Abstract
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.