_2 (\alpha_4 - 1) + \left( \frac{\alpha_2}{\alpha_4} - \alpha_2 \right)(\beta_{4a} - 1) + \beta_5 \left( \frac{1}{B} - \frac{\alpha_2}{\alpha_4} \right) \left( \beta_{4b} - \beta_5 \right) \right] , \end{align}

which, together with (2.1) and (2.7), leads to

(2.25)\begin{align} & \left( 1-B \right)\beta_{4a}^{2} - 2B \left( \frac{\alpha_2}{\alpha_4}-\alpha_2 \right) \beta_{4a} - 2B \left( \alpha_2 \alpha_4 - \frac{1}{2}\alpha_4^{2} - \frac{\alpha_2}{\alpha_4} \right) \nonumber\\ &\quad -2\beta_5 \left( \beta_{4b} - \beta_5 \right) \left( 1-B \frac{\alpha_2}{\alpha_4} \right) - 1 = 0. \end{align}

For $\beta _5$, we need to consider the effect of mixing in the viscous zone. By employing the same approach as in the staggered case, we obtain (2.16) for the aligned case as well. Note, however, that $\psi$ and $\beta _{4m}$ are different for the aligned case, which are

(2.26)\begin{equation} \psi = B \left[ \frac{\alpha_2}{\alpha_4} + \beta_5 \left( \frac{1}{B} - \frac{\alpha_2}{\alpha_4} \right) \right] , \end{equation}

and

(2.27)\begin{equation} \beta_{4m} = \frac{\left( \dfrac{\alpha_2}{\alpha_4} - \alpha_2 \right) + \beta_5 \left( \dfrac{1}{B} - \dfrac{\alpha_2}{\alpha_4} \right)}{\dfrac{1}{\beta_{4a}} \left( \dfrac{\alpha_2}{\alpha_4} - \alpha_2 \right) + \dfrac{\beta_5}{\beta_{4b}} \left( \dfrac{1}{B} - \dfrac{\alpha_2}{\alpha_4} \right)} . \end{equation}

We also need (2.14) together with the above equations to conserve the total mass in the viscous zone, but this is automatically satisfied since here we enforce $\alpha _5=\alpha _1=1$ due to the periodicity of the flow. Finally, $C_T$, $C_P$ and $K$ are all defined as in the staggered case, yielding the same equations (2.18) to (2.20) and, thus, (2.21) for $\alpha _2$.

In summary, the theoretical model for the aligned case consists of three non-dimensional input parameters ($B$, $K$, $m$), seven non-dimensional unknowns ($\alpha _2$, $\alpha _4$, $\beta _{4a}$, $\beta _{4b}$, $\beta _{4m}$, $\beta _5$, $\psi$) and a set of seven equations to be solved numerically: (2.4), (2.16), (2.21), (2.23), (2.25), (2.26) and (2.27).

2.3. Example solutions

Here we present some example solutions of the above theoretical model, first for the aligned case and then for the staggered case. As this is a three-parameter ($B$–$K$–$m$) problem, we start with fixing the disc resistance coefficient, $K$, at its optimal value for the complete mixing case, i.e. $m=1$. When $m=1$, our problem (for both aligned and staggered cases) reduces to the two-parameter ($B$–$K$) problem of Garrett & Cummins (Reference Garrett and Cummins2007) (hereafter referred to as GC07); hence, this optimal value for $m=1$ is known to be $K=2(1+B)^{3}/(1-B)^{2}$, which we refer to as $K_\mathrm {GC07}$.

Figure 3 shows how the performance of aligned rows of such ‘suboptimal’ actuator discs (with $K=K_\mathrm {GC07}$) changes with $m$, at two different blockage ratios, $B=0.05$ and $0.2$. As can be expected intuitively, the power coefficient $C_P$ of aligned discs decreases monotonically as the mixing factor $m$ decreases (regardless of the blockage ratio). This agrees with the common observation that the power of aligned rows of turbines tends to decrease as we reduce the streamwise spacing between the rows, noting that the mixing tends to be less complete (i.e. $m$ tends to be small) when the spacing is small.

Figure 3. Theoretical prediction of the effect of mixing on the performance of aligned rows of actuator discs at two different blockage ratios: (a) $B=0.05$ and (b) $B=0.2$. The disc resistance coefficient is fixed at the optimal value for the case with complete mixing ($m=1$), i.e. $K=2(1+B)^{3}/(1-B)^{2}$.

We also present in figure 3 the values of $\alpha _2(\beta _{4a}^{2}-\alpha _4^{2})$ and $1/\psi ^{3}$, to explain why $C_P$ decreases as $m$ decreases. As can be seen from (2.19), $C_P$ is equivalent to the product of these two values; the former is a different power coefficient defined using the velocity upstream of the disc ($u_{ref}$) instead of the cross-sectional average velocity ($u_{av}$), and the latter represents the effect of the difference between $u_{ref}$ and $u_{av}$. Here we can see that the value of $\alpha _2(\beta _{4a}^{2}-\alpha _4^{2})$ actually increases as $m$ decreases. This is essentially because the local blockage effect is enhanced (or the ‘effective’ blockage ratio increases) when the upstream core flow is slower than the upstream bypass flow (Draper et al. Reference Draper, Nishino, Adcock and Taylor2016). This enhancement of the local blockage effect caused by the incomplete wake mixing of the upstream disc, however, is not strong enough to compensate for the ‘loss’ of power possessed by the upstream core flow compared with the ‘original’ power possessed by the cross-sectionally averaged flow (i.e. the increase rate of $\alpha _2(\beta _{4a}^{2}-\alpha _4^{2})$ is smaller than the decrease rate of $1/\psi ^{3}$); therefore, $C_P$ eventually decreases as $m$ decreases.

Although the above analysis was for ‘suboptimal’ discs with $K=K_\mathrm {GC07}$, the same relationship between $C_P$ and $m$ is generally found for aligned discs with any given $K$ values. Figures 4(a) and 4(b) show $C_P$ vs $K$ curves for five different $m$ values, again at $B=0.05$ and $0.2$, respectively, demonstrating the trend. It can also be seen that the optimal $K$ value (to maximise $C_P$) tends to decrease as $m$ decreases, although the $C_P$ obtained at $K=K_\mathrm {GC07}$ is still close to the maximum $C_P$ for a given $m$ (especially when the blockage ratio is high). Figure 4(c) shows how the maximum $C_P$, or $C_{P{max}}$, changes with $m$. As can be expected, for aligned rows of actuator discs, $C_{P{max}}$ also decreases monotonically as $m$ decreases. Figure 5 shows $C_P$ vs $C_T$ curves for the same five different $m$ values as in figures 4(a) and 4(b), again at $B=0.05$ and $0.2$. The same trend can be confirmed here as well.

Figure 4. Theoretical $C_P$ vs $K$ for the aligned case at (a) $B=0.05$ and (b) $B=0.2$; and (c) the maximum $C_P$ obtained by varying $K$ for a given set of $B$ and $m$.

Figure 5. Theoretical $C_P$ vs $C_T$ for the aligned case at (a) $B=0.05$ and (b) $B=0.2$.

Next, we focus on staggered rows of actuator discs. Similarly to the aligned case, we start with the performance of ‘suboptimal’ discs with $K=K_\mathrm {GC07}$, which is shown in figures 6(a) and 6(b) for $B=0.05$ and $0.2$, respectively. At $B=0.05$, the value of $1/\psi ^{3}$ slightly increases (meaning that the upstream core-flow velocity becomes slightly higher than the cross-sectional average velocity) as $m$ decreases from 1 to about 0.95, and then decreases as $m$ further decreases. In contrast, the value of $\alpha _2(\beta _{4a}^{2}-\alpha _4^{2})$ first decreases slightly and then increases as $m$ decreases from 1; this is again due to changes in the ‘effective’ blockage ratio, i.e. the local blockage effect is enhanced (or diminished) when the upstream core flow is surrounded by a faster (or slower) bypass flow. However, the increase (or decrease) rate of $\alpha _2(\beta _{4a}^{2}-\alpha _4^{2})$ tends to be smaller than the decrease (or increase) rate of $1/\psi ^{3}$, and, hence, $C_P$ follows the trend of $1/\psi ^{3}$, i.e. $C_P$ slightly increases first and then decreases as $m$ decreases from 1. This initial increase in $1/\psi ^{3}$ (and, thus, in $C_P$) is more clearly seen at $B=0.2$, demonstrating that the beneficial effect of the staggered layout (allowing the velocity upstream of the disc to become higher than the cross-sectional average velocity) is enhanced at a higher blockage ratio.

Figure 6. Theoretical prediction of the performance of staggered rows of actuator discs: the effect of mixing at (a) $B=0.05$ and (b) $B=0.2$; and (c) comparison of $C_P$ vs $B$ between the complete mixing case ($m=1$) and optimal mixing case ($m=m_{C_P}$). Plotted together are $m_{C_P}$ and $m_{\psi }$, which are the values of $m$ to maximise $C_P$ and to minimise $\psi$, respectively. The disc resistance coefficient is fixed at the optimal value for $m=1$, i.e. $K=2(1+B)^{3}/(1-B)^{2}$.

Figure 6(c) compares $C_P$ vs $B$ curves for the complete mixing case ($m=1$) and for the ‘optimal mixing’ case ($m=m_{C_P}$, where $m_{C_P}$ is the value of $m$ that maximises $C_P$), again for staggered rows of ‘suboptimal’ discs ($K=K_\mathrm {GC07}$). Also plotted together are $m_{C_P}$ and $m_{\psi }$, the latter of which represents the value of $m$ that minimises $\psi$ (and, thus, maximises $1/\psi ^{3}$). It can be seen how the increase in $C_P$ due to the beneficial effect of the staggered layout is enhanced, and how the optimal level of mixing decreases, as we increase the blockage ratio. It is also worth noting that the difference between $m_{C_P}$ and $m_{\psi }$ is small, especially at low blockage ratios. This reflects the dominant influence of $1/\psi ^{3}$ on $C_P$ as described earlier in figures 6(a) and 6(b).

The $C_P$ values presented above are for ‘suboptimal’ discs (with $K=K_\mathrm {GC07}$) and can therefore be increased further by adjusting $K$ for a given $m$ (or for a given streamwise distance between the rows). However, this additional increase in $C_P$ achieved by varying $K$ is rather small for the staggered case. Figures 7(a) and 7(b) show $C_P$ vs $K$ curves for five different $m$ values, again at $B=0.05$ and $0.2$, respectively. When $B$ is small, $K_\mathrm {GC07}$ tends to be already close to the optimal $K$ value for a given $m$; hence, the benefit of further adjusting $K$ is small. When $B$ is large, $K_\mathrm {GC07}$ is not so close to the optimal $K$ for a given $m$, but the $C_P$ vs $K$ curve tends to have a flatter peak; hence again, the additional gain in $C_P$ by adjusting $K$ is small. Figure 7(c) shows the maximum $C_P$ (achieved by adjusting $K$) vs $m$ curves for five different blockage ratios. The curves for $B=0.05$ and $0.2$ are in fact very similar to the $C_P$ vs $m$ curves for $K=K_\mathrm {GC07}$ plotted earlier in figures 6(a) and 6(b). At $B=0.2$, for example, the highest $C_P$ value achieved with $K=K_\mathrm {GC07}$ is only 1.1 % lower than that achieved by optimising $K$. The impact of $m$ on $C_P$ can also be confirmed from the $C_P$ vs $C_T$ curves in figure 8, plotted for the same sets of $m$ and $B$ as in figures 7(a) and 7(b).

Figure 7. Theoretical $C_P$ vs $K$ for the staggered case at (a) $B=0.05$ and (b) $B=0.2$; and (c) the maximum $C_P$ obtained by varying $K$ for a given set of $B$ and $m$.

Figure 8. Theoretical $C_P$ vs $C_T$ for the staggered case at (a) $B=0.05$ and (b) $B=0.2$.

In summary, these example solutions of the simple three-parameter ($B$–$K$–$m$) theoretical model suggest the following. (i) For an infinitely large staggered array of tidal turbines, there is an optimal streamwise turbine spacing (for a given cross-sectional blockage ratio or a given lateral turbine spacing) to maximise the power of each turbine relative to the power of cross-sectional average flow. This is an outcome of the combined effects of local blockage and wake mixing, i.e. this optimum exists because the local flow velocity upstream of each disc can become higher than the cross-sectional average velocity (and this does help to increase the relative power despite reducing the ‘effective’ blockage ratio) when the streamwise turbine spacing is reasonably (not excessively) large for the wake mixing behind each disc to be largely (but not entirely) completed. (ii) The optimal turbine resistance to maximise this relative power also depends on both lateral and streamwise turbine spacing, but the optimal value for a single row, namely $K=2(1+B)^{3}/(1-B)^{2}$, is expected to yield close to the maximum relative power. (iii) For an infinitely large aligned array of tidal turbines, however, this relative power is maximised simply when the streamwise turbine spacing is large enough for the wake mixing to be entirely completed; hence, the optimal turbine resistance becomes identical to that for a single row.

Finally, it should be remembered that, in a real tidal array to be built in the future, the cross-sectional average velocity (which was considered as a fixed parameter in the above analysis) would depend on how the flow resistance caused by the entire array alters the natural tidal channel-scale momentum balance (see, e.g. Vennell Reference Vennell2012; Vennell et al. Reference Vennell, Funke, Draper, Stevens and Divett2015; Gupta & Young Reference Gupta and Young2017). As the additional resistance caused by the array would reduce the mass flow rate through the array, the turbine resistance required to maximise the power for a given ‘natural’ mass flow rate (observed in the ‘natural’ channel without the array) would be lower than the optimal resistance discussed above. It should also be noted that, in the real world, tidal currents are oscillatory and, thus, it would be beneficial to vary the turbine resistance during the tidal cycle (Vennell & Adcock Reference Vennell and Adcock2014).

4.1. Streamwise velocity field

We present in figure 10 contours of the normalised time-averaged streamwise velocity ($\langle u \rangle /U_0$) comparing aligned and staggered layouts with the same streamwise spacing of $S_x/D=9$ and lateral spacing of $S_y/D = 6$, 4 and 3. For aligned arrays, reducing the lateral separation between devices in the same row leads to an increased local blockage that induces larger flow acceleration in the bypass flow between them, which is well observed for AL-$9\times 3$ with the bypass-flow velocity being approximately 20 % higher than the bulk velocity. In perfectly staggered arrays the wake generated behind each turbine is mostly recovered when reaching the following row at a downstream distance of $S_x$. Then, due to the lateral blockage caused by the turbines located on both sides, the mostly recovered wake accelerates further and then impinges the turbine located in the next row, i.e. at 2$S_x$ downstream of the turbine that originally generated the wake. For layouts with low lateral blockage, i.e. $S_y/D = 6$ and 4, there is a wider lateral spread of the wakes, which is well observed in figure 10 for the aligned cases. However, in staggered configurations this lateral wake expansion appears limited compared with their aligned counterparts.

Figure 10. Contours of time-averaged streamwise velocity $\langle u \rangle /U_0$ at hub height for aligned and staggered cases with $S_x/D = 9$ and lateral spacing $S_y/D = 6$ (a,b), 4 (c,d) and 3 (e,f).

The different spreading rates of the time-averaged turbine wakes are partly explained by their instantaneous behaviour. In figure 11 we present contours of instantaneous streamwise velocities ($u/U_0$) for the previous cases shown in figure 10 with $S_x/D = 9$ and changing $S_y$, which reveals a pronounced meandering nature of the wakes with large oscillation amplitudes (Foti et al. Reference Foti, Yang, Shen and Sotiropoulos2019). In both aligned and staggered configurations, the wake meandering is influenced by the lateral distance between turbines in each row, i.e. reducing $S_y$ leads to a smaller meandering amplitude. This trend seems clearer in the staggered cases, e.g. the wake meandering almost disappears in the staggered cases with $S_y/D=4$ and $S_y/D=3$, explaining the narrow time-averaged wakes shown earlier in figure 10. This is presumably because, in the aligned cases, the flow going through the lateral gaps between turbines creates high-speed streaks, resulting in a larger velocity difference between the wake (core) and surrounding (bypass) flows (Stevens et al. Reference Stevens, Gayme and Meneveau2014). Consequently, there is a stronger instability in the shear layer of turbine wakes that enhances the wake meandering (Yang & Sotiropoulos Reference Yang and Sotiropoulos2019a) more in aligned arrays compared with their staggered counterparts. It is worth noting that the vertical confinement and shorter lateral spacing (commonly expected for future tidal arrays) both lead to a larger velocity difference in aligned arrays. Hence, tidal turbine wakes in a large array could be more sensitive to their arrangement than wind turbine wakes in a large wind farm.

Figure 11. Contours of instantaneous streamwise velocity $u/U_0$ at hub height for aligned and staggered cases with $S_x/D = 9$ and lateral spacing $S_y/D = 6$ (a,b), 4 (c,d) and 3 (e,f). Same legend as 10.

4.2. Turbulence statistics

We further analyse the flow field in figures 12 and 13 with contours of turbulence intensity in the streamwise ($\sigma _u/U_0$, with $\sigma _u = \langle u'u' \rangle ^{0.5}$) and transverse directions ($\sigma _v/U_0$, with $\sigma _v = \langle v'v' \rangle ^{0.5}$), again for the cases with $S_x/D = 9$. These second-order statistics indicate that having turbines in an aligned layout leads to a notably stronger flow unsteadiness both inside and outside the wake of each turbine compared with the staggered cases. This agrees with the earlier observation that the wake meandering is stronger in the aligned cases. In the near-wake region, in which tip vortices are still coherent and maintain a shear layer between the core flow and the bypass flow (Ouro et al. Reference Ouro, Ramírez and Harrold2019c), both streamwise and spanwise turbulence intensities are substantially higher in the aligned cases than in the staggered cases. Low turbulence intensity regions in the staggered cases are evident in the flow bypassing each turbine, coinciding with the regions in which the wake of an upstream turbine is almost fully recovered as shown earlier in figures 10 and 11. The results also show that increasing the blockage ratio $B$ (or reducing the distance $S_y$ between turbines in the same row) reduces the turbulence intensity irrespective of the overall turbine arrangement, as a consequence of constraining the formation of high-momentum streaks and, thus, meandering motion.

Figure 12. Contours of streamwise turbulence intensity ($\sigma _u/U_0$) at hub height for aligned and staggered cases with $S_x/D = 9$ and lateral spacing $S_y/D = 6$ (a,b), 4 (c,d) and 3 (e,f).

Figure 13. Contours of spanwise turbulence intensity ($\sigma _v/U_0$) at hub height for aligned and staggered cases with $S_x/D = 9$ and lateral spacing $S_y/D = 6$ (a,b), 4 (c,d) and 3 (e,f).

As can be expected from the above results, the resulting unsteady meandering wake also notably affects the distribution of turbulent momentum exchange, which is described in figure 14 with contours of the Reynolds shear stress ($-\langle u'v' \rangle /U_0^{2}$) at hub height. For each array layout, narrowing the lateral spacing reduces the magnitude of Reynolds shear stress, indicating that there is a lower level of momentum exchange between the wakes and the surrounding bypass flows. Staggered layouts consistently attain lower magnitudes of shear stresses compared with the aligned layouts irrespective of the number of turbines deployed.

Figure 14. Contours of Reynolds shear stress ($-\langle u'v' \rangle /U_0^{2}$) obtained at hub height for aligned and staggered cases with $S_x/D = 9$ and lateral spacing $S_y/D = 6$ (a,b), 4 (c,d) and 3 (e,f).

4.3. Centreline velocities

Next, we quantitatively compare the mean streamwise velocity ($\langle u \rangle /U_0$) distribution along the wake centreline in figure 15 for every configuration simulated (see table 1). Note that, for $S_x/D = 18$, 9 and 6, we plot the velocity distribution over two rows of turbines, including as a shaded area the location of the intermediate row of turbines. When the streamwise spacing is relatively small ($S_x/D = 9$ and 6), the values of $\langle u \rangle /U_0$ behind the turbine nacelles increase faster for the lower lateral blockage cases, i.e. $S_y/D = 6$ and 12. This quicker wake recovery in the lower blockage cases results from a larger entrainment of ambient flow into the wake, which can be enhanced by a stronger meandering behaviour as seen in the distribution of Reynolds shear stresses in figure 14.

Figure 15. Distribution of time-averaged streamwise velocities ($\langle u\rangle /U_0$) along the wake centreline at the hub height for aligned (a,b,d,f) and staggered (c,e,g) configurations with $S_x/D = 36$ (a), 18 (b,c), 9 (d,e) and 6 (f,g), and $S_y/D = 12$ (black), 6 (blue), 4 (green) and 3 (orange).

Although lower blockage ratios (or larger $S_y/D$ values) tend to result in higher wake recovery rates in the near-wake region, this trend does not hold in the far-wake region. In the aligned cases with $S_x/D = 18$, the values of $\langle u\rangle /U_0$ are slightly over unity shortly upstream of the turbines for configurations with $S_y/D \leq 6$, whereas for $S_y/D = 12$, the wake velocity is not fully recovered to the bulk velocity value despite the high wake recovery rate in the near-wake region. Further decreasing $S_x/D$ to 9 in aligned cases reduces the amount of wake recovery for every $S_y/D$, resulting in $\langle u\rangle /U_0$ of about 0.9 to 0.95 at one rotor-diameter upstream of the turbines (i.e. at $x/D = 8$ and 17). It is clearly seen that the wakes in the cases with a higher blockage (AL-$9\times 4$ and AL-$9\times 3$) have a lower recovery rate in the near-wake region but a higher recovery rate in the far-wake region. The same trend can be observed in the cases with $S_x/D = 6$ in which the streamwise velocities upstream of the turbines are even lower due to the lack of space for the wake to recover. Overall, for aligned arrays, we may conclude that $S_x/D$ is the main design parameter that determines how much the wake recovers before approaching the turbine downstream, with $S_y/D$ contributing to a lower extent by changing the turbulent wake characteristics.

In staggered arrays with $S_x/D = 18$, the centreline mean velocity notably increases at around $x/D \approx 18$ due to the local blockage caused by the turbines in the intermediate second row. Such flow acceleration is more noticeable when the lateral spacing $S_y/D$ is small, e.g. for ST-$18\times 3$ and ST-$18\times 4$, the maximum velocity is about 1.1$U_0$ whereas, for ST-$18\times 6$ or ST-$18\times 12$, it is approx. 1.02$U_0$. This agrees qualitatively with the theoretical analysis presented earlier in § 2.3, i.e. the local flow velocity upstream of each turbine may exceed the cross-sectional average velocity in staggered arrays. Reducing the streamwise spacing $S_x/D$ makes this additional local flow acceleration less noticeable, but the rate at which the wake velocity recovers still varies with $S_y/D$, being higher for the larger blockage cases with $S_y/D = 4$ and 3. Another distinct feature of time-averaged wakes in staggered arrays is that after passing through the streamwise location of the second row (i.e. $x \geq S_x$) the centreline mean velocity remains fairly constant until about two rotor-diameters upstream of the next turbine.

Finally, comparing aligned and staggered cases for a given streamwise spacing, we can confirm that the velocity recovery rate in the near-wake region is higher in the former configuration, due to stronger turbulent mixing enhanced by the wake meandering as shown earlier.

4.4. Array efficiency

As the impact of array layout and turbine spacing on the flow field has been confirmed, we now focus on the thrust and power coefficients of turbines to analyse their efficiency in the simulated infinitely large tidal arrays. For each configuration, time-averaged hydrodynamic coefficients are further averaged for all turbines comprising the array, and the results are presented in figure 16 with error bars indicating the standard deviation of array-averaged values. For aligned layouts with a given $S_y/D$, both thrust and power coefficients tend to have their maximum values at the largest streamwise spacing of $S_x/D = 36$, but the results at $S_x/D = 18$ are similar as this streamwise spacing is still large enough for the wakes to almost fully recover. However, when $S_x/D$ is further reduced, both $C_T$ and $C_P$ drop considerably in line with the decrease in the mean streamwise velocity upstream of each turbine (Ali et al. Reference Ali, Hamilton, DeLucia and Cal2018), as shown earlier in figure 15. In contrast, the performance of turbines in staggered configurations appears less sensitive to $S_x/D$.

Figure 16. Results of $C_T$ (a–d) and $C_P$ (e–h) obtained from the LES for the aligned ($\square$) and staggered ($\bigcirc$) layouts.

Comparing arrays with a large lateral spacing ($S_y/D = 12$), staggered configurations consistently provide higher $C_T$ and $C_P$ values than the aligned counterparts, with larger differences attained at shorter streamwise spacing. For cases with $S_y/D = 6$ and 4, staggered configurations still tend to provide higher $C_T$ and $C_P$ than the aligned counterparts, although the differences are negligibly small at $S_x/D = 18$. Further reducing the lateral spacing to $S_y/D = 3$ leads to higher $C_T$ in the aligned case than in the staggered case at $S_x/D = 18$, but again $C_P$ values are consistently higher in the staggered cases. These results suggest that, as expected from earlier wind farm studies, staggered configurations are in general more efficient than the aligned counterparts (Tian, Ozbay & Hu Reference Tian, Ozbay and Hu2018).

4.5. Turbine loading unsteadiness

Next, we analyse the temporal fluctuations of thrust and power from their root-mean-square (r.m.s.) values. The results averaged over all turbines comprising the array are presented in figure 17, which shows that decreasing $S_x/D$ leads to an increase in the fluctuations of both thrust and power, as expected from an enhanced unsteadiness of the approaching flow. Adopting a large lateral spacing also increases the load fluctuations in comparison to cases with the same streamwise spacing and a smaller lateral spacing. This is in line with the strength of wake oscillations increasing with $S_y/D$, as shown earlier in figure 11. It is also worth noting that turbines in aligned configurations tend to have larger load variations than the staggered counterparts, especially when decreasing $S_x/D$. These results suggest that staggered configurations not only enhance the array's efficiency (i.e. mean power coefficient) but can also reduce the load fluctuations and, thus, the long-term fatigue damage of turbine rotors.

Figure 17. Results of root-mean-squared temporal fluctuations of $C_T$ (a) and $C_P$ (b) obtained from the LES of the considered array layouts.

To further link the fluctuation of turbine forces with the wake meandering, we present in figure 18 the power spectral density of the power signal from a single turbine comparing the same sets of staggered and aligned layouts discussed earlier in figures 10 to 14. The spectra show distinct peaks at the blade passing frequency $f_p=3f_0$, where $f_0=\varOmega /(2{\rm \pi} )$ is the rotational frequency, and its harmonics (6$f_0$ and 9$f_0$) which indicate turbine-induced high-frequency changes in the power generation (Ouro et al. Reference Ouro, Ramírez and Harrold2019c). Wake meandering is, however, a low-frequency phenomenon that is identified by the peaks in these spectra at $f_{wm} = 0.08f_0$, i.e. its undulating motion has a period of about 12 times that of the turbines’ rotation. These peaks are well observed in the aligned cases irrespective of $S_y$, whilst in the staggered cases the signal at those low frequencies reduces with lower $S_y$ and, for $S_y/D = 4$ and 3, the peak due to the wake meandering disappears. This agrees well with our earlier observation of the disappearance of wake meandering from the instantaneous velocity fields presented in figure 11.

Figure 18. Power spectral density of the instantaneous power signal from a single turbine. Comparison between aligned (red line) and staggered (black line) configurations with $S_x/D = 9$ and $S_y/D = 6$ (a), 4 (b) and 3 (c). Frequencies are normalised by the rotational frequency $f_0$. Here $f_{wm}$ denotes the wake meandering frequency and $f_p$ is the blade passing frequency.

The power spectra from the aligned cases appear to follow a $-7/3$ decay after the peak at $f_{wm}$ irrespective of $S_y$, and this is also observed for the staggered array with $S_y/D = 6$. Narrowing the lateral spacing for the ST-$9\times 4$ and ST-$9\times 3$ cases leads to three major changes: a progressive transition in the decay slope from $-7/3$ to $-5/3$, a decrease in the energy associated to the low-frequency (inertial) range, and the absence of a distinct peak at $f_{wm}$. Earlier studies with tidal turbines reported a slope of $-5/3$ from the spectrum of the power output signal (e.g. Deskos et al. Reference Deskos, Payne, Gaurier and Graham2020) resulting from the ambient turbulence governing the power fluctuation. However, our power spectra in figure 18 suggest that low-frequency changes in the power output of turbines in a large tidal array could be strongly affected by the wake meandering, depending on the turbine spacing and array layout. The Strouhal number ($St=f_{wm}D/U_0$) associated with the wake meandering frequency is found to be approximately 0.11, which is similar to the values reported for wind turbines, e.g. 0.15 obtained by Foti et al. (Reference Foti, Yang, Guala and Sotiropoulos2016).

The observed variability in the decay slope for different array layouts is presumably because the dynamics of flow through a large tidal array is mostly driven by the turbine-induced changes (unlike wind farms, where low-frequency dynamics are also impacted by large-scale flow structures in the atmospheric boundary layer; see, e.g. Stevens, Gayme & Meneveau Reference Stevens, Gayme and Meneveau2016; Bossuyt et al. Reference Bossuyt, Meneveau and Meyers2017). Further investigations into these spectra for cases with irregular turbine layouts and lower vertical confinements (larger water depths) would be helpful in future studies.