A new theoretical model is proposed to explore the efficiency of a long array of tidal turbines partially blocking a wide channel cross-section. An idea of scale separation is introduced between the flow around each device (or turbine) and that around the entire array to assume that all device-scale flow events, including ‘far-wake’ mixing behind each device, take place much faster than the horizontal expansion of the flow around the entire array. This assumption makes it possible to model the flow as a combination of two quasi-inviscid problems of different scales, in both of which the conservation of mass, momentum and energy is considered. The new model suggests the following: when turbines block only a small portion of the span of a shallow channel cross-section, there is an optimal intra-turbine spacing to maximize the efficiency (limit of power extraction) for a given channel height and width. The efficiency increases as the spacing is reduced to the optimal value due to the effect of local blockage, but then decreases as the spacing is further reduced due to the effect of array-scale choking, i.e. reduced flow through the entire array. Also, when the channel is infinitely wide, the efficiency depends solely on the local area blockage rather than on the combination of the intra-turbine spacing and the channel height. As the local blockage is increased, the efficiency increases from the Lanchester–Betz limit of 0.593 to another limiting value of 0.798, but then decreases as the local blockage is further increased.

The characteristics of flow past a partial cross-stream array of (idealized) tidal turbines are investigated both analytically and computationally to understand the mechanisms that determine the limiting performance of partial tidal fences. A two-scale analytical partial tidal fence model reported earlier is further extended by better accounting for the effect of array-scale flow expansion on device-scale dynamics, so that the new model is applicable to short fences (consisting of a small number of devices) as well as to long fences. The new model explains theoretically general trends of the limiting performance of partial tidal fences. The new model is then compared to three-dimensional Reynolds-averaged Navier–Stokes (RANS) computations of flow past an array of various numbers (up to 40) of actuator disks. On the whole, the analytical model agrees well with the RANS computations, suggesting that the two-scale dynamics described in the analytical model predominantly determines the fence performance in the RANS computations as well. The comparison also suggests that the limiting performance of short partial fences depends on how much of device far-wake mixing takes place within the array near-wake region. This factor, however, depends on the structures of the wake and therefore on the type/design of devices to be arrayed.

The efficiency of tidal stream turbines in a large array depends on the balance between negative effects of turbine-wake interactions and positive effects of bypass-flow acceleration due to local blockage, both of which are functions of the layout of turbines. In this study we investigate the hydrodynamics of turbines in an infinitely large array with aligned or staggered layouts for a range of streamwise and lateral turbine spacing. First, we present a theoretical analysis based on an extension of the linear momentum actuator disc theory for perfectly aligned and staggered layouts, employing a hybrid inviscid-viscous approach to account for the local blockage effect within each row of turbines and the viscous (turbulent) wake mixing behind each row in a coupled manner. We then perform large-eddy simulation (LES) of open-channel flow for 28 layouts of tidal turbines using an actuator line method with doubly periodic boundary conditions. Both theoretical and LES results show that the efficiency of turbines (or the power of turbines for a given bulk velocity) in an aligned array decreases as we reduce the streamwise turbine spacing, whereas that in a staggered array remains high and may even increase due to the positive local blockage effect (causing the local flow velocity upstream of each turbine to exceed the bulk velocity) if the lateral turbine spacing is sufficiently small. The LES results further reveal that the amplitude of wake meandering tends to decrease as we reduce the lateral turbine spacing, which leads to a lower wake recovery rate in the near-wake region. These results will help to understand and improve the efficiency of tidal turbines in future large arrays, even though the performance of real tidal arrays may depend not only on turbine-to-turbine interactions within the array but also on macro-scale interactions between the array and natural tidal currents, the latter of which are outside the scope of this study.

In this paper we extend linear momentum actuator disc theory to consider two rows of tidal turbines placed in a centred or staggered arrangement. The extensions assume a streamwise spacing between rows that is sufficient for pressure equalization, but is not too large for significant mixing of the upstream turbine wake before the second row. We first consider a given number of turbines in a tidal channel; in this case the average power for a staggered arrangement over two rows is found to be higher than that for a centred arrangement, but lower than can be obtained by placing all turbines side-by-side in one row (if all turbines have the same local resistance). Furthermore, staggered arrangements extract power more efficiently than centred arrangements, but less efficiently than a single row with the same number of turbines, and this has implications for ranking different arrangements of tidal turbines. We also use the extended actuator disc models (together with an argument of scale separation) to consider some example arrangements of tidal turbines in laterally unconfined flow. Specifically, it is shown that locally staggering a fixed number of turbines in an array to form a tidal farm generates less power than placing the same number of turbines side-by-side. However, if more than one row of turbines is adopted (perhaps to keep the farm spatially compact) then the optimum turbine spacing within a row increases significantly with addition of a second row. This trend suggests that multi-row tidal turbine farms would require wide turbine spacing within each row to maximize the power per turbine, similarly to existing offshore wind farms.