Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Abstract. This paper presents limits for stability of projection type schemes when using high order pressure-velocity pairs of same degree. Two high order h/p varia-tional methods encompassing continuous and discontinuous Galerkin formulations are used to explain previously observed lower limits on the time step for projection type schemes to be stable [18], when h- or p-refinement strategies are considered. In addition, the analysis included in this work shows that these stability limits do not depend only on the time step but on the product of the latter and the kinematic vis-cosity, which is of particular importance in the study of high Reynolds number flows. We show that high order methods prove advantageous in stabilising the simulations when small time steps and low kinematic viscosities are used. Drawing upon this analysis, we demonstrate how the effects of this instability can be reduced in the discontinuous scheme by introducing a stabilisation term into the global system. Finally, we show that these lower limits are compatible with Courant-Friedrichs-Lewy (CFL) type restrictions, given that a sufficiently high polynomial or

An experimental investigation of the influence of inter-turbine spacing on the loads and performance of a co-planar tidal turbine fence

Multi-rotor tidal turbine systems offer engineering benefits through shared infrastructure and improved opportunities for maintenance. Additionally, the ability to specify accurately the inter-turbine spacing in co-planar arrays allows rotors to be designed and deployed to benefit from the constructive interference effects available from neighbouring rotors. In this work we consider the effect of inter-turbine spacing and control of this spacing for a fence of turbines in low overall levels of global blockage (4.5%). We conduct experiments in a towing tank using two tidal turbine models that were previously designed to benefit from constructive interference effects at high local blockage, i.e. close inter-turbine spacing. The turbines were towed in a side-by-side configuration by suspending them from above. By making use of the tank’s side wall to act as a symmetry plane we were able to emulate a fence of four laterally arrayed turbines. As indicated by theory, decreasing inter-turbine spacing is shown to have a positive effect on fence performance. By reducing the inter-turbine spacing from one diameter to a quarter of a diameter, we observe an overall 1.4% performance increase, which is driven by a 2.8% increase in the inboard turbine’s power coefficient. This research is a first attempt to quantify constructive interference effects for a four turbine fence; the methods and results will be used to instruct further studies to aid the development of such multi-rotor tidal turbine systems

Investigation of wind turbine wake superposition models using Reynolds‐averaged Navier‐Stokes simulations

It is well accepted that the wakes created by upstream turbines significantly impact on the power production and fatigue loading of downstream turbines and that this phenomenon affects wind farm performance. Improving the understanding of wake effects and overall efficiency is critical for the optimisation of layout and operation of increasingly large wind farms. In the present work, the NREL 5‐MW reference turbine was simulated using blade element embedded Reynolds‐averaged Navier‐Stokes computations in sheared onset flow at three spatial configurations of two turbines at and above rated flow speed to evaluate the effects of wakes on turbine performance and subsequent wake development. Wake recovery downstream of the rearward turbine was enhanced due to the increased turbulence intensity in the wake, although in cases where the downstream turbine was laterally offset from the upstream turbine this resulted in relatively slower recovery. Three widely used wake superposition models were evaluated and compared with the simulated flow‐field data. It was found that when the freestream hub‐height flow speed was at the rated flow speed, the best performing wake superposition model varied depending according to the turbine array layout. However, above rated flow speed where the wake recovery distance is reduced, it was found that linear superposition of single turbine velocity deficits was the best performing model for all three spatial layouts studied

The efficiency of an array of tidal turbines partially blocking a wide channel

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. © 2012 Cambridge University Press

Designing multi-rotor tidal turbine fences

An embedded Reynolds-Averaged Navier-Stokes blade element actuator disk model is used to investigate the hydrodynamic design of tidal turbines and their performance in a closely spaced cross-stream fence. Turbines designed for confined flows are found to require a larger blade solidity ratio than current turbine design practices imply in order to maximise power. Generally, maximum power can be increased by operating turbines in more confined flows than they were designed for, although this also requires the turbines to operate at a higher rotational speed, which may increase the likelihood of cavitation inception. In-array turbine performance differs from that predicted from single turbine analyses, with cross-fence variation in power and thrust developing between the inboard and outboard turbines. As turbine thrust increases the cross-fence variation increases, as the interference effects between adjacent turbines strengthen as turbine thrust increases, but it is observed that cross-stream variation can be mitigated through strategies such as pitch-to-feather power control. It was found that overall fence performance was maximised by using turbines designed for moderately constrained (blocked) flows, with greater blockage than that based solely on fence geometry, but lower blockage than that based solely on the turbine and local flow passage geometry to balance the multi-scale flow phenomena around tidal fences