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 crosssection, 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.

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We present a comprehensive set of two-dimensional (2D) unsteady Reynolds-averaged Navier-Stokes (URANS) simulations of flow around a pair of counter-rotating vertical-axis wind turbines (VAWTs). The simulations are performed for two possible configurations of the counter-rotating VAWT pair, with various gaps between the two turbines, tip-speed-ratios and wind directions, in order to identify key flow mechanisms contributing to the enhanced performance of a pair of turbines compared to an isolated turbine. One of the key mechanisms identified, for the case of two turbines arrayed side-by-side with respect to the incoming wind, is the change of lateral velocity in the upwind path of each turbine due to the presence of the neighbouring turbine, making the direction of local flow approaching the turbine blade more favourable to generate lift and torque. The results also show that the total power of a staggered pair of turbines cannot surpass that of a side-by-side pair of turbines. Some implications of the present results for the prediction of the performance of single and multiple rows (or a farm) of VAWTs are also discussed. The local flow mechanisms identified in the present study are expected to be of great importance when the size of the farm is relatively small