3.1. Performance of a multi-scale device in an infinitely wide channel

We start by presenting results for the case of an infinitely wide channel $(B_n\to 0)$, which is slightly simpler than the finite global blockage case. The optimal global power coefficient for an $n$-scale energy extraction device is plotted in figure 3. An analytic curve fit approximation to the optima, with the relative percentage error shown in figure 3(b), is suggested to be

(3.1)\begin{equation} C_{P_G,{{Max}}} (n) = \frac{C_{P,\textit{Betz}}-1}{n}+1, \end{equation}

where $C_{P,\textit{Betz}}$ is the Betz limit of $16/27$. Note that the function is concave for all $n>1$, is always a lower bound on the numerical optima, and that

(3.2)\begin{equation} \lim_{n\to\infty} C_{P_G,{{Max}}}(n) = 1, \end{equation}

showing that the optimal efficiency of a multi-scale energy extraction device which makes use of constructive interference (local blockage) in an unbounded fluid is 100 % of the undisturbed kinetic energy flux through the device area.

Figure 3.

The maximum power coefficient for a multi-scale device in an infinitely wide channel, here shown for all scales to $n\leqslant 50$. (a) The circles indicate the numerical solutions, and the solid line gives an analytical approximation to the optimal $C_P$. (b) The relative percentage error between the approximation and numerical solution.

Of particular interest is why the maximum power extraction asymptotes to a $C_{P_G}$ of unity, whether this is the maximum for any energy extraction device, and how the properties of the multi-scale energy extraction device are distributed across the scales. The following discussion addresses these questions.

3.2. Properties of optimal unbounded multi-scale devices

For the case of an infinitely wide channel $(B_n\to 0)$, the theoretical upper bound of energy extraction is limited by $C_{P_G}^*=1$ as the static pressure upstream and downstream of the device must be the same. The only energy that can therefore be extracted must be derived from the upstream kinetic flux, which is analogous to the single-scale unblocked energy extractor considered by Betz. In the limit, the multi-scale device clearly approaches this limit of energy extraction in an unbounded steady fluid. This section clarifies the physics of the convergence to this optimal condition for large $n$.

For the single-scale Betz-like device, increasing the static pressure drop across the device by increasing the resistance to the flow comes at the cost of modifying the upstream stream-tube area. Additional scales allow for a decoupling of these two phenomena, and as the total number of scales becomes large, these conditions, namely the optimal static pressure drop across the device and no outer stream-tube modification can be simultaneously realised. At large $n$, the smallest scales are able to achieve blocked flow conditions, and can remove power with essentially no local flow divergence (hence large kinetic efficiency). At the same time, the entire array is becoming infinitely large, such that $B_D$ tends to zero and so the discs at the smallest scale cannot cause large-scale flow diversion. Numerical evidence is presented to illustrate the convergence to these two optimal conditions.

Figure 4(a) illustrates the convergence of $\alpha _G$ to $1/2$ for large $n$, allowing for an optimal pressure drop, normalised on the upstream dynamic pressure $\rho u_\infty ^2/2$, of $\varDelta c_p^*=2$ across the smallest scale of the device. Figure 4(c) likewise illustrates the convergence of $\alpha _n\to 1$, implying that the stream-tube area remains unmodified at the largest scale without violation of the velocity constraint, namely that the velocity reduction factor in the far wake, $(1-\gamma _n)$, is twice the disc plane's reduction factor, $(1-\alpha _n)$.

Figure 4.

Global parameters exhibiting convergent behaviour for large $n$ (markers indicating numerical solutions with circles for $B_G = 0$, crosses for $B_G = 0.1$, pluses for $B_G = 0.15$, stars for $B_G = 0.2$ and filled circles for $B_G = 0.25$; every fourth simulated data point plotted). (a) The global velocity induction factor $\alpha _G$, (b) the wake tuning velocity factor $\gamma _G$, the velocity factors at the largest scale (c) $\alpha _n$ (logarithmic scale for $x$-axis) and (d) $\gamma _n$.

In the limit as ${n\to \infty }$, the multi-scale energy extraction device therefore approaches the theoretical upper bound of energy extraction from an unbounded fluid, by simultaneously maximising the pressure drop across the discs at the smallest scale, and leaving the stream-tube unmodified at the largest scale. The distributions of the parameters that maximise the power extraction for an $n=100$ multi-scale device are shown in figure 5, as a close approximation for the limiting behaviour of an optimal (infinite-scale) device. Figure 5(a,b) demonstrates the gradual deceleration of the flow across each scale, with all $\gamma _s$ and $\alpha _s$ nearing unity, and with the locally normalised static pressure drop gradually (and optimally) building up to the smallest scale (figure 5c). The distribution of the local blockage ratios, shown in figure 5(d), satisfies the optimal constraints on velocity and static pressure drop, with the blockage approaching near unity at the smallest scale. Figure 6 likewise shows the distribution of Lagrangian multipliers across scales, giving indication of the relative importance of a given thrust constraint between consecutive scales. This illustrates the relative decoupling of the critical thrust constraint at the largest (stream-tube) scale and the constraints between the smallest (pressure-drop) scales. In the limit, the thrust constraint on the smallest scale becomes negligible $(\lambda _1\to 0)$, and the smallest scale therefore decouples. All of the intermediate scales allow the flow to achieve both outer and inner-scale optima subject to the physical constraints of the system. Variations in the constraint at the largest scale dictate the energy capture of the entire device; changes in outer-scale blockage will therefore have a significant impact on the energy extraction efficiency.

Figure 5.

The solution for an $n=100$ multi-scale energy extraction device illustrating the approximate limiting behaviour of an infinite-scale device (markers indicating numerical solutions with circles for $B_G = 0$, crosses for $B_G = 0.1$, pluses for $B_G = 0.15$, stars for $B_G = 0.2$ and filled circles for $B_G = 0.25$; every fourth simulated data point plotted). The velocity induction factors (a) $\alpha _s$ (logarithmic scale for $x$-axis) and (b) $\gamma _s$, (c) the scale thrust coefficients $C_{T_s}$, (d) the local blockage ratios $B_s$, are shown for all scales $1 \leqslant s \leqslant 100$.

Figure 6.

Lagrangian multiplier, $\lambda _s$, distribution across scales, $1 \leqslant s \leqslant n-1$, for an $n=100$ unbounded multi-scale device.

Additional observations on the limiting parameters can be made with the aid of the governing equations, constraints and numerics. Consider a substitution of the constraints (2.8) into the objective function (2.7)

(3.3)\begin{equation} C_{P_G} = \frac{\alpha_1 C_{T_j}\prod\limits_{s=2}^n \alpha_s^3}{\prod\limits_{s=2}^{j} \alpha_s^2 B_{s-1}}, \end{equation}

which is true for all $j\geqslant 2$. By considering the large-scale constraint $j=n$, we can write

(3.4)\begin{equation} C_{P_G} = \frac{(1-\gamma_n)(1+\gamma_n)}{\prod\limits_{i=1}^{n-1} B_i}\prod_{i=1}^n \alpha_i. \end{equation}

As the number of scales becomes large $(n\to \infty )$, the optimal induction factor at the outermost scale becomes $(1-\alpha _n)\to 0$. This condition is logical for optimal extraction as in the limit it implies that, by mass conservation, no flow is diverted around the device. Additionally, by considering the constraint on velocity at the largest scale, this implies that $(1-\gamma _n)\to 0$. Substituting these optimal conditions into (3.4) allows for a consideration of the individual terms of the power coefficient

(3.5)\begin{equation} C^*_{P_G}=\lim_{\gamma_n\to1} \frac{(1-\gamma_n)(1+\gamma_n)}{\prod\limits_{i=1}^{n-1} B_i}\prod_{i=1}^n \alpha_i, \end{equation}

where for large $n$ we observe numerically that the optimal $\alpha _G$ for energy extraction is (see figure 4a)

(3.6)\begin{equation} \lim_{n\to\infty} \prod_{i=1}^n \alpha_i = \alpha_G =\frac{1}{2}. \end{equation}

For the maximum $C_{P_G}$ to be equal to the upper bound of 1, we must therefore have

(3.7)\begin{equation} (1-\gamma_n)\sim \prod_{i=1}^{n-1} B_i= B_D\end{equation}

and

(3.8)\begin{equation} \lim_{n\to\infty} \prod_{i=1}^{n-1} B_i = 0, \end{equation}

which is confirmed by the numerical results. Considering the final constraint, we can also show that

(3.9)\begin{equation} (1-\gamma_n)(1+\gamma_n)=C_{T_1}\prod_{i=2}^n \alpha_i^2 B_{i-1}, \end{equation}

so that for large $n$

(3.10)\begin{equation} \lim_{n\to\infty} C_{T_1} = \lim_{n\to\infty} \left(\frac{(1-\gamma_n)}{\prod\limits_{i=1}^{n-1}B_i}\frac{\alpha_1^2(1+\gamma_n)}{\prod\limits_{i=1}^n\alpha_i^2}\right) = \frac{2\alpha_1^2}{\left(\dfrac{1}{2}\right)^2}=8\alpha_1^2, \end{equation}

from which it is clear that to maximise the objective function (2.7), the product $\alpha _1 C_{T_1}$ is maximised (and $C_{P_G}\to 1$) when $\alpha _1$ tends to $1$ as $n\to \infty$. This shows that $\lim _{n\to \infty } C_{T_1}=8$, which can clearly be observed in figure 5(c). The constituent components for optimal energy extraction are therefore

(3.11)\begin{equation} C_{P_G}^* = \alpha_1^* C_{T_1}^* \prod_{i=2}^n \left(\alpha_i^*\right)^3 = 1 \times 8 \times \left(\tfrac{1}{2}\right)^3 = 1, \end{equation}

where $\alpha _1^*=1$ physically represents (through mass conservation) the lack of inner stream-tube modification (note that $B_1\to 1$ also), $C_{T_1}^*=8$ (normalised on $1/2\rho u_2^2)$ represents the optimal thrust (or alternatively the pressure drop $\Delta c_p^*=2$ normalised on $1/2\rho u_\infty ^2$) upon extraction and $\alpha _G^*=1/2$ the optimal deceleration of the potential flow upstream of the extraction plane. The convergence to all of these conditions is observed in the numerical results. In the limiting case, therefore, a multi-scale energy extraction device approaches this upper bound of energy extraction without violation of conservation laws, providing a new theoretical limit of energy extraction from a steady unbounded fluid.

The condition that $(1-\gamma _n)=0$ as $n\to \infty$ also implies that

(3.12)\begin{equation} \lim_{n\to\infty} C_{T_n} = 0, \end{equation}

i.e. the thrust coefficient when referenced to the area of the $n\textrm {th}$-scale device tends to zero, whilst the global thrust coefficient $C_{T_G}$, which is referenced to actuator disc area, must of course remain non-zero. That $C_{T_n} \to 0$ is of particular interest to the mass-flux and blockage coupling considerations in both tidal and wind engineering flows. In a head driven system, resistance to the flow will reduce the mass flux through the system (see Vennell Reference Vennell2010), with that reduction being a function of both the large-scale blockage and thrust. The limiting behaviour of the multi-scale device, namely that $C_{T_n}\to 0$ and that $B_D\to 0$ removes the impact on the flux through the system. In the limit, therefore, the assumption that the mass fluxes far upstream and downstream of the energy extracting device are equal is implicit to the model for the case of an infinitely wide channel.

3.3. Fractal approximation to unbounded multi-scale devices

We stress here that, although the multi-scale device approaches the optimal conditions for energy extraction, it is not necessarily the only method of optimal extraction. Other device configurations which satisfy the unmodified stream-tube at largest scale, optimal pressure drop at smallest scale and conservation law constraints remains an open question, and further investigations into satisfying these constraints could uncover both novel and practical methods for extracting power above the Betz limit.

A simple example of this which helps aid in the visualisation of a multi-scale device, is a fully fractal arrangement where the local blockage remains the same across all scales, i.e. $B_L=B_s$ for $1\leqslant s \leqslant n-1$. Figure 7 demonstrates that such an array does not significantly under-perform the optimal arrangement, with the large and small-scale constraints also decoupling in the limit.

Figure 7.

Properties of a fractal multi-scale energy extracting device where the local blockage is held constant for all scales. (a) A comparison of the optimal (solid line) and fractal device energy (crosses) extraction efficiency and (b) the local blockage ratio required for optimal extraction for an $n$-level fractal device.

Figure 8 further illustrates the relationship between the local blockage ratio (which once again remains the same across scales for the fractal device) and the resultant power coefficient. Given the small under-performance when compared with the optimal configuration, figure 8 condenses the complex multi-dimensional data of the optimal configuration and figuratively describes many of its conclusions. The maximum power extraction efficiency is seen to asymptote to 1 with increasing number of scales. This makes clear the diminishing returns of additional scales, as well as the increasing local blockage ratio required to realise optimal extraction as the layers of the fractal increase. Note that for equally sized discs, the maximum local blockage achievable at the innermost scale is ${\rm \pi} /4=0.785$ for circular discs arrayed in a rectangular grid, and ${\rm \pi} /2\sqrt {3}=0.907$ for circular discs arrayed in rows staggered above and below each other. This provides useful upper physical bounds to the problem indicating that in practical terms no further power coefficient increase can be achieved beyond approximately 10 and 20 fractal layers for rectangular and staggered grid arrangements of equally sized circular actuators.

Figure 8.

The relationship between local blockage and power extraction for multi-scale fractal devices in an unbounded flow ($B_n=0$). The dashed line provides the locus connecting the optimal power coefficients for fractal devices up to $n=100$ scales. Maximum achievable local blockage ratios are shown for both rectangular $(B_L=0.785)$ and staggered $(B_L=0.907)$ arrays of equally sized circular discs.

An approximation to the physical arrangement of the discs for the $n=3$ fractal case is shown in figure 9, which illustrates both the conceptually fractal-like construction of such a device, and the final device with a $C_{P_G}\sim 87\ \%$ extraction efficiency (see optimum locus with $n=3$ in figure 8). We note that the number of arrays and sub-arrays $d_s$ each subsequent fractal iteration splits into would need to be greater than the four shown in the figure to fully realise the extraction limit (see Nishino & Willden (Reference Nishino and Willden2013) for details on the influence of the number of actuators required for a two-scale device).

Figure 9.

A schematic of a fractal arrangement of discs, which may be visually interpreted by considering the recurrent division of discs into $d$ new sub-discs per disc with identical spacing between the sub-discs. Here, the array is split $n-1=2$ times into $d=4$ discs each time, with a constant local blockage ratio of $B_L\sim 0.55$, which is near optimal for the fractal arrangement with $n=3$. The final arrangement is shown in black.

3.4. Properties of optimal unbounded finite-scale devices

Although an optimal pressure coefficient is only realised for an infinite number of scales, the maximum power coefficient (see figure 3) rapidly asymptotes to the optimal condition for relatively small $n$. Furthermore, in practice, it appears likely that the maximum number of scales will be limited to small $n$. This section therefore considers comparisons of parameter distributions for finite $n$, to highlight the differences in these distributions prior to convergence.

Figure 10(a,b) illustrates the velocity factor distributions across scales for various $n$. For increasing numbers of device scales, the velocity factors always increase at any given normalised scale, with $\alpha _s$ and $\gamma _s$ both nearing unity across all scales for $n=100$. It is clear that as the number of scales increases, the flow across scales decelerates through each subsequent scale more gradually. The reduction in the differential between core and bypass flow velocities at any given scale minimises mixing losses at that scale, whilst keeping $\alpha _G$ as close to 1/2 as possible. For a very small number of scales the flow must be rapidly decelerated to ensure a large enough static pressure drop can occur across the energy extraction device, resulting in substantially larger stream-tube deformation.

Figure 10.

Distributions of (a) $\alpha _s$, (b) $\gamma _s$, (c) $C_{T_s}$ and (d) $B_s$ for an $n = 2$ (circles connected by dotted line), $n = 5$ (pluses connected with a dash-dot line), $n = 10$ (triangles connected with a dashed line) and $n = 100$ (continuous line) multi-scale device for optimal power in an infinitely wide channel $B_n = 0$. For clarity, the scale in each device has been normalised by the total number of scales for that device.

Likewise, the thrust coefficient gradually builds up across scales to the smallest (disc) scale (figure 10c) for all numbers of device scales considered. Further, the thrust coefficient distribution appears relatively universal for different numbers of device scales, and approaches the limiting case ($n\to \infty$) from above as the number of problem scales is increased (i.e. at all normalised scales the thrust coefficient is slightly higher for a lower number of device scales). The physical interpretation of the shape of the $C_{T_s}$ curve is analogous to the intuition of the optimum power extraction of the Betz actuator disc model. As a larger thrust is applied to the flow to attempt to increase power extraction, more fluid is ‘diverted’ around the disc. Likewise, with too little thrust, more flow passes through the disc but the power is poorly extracted from the fluid. In the multi-scale device, as $n$ becomes larger, the device better decouples these scales such that at the largest scale little resistance is applied to the flow, and at the smallest scale the largest possible resistance is applied to the flow. The near universality of the curve with respect to $s/n$ supports the observed higher optimal power coefficient as $n$ increases; if the thrust distribution is bound to a universal curve then a greater range of scales from $1/n\leqslant s/n\leqslant 1$ is required to achieve greater pressure drop at the smallest scale, and least stream-tube deformation (lowest thrust) at the largest scale. This is achieved as increasing $n$ provides greater scale separation along the universal curve, leading to higher optimal power.

The value of $C_{T_n}$, i.e. $C_{T_s}$ at $s/n=1$, is observed to decrease as $n$ increases (less impact on outermost scale), and so, despite the power extraction efficiency increasing, the impact on global mass-flux reduction is reduced. Interestingly, although the blockage ratios increase at all scales with increasing $n$ (figure 10d), the device blockage ratio actually decreases (see figure 11). So that for optimal power extraction in globally unblocked flow, as the number of scales increases, the devices occupy a vanishingly small proportion of the spatial extent covered by the devices.

This is particularly relevant to both tidal (Vennell Reference Vennell2013) and wind (Bleeg et al. Reference Bleeg, Purcell, Ruisi and Traiger2018) energy extraction devices, where the coupling of mass flux and device blockage was observed to play an important role in limiting power extraction. In short, multi-scale devices operating at their optimal energy extraction efficiency, decrease the impact of this detrimental effect. For a fixed channel width, this does, however, come at the expense of reduced power generation, as the discs occupy a smaller fraction of the device area with increasing number of scales. Finally, note that the global thrust coefficient for an $n=50$-scale device is near $2.5$-fold larger than the $n=1$ scale device (figure 12), showing that discs need to sustain higher thrust levels for devices with power extraction exceeding the Betz limit.

The distributions of blockage ratios shown in figure 10(d) further relate the geometry of the device to the physics. Note that presently the largest scale is not included, as the blockage for the largest $n\textrm {th}$-scale is zero by definition. Across the entire $s-n$ parameter space, the blockage is bounded by $B_s\geqslant 0.4$, with the blockage at the largest interior ($n-1\textrm {th}$)-scale converging to 1/2 and at the smallest scale to 1. Similar to the velocity factors, the blockage ratio for a given normalised scale monotonically increases with increasing $n$. As in the Garrett & Cummins (Reference Garrett and Cummins2007) model (see Appendix A), blockage at each scale is the physical mechanism by which the velocity factors are modified to optimise the power coefficient of the device. This is clear by consideration of the constraint equation (2.6), whereby the blockage ratio is used to constrain the thrust coefficient rise across scales.

3.5. Finite blockage multi-scale devices

We next consider finite global blockage ratios. For practical purposes we limit our investigation to global blockages less than 0.25. It is not clear that, unless a free surface is introduced into the problem, any new physics emerges for blockages greater than our chosen value.

Figure 13(a) presents the optimal $C_{P_G}$ for different blockage ratios at different problem scales. Numerical results (indicated by the markers) are compared with an empirical fit given by

(3.13) \begin{equation} C_{P_{G, {{Max}}}} \approx \frac{1}{n} {\left({ 1-B_G }\right)}^{{-}2} {[{ C_{P,{\textit{Betz}}} + {\left({n-1}\right)} {\left({ 1-B_G }\right)}^{4/9} }]} , \end{equation}

where we note that the leading term $C_{P,\textit{Betz}} {(1-B_G)}^{-2}$ is the limit determined by Garrett & Cummins (Reference Garrett and Cummins2007) for a (single scale) array of discs homogeneously arrayed across a channel so as to present blockage $B_G$. Analogous to the unbounded case, as $n \to \infty$, $C_{P_{G, {{Max}}}}$ converges to the limit ${(1-B_G)}^{-14/9}$, which increases with global blockage as observed in figure 13(a) for $n=100$. Even a modest global blockage of 20 % is seen to increase the maximum power coefficient to 1.41 as $n\to \infty$. Consequently greater power than the upstream kinetic flux passing through an area equal to the net actuator frontal area may be extracted for $B_G > 0$. The relative percentage error between the numerical solution and the empirical fit is shown in figure 13(b) and is below 0.5 % over the range of global blockage ratios and number of scales considered. We conclude therefore that the power coefficient limit for an infinite-scale problem in arbitrary global blockage is then $C_{P_{G, {{Max}}}}={(1-B_G)}^{-14/9}$, over the range of global blockage considered herein $(0\leqslant B_G \leqslant 0.25)$.

Figure 11.

The device blockage ratio $B_D$ for optimal power as a function of the number of scales of a multi-scale device (markers indicate numerical solutions with circles for $B_G = 0$, crosses for $B_G = 0.1$, pluses for $B_G = 0.15$, stars for $B_G = 0.2$ and filled circles for $B_G = 0.25$; every fourth simulated data point plotted).

Figure 12.

The relationship between global thrust coefficient $C_{T_G}$ and the number of scales for $n\leqslant 50$ for optimal power in globally unblocked flow $B_G=0$. The circles indicate the numerical solutions.

Figure 13.

The maximum power coefficient for a multi-scale device with different global blockage ratios (circles for $B_G = 0$, crosses for $B_G = 0.1$, pluses for $B_G = 0.15$, stars for $B_G = 0.2$ and filled circles for $B_G = 0.25$; every other simulated data point plotted), here shown for all scales to $n\leqslant 100$. (a) The markers indicate the numerical solutions, and the solid line gives the empirical fit to the optimal $C_{P_G}$ (3.13). (b) The relative percentage error between the numerical solution and the empirical fit.

Figure 14 illustrates how different effects contribute to enabling a higher optimal power coefficient. Adding more problem scales can be visualised as a vertical traverse in the direction of the blue arrow, whilst increasing global blockage can be visualised as a horizontal traverse in the direction of the red arrow. Starting at the Betz limit ($B_G=0$ on the thin continuous line representing a single-scale device), both multi-scale mixing (vertical traverse) or streamwise pressure gradients resulting from finite global blockage (horizontal traverse) may be used to increase the optimal power coefficient. Although it is tempting to think that very high power coefficients in excess of unity may be achievable, in all practical terms global blockages are likely not to exceed $B_G\sim 0.1$ aside from at exceptional locations, and it is difficult to see how more than $2$ or $3$ device scales may be realised. Thus the range of practical importance is confined to the lower left quadrant of figure 14. Nevertheless, this figure provides a useful guide to the interplay between the two effects that serve to increase the optimal energy extraction efficiency.

Figure 14.

The maximum power coefficient as a function of global blockage (lines indicate the empirical fit (3.13) whilst markers indicate the numerical data points) for an $n = 1$ (continuous line with circles), $n = 2$ (dashed line with pluses), $n = 10$ (dash-dot line with triangles) and $n = \infty$ (thick continuous line) multi-scale device. Arrows indicate the effects that contribute to an increase in $C_{P_G,{{Max}}}$.

The aforementioned decoupling of the inner-scale pressure drop and the outermost scale stream tube modification is also evident for finite global blockage cases. Figure 4(a) shows that, in these cases, the limit for $\alpha _G$ for large $n$ decreases with increasing $B_G$ – leading to a higher static pressure drop across the smallest scale – while the stream-tube area remains unmodified ($\alpha _n\to 1$ for large $n$, figure 4c).

Figure 5 also presents the relevant parameters for an $n = 100$ multi-scale device with finite global blockages. Compared with the unbounded case the flow is further decelerated across all scales for finite global blockages (figure 5a), which leads to a steeper increase in static pressure towards the smallest scale (figure 5c). The optimal distribution of local blockage ratios on the other hand flattens with increasing global blockage, with even the local blockage at the larger scales approaching near unity. This, together with $\alpha _n \to 1$ for $n=100$ (figure 4c), infers that even relatively low finite outer-scale blockage is a very effective mechanism to constrain the outer stream-tube and prevent its significant expansion; by contrast, when the flow is globally unblocked, only modest levels of blockage can be imposed across the largest scales so as to limit outer stream-tube expansion (figure 5d). This observation is also captured through examination of the device blockage $B_D$ (figure 11); for globally unblocked flows very low device blockage is permissible for optimal power efficiency, and $B_D\to 0$ as $n\to \infty$, whilst for even small but non-zero outer-scale blockage, significant levels of local blockage $B_D\geqslant 0.1$ are permissible at even large $n$ for optimal power efficiency. This infers that the optimal energy extractor is far more compact in flows that are constrained at the outermost scale, i.e. for $B_n>0$, as compared with globally unblocked $(B_n=B_G=0)$ flows.

Finally, we present the parameter distributions and trends across scales for devices of different numbers of scales for $B_G=0.1$ (figure 15). Generally the results follow those of the unbounded case (figure 10), with slightly reduced velocity factors $\alpha _s$ due to the greater upstream deceleration permissible with a finite outer-scale blockage constraint. As already discussed, finite outer scale blockage permits greater local blockage across the scales; compare figure 15(c) with figure 10(c). As in the unbounded case, the local thrust coefficient variation approximately collapses to a single curve with peak $C_{T_s}$ at the innermost scale of the device with the largest number of scales. This device local thrust coefficient is then significantly in excess of the limiting value, $C_{T_s}=8.0$, for unbounded flows.

3.6. Non-deforming free-surface assumption

We can now consider the implicit assumption that the cross-sectional area of the flow remains approximately uniform, i.e. that the head loss is significantly smaller than the channel depth. The disc configuration shown in figure 2, namely a wide row of discs in a shallow channel where the channel height is smallest, represents the critical case for head loss considerations. Other configurations, such as that shown in figure 1, will be necessarily less restrictive. Although at first it may appear that the bypass flow velocity at the inner scale becomes infinitely large in the limit due to $B_1\to 1$, we note that actually $\beta _1$ remains finite and asymptotic as evidenced by $C_{T_1}$ remaining finitely bounded (this can only occur if $\alpha _s,\gamma _s\to 1$). At any given scale it can readily be shown by conservation that $C_{T_s}=\beta _s^2-\gamma _s^2$. As $\gamma _s\to 1$ in the limit of $n\to \infty$, this implies that $\beta _1$ asymptotes to $\beta _1=\sqrt {C_{T_1}+1}$, with figure 16 demonstrating this from the numerical results. As $C_{T_s}$ always decreases with increasing $s$ (see for instance figure 10), it follows that $\beta _1$ will always be the largest bypass flow induction factor, and that the region of flow by-passing the inner disc scale therefore represents the critical spanwise location for maximum head loss in the critical case of the wide and shallow channel realisation of the multi-scale problem. Applying Bernoulli's principle along the bypass, the pressure drop $\Delta p_{\infty ,1}$ between the far upstream pressure and the bypass of the innermost scale is therefore

(3.14)\begin{equation} \Delta p_{\infty,1} = \frac{1}{2}\rho u_\infty^2\left(\beta_1^2\left(\prod_{i=2}^n\alpha_i^2\right)-1\right) = \frac{1}{2}\rho u_\infty^2\left(\left(C_{T_1}+1\right)\left(\prod_{i=2}^n\alpha_i^2\right)-1\right), \end{equation}

and the head loss is given by $\Delta h_{\infty ,1} = \Delta p_{\infty ,1}/(\rho g)$. Following Garrett & Cummins (Reference Garrett and Cummins2007), the assumption that the channel cross-sectional area remains unchanged requires that the head loss is significantly smaller than the channel height $H$, and so a condition on the Froude number may be formulated as follows:

(3.15)\begin{gather} H \gg \frac{u_\infty^2}{2g}\left(\left(C_{T_1}+1\right)\left(\prod_{i=2}^n\alpha_i^2\right)-1\right), \end{gather}

(3.16)\begin{gather}\frac{u_\infty^2}{gH} \ll \frac{2}{\left(C_{T_1}+1\right)\left(\prod\limits_{i=2}^n\alpha_i^2\right)-1}, \end{gather}

(3.17)\begin{gather}Fr \ll 2^{1/2}{\left[{\left(C_{T_1}+1\right)\left(\prod_{i=2}^n\alpha_i^2\right)-1}\right]}^{{-}1/2}. \end{gather}

This condition is well satisfied for tidal flows which typically have Froude numbers in the range of $0.05\leqslant Fr \leqslant 0.25$, and almost always satisfied for wind farm flows. For instance, for the unbounded case where $\prod _{i=2}^n\alpha _i\to 1/2$ and $C_{T_1}\to 8$, the condition reduces to $Fr \ll 1.26$. For a large global blockage case of $B_G=0.25$, where $\prod _{i=2}^n\alpha _i\to 0.47$ (see figure 4) and $C_{T_1}\to 15$ (see figure 5), the condition reduces to $Fr \ll 0.93$. It is clear therefore that the multi-scale model broadly satisfies the non-deforming free surface assumption for modest global blockage ratios. Analogous to the single-scale model of Garrett & Cummins (Reference Garrett and Cummins2007), the validity of this assumption for the multi-scale model is primarily dependent on the global blockage ratio, and is increasingly restrictive as the global blockage increases.

Figure 15.

Distributions of (a) $\alpha _s$, (b) $\gamma _s$, (c) $C_{T_s}$ and (d) $B_s$ for an $n = 2$ (circles connected by dotted line), $n = 5$ (pluses connected with a dash-dot line), $n = 10$ (triangles connected with a dashed line) and $n = 100$ (continuous line) multi-scale device for optimal power for a global blockage ratio $B_G = 0.1$. For clarity, the scale in each device has been normalised by the total number of scales for that device.

Figure 16.

The inner-scale bypass flow velocity $\beta _1$ for optimal power as a function of the number of scales of a multi-scale device (markers indicate numerical solutions with circles for $B_G=0$, crosses for $B_G=0.1$, pluses for $B_G=0.15$, stars for $B_G=0.2$ and filled circles for $B_G=0.25$; every fourth simulated data point plotted.

3.7. Numerical demonstration of multi-scale performance uplift

We now present three-dimensional Reynolds-averaged Navier–Stokes (RANS) computations of the $n=2$ and $n=3$ optimal configurations, with the focus on demonstrating both the uplift in power coefficient and scale-separation arguments. We simulate the configuration shown in figure 2, namely a wide row of discs in a shallow channel, where we compare the theoretical results with a set of four configurations (table 1), all with the same global blockage ratio. The methodology closely follows that of Nishino & Willden (Reference Nishino and Willden2013), where the governing RANS equations are solved assuming steady and incompressible flow, and turbulence closure is provided using the $k-\epsilon$ model of Launder & Spalding (Reference Launder and Spalding1974) (k being the turbulent kinetic energy density and $\epsilon$ the rate of dissipation). The height of the channel is set to $1.2D$, where $D$ is the actuator disc diameter, and the width of the channel is set such that the global blockage ratio is $B_G=0.0785$. The Reynolds number (Re, based on the turbine diameter) is set to be large enough ($Re = 1.5\times 10^6$) such that viscous forces are negligible, and the inlet turbulence model quantities are set to be very low ($k=1.5\times 10^{-6}\ \textrm {m}^2\ \textrm {s}^{-2}$ and $\epsilon = 1.5\times 10^{-10}\ \textrm {m}^2\ \textrm {s}^{-3}$, where the inlet velocity is set to unity). The actuator discs are likewise modelled following Nishino & Willden (Reference Nishino and Willden2013), where a change in momentum flux equal to $S_U=K(U_d^2/{2}$), where $U_d$ is the local streamwise velocity, is added to the momentum equation at the location of the discs to the momentum equation, and $K$ is assumed to be uniform across the surface of all discs. The resultant disc averaged global thrust and power coefficients are then calculated by $\langle C_{TG} \rangle =K\langle U_d^2\rangle / u_\infty ^2$ and $\langle C_{PG}\rangle =K\langle U_d^3 \rangle / u_\infty ^3$, noting that the reference flow speed is unity and where the angle brackets represent face averaging over all discs. See Nishino & Willden (Reference Nishino and Willden2013) for further details and numerical model validation.

Table 1.

Summary of flow configurations studied in the numerical framework.

As demonstrated by Nishino & Willden (Reference Nishino and Willden2013), the performance uplift for this specific configuration of discs is primarily limited by the mixing rate, which can be controlled by variations in the turbulence model coefficient $C_{\epsilon 1}$ and the number of discs. Here, we leave $C_{\epsilon 1}$ at its widely used value of $1.44$, although we note that lower values would promote mixing, and choose to vary the number of discs to better isolate the multi-scale effect.

The blockage ratios presented in table 1 are the theoretically optimal blockage ratios for the given global blockage ratio. The $n=2$-scale studies can be readily compared with those of Nishino & Willden (Reference Nishino and Willden2013). The $6$-group $n=3$ study can be readily compared against the $6$ disc $n=2$ study, and the $240$ disc $n=3$ study can be compared against the three-scale theoretical optima.

The summary plot in figure 17 clearly demonstrates the uplift in performance and provides numerical validation for the theoretical model. The $n=2$ device optima fall between the single-scale (Garrett & Cummins Reference Garrett and Cummins2007) limit and the theoretical two-scale limit. As anticipated, the fence underperforms the theoretical limit due to finite fence length effects, with the $48$ disc fence significantly outperforming the $6$ disc fence. For $48$ discs, the three-scale optimal configuration very slightly outperforms the two-scale $48$ disc configuration; this is the result of the interplay between compounding of deleterious finite fence effects with only $8$ discs at the inner scale and $6$ discs at the outer scale, and uplift due to the additional scale. Finally, for a large number of discs, it is clear that the three-scale array outperforms the two-scale arrays, and the efficiency instead approaches the three-scale theoretical limit.

The assumption underpinning the theoretical analysis is that of scale separation between each sequential scale. Figure 18 shows how the mixing process across each scale occurs sequentially, and that the far wake mixing of a given scale takes place much faster than the horizontal expansion of the flow around the next scale. For this reason, the near wake (from say $z/D \sim 5$) of the $n=2$ scale simulation resembles a single-scale wake, and the near wake (from $z/D \sim 5$) of the $n=3$-scale device resembles that of an $n=2$-scale $6$ disc array (which further mixes to resemble a single-scale wake far downstream). With the aid of figure 17, one can inductively picture how with each additional scale the scale-separation assumption is satisfied. For instance, the near wake of an $n=4$-scale device will resemble an array of $n=3$-scale devices (with the accelerated bypass between each array) and so forth.

Figure 17.

Power curves for the numerical configurations detailed in table 1, illustrating power uplift under multi-scale effects for a global blockage ratio $B_G=0.0785$. The dashed line shows the single-scale (Garrett & Cummins Reference Garrett and Cummins2007) limit, and the dash-dot and solid lines show both the $n=2$ and $n=3$ scale limits, respectively, alongside their corresponding power vs thrust variations. Symbols indicate numerical results; squares, $6$ disc, $n=2$ configuration; diamonds, $48$ disc, $n=2$ configuration; circles, $48$ disc, $n=3$ configuration; triangles, $240$ disc, $n=3$ simulations.

Figure 18.

Normalised streamwise velocity field contours at hub height qualitatively illustrating scale separation for the $6$ disc $n=2$ scale (a) and $6$ group of $8$ disc $n=3$ (b) configurations. The flow is moving left to right, with bypass and core flow regions clearly visible. Note the differences in geometric scale between the two simulations.

We note that although the multi-scale shallow water array numerically approaches the theoretical optima, other configurations that encourage wake mixing may also be useful for extracting power above the Betz limit. The theoretical optimum provides an upper bound on the energy extraction efficiency (which will always hold), and sets a benchmark against which the multi-faceted effects of finite mixing lengths, real array inefficiencies, number of devices, turbulence effects and other flow physics can be evaluated.