Proceedings of the ASME nd International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 9-14, 2013, Nantes, France

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1 Proceedings of the ASME nd International Conference on Ocean, Offshore and Arctic Engineering OMAE203 Jne 9-4, 203, Nantes, France OMAE THE EFFICIENCY OF TIDAL FENCES: A BRIEF REVIEW AND FURTHER DISCUSSION ON THE EFFECT OF WAKE MIXING Takafmi Nishino Department of Engineering Science University of Oxford Oxford, OX 3PJ United Kingdom takafmi.nishino@eng.ox.ac.k Richard H. J. Willden Department of Engineering Science University of Oxford Oxford, OX 3PJ United Kingdom richard.willden@eng.ox.ac.k ABSTRACT Recent discoveries on the limiting efficiency of tidal fences are reviewed, followed by a new theoretical investigation into the effect of wake mixing on the efficiency of fll tidal fences (i.e. trbines arrayed reglarly across an entire channel span). The new model is based on the momentm and energy balance eqations bt incldes several nclosed terms, which depend on the actal (three-dimensional) characteristics of trbine nearwake mixing and therefore need to be modelled empirically. The new model agrees well with three-dimensional actator disk simlations when those nclosed terms are assessed based on the simlations themselves, sggesting that this low-order model cold serve as a basis to analyse how varios physical factors (sch as the design of trbines) affect the limiting efficiency of tidal fences via changes in those terms describing the characteristics of trbine near-wake mixing. Also discssed is the effect of wake mixing on the efficiency of partial tidal fences. INTRODUCTION The development of efficient and sstainable energy technologies is amongst the most pressing isses facing the world today. Tidal-stream power generation is, albeit its basic concept has been known for many years, one of the emerging technologies in this field. Varios types of tidal-stream devices (the majority can be described as nderwater versions of wind trbines) are crrently being proposed and tested; however, it is generally recognised that, regardless of the exact design of devices to be employed, a considerably large nmber of devices wold be reqired to make a meaningfl contribtion to the ftre energy spply. As reported in e.g. [], leading indstrial tidal energy developers have already been demonstrating the performance of fll-scale devices and are in the process of planning the first tidal power farm deployment. One of the biggest challenges here is how to design efficient tidal power farms, i.e., how to array sch a large nmber of tidal-stream devices to maximise the overall power generation whilst keeping their impact on the natral environment to an acceptable level. The so-called tidal fences (spanwise array of tidal-stream trbines) are promising on their own and also as a key component of large tidal power farms to be deployed in the ftre. RECENT DISCOVERIES A series of important scientific (flid-mechanical) theories on the efficiency of tidal trbines has been derived dring the last several years. Among others, Garrett and Cmmins [3] have reported an extension of the classical Lanchester-Betz theory (on the efficiency of wind power generation based on the balances of mass, momentm and energy; see e.g. [4]). An important finding was the significance of the effect of tidal channel blockage; the hydrodynamic limit of power extraction by trbines placed in a tidal channel is proportional to ( B) 2, where B is the channel blockage ratio (ratio of the frontal-projected area of tr- Copyright c 203 by ASME Downloaded From: on /26/204 Terms of Use:

2 ( a) ( b) U C U UC U U U + A A A D n ( d+s ) ( c) d+s h w FIGURE. SCHEMATIC OF A PARTIAL TIDAL FENCE MODEL (FOR COMMON HORIZONTAL-AXIS TURBINES): (a) ARRAY-SCALE FLOW EXPANSION AND MIXING, (b) DEVICE-SCALE FLOW EXPANSION AND MIXING, AND (c) CROSS-SECTIONAL VIEW OF THE CHANNEL. THE GLOBAL BLOCKAGE IS DEFINED AS nπd 2 /4hw (WHERE n IS THE NUMBER OF ROTORS AND d THE ROTOR DIAME- TER), WHEREAS THE LOCAL BLOCKAGE IS DEFINED AS πd 2 /4h(d + s). REPRODUCED FROM [2]. bines to the channel cross-sectional area). Holsby et al. [5] and Whelan et al. [6] frther extended this theory, explaining that an effective channel blockage may change de to the free-srface effect (the effect of changes in water depth accompanied by the power extraction) and hence the limit of power extraction may also change, depending on the Frode nmber of the flow. It shold be noted here that the above theories/models by Garrett and Cmmins [3], Holsby et al. [5] and Whelan et al. [6] are all concerned with the local efficiency of tidal trbines; in other words, these theories assme that the installation of trbines into a tidal channel does not change the amont of flow coming into the channel. In reality, however, the flow coming into the channel redces nless the hydrodynamic drag cased by the installation of trbines is negligibly small compared to other flow resistances throgh the channel, e.g., sea-bed friction (Garrett and Cmmins [7]). Vennell [8,9] combined a simplified model of this effect (namely a channel dynamics model) with the local theory of Garrett and Cmmins [3] to estimate the efficiency of a nmber of trbines arrayed across the cross-sections of varios types of tidal channels. Also, Vennell [0] has taken into accont the effect of drag acting on trbine spporting strctres in his tidal farm model. A drawback to the above models stemming from the local theory of Garrett and Cmmins [3] (regardless of whether the channel dynamics effects are considered or not), however, is that they are not directly applicable to trbine arrays blocking only a part of tidal channel cross-section. They assme that trbines are reglarly arrayed across the entire channel cross-section, i.e., they are concerned with fll (rather than partial ) tidal fences. (This is ltimately becase they employ a simple qasi-inviscid assmption allowing the mixing of flow behind trbines only in a single far-wake (or far-downstream) region; in this sense, all theories/models mentioned above can be described as being based on a single-scale wake mixing assmption.) Two major examples that highlight the significance of considering partial tidal fences are: (i) channels where a considerable portion of their cross-section needs to be nblocked in order to allow for navigation of vessels and so forth, and (ii) headland sites, where the channel cross-section is semi-infinitely wide (and may also be deep) and ths cannot be flly blocked by the fence. In recognition of this, Nishino and Willden [2] have derived another important extension of the Lanchester-Betz theory to explore the efficiency of a partial tidal fence by introdcing an idea of scale separation between the flow arond each trbine (or device in general) and that arond the entire array (Fig. ). One of the key findings from the new theory was the existence of optimal intra-device spacing to maximize the efficiency of a partial tidal fence installed in a wide channel; for an infinitely wide channel, for example, the limit of energy extraction increases from the Lanchester-Betz limit of 59.3% (of the kinetic energy of ndistrbed incoming flow) to another limit of 79.8% if the devices are optimally spaced. The new model acconts for the effects of device- and array-scale far-wake mixing separately and may therefore be described as the first (and presmably the simplest) tidal-stream power generation model employing a mlti-scale wake mixing concept. 2 Copyright c 203 by ASME Downloaded From: on /26/204 Terms of Use:

3 More recently, Nishino and Willden [] have extended the partial tidal fence model by better acconting for the effect of array-scale flow expansion on device-scale dynamics, so that the extended model is applicable to short partial fences (consisting of a small nmber of devices, where the two scales are not completely separated) as well as to long partial fences. The extended model was compared with three-dimensional Reynolds-averaged Navier-Stokes (RANS) simlations of an array of varios nmbers of actator disks; on the whole, the extended model agreed fairly well with RANS reslts. The partial fence model has also been combined with a channel dynamics model to accont for the redction of inflow de to the installation of devices (Willden and Nishino [2]). Also, Vogel et al. [3] have recently extended the partial fence model to accont for the free-srface effect. ISSUES AND CHALLENGES Althogh the tidal fence model of Nishino and Willden [2, ] has provided many new insights into the efficiency of tidal fences, the crrent model still neglects the effects of several important physical factors, leaving a gap between or nderstanding and reality and ths preventing s from designing trly efficient tidal fences or farms. Presming that the crrent tidal fence model is going to be frther extended to (either analytically or empirically) incorporate the effects of those missing physical factors, most of their effects on the efficiency of tidal fences/farms cold be essentially classified into one or more of the following three categories: (i) effects by changing the characteristics of device nearwake mixing, which has not been taken into accont in any of the theories/models described above; (ii) effects by changing the interaction of device-scale and array-scale wake mixing, which mst also be taken into accont to flly nderstand the efficiency of mltiple and/or short partial fences (rather than a long partial fence); and (iii) effects by changing the tidal channel dynamics. As concerns the first category, Nishino and Willden [4] recently performed three-dimensional RANS simlations of confined flow past an actator disk (representing an ideal tidalstream device). This stdy, sing a so-called blade-indced trblence model to simlate varios levels of device wake mixing, has demonstrated that the rate of mixing in the device near-wake region affects the limit of power extraction by the device. As will be presented later in this paper, we are crrently developing a new analytical model of device-scale power extraction inclding the effect of device near-wake mixing. The new model is based on momentm and energy balances bt incldes several nclosed terms, which depend on actal (three-dimensional) characteristics of near-wake mixing and ths need to be modelled empirically. The model agrees well with three-dimensional actator disk simlations when those nclosed terms are assessed based on the simlations themselves, sggesting that this low-order model cold serve as a basis to analyse how varios physical factors affect the limit of power extraction (via changes in those terms describing the characteristics of device near-wake mixing) if those terms are modelled properly based on practical device wake data rather than actator disk simlations. The challenge here is how to obtain sch practical device wake data (inclding the effects of sea-srface, sea-bed and differences in the design of devices). Apart from condcting experiments, a possible soltion is to se high-fidelity simlations, sch as Large-Eddy simlations (LES), of flow arond a device of interest. Althogh the LES of rotating trbine blades (at practical Reynolds nmbers) are infeasible de to hge comptational cost, recent LES stdies sing a so-called actator-line method (e.g. [5]) have shown some promising reslts. Note that sch simlations are valable not only for the above prpose bt also for (and are sally intended for) examining the performance of device itself, i.e., how mch power a device of a specific design can actally extract compared to the limit of power extraction. Of importance here is that differences in the design of devices affect the limit of power extraction and the performance of the device itself at the same time. The second category, the interaction of device- and arrayscale wake mixing, is a crcial sbject for designing efficient tidal fences and farms. Several experimental and comptational stdies have been reported in recent years on the characteristics of flow arond varios types of trbine arrays (e.g. [5 7]); nevertheless the knowledge obtained so far is still limited. The analytical tidal fence model of Nishino and Willden [2, ] has a potential to provide s with a more comprehensive view on the efficiency of trbine arrays; however its applicability to practical tidal farm design is still limited mainly becase the effect of the interaction of device- and array-scale mixing has not been considered sfficiently. The interaction is important for (i) short partial fences, where array-scale mixing initiates before the completion of device-scale mixing, and (ii) mltiple partial fences, where the interaction is inevitable nless the streamwise gap between each fence is sfficiently large (relative to the width of fences). Hence the challenge here is to frther investigate, nderstand and model the effect of sch interactions. As will be briefly discssed later in this paper, it seems possible to evolve the aforementioned tidal fence model into a more generic loworder model that sfficiently acconts for the effect of deviceand array-scale interaction and thereby predicts the efficiency of many different configrations of trbine arrays. Sch a robst low-order model wold be sefl not only to nderstand basic mechanisms bt also to optimise the design of large tidal power farms to be constrcted in the ftre. The third category, tidal channel (or basin) dynamics, is another important sbject when the magnitde of hydrodynamic 3 Copyright c 203 by ASME Downloaded From: on /26/204 Terms of Use:

4 drag indced by the installation of trbine arrays is not negligible compared to other flow resistances throgh the tidal channel/basin itself [7]. This is important not only for correctly estimating the performance of large tidal power farms to be installed bt also for predicting their impact on the natral environment. Since neither experiments nor high-fidelity simlations of flow arond sch large tidal power farms are possible, a different type of approach is reqired here to address this sbject. As already mentioned earlier, one possible approach is to se an analytical channel dynamics model, sch as those by Garrett and Cmmins [7], Vennell [8 0] and Willden and Nishino [2]. More recently, Cmmins [8] has reported a new analytical model that is applicable to mltiple branching channels, sggesting a possible frther extension of this type of approach. Another type of approach is to incorporate a low-order model of flow arond trbine arrays into an ocean flow solver (based on the shallow-water eqations, for example). A prototype of this approach has been reported, e.g., by Draper et al. [9], who incorporated the local power extraction model of Holsby et al. [5] into a discontinos Galerkin shallow-water solver. A key problem here is that the model to be incorporated mst properly accont for the effects of device near-wake mixing and the interaction of device- and array-scale mixing; hence the challenges in the preceding two categories are crcial here as well. Sch large scale simlations sing a robst low-order model of trbine arrays wold provide more reliable predictions of the impact/efficiency of large tidal power farms than the simlations crrently sed in the commnity, e.g. [20]. EFFECT OF NEAR-WAKE MIXING ON THE EFFICIENCY OF A FULL TIDAL FENCE Below we present or new theoretical investigation into the effect of device near-wake mixing. The effect on a single device, or eqivalently a fll tidal fence (i.e. devices reglarly spaced across an entire channel span) is discssed first by extending the local power extraction model of Garrett and Cmmins [3], followed by some comments on the effect on a partial tidal fence before conclding the paper. Figre 2 shows a schematic of the new device-scale power extraction model. This is a modified version of a schematic sed by Holsby et al. [5] for their actator disk in a parallel-sided tbe model, which is practically the same as the model of Garrett and Cmmins [3] bt ses somewhat different notations to describe the system; we mostly follow their notations. Let s consider a single device (or actator disk) of area A (= πd 2 /4, where d is the disk diameter) placed in a parallelsided flow passage (channel or tbe). The cross-sectional area of the flow passage is AR, i.e., the channel blockage B = /R. The streamtbe that contacts the disk edge divides the flow passage into two sb-passages, namely bypass and core flow passages. Here we define five streamwise locations (stations to 5): (Normalized) power removed: Bypass ( to 4) Core ( to 4) Total ( to 4) Total ( to 5) p C SB C + C P C + C P T / 2 SC H 2 S B S C 4 4 p 4 p 2 p 3 p 4 3 A = C T 4 p 4 p 5 ( CSB > 0 ) ( CSC < 0 ) ( C H = C SB + CSC) ( C T = C P / 2) FIGURE 2. SCHEMATIC OF A NEW DEVICE-SCALE TIDAL- STREAM POWER EXTRACTION MODEL INCLUDING THE EF- FECT OF DEVICE NEAR-WAKE MIXING. : far pstream, where pressre and velocity are niform; 2: immediately pstream of the disk; 3: immediately downstream of the disk; 4: so-called pressre-eqilibrim location, where pressre eqilibrates between the bypass and core flow passages; and 5: far downstream, where not only pressre bt also velocity retrns to be niform. Also, we describe (cross-sectionally averaged) bypass and core flow velocities at station i as β i and α i, respectively, where is the velocity far pstream and far downstream. Similarly, we define local pressres p i as shown in Fig. 2. In the models of Garrett and Cmmins [3] and Holsby et al. [5], viscos (or trblent) mixing is allowed to take place only between stations 4 and 5, i.e., they consider the effect of far-wake mixing bt neglect near-wake mixing. In the new model, we consider the effect of near-wake mixing (between stations 3 and 4) as well. Specifically, we consider the following two factors: (i) energy transfer between the bypass and core flow passages de to near-wake mixing and the attendant energy loss; and (ii) non-niformity of the velocity within each flow passage at station 4. Frther details are described below. Here we se the terminology near-wake to specifically indicate the region between stations 3 and 4. Note that the extent of this region depends not only on device operation conditions bt also on the channel blockage; hence this nearwake region does not necessarily agree with what near-wake sally represents for wind trbines, e.g. [2]. 4 Copyright c 203 by ASME Downloaded From: on /26/204 Terms of Use:

5 Energy Balance We first consider the energy balance within the bypass and core flow passages between stations and 4. The main difference from the models of Garrett and Cmmins [3] and Holsby et al. [5] is that, de to near-wake mixing between stations 3 and 4, the bypass flow loses a certain amont of energy and the core flow gains a part of the energy that the bypass flow loses (as the mixing decelerate the higher-speed bypass flow and accelerate the lower-speed core flow). Here we define the power removed from the bypass and core flows de to near-wake mixing as S B (> 0) and S C (< 0), respectively (therefore S C is the power added to the core flow). In general S B is larger than S C, and the difference, S B ( S C ), represents the attendant power loss (i.e. heat generation de to mixing). Another isse to be considered is that the near-wake mixing generates a mixing layer of a certain thickness; hence the crosssectional velocity profile (within each flow passage, i.e., bypass and core) is not niform at station 4. However, here we temporarily assme that the velocity profile (within each flow passage) is niform at station 4. The non-niformity of the velocity profile will be considered later. In order to inclde S B and S C in the new model, we consider each flow passage as a control volme and derive the energy conservation eqation for it (rather than the Bernolli eqation). For the bypass flow passage (between stations and 4), the energy eqation can be written as ( ) 2 ρ3 + p A(R α 2 ) ( = 2 ρ3 β4 3 + p 4β 4 ) ( A R α ) 2 + S B () α 4 where β 4 = (R α 2 )/(R α 2 /α 4 ) is derived from the conservation of mass (note that the cross-sectional areas of bypass flow passage at stations and 4 are A(R α 2 ) and A(R α 2 /α 4 ), respectively, which are also derived from mass conservation). Hence Eqn. () can be simplified to 2 ρ2 ( β4 2 S B ) + p p 4 = A(R α 2 ) Meanwhile, for the core flow passage (between stations and 4) the energy eqation can be written as ( ) ( ) 2 ρ3 + p Aα 2 = 2 ρ3 α4 3 + p 4α 4 A α 2 + P + S C (3) α 4 where P is the power removed by the actator disk (at the disk location). It is assmed that P = α 2 T, where T = (p 2 p 3 )A is (2) the thrst on the disk. Hence Eqn. (3) can be simplified to 2 ρ2 ( α 2 4 ) + p p 4 = T A + S C Aα 2 (4) Thrst, Power and Other Coefficients Here we can define the thrst and power coefficients, C T and C P, as well as two new coefficients representing the power removed de to near-wake mixing, C SB for the bypass and C SC for the core flow passages: C SB = C SC = S B 2 ρ3 A S C 2 ρ3 A (5) (6) C T = T 2 ρ2 A = β 4 2 α4 2 + C SB C SC (7) R α 2 α 2 C P = P 2 ρ3 A = α 2C T (8) Note that Eqn. (7) is derived from Eqns. (2) and (4). Also, we can define the near-wake heat loss coefficient, C H, as follows: C H = S B ( S C ) 2 ρ3 A = C SB +C SC (9) Momentm Balance Next, we consider the momentm balance for the entire flow passage (i.e., both bypass and core flow passages) between stations and 4. Again, we temporarily assme that the velocity profile (within each flow passage) is niform at station 4. Since the mixing between the bypass and core flow passages (or more precisely, viscos/trblent shear stress acting on the streamtbe dividing the two passages) does not affect the momentm balance for the entire channel, the momentm eqation to be considered here is the same as that sed for the model of Holsby et al. [5], i.e.: ( p AR p 4 AR T = ρ 2 β4 2 A R α ) 2 ρ 2 A(R α 2 ) α 4 which can be simplified to + ρ 2 α 2 4 A α 2 α 4 ρ 2 Aα 2 (0) (p p 4 )R 2 ρ2 C T = 2(R α 2 )(β 4 ) + 2α 2 (α 4 ) () 5 Copyright c 203 by ASME Downloaded From: on /26/204 Terms of Use:

6 Note that the left-hand side of Eqn. () represents the (nondimensionalized) force acting on the entire channel between stations and 4, whereas the right-hand side represents the corresponding momentm change throgh this control volme. We then sbstitte Eqns. (2) and (7) into the above Eqn. () to remove p p 4 and C T, reslting in the following eqation to be satisfied: R(β 2 4 ) (β 2 4 α 2 4 ) + R R α 2 C SB + α 2 C SC = 2(R α 2 )(β 4 ) + 2α 2 (α 4 ) (2) Eqation (2) basically means that α 2 (core flow velocity at the disk) and α 4 (core flow velocity at the pressre-eqilibrim location) are not independent (note that β 4 is a fnction of α 2 and α 4 as already noted). For a given set of R (= /B), C SB and C SC, Eqn. (2) provides the relationship between α 2 and α 4, i.e., we can obtain α 4 for a given α 2 and vice versa. Usally, giving α 2 is more straightforward to nderstand since the axial indction factor of the disk is defined as a = α 2. Near-Wake Mixing Loss Factors Before examining the soltion of Eqn. (2), we discss a little frther the near-wake power removal coefficients, C SB and C SC, and the near-wake heat loss coefficient, C H (= C SB +C SC ). Mathematically, any two of the above three coefficients can be considered independent from each other; physically, however, they are determined by the actal (three-dimensional) characteristics of near-wake mixing. (Althogh not presented here, this can be seen by considering the momentm balances within the bypass and core flow passages separately, where velocity, pressre and shear stress distribtions along the streamtbe srface and also the shape of the streamtbe determine the relationship among the three coefficients). To better describe the basic characteristics of near-wake mixing (withot knowing sch details of the mixing that depends on the type/design of devices), here we introdce the following two parameters, namely the near-wake mixing loss factors, as follows: η = η 2 = power lost in the near wake power lost in the (near and far) wake = C H ac T (3) power lost in the near wake power removed from the bypass flow = C H C SB (4) Note that since the power removed from the entire flow passage (between stations and 5) is (p p 5 )AR = T, the power lost in the (near and far) wake is T P = ( α 2 )T = at. η is the ratio of near-wake mixing loss to total mixing loss ; ths FIGURE 3. EFFECTS OF THE NEAR-WAKE MIXING LOSS FAC- TORS η and η 2 ON THE POWER COEFFICIENT C P FOR A LOW BLOCKAGE (B = 0.) AND HIGH AXIAL INDUCTION FACTOR (a = 0.4) CASE. NOTE THAT UNIFORM BYPASS AND CORE FLOW VELOCITY PROFILES ARE ASSUMED AT STATION 4 AND HENCE THE RESULTS ARE ONLY QUALITATIVELY RELIABLE. η = 0 means that the mixing takes place only in the far-wake region (i.e., the present model retrns to the model of Garrett and Cmmins [3]), whereas η = means that the mixing is flly completed within the near-wake region. Meanwhile, η 2 represents the significance of energy loss dring the process of near-wake mixing, i.e., η 2 = 0 means that all power removed from the bypass flow is transferred (or added) to the core flow (C SB = C SC ), whereas η 2 = means that all power removed from the bypass flow is lost (to generate heat). Now we present examples of the soltion of Eqn. (2). Note that once α 4 has been obtained by nmerically solving Eqn. (2) for a given set of α 2 (= a), R (= /B), C SB and C SC, we can obtain the thrst and power coefficients, C T and C P, from Eqns. (7) and (8). Here we consider a low blockage (B = 0.) and high indction factor (a = 0.4) case with varios C SB and C SC, and the vales of C P obtained are plotted in Fig. 3 (with respect to the near-wake mixing loss factors η and η 2 defined above rather than C SB and C SC ). As can be seen from the figre, the crrent model shows that the power coefficient depends not only on η (the ratio of the near-wake mixing loss to the total mixing loss) bt also on η 2 (the ratio of the near-wake mixing loss to the power actally removed from the bypass flow de to the near-wake mixing). Interestingly, the model sggests that C P can even decrease as η increases if η 2 is very high, i.e., if the near-wake mixing is very inefficient (althogh sch inefficient mixing is nlikely to happen in reality). 6 Copyright c 203 by ASME Downloaded From: on /26/204 Terms of Use:

7 Momentm and Energy Correction Factors So far we have temporarily assmed that the bypass and core flow velocity profiles are niform (for each flow passage) at station 4. This means that the bypass flow loses energy and the core flow gains energy de to near-wake mixing bt no mixing layer develops between the two flow passages. Now we consider the development of the mixing layer in the near-wake region, or in other words, we consider the non-niformity of the bypass and core flow velocity profiles at station 4, as described in Fig. 4. Of importance here is that, for each flow passage, the cross-sectional integrals of the sqare and cbe of the actal (non-niform) velocity are larger than those of the cross-sectionally-averaged (niform) velocity. In order to accont for the differences between them, we introdce the so-called momentm and energy correction factors (see, e.g. Massey and Ward-Smith [22]). Using to denote the actal streamwise velocity, the momentm correction factors for the bypass and core flow passages at station 4, ψ β4 and ψ α4, are defined as follows: bypass 2 ds A(R α 2 /α 4 ) = ψ β4(β 4 ) 2 (5) core 2 ds Aα 2 /α 4 = ψ α4 (α 4 ) 2 (6) Similarly, the energy correction factors for the bypass and core flow passages at station 4, φ β4 and φ α4, are defined as follows: bypass 3 ds A(R α 2 /α 4 ) = φ β4(β 4 ) 3 (7) core 3 ds Aα 2 /α 4 = φ α4 (α 4 ) 3 (8) By replacing (β 4 ) 2, (α 4 ) 2, (β 4 ) 3 and (α 4 ) 3 in Eqns. (), (3) and (0) with ψ β4 (β 4 ) 2, ψ α4 (α 4 ) 2, φ β4 (β 4 ) 3 and φ α4 (α 4 ) 3, we can obtain the following eqations that replace Eqns. (2) (4) and (2): 2 ρ2 ( φ β4 β4 2 S B ) + p p 4 = A(R α 2 ) (9) 2 ρ2 ( φ α4 α 2 4 ) + p p 4 = T A + S C Aα 2 (20) Bypass Core Bypass 4 4 AR A 2 4 s ds = A ( R ) * 2 bypass s ds = A ( R ) * 3 bypass s * 2 ds core 2 = A s 2 4 * 3 ds core 3 = A FIGURE 4. SCHEMATIC OF THE EFFECT OF NEAR-WAKE MIXING ON THE STREAMWISE VELOCITY PROFILE AT STA- TION 4. RED CURVE REPRESENTS THE ACTUAL STREAMWISE VELOCITY ( ) PROFILE. and R(φ β4 β 2 4 ) (φ β4 β 2 4 φ α4 α 2 4 ) + R R α 2 C SB + α 2 C SC = 2(R α 2 )(ψ β4 β 4 ) + 2α 2 (ψ α4 α 4 ) (2) Eqations (9) and (20) are the final forms of the energy balance eqations for the bypass and core flow passages, respectively, and Eqn. (2) is the final form of the momentm balance eqation to be solved nmerically (for a given set of ψ β4, ψ α4, φ β4 and φ α4 as well as α 2, R, C SB and C SC ). Also, we can obtain the following eqation for C T that replaces Eqn. (7): C T = φ β4 β 2 4 φ α4 α C SB R α 2 C SC α 2 (22) Comparisons with 2-D and 3-D Simlations Now we compare the above analytical model with two- and three-dimensional actator disk simlations. The simlations are similar to those reported earlier in [4] bt different in terms of the following points:. The new 3-D simlations are performed for flow throgh a tbe rather than a rectanglar channel; 2. In addition to the 3-D simlations, 2-D simlations are also performed for comparison; 3. Instead of sing a RANS trblence model, all simlations are performed as (steady) Navier-Stokes simlations with an artificial kinematic viscosity profile defined in the wake region (described later in Fig. 5) to simlate the (simplified) effect of wake mixing; and 7 Copyright c 203 by ASME Downloaded From: on /26/204 Terms of Use:

8 ν ν FIGURE 5. STREAMWISE VARIATIONS OF ARTIFICIAL KINE- MATIC VISCOSITY FOR 2-D AND 3-D NAVIER-STOKES SIMU- LATIONS. ACTUATOR DISK IS LOCATED AT x/d = Instead of sing symmetry bondary conditions, inviscid (or slip) bondary conditions are applied at the tbe or channel walls. All simlations are performed for a relatively low blockage of B = 0., which means that the tbe diameter for 3-D simlations is 3.62d (where d is the disk diameter), whereas the channel width for 2-D simlations is 0d (where d is the 2-D actator fence width). Only a qarter of the tbe cross-section is simlated for 3-D simlations (as the flow field is axi-symmetric) and only a half of the channel cross-section for 2-D simlations (as the flow field is symmetric). Grid resoltion is similar to the medim level sed in [4], which has been confirmed to be fine enogh to obtain grid-independent reslts. All other conditions are the same as those sed in [4]. Figre 5 shows two different profiles of artificial kinematic viscosity given in the disk wake region. Note that the actator disk is located at x/d = 0 as defined in [4]. For Case A, the flow simlated is practically inviscid pstream and immediately downstream of the disk (x/d 0.5) bt then the kinematic viscosity gradally increases p to a given vale of ν w at x/d = ; the variation in the transient region, 0.5 x/d, is modelled sing a half period of a sinsoid. For Case B, the transient region is located more downstream of the disk, 3.5 x/d 4. (Here the transient region conceptally represents the region where the breakdown of large-scale wake strctres takes place, althogh these viscosity profiles are rather simplified.) Figre 6 shows the effect of the wake kinematic viscosity ν w on the power coefficient C P obtained for for different cases, namely Cases A and B for the two- and three-dimensional simlations, respectively. The axial indction factor, a, is fixed at 0.4. The vales of ν w tested here are, roghly speaking, comparable to the kinematic eddy viscosity obtained in the previos RANS stdy [4]. It can be seen that essentially the power increases as the wake kinematic viscosity increases (except for 3-D Case B, where the power does not change). Of interest here is that the increase in power is larger for Case A than for Case B (for both 2-D and 3-D); this is becase the vale of ν w in Case A affects the strength of near-wake mixing more than that in Case B. ν FIGURE 6. EFFECT OF WAKE KINEMATIC VISCOSITY ν w ON THE POWER COEFFICIENT C P (FOR B = 0., a = 0.4). Althogh not shown here, the pressre-eqilibrim location was fond to be arond 5.5 < x/d < 0 for 2-D and.5 < x/d < 3 for 3-D cases, sggesting the reason why ν w does not affect C P in 3-D Case B (becase mixing takes place only in the far-wake region for this case). The difference in the pressre-eqilibrim location between the 2-D and 3-D cases is presmably related to the difference in the width of the bypass flow passage, which is mch wider in the 2-D cases than in the 3-D cases (even thogh the area blockage, B = 0., is the same). In order to compare these simlations with the new analytical model, simlation reslts were frther analysed to assess the vales of nknowns in the analytical model. Specifically, the pressre-eqilibrim location (or station 4) was defined in the simlated flow field data as the first (most pstream) streamwise location in the wake where the cross-sectionally-averaged pressre eqilibrates between the bypass and core flow passages; then ψ β4, ψ α4, φ β4 and φ α4 were calclated from the velocity profiles at that location. Also, C SB and C SC were calclated from the change of energy flx between stations (far pstream of the disk) and 4 for each flow passage (note that the energy removal from the bypass flow corresponds to C SB, whereas that from the core flow corresponds to the sm of C P and C SC ; cf. Fig. 2). Using the obtained set of ψ β4, ψ α4, φ β4, φ α4 C SB and C SC (and also the given blockage B = 0. and axial indction factor a = 0.4), we first attempted to calclate α 4 from Eqn. (2) and ths C T and C P from Eqns. (22) and (8). This yielded satisfactory reslts for some cases (an example is shown in Table ) bt not for other cases, presmably becase Eqn. (2) is rather sensitive to the vales of momentm and energy correction factors, i.e., small nmerical errors in the comptation of ψ and φ reslt in large errors in α 4 and ths in C T and C P. To avoid this difficlty, 8 Copyright c 203 by ASME Downloaded From: on /26/204 Terms of Use:

9 TABLE. EXAMPLES OF FLOW COEFFICIENTS (3-D Case A, ν w /d = 0.025, B = 0., a = 0.4). ψ β4 ψ α4 φ β4 φ α4 C SB C SC C H C P C P from Eqns. (2), (22) and (8) X X X X a basis for analysing how varios physical factors (sch as the type/design of devices) affect the limit of power extraction via changes in the characteristics of device near-wake mixing. η FIGURE 7. COMPARISON OF THE VARIATION OF FLOW CO- EFFICIENTS (FOR B = 0., a = 0.4). LINES: OBTAINED FROM SIMULATED FLOW FIELD DATA (3-D Case A); SYMBOLS X : POWER COEFFICIENTS OBTAINED FROM EQNS. (22) AND (8) here we calclate α 4 from the simlated velocity data at station 4 and then calclate C T and C P from Eqns. (22) and (8), i.e., we avoid the se of Eqn. (2). This means that the reslts shown below will confirm the validity of the energy balance eqations (9) and (20) and also the analytical C T and C P given by (22) and (8), bt not the momentm balance eqation (2). Figre 7 shows a comparison of C P obtained directly from the simlated flow at the disk location (3-D Case A; for B = 0., a = 0.4) and indirectly from Eqns. (22) and (8). Also plotted here are the vales of C SB, C H and η 2 derived from the simlations. It can be seen that the vales of C P obtained from Eqns. (22) and (8) (sing the simlation-derived vales of φ β4, φ α4 C SB and C H ) agree well with those obtained directly from the simlations. It shold be reminded that the crrent simlations are simplified actator disk simlations; hence the characteristics of near wake mixing (and ths the coefficients describing its characteristics, sch as φ, C SB and C SC ) might be rather different from those for practical trbine/device wakes. Of importance here, however, is that the present analytical model explains the basic relationship between the characteristics of device near-wake mixing and the limit of power extraction; therefore the model cold be sefl as η EFFECT OF NEAR-WAKE MIXING ON THE EFFICIENCY OF A PARTIAL TIDAL FENCE Finally, we briefly discss the effect of near-wake mixing on the efficiency of partial tidal fences. For a partial tidal fence we can consider two different types of near-wake regions, namely (i) device near-wake region, which is the region pstream of the device-scale pressre eqilibrim location, and (ii) array nearwake region, which is the region pstream of the array-scale pressre eqilibrim location. Hence it is expected that the limiting efficiency of a partial tidal fence generally depends on the following three different types of near-wake mixing: A. Device-scale mixing in the device near-wake B. Device-scale mixing in the array near-wake C. Array-scale mixing in the array near-wake It shold be noted that the tidal fence model of Nishino and Willden [2, ] does not accont for Type A or C of near-wake mixing (and Type B has been considered only in sch a special way that the device-scale mixing is flly completed before the array-scale wake expansion initiates [2] or before the array-scale bypass and core flow pressre eqilibrates []). We however expect that it is possible to develop a more generic tidal fence model that acconts for all three types of mixing listed above; the reslting model shold hence encompass the device-scale model reported in this paper and the crrent tidal fence model of Nishino and Willden [2, ]. Frther investigations are crrently in progress and will be reported in the ftre. CONCLUDING REMARKS Leading indstrial tidal energy developers are crrently in the process of planning the first tidal power farm deployment, where the challenge is how to array a large nmber of devices to maximise their overall power generation whilst keeping their impact on the natral environment to an acceptable level. In the first part of the paper we have reviewed recent discoveries on the efficiency of tidal fences (spanwise array of tidal-stream trbines), which are promising on their own and may also become a key component of large tidal power farms to be deployed in the 9 Copyright c 203 by ASME Downloaded From: on /26/204 Terms of Use:

10 ftre. Differences between fll and partial tidal fences have been clarified; for the latter of which the interaction of deviceand array-scale wake mixing is expected to play a key role. (Postscript: it shold be noted that fll tidal fences discssed in this paper are described as partial fences in [3] from a different viewpoint.) We have also discssed some remaining challenges to better nderstand the efficiency of tidal fences and farms. In the latter part of the paper we presented a new theoretical investigation into the effect of device near-wake mixing on the efficiency of fll tidal fences (to tackle one of those remaining challenges). The new theoretical model is based on the momentm and energy balance eqations acconting for the following two factors: (i) energy transfer between the bypass and core flow passages de to near-wake mixing and the attendant heat loss; and (ii) development of a mixing layer in the near-wake region. The new model agrees well with three-dimensional actator disk simlations when its nclosed terms (arising from the inclsion of the above two factors) are assessed based on the simlations themselves, sggesting that this low-order model cold serve as a basis to analyse how varios physical factors, sch as the design of devices, affect the fence efficiency via changes in those terms describing the characteristics of device near-wake mixing. A similar approach may also be applicable to the modelling of array near-wake mixing for partial tidal fences; frther investigations are in progress and will be reported in the ftre. ACKNOWLEDGMENT The athors acknowledge the spport of the Oxford Martin School, University of Oxford, who have fnded this research. REFERENCES [] CarbonTrst, 20. Accelerating marine energy. Carbon Trst, UK. [2] Nishino, T., and Willden, R. H. J., 202. The efficiency of an array of tidal trbines partially blocking a wide channel. J. Flid Mech., 708, pp [3] Garrett, C., and Cmmins, P., The efficiency of a trbine in a tidal channel. J. Flid Mech., 588, pp [4] van Kik, G. A. M., The lanchester betz jokowsky limit. Wind Energy, 0, pp [5] Holsby, G. T., Draper, S., and Oldfield, M. L. G., Application of linear momentm actator disc theory to open channel flow. Tech. rep., OUEL 2296/08, Dept. Engineering Science, University of Oxford. [6] Whelan, J. I., Graham, J. M. R., and Peiró, J., A freesrface and blockage correction for tidal trbines. J. Flid Mech., 624, pp [7] Garrett, C., and Cmmins, P., The power potential of tidal crrents in channels. Proc. R. Soc. A, 46, pp [8] Vennell, R., 200. Tning trbines in a tidal channel. J. Flid Mech., 663, pp [9] Vennell, R., 20. Tning tidal trbines in-concert to maximise farm efficiency. J. Flid Mech., 67, pp [0] Vennell, R., 202. The energetics of large tidal trbine arrays. Renewable Energy, 48, pp [] Nishino, T., and Willden, R. H. J., (sbmitted). Two-scale dynamics of flow past a partial cross-stream array of tidal trbines. J. Flid Mech. [2] Willden, R. H. J., and Nishino, T., (to be sbmitted). A channel dynamics model for partial tidal fences. [3] Vogel, C. R., Holsby, G. T., and Willden, R. H. J., (to be sbmitted). On the extractable power of an array of trbines partially spanning a wide channel with a free srface. [4] Nishino, T., and Willden, R. H. J., 202. Effects of 3-d channel blockage and trblent wake mixing on the limit of power extraction by tidal trbines. Int. J. Heat Flid Flow, 37, pp [5] Chrchfield, M. J., Li, Y., and Moriaty, P. J., 20. A largeeddy simlation stdy of wake propagation and power prodction in an array of tidal-crrent trbines. In Proc. 9th Eropean Wave and Tidal Energy Conference (EWTEC 20), 5-9 September, Sothampton, UK. [6] Stallard, T., Collings, R., Feng, T., and Whelan, J. I., 20. Interactions between tidal trbine wakes: experimental stdy of a grop of 3-bladed rotors. In Proc. 9th Eropean Wave and Tidal Energy Conference (EWTEC 20), 5-9 September, Sothampton, UK. [7] Trnock, S. R., Phillips, A. B., Banks, J., and Nicholls-Lee, R., 20. Modelling tidal crrent trbine wakes sing a copled rans-bemt approach as a tool for analysing power captre of arrays of trbines. Ocean Eng., 38, pp [8] Cmmins, P. F., 203. The extractable power from a split tidal channel: An eqivalent circit analysis. Renewable Energy, 50, pp [9] Draper, S., Holsby, G. T., Oldfield, M. L. G., and Borthwick, A. G. L., 200. Modelling tidal energy extraction in a depth-averaged coastal domain. IET Renew. Power Gener., 4, pp [20] Wilson, S., Borban, S., and Coch, S., 202. Understanding the interactions of tidal power projects across the k continental shelf. In Proc. 4th International Conference on Ocean Energy (ICOE), 7-9 October, Dblin, Ireland. [2] Vermeer, L. J., Sørensen, J. N., and Crespo, A., Wind trbine wake aerodynamics. Prog. Aerosp. Sci., 39, pp [22] Massey, B., and Ward-Smith, J., Mechanics of Flids (8th Edition). Taylor & Francis. 0 Copyright c 203 by ASME Downloaded From: on /26/204 Terms of Use: