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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The wake transition of the flow around two circular cylinders placed in staggered arrangements with fixed streamwise separation of 5D and cross stream separation varying from 0D to 3D has been studied. The wake transition is compared to that of a single isolated cylinder. Linear stability analysis utilising Floquet theory and direct numerical simulations, using the spectral/hp element method were carried out. It is found that besides modes A and B, mode C can also appear in the wake transition, depending on the relative positioning of the cylinders. The structure of mode C is analysed and the non-linear character of the bifurcation for this mode is investigated. IntroductionThe flow around circular cylinders has been extensively studied due to its practical importance in engineering and scientific relevance in fluid mechanics. On the engineering side, there are a number of applications in mechanical, civil and naval engineering that employ circular-cylindrical structures, such as heat exchangers, chimneys and off- shore platforms. In scientific terms, the flow around circular cylinders presents various important physical phenomena, such as separation, vortex shedding and transition.In the next two subsections, some of the relevant research that has been carried out on two different aspects of this flow is reviewed. These aspects are the wake transition in bluff body flows and the flow around circular cylinders in staggered arrangements, illustrated in figure 1. Wake transitionIn the last two decades, much effort has focused on the study of the three-dimensional transition that takes place in the von Kármán wake that appears in the flow downstream of a single cylinder. This line of research in its contemporary form was instigated by the seminal work of Williamson (1988), in which two distinct stages were identified in the wake transition, spanning 160 Re 300. The limits of these stages were identified by discontinuities in the curve obtained when the Strouhal number (St) was plotted against the Reynolds number (Re) and these discontinuities correspond to the manifestation of different three-dimensional structures in the wake. The first to appear is called mode A, and it has a spanwise wavelength of approximately 4 diameters. The second is called mode B, and its spanwise wavelength is close to 1 diameter. A number of papers on additional experiments and direct numerical simulations (DNS) for this Re range have Wake transition in staggered arrangements 3 been published since then with the aim of reproducing these findings, see for example Wu et al. (1994), Zhang et al. (1995) and Thompson et al. (1996). A great leap forward occurred with the work of Barkley & Henderson (1996), who performed high accuracy Floquet stability analyses of two-dimensional time periodic base flows, precisely identifying the critical Reynolds numbers and also characterising each of the modes that take part in the wake transition. Williamson (1996) presented an extensive study on the wake transition, setting the most significant results pre...