water Article
Study of Hydrodynamic Interference of Vertical-Axis Tidal Turbine Array Guangnian Li 1, *, Qingren Chen 2 1 2 3
*
and Hanbin Gu 3, *
Faculty of Maritime and Transportation, Ningbo University, Ningbo 315611, China Wuhan Rules and Research Institute, China Classification Society, Wuhan 430022, China;
[email protected] School of Naval Architecture and Mechanical-electrical Engineering, Zhejiang Ocean University, Zhoushan 316022, China Correspondence:
[email protected] (G.L.);
[email protected] (H.G.); Tel.: +86-574-87600505 (G.L.); +86-580-2550051 (H.G.)
Received: 12 August 2018; Accepted: 4 September 2018; Published: 12 September 2018
Abstract: The hydrodynamic interference between tidal turbines must be considered when predicting their overall hydrodynamic performance and optimizing the layout of the turbine array. These factors are of great significance to the development and application of tidal energy. In this paper, the phenomenon of hydrodynamic interference of the tidal turbine array is studied by the hydrodynamic performance forecast program based on an unsteady boundary element model for the vertical-axis turbine array. By changing the relative positions of two turbines in the double turbine array to simulate the arrangement of different turbines, the hydrodynamic interference law between the turbines in the array and the influence of relative positions on the hydrodynamic characteristics in the turbine array are explored. The manner in which the turbines impact each other, the degree of influence, and rules for turbine array arrangement for maximum efficiency of the array will be discussed. The results of this study will provide technical insights to relevant researchers. Keywords: tidal current energy; vertical axis turbine; hydrodynamic interference; array arrangement
1. Introduction Due to an increasingly severe energy crisis, environmental damage, and other issues, China must focus on its energy infrastructure by developing renewable energy sources. Therefore, promoting the development and utilization of China’s clean and renewable energy resources in the future will be an important strategy [1,2]. Tidal current energy is one viable option and compared with other renewable energy approaches, it is more concentrated. The energy density is about thirty times that of solar energy and four times that of wind energy [3]. The energy flux density of tidal current energy is huge and this means that the geometric scale of the transducer is relatively small, the scale of the project is controlled, and the influence on the ecological environment is relatively small. Because tidal energy has the advantages above, many countries are competing to develop technology to harness tidal current energy. An efficient transducer requires a simple structure, easy maintenance, and small impact on the ecological environment and human activities. The design and large-scale engineering of efficient transducers have become the focus of tidal current energy research at home and abroad. Hydraulic turbines, which are wheel-type ocean tidal current transducers, are widely used in current energy applications. According to the relationship between the main shaft of the hydraulic turbine and the direction of the flow, it can be divided into a horizontal axis turbine and vertical axis turbine. The hydraulic turbine whose main shaft is parallel to the direction of the flow is called the horizontal axis turbine, and the hydraulic turbine whose main shaft is perpendicular to the direction of the flow is called the vertical axis hydraulic turbine. For example, some of the horizontal axis turbines Water 2018, 10, 1228; doi:10.3390/w10091228
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are the British AK-1000TM [4], SeaGen [5], Tidel [6], and Irish Open Centre [7]. While Canada’s Davis Hydro Turbine [8], EnCurrent [9], America’s Gorlov [10], OCGen [11], Italian Kobold [12], Wanxiang No. 1, Wanxiang No. 2, and Haineng No. 1 [3] are vertical axis turbines. The vertical axis turbine has the characteristic of not being subjected to flow direction restrictions and has a simple blade structure, low working tip-speed ratio, and low noise. The power generation system can be installed on the water surface, which is an attractive feature for researchers. Like wind power generation, the centralized layout of the large-scale hydraulic turbine can better reflect its engineering application value. The arrangement of the hydraulic turbine is very important, and the arrangement of the high-volume hydro-generators affects the power output of the power station directly. Considering the hydrodynamic interaction between turbines in an array, it is of great practical significance to accurately predict the overall hydrodynamic performance of the hydraulic turbine array and further optimize the arrangement of the array. This can scale-up the tidal current power station and industrialize it. The arrangement of hydraulic turbines is generally in one of the following three ways: (1) the hydraulic turbines are only installed in a row, and the single row of hydraulic turbines is perpendicular to the direction of the ocean tidal current; (2) the hydraulic turbine array consists of two rows, and these two rows of hydraulic turbine are perpendicular to the direction of the ocean tidal current, and the two hydraulic turbines are staggered along the ocean flow; and (3) the turbine array which is perpendicular to the ocean’s current direction consists of three or more rows of turbines [13]. Many researchers at home and abroad have launched a preliminary study on the arrangement of turbines. Bai et al. [14] used the computational fluid dynamics (CFD) method to study the hydrodynamic performance of a horizontal axis turbine. Churchfield et al. [15] used large-eddy simulation (LES) to study the power and wake of the horizontal axis turbine. Turnuck et al. [16] used the blade element momentum theory (BEMT) and Reynolds-averaged Navier-Stokes (RANS) models to analyse the wake characteristics and overall efficiency of the horizontal axis turbine. Antheaume et al. [17] contrasted the efficiency of the vertical axis turbine and turbine array. Li et al. [18] studied the array of turbines containing two vertical axis turbines. Results showed that the spacing of the turbine along the flow direction was the main factor that impacted the output power of the turbine. Goude et al. [19] studied the efficiency of the fleet containing five vertical axis turbines, and analysed the advantages and disadvantages of “-” type and “Z” type turbine arrangement. Gebreslassie et al. [20] investigated the influence of wake interaction and blockage on the performance of individual turbines in a staggered configuration in a tidal stream farm using the CFD-based Immersed Body Force turbine modelling method. Zanforlin et al. [21] investigated the hydrodynamic interactions between three closely-spaced vertical axis tidal turbines using CFD. Giorgetti et al. [22] used the CFD to study the interference between medium-solidity Darrieus vertical axis turbines and analysed the physical mechanism of power enhancement of the turbines. Fengshan [13] used numerical simulation techniques to study the vertical axis tidal turbine array arrangement problem. At present, there is no unified conclusion on the arrangement of turbines in tidal power plants, and there are some differences in the conclusions reached in different studies. Due to the complexity of the interaction problem between turbines, the previous studies were limited to a single parameter, such as the impact range of the turbine wake or the influence of the turbine transverse spacing on the energy conversion efficiency. Because of the above two points, the study of the tidal energy station turbine array arrangement is very urgent and important. Based on boundary element theory, this paper developed a simplified algorithm that can quickly predict the relative power coefficient of a vertical axis turbine array. The hydrodynamic characteristic prediction program for a vertical axis turbine array was developed by adopting the method. The influence of the relative position of the turbines on the hydrodynamic characteristics in the turbine array operation and the hydrodynamic disturbances between the turbines in the turbine array were studied.
in the turbine array operation and the hydrodynamic disturbances between the turbines in the turbine array were studied. 2. Methods Water 2018,In 10,this 1228 paper,
3 of 15 the overall hydrodynamic characteristics of a vertical axis turbine array were studied by analysing the influence of changing the relative position of the two turbines. Figure 1 shows a turbine array containing two separate vertical axis turbines. The coordinates for the 2. Methods rotation centre for turbine A were defined as (x1, y1) and as (x2, y2) for turbine B. The relevant In this paper, the overall parameters in Figure 1 arehydrodynamic expressed as characteristics of a vertical axis turbine array were studied by analysing the influence of changing the relative position of the two turbines. Figure 1 shows a Xdturbines. = x1 − xThe turbine array containing two separate vertical axis coordinates for the rotation centre for(1) 2 turbine A were defined as (x1 , y1 ) and as (x2 , y2 ) for turbine B. The relevant parameters in Figure 1 are Yd = y1 − y2 (2) expressed as
Xd = x1 − x2
Y ψ= tan ( d ) Yd = y1 − y2 X d −1
Yd ) ( XXd2d + Yd2 )
ψ = tan−1 (
L=
where
ψ
(1) (2)(3) (3) (4)
q L = ( Xd2 spacing, + Yd2 ) and d is the distance between (x1, y1) (4) is the deflection angle, L is the turbine and
). the deflection angle, L is the turbine spacing, and d is the distance between (x1 , y1 ) and (x2 , y2 ). (x2,ψy2is where
Figure 1. The model of twin-turbine system and the important parameters. Figure 1. The model of twin-turbine system and the important parameters.
A close-up of the blade motion in the dotted circle in Figure 1 is shown in Figure 2, in which A close-up of the blade motion in the dotted circle in Figure 1 is shown in Figure 2, in which a a moving coordinate system, ξoη, was fixed on each blade. The motion of the blade at a time, t, moving coordinate system, ξ oη , was fixed on each blade. The motion of the blade at a time, t, can can be represented by two components, the rotating angular velocity, Ω(t), about the origin, o, and the be represented by U, two angular Ω(t), the that origin, and the translational velocity, ofcomponents, the origin, o. the Therotating azimuthal angle velocity, θ is defined asabout the angle theo,blade translational U, ofcenter the origin, azimuthal is defined angle that the rotates relative tovelocity, the rotation O1 of o. theThe turbine in timeangle t, θ =θωt, in whichasωthe is the angular blade rotates relative to the rotation center O 1 of the turbine in time t, θ = ωt, in which ω is the velocity about the PEER rotation center O1 . Water 2018, 7, x FOR REVIEW 4 of 15 angular velocity about the rotation center O1.
Figure 2. Schematic diagram of blade motion and the local coordinate system. Figure 2. Schematic diagram of blade motion and the local coordinate system.
The velocity, V , of a uniform incoming flow at infinity is
V = (Vx , Vy ) where
(5)
Φ ( p, t ) is assumed to be the velocity potential at point, p, in the fluid domain, τ e , and time
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The velocity, V, of a uniform incoming flow at infinity is V = Vx , Vy
(5)
where Φ( p, t) is assumed to be the velocity potential at point, p, in the fluid domain, τe , and time t, which meets the Laplace equation:
∇2 Φ( p, t) = 0 ( p ∈ τe )
(6)
The non-penetration boundary condition on the blade surface is ∂Φ = (U + Ω × r ) · nb ∂n Sb
(7)
where nb is the normal of the blade surface, Sb ; r is the vector from the origin, o, to any surface point in the coordinate system, ξoη. At infinity, Φ( p, t) satisfies the velocity potential for uniform incoming flow according to Φ( p, t) = xVAx + yVAy ( p → ∞) (8) If φ is the perturbation velocity potential, then Φ can be decomposed into two parts: Φ = xVAx + yVAy + φ
(9)
Assuming a zero initial perturbation, φ should satisfy the following equation and boundary conditions: ∇2 φ( p, t) = 0 ( p ∈ τe ) (10) ∂φ = (U − V + Ω × r ) · nb (11) ∂n Sb
∇φ → 0 ( p → ∞)
(12)
∇ φ → 0 ( t = t0 )
(13)
where t0 is the initial time. The blade force is expressed as f=
Z
pndS
(14)
Sb
where S represents the blade surface and p is the pressure on the blade surface. The moment of the blade to reference point, o, is Z p(r × n)dS
q=
(15)
Sb
The torque coefficient of a single blade in a turbine rotor is formulated as Cq =
q 0.5ρV 2 Db ·
R
(16)
where D is the diameter of turbine, b length of blade and R is the radius of the turbine. The force of each blade on a single turbine is superimposed to obtain the force of the whole rotor. If the force of the turbine is F and the torque is Q, then Z
F=
∑ fi
i =1
(17)
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Z
Q=
∑ qi
(18)
i =1
where Z is the blade number of a single turbine. The resultant force coefficient of the turbine rotor is formulated as F F= (19) 0.5ρV 2 Db And the average moment coefficient of the turbine rotor shaft is formulated as CQ =
Q 0.5ρV 2 DbR
(20)
Q·ω 0.5ρV 3 Db
(21)
The power coefficient can be defined as Cp =
The power coefficient of turbine A is defined as Cp1 , the power coefficient of turbine B as Cp2 and the power coefficient of a single turbine under the same conditions as Cp . The relative power coefficient of double turbines array is Cp 0 =
C p1 + C p2 2C p
(22)
When Ψ = 90◦ , the turbine array is in parallel and the connecting line between the rotating axis of the turbines is perpendicular to the direction of tidal current. When Ψ = 0◦ , the turbine array is in series and the connecting line is parallel to the tidal current. A variety of different turbine arrangements can be simulated by changing L and Ψ. At the same time, the influence of the turbine steering on the turbine array is simulated by changing the relative rotation direction of the turbine. 3. Results 3.1. Issues of Turbine Array Figure 3a,b show the central path of trailing vortices in the turbine wake. It can be seen from Figure 3 that the tail vortex of the “-” type arrangement of the turbine array is distinct from that of the “Z” type arrangement. In a turbine array, there is a problem of mutual interaction between adjacent turbines because the rear part of the turbine array is often in the wake of the front turbine. Considering the spatial structure of the tidal current, if a multi-row turbine must be arranged, then it is necessary to arrange the horizontal and vertical spacing of the turbines reasonably. Theoretically, there must be an optimal layout mode that makes the power input and output ratio better.
the “Z” type arrangement. In a turbine array, there is a problem of mutual interaction between adjacent turbines because the rear part of the turbine array is often in the wake of the front turbine. Considering the spatial structure of the tidal current, if a multi-row turbine must be arranged, then it is necessary to arrange the horizontal and vertical spacing of the turbines reasonably. Water 2018, 10, 1228 6 of 15 Theoretically, there must be an optimal layout mode that makes the power input and output ratio better.
(a)
(b) Figure 3. Central path of trailing vortices of turbine array; tip speed ratio λ = 2. The mutual interference Figure 3. Central path of trailing vortices of turbine array; tip speed ratio λ = 2. The mutual of wake of the turbines is shown. (a) Three turbines in “-” type arrangement; (b) Three turbines in “Z” interference of wake of the turbines is shown. (a) Three turbines in “-” type arrangement; (b) Three type arrangement. turbines in “Z” type arrangement.
3.2. Validation of the Method 3.2. Validation of the Method In this paper, the numerical simulation results are compared with models tested in the towing paper, thetonumerical simulation are compared with models tested in the towing tank.In A this torque meter test the total torque ofresults two turbines was used in the model test that was done tank. A torque meter to test the total torque of two turbines was used in the model test that was in laboratory conditions [23,24]. This section covers the double turbine in parallel case conditions; at done in laboratory conditions [23,24]. This section covers the double turbine in parallel case this moment the mutual interaction of the two turbines are basically the same. Figure 4 shows the conditions; this moment thecomparison mutual interaction the two the same. Figure comparativeatresults. From the of Figureof4a,b, it canturbines be seen are thatbasically the two kinds of different 4tipshows the comparative results. From the comparison of Figure 4a,b, it can be seen that the two speed ratios, λ = ωR/V, are similar, and the numerical simulation is approximately in the same kinds of different tip speed ratios, λ = ωR/V, are similar, and the numerical simulation is direction as the experimental results. Although there is a difference with a maximum error of less than approximately in the same direction the experimental Although there is a difference with 10%, the numerical simulation wouldas generally reflect theresults. actual situation. a maximum error of less than 10%, the numerical simulation would generally reflect the actual situation.
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(a) (a)
(b) (b) Figure The variation of relative power coefficient turbine spacing of double turbines in Figure 4. 4. The variation of relative power coefficient withwith turbine spacing of double turbines in parallel Figure 4. The variation of relative power coefficient with turbine spacing of double turbines in parallel connection. (a) λ = 1.7; (b) λ = 2.75. connection. (a) λ = 1.7; (b) λ = 2.75. parallel connection. (a) λ = 1.7; (b) λ = 2.75.
3.3. AnalysisofofResults Results 3.3. Analysis 3.3. Analysis of Results
3.3.1. Cases 3.3.1. CasesofofΨΨ= =0◦0° 3.3.1. Cases of Ψ = 0° double turbine series means the downstream turbine is situated the of wake of the AA double turbine series means thatthat the downstream turbine is situated in the in wake the upstream A double turbine series means that the downstream turbine is situated in the wake of the upstream one. In case, the upstream hasimpact an apparent impact on the downstream one. In this case, thethis upstream turbine has anturbine apparent on the downstream turbine. Figure 5 upstream one. In this case, the upstream turbine has an apparent impact on the downstream turbine. Figure 5 shows the way that the relative of a changes double turbine shows the way that the relative power coefficient of apower doublecoefficient turbine series with theseries turbine turbine. Figure 5 shows the way that the relative power coefficient of a double turbine series changes with the turbine spacing, L. The turbine blade profile of the calculation model spacing, L. The blade spacing, profile ofL.theThe calculation was NACA0018; the chordmodel lengthwas of the changes withturbine the turbine turbine model blade profile of the calculation was NACA0018; the chord length of the blade, C, was 0.06 m; the turbine radius, R, was 0.25 m and power Z= blade, C, was 0.06 m; the turbine radius, R, was 0.25 m and Z = 1. As shown in Figure 5, the relative NACA0018; the chord length of the blade, C, was 0.06 m; the turbine radius, R, was 0.25 m and Z = 1. As shown Figurearray 5, the relative power coefficient of the turbine the spacing coefficient of thein as the spacing increased, where itarray rose increases from 0.6 atas = 2.5R to near 1. As shown inturbine Figure 5, theincreases relative power coefficient of the turbine array increases asL the spacing increased, where it rose from 0.6 at L = 2.5R to near 0.75 at L = 30R. There are ups and downs in the 0.75 at L = 30R. There are ups and thetodata and the are turbine spacing is, the more increased, where it rose from 0.6downs at L = in 2.5R nearset, 0.75 at the L = smaller 30R. There ups and downs in the data set, and the smaller the turbine spacing is, the more intense the fluctuation. In the case of a intense the fluctuation. In the case of a turbine in series, the influence of the upstream turbine wake on data set, and the smaller the turbine spacing is, the more intense the fluctuation. In the case of athe turbine in series, the influence of the upstream turbine wake on the downstream one is very large. downstream is very large. When the upstream spacing, L,turbine equalled 30R,on the overall energy conversion turbine in one series, the influence of the wake the downstream one is veryefficiency large. When the spacing, L, equalled 30R, the overall energy conversion efficiency of the turbine was 75%, of When the turbine was 75%, is equal to that of the two separate turbines. the spacing, L, which equalled 30R, the overall energy conversion efficiency of the turbine was 75%, which is equal to that of the two separate turbines. which is equal to that of the two separate turbines.
Figure5.5.The Thechanges changesof ofpower power coefficient coefficient with ratio λ =λ 2. Figure with spacing spacingwhere wherethe thetip tipspeed speed ratio = 2. Figure 5. The changes of power coefficient with spacing where the tip speed ratio λ = 2.
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Figure 6 shows the change law for the front and rear turbine torque coefficient as a function of Figure 6 shows the change law for the front and rear turbine torque coefficient as a function the azimuthal angle and for that of a single turbine torque coefficient under the same conditions. of the azimuthal angle and for that of a single turbine torque coefficient under the same conditions. According to the figure, the loss of the total power coefficient of the double turbine array in series According to the figure, the loss of the total power coefficient of the double turbine array in series was was mainly due to the decrease of the downstream turbine efficiency. Figure 6a shows that at the mainly due to the decrease of the downstream turbine efficiency. Figure 6a shows that at the distance of distance of L = 7R, the torque coefficient of the downstream turbine is only about 30% of the single L = 7R, the torque coefficient of the downstream turbine is only about 30% of the single turbine torque turbine torque coefficient under the same operating conditions, whereas the spacing of the turbine coefficient under the same operating conditions, whereas the spacing of the turbine is different and the is different and the torque coefficient is different from the distribution of the position angle. torque coefficient is different from the distribution of the position angle. Meanwhile, the downstream Meanwhile, the downstream turbine will also have an impact on the upstream turbine. This effect turbine will also have an impact on the upstream turbine. This effect mainly comes from the presence mainly comes from the presence of downstream turbines that impede the evolution of upstream of downstream turbines that impede the evolution of upstream turbine wake. As is shown in Figure 6b, turbine wake. As is shown in Figure 6b, the torque coefficients of the upstream turbines vary at the torque coefficients of the upstream turbines vary at different intervals. This difference is relatively different intervals. This difference is relatively small compared with the impact of upstream small compared with the impact of upstream turbines on downstream turbines. The above is for the turbines on downstream turbines. The above is for the case of single-blade turbines; with the case of single-blade turbines; with the number of turbine blades increases, the wake produced by a number of turbine blades increases, the wake produced by a single turbine becomes more complex single turbine becomes more complex and the hydrodynamic interaction between turbines increases. and the hydrodynamic interaction between turbines increases. Figure 7 shows the change law of the Figure 7 shows the change law of the torque coefficient changing with the azimuthal angle in the case torque coefficient changing with the azimuthal angle in the case of the same turbine size with three of the same turbine size with three blades. As illustrated, the torque coefficient of the upstream turbine blades. As illustrated, the torque coefficient of the upstream turbine is significantly smaller than is significantly smaller than that of the single turbine, and the torque coefficient of the downstream that of the single turbine, and the torque coefficient of the downstream turbine is lawless with a turbine is lawless with a very small value close to zero. very small value close to zero.
(a)
(b) Figure of of spacing on turbine moment coefficient where Zwhere = 1. (a)ZThe on downstream Figure 6.6.Influence Influence spacing on turbine moment coefficient = impact 1. (a) The impact on turbine; (b) The impact(b) on The upstream turbine. downstream turbine; impact on upstream turbine.
Compared the turbine turbine rotor rotor force force of of aa double double turbine turbine is is also also different. different. Compared with with aa single single turbine, turbine, the Figure 8 shows the contrast between the upstream and downstream turbine rotor and how Figure 8 shows the contrast between the upstream and downstream turbine rotor and how the the force force coefficient of the single turbine rotor changes with the azimuthal angle for the case of a three-blade coefficient of the single turbine rotor changes with the azimuthal angle for the case of a three-blade turbine ofof thethe upstream turbine is turbine with with aa tip tipspeed speedratio ratioofofλλ==2.2.As Asillustrated, illustrated,the thetorque torquecoefficient coefficient upstream turbine smaller than that of the single turbine under the same operating conditions, and the force coefficient of is smaller than that of the single turbine under the same operating conditions, and the force
coefficient of the downstream turbine is significantly smaller than a single turbine in the same
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2018, 7, x FOR PEER REVIEW 9 of 15 theWater downstream turbine is significantly smaller than a single turbine in the same working conditions. Water 2018, 7, x FOR PEER REVIEW 9 of 15 The variation of the peak and valley values of the downstream turbine torque coefficient with the working conditions. The variation of the peak and valley values of the downstream turbine torque curve of theconditions. azimuthal The angle is clear. of The corresponding to the peak and valley values working theazimuthal peak andisangle valley values of the downstream turbine torque coefficient with the curvevariation of the azimuthal angle clear. The azimuthal angle corresponding to the is substantially the same as that of the upstream turbine. However, the whole curve is not smooth, coefficient curveisofsubstantially the azimuthalthe angle is clear. The angleturbine. corresponding to the peak and with valleythe values same as that of azimuthal the upstream However, the unlike the upstream turbine torque coefficient curve, which isofsmooth and regular. This shows that peak and valley values is substantially the same as that the upstream turbine. However, thethe whole curve is not smooth, unlike the upstream turbine torque coefficient curve, which is smooth rotor of the turbine was irregular. whole curve is downstream notshows smooth, the upstream torqueturbine coefficient curve, which is smooth andforce regular. This thatunlike the rotor force of theturbine downstream was irregular. and regular. This shows that the rotor force of the downstream turbine was irregular.
Figure 7. Influence of spacing on turbine moment coefficient where Z = 3 and λ = 2. Figure 7. Influence of spacing on turbine moment coefficient where Z = 3 and λ = 2. Figure 7. Influence of spacing on turbine moment coefficient where Z = 3 and λ = 2.
Figure 8. The force of the rotor varies with the azimuthal angle in the case of double turbines in Figure 8. TheZforce of the rotor varies with the azimuthal angle in the case of double turbines in series where = 3 and λ = 2. Figure 8. The force of the rotor varies with the azimuthal angle in the case of double turbines in series series where Z = 3 and λ = 2. where Z = 3 and λ = 2.
3.3.2. Cases of Ψ = 90° ◦ 3.3.2. Cases 3.3.2. Cases ofofΨΨ = =9090° A double turbine parallel means that the axial connections of the two turbines are double parallel that axial ofturbines the two are perpendicular toturbine theparallel direction ofmeans flow. In thisthe case, the connections external environment of the two AA double turbine means that the axial connections of theload two areturbines perpendicular perpendicular to the direction of flow. In this case, the external load environment of the two turbines was of basically andexternal the double were of affected the hydrodynamic to the direction flow. Inthe thissame, case, the loadturbines environment the twobyturbines was basically turbines wasfrom basically the same, and9 the double turbines wererotation affectedefficiency by the hydrodynamic interactions each other. Figure shows how the relative of theeach double the same, and the double turbines were affected by the hydrodynamic interactions from other. interactions from each other. Figure 9 showsofhow the relative rotation efficiency of the double turbine in parallel changes with the variation the turbine spacing. The turbine blade profile Figure 9 shows how the relative rotation efficiency of the double turbine in parallel changeswas with turbine in parallel changes with the variation of0.06 the m; turbine spacing. The turbine blade profile was NACA0018; the chord length of blade, C, was the turbine radius, R, was 0.25 m and Z = 3. of the variation of the turbine spacing. The turbine blade profile was NACA0018; the chord length NACA0018; the chord C, was the 0.06turbine m; the turbine R, wasthe 0.25relative m and power Z = 3. As can be seen from length Figureof9,blade, the smaller spacing,radius, the greater blade, C, was 0.06 m; the turbine radius, R, was 0.25 m and Z = 3. As can be seen from Figure 9, As can be of seen Figure 9, thearray. smaller the the turbine spacing, the the greater the relative power coefficient thefrom parallel turbine When distance between turbines decreased, the thecoefficient smaller the turbine spacing, the greater the relative power coefficient of the parallel turbine array. of thecoefficient parallel turbine array. When the distance the coefficient turbines decreased, the relative power increases steadily. When L = 6.5R,between the power of the double When the distance between increases the turbines decreased, the power coefficient increases steadily. relative power coefficient steadily. L =relative 6.5R, the power coefficient of theLdouble turbine rose about 13% compared with the twoWhen that were independent of each other. When = 2.3R, When L = 6.5R, the power coefficient of the double turbine rose about 13% compared with two turbine roseincreased about 13%tocompared withthe thecalculation two that were independent of each other. When L = the 2.3R, this value 31%. From results, when the turbine was connected in that were independent of each When L =power 2.3R, this valuewhen increased to 31%.array From the calculation this value 31%.other. From calculation results, theturbine turbine was when connected in parallel, it increased effectivelytoimproved the the overall coefficient of the the two results, when the turbine was connected in parallel, it effectively improved the overall power coefficient parallel, it effectively improved the overall power coefficient of the turbine array when the two turbine trajectories were very close to each other. The improvement of energy efficiency was of turbine the turbine twoclose turbine trajectories close to each other. The improvement trajectories werethe very eachenergy other.were The very improvement of energy efficiency beneficial inarray the when development of thetotidal project. However, the gap between the was two of beneficial energy was beneficial inthe the development of the tidalturbine energy project. However, the gap the small development tidal energy project. However, the gap between theliving two turbinesefficiency isin too to affectofthe daily maintenance of the generator and the turbines is too small to affect the daily maintenance of the turbine generator and the living between the two turbines is too small to affect the daily maintenance of the turbine generator and environment of marine life. Therefore, when developing tidal power, the arrangement of thethe environment of marine life.belife. Therefore, when developing tidaltidal power, thethe arrangement of of thethe living environment ofshould marine Therefore, when developing power, arrangement vertical axis turbine fully considered. vertical axis turbine should be fully considered. vertical axis turbine should be fully considered.
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Figure 9. The relative power coefficient varies with the spacing in parallel connection where λ = 2.5. Figure 9. The relative power coefficient varies with the spacing in parallel connection connection where where λλ == 2.5.
Figure 10 shows the variation curve of the single blade moment coefficient as a function of the 10 shows the variation curve of the single blade moment coefficient as a function of the Figureangle azimuthal of the parallel turbine array. As shown, if the blade in the upstream disk had an parallel turbine array. As shown, shown, if the blade in the upstream disk had an azimuthal angle azimuthal angle, of θ, the from about from 1200° 1290°, blade torque coefficient is slightly larger ◦ to ◦ , the azimuthal angle, θ, from about from 1200 to 1290 the blade torque coefficient is slightly larger 1200° 1290°, than that of the single turbine. In the downstream disk, the turbine torque coefficient is negative in of the single turbine. In the downstream disk, the turbine torque coefficient is negative in than that parallel and stand-alone situations. In parallel, the absolute value of the moment coefficient is small. parallel and stand-alone situations. In parallel, parallel, the the absolute absolute value value of the moment moment coefficient coefficient is is small. small. Compared with the single-blade torque coefficient of the turbineofinthe both parallel and stand-alone Compared with the single-blade torque coefficient of the turbine in both parallel and stand-alone situations, it can be seen that the azimuthal angle corresponding to the output area in parallel is it can thethe azimuthal angle corresponding to the area in parallel is larger situations, canbe beseen seenthat that azimuthal corresponding to output the area in parallel is larger thanitthat in the single case, and the angle azimuthal angle which theoutput peak corresponds to is than that in the single case, and the azimuthal angle which the peak corresponds to is hysteretic. larger than that in the single case, and the azimuthal angle which the peak corresponds to is hysteretic. hysteretic.
Figure 10. The Thevariation variationlaw law single blade moment coefficient with azimuthal angle in parallel Figure 10. of of single blade moment coefficient with azimuthal angle in parallel turbine Figure 10. Theλ variation law of single blade moment coefficient with azimuthal angle in parallel turbine where = 2.5. where λ = 2.5. turbine where λ = 2.5.
Figure Figure 11 11 shows shows the the comparison comparison curve curve for for the the turbine turbine rotor rotor torque torque coefficient coefficient as as aa function function of of Figure 11angle showsin comparison curve forsituations. the turbineAs rotor as 11, a function of the parallel and can be from the the azimuthal azimuthal angle inthe parallel and stand-alone stand-alone situations. As can torque be seen seencoefficient from Figure Figure 11, the valley valley the azimuthal angle in parallel and stand-alone situations. As can be seen from Figure 11, the valley value value of of the turbine turbine rotor rotor curves curves in in parallel parallel is is significantly significantly larger larger than than that that in in stand-alone stand-alone situation. situation. value of the turbine rotor curves in parallel is significantly larger than that in stand-alone situation. In In the the stand-alone stand-alone situation, situation, the the valley valley value value is is around around 0.07, 0.07, and and the the corresponding valley is close to In stand-alone situation, valley value is around andthe the stand-alone corresponding valley isand closethe to 0.2 in parallel. parallel.The The peak inthe parallel is slightly higher than 0.2the in peak in parallel is slightly higher than0.07, the stand-alone situation,situation, and the azimuthal 0.2 in parallel. The peak in parallel is slightly higher than the stand-alone situation, and the azimuthal angle corresponding todelayed. the peakFigure is delayed. Figure 12 shows the information given by angle corresponding to the peak is 12 shows the information given by the rotor torque azimuthal angle corresponding to the peak is delayed. Figure 12 shows the information given by the rotor torque curve ofturbine the four-blade turbine parallel and in case, the stand-alone case, coefficient curvecoefficient of the four-blade in parallel and ininthe stand-alone which is similar the rotor torque coefficient curve of the four-blade turbine in parallel and in the stand-alone case, which is similar Figure of 11.peak Theand number valley of curve of turbine rotorduring torquea to Figure 11. Thetonumber valleyofofpeak curveand of turbine rotor torque coefficient which is similar to Figure 11. The number of peak and valley of curve of turbine rotor torque coefficient during a rotation period is equal to the number Z. Figures 11 and 12 show that the rotor rotation period is equal to the number Z. Figures 11 and 12 show that the rotor torque coefficient curve coefficient during a rotation period is equal to the number Z. Figures 11 and 12 show that the torque coefficient curve in the situation is smoother than the parallel case curve. Inrotor case in the stand-alone situation is stand-alone smoother than the parallel case curve. In case of the double machine, torque coefficient curve in the stand-alone situation is smoother than the parallel case curve. In case of doublecurve machine, the moment curve has more “burr”. shouldbetween be caused by interaction thethe moment has more “burr”. This should be caused by This interaction turbines, and this of the double machine, thewill moment curve has more should be caused by interaction between and of this affecttorque the stability of the“burr”. turbineThis torque output. will affectturbines, the stability the turbine output. between turbines, and the this variation will affectlaw the stability of the turbine torquesingle output. Figure shows parallel turbine turbine resultant Figure 13 shows the variation law of of thethe parallel turbine andand single turbine rotorrotor resultant force Figure 13 shows the variation law of the parallel turbine and single rotor resultant force coefficient as a function the azimuthal As shown in Figure 13,turbine when in parallel, the coefficient as a function of theofazimuthal angle.angle. As shown in Figure 13, when in parallel, the rotor force coefficient as a function of the azimuthal angle. As shown in Figure 13, when in parallel, the rotor force curve with fluctuations small fluctuations more than stable thecase, single case, the larger force curve with small is moreisstable thethan single and the and mean is mean larger is than the rotor force curve with small fluctuations is more stable than the single case, and the mean is larger than the single. The peak value is larger than that of the turbine in the stand-alone situation, and the than the single. The peak value is larger than that of the turbine in the stand-alone situation, and the
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valley is larger than that ofthan the that rotorofcurve. The azimuthal angle of the peak and valley value single.value The peak value is larger the turbine in the stand-alone situation, and the valley valley value is rotor largerthat than that of thecurve. rotor curve. The azimuthal angle ofcorresponding theand peak and valley value of the turbine force in parallel is higher than the azimuthal angle to the peak value is larger than of the rotor The azimuthal angle of the peak valley value of the valley value is larger than that of the rotor curve. The azimuthal angle of the peak and valley value of the turbine rotor force in parallel is higher than the azimuthal angle corresponding to the peak and valley value in the case of single machine. When the strength of the turbine array is to be turbine rotor force parallel is higheristhan the than azimuthal angle corresponding to the peaktoand of the turbine rotorinforce in parallel higher the azimuthal angle corresponding thevalley peak and valley value in the case of single machine. When the strength of the turbine array is to be checked, thevalue above situation should be fully value in the case ofinsingle machine. When theconsidered. strength of the array is toturbine be checked, and valley the case of single machine. When theturbine strength of the arraythe is above to be checked, the above situation should be fully considered. situation the should be situation fully considered. checked, above should be fully considered.
Figure 11. The comparison of the turbine torque coefficient in the parallel and single situation where Figure Z = 3. 11. The comparison of the turbine torque coefficient in the parallel and single situation where Figure 11. The comparison of the turbine torque coefficient in the parallel and single situation where Figure Z = 3. 11. The comparison of the turbine torque coefficient in the parallel and single situation where Z = 3. Z = 3.
Figure 12. The comparison of the turbine torque coefficient in the parallel and single situation where The comparison comparisonofofthe theturbine turbine torque coefficient in the parallel and single situation Figure torque coefficient in the parallel and single situation wherewhere Z = 4. Z = 4. 12. The Figure 12. The comparison of the turbine torque coefficient in the parallel and single situation where Z = 4. Z = 4.
Figure 13. The Thecomparison comparison turbine resultant coefficient in the parallel and single Figure 13. of of thethe turbine resultant force force coefficient in the parallel and single situation Figure 13. The Zcomparison of the turbine resultant force coefficient in the parallel and single situation where = 3 and λ = 2.5. where = 3The and λ = 2.5. FigureZ13. comparison of the turbine resultant force coefficient in the parallel and single situation where Z = 3 and λ = 2.5. situation where Z = 3 and λ = 2.5.
Case of Arbitrary Ψ 3.3.3. Arbitrary Ψ 3.3.3. Arbitrary Ψ vertical turbines mentioned mentioned above above in in series series and and parallel parallel are are two two special special cases. cases. 3.3.3.The Arbitrary Ψ axis turbines The vertical axis axis turbines mentioned above in series andincoming parallel are two cases. the turbine turbine connected direction of the at a special certain angle. Normally, the isisconnected toto thethe direction of the incoming flowflow at atwo certain angle. The The vertical axis axis turbines mentioned above in series and parallel are special cases. Normally, the turbine axis is connected to the direction of the incoming flow at a certain angle. The relative efficiency of the sets with different relative by angle. fixing The the Normally, the turbine axisturbine is connected to the direction of the positions incomingwas flowanalysed at a certain relative efficiency of the turbine sets with different relative positions was analysed by fixing the relative efficiency of the turbine sets with different relative positions was analysed by fixing the
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The relative efficiency of the turbine sets with different relative positions was analysed by fixing the distance between between the the axes axes of of the the two two turbines turbines and and changing changing the the angle angle between between the the connecting connecting line line distance of the the axis axis and and the the direction direction of of the the flow. flow. Figure 14 shows the way that the relative efficiency of the of flow. Figure 14 shows the way that the relative efficiency of the three-blade turbine array changed with the declination. declination. The turbine blade was NACA0018; NACA0018; three-blade turbine array changed with the The turbine turbine blade blade profile profile was The NACA0018; the chord length of blade, C, was 0.25 m; and the radius of the blade, R, was 1m. The calculated the chord 0.25 m;m; andand thethe radius of the R, was work chord length lengthofofblade, blade,C,C,was was 0.25 radius of blade, the blade, R, 1m. wasThe 1m.calculated The calculated work condition was λ = 2, the axis spacing was L = 4R and the deflection angle changed from Ψ 0° condition was λwas = 2,λthe was was L = 4R the the deflection angle changed from Ψ =Ψ0=◦= to work condition = 2,axis the spacing axis spacing L =and 4R and deflection angle changed from 0° to Ψ = 90°. Figures 15 and 16 show the way that the relative power coefficient of the two-blade ◦ Ψ Figures 15 and 16 show the way relative power power coefficient of the two-blade turbine to =Ψ90= .90°. Figures 15 and 16 show the that waythe that the relative coefficient of the two-blade turbine array changed with the deflection angle. The turbine blade profile was NACA0015; the array changed with the deflection angle. Theangle. turbineThe blade profileblade was NACA0015; chord length turbine array changed with the deflection turbine profile was the NACA0015; the chord length of blade, C, was 0.125 m; and the radius ofR,the blade, R, wascalculated 0.5m. The calculated work of blade, C, was 0.125 C, m;was and0.125 the radius theradius blade,of was 0.5m. work condition chord length of blade, m; andofthe the blade, R, The was 0.5m. The calculated work condition was λ = 2.4; the axis spacing was L = 2.5R, 3R, 3.5R, 4R, 4.5R, or 5R; and the deflection was λ = 2.4; theλaxis spacing wasspacing L = 2.5R, 3R,L3.5R, 4R, 3R, 4.5R,3.5R, or 5R; theor deflection changed condition was = 2.4; the axis was = 2.5R, 4R,and 4.5R, 5R; and angle the deflection angle changed from Ψ = 0° to Ψ = 90°. It can be seen from Figure 14 that when the distance between ◦ ◦ from = 0 to Ψ = 90Ψ.=It0°can 14from that when between the axis of the angleΨchanged from to be Ψ =seen 90°.from It canFigure be seen Figurethe 14 distance that when the distance between the axis of the two turbines was fixed, the change of relative power coefficient can be divided into two turbines was fixed, the change of relative power coefficient can be divided into two stages, the axis of the two turbines was fixed, the change of relative power coefficient can be dividedwith into two stages, with the deflection angle changing from Ψ = 0° to Ψ = 90°. From Ψ = 0° to Ψ = 30°, the ◦ . From Ψ = 0◦ to Ψ = 30◦ , the relative power the angle = 0◦ to Ψfrom = 90Ψ twodeflection stages, with the changing deflectionfrom angleΨchanging = 0° to Ψ = 90°. From Ψ = 0° to Ψ = 30°, the relative power coefficient increases rapidly from 0.45 to301.05. From Ψ = 30° to Ψ = 90°, the relative ◦ to Ψ ◦ , the coefficient increases rapidly from 0.45 to 1.05.from From Ψ =to = 90Ψ relative coefficient relative power coefficient increases rapidly 0.45 1.05. From = 30° to Ψ =power 90°, the relative power coefficient slowly increases, and the relative efficiency of the turbine in parallel state of the slowly the relative efficiency ofrelative the turbine in parallel state of the in two turbines reaches a power increases, coefficientand slowly increases, and the efficiency of the turbine parallel state of the two turbines reaches a maximum of 1.18. In Figures 15 and 16, the variations of the relative maximum of 1.18. In Figures 15 andof 16,1.18. the variations theand relative efficiency of theofturbine array two turbines reaches a maximum In Figuresof15 16, the variations the relative efficiency of the turbine array with the deflection angle inspacings the case of six different shaft spacings with the deflection anglearray in thewith casethe of six differentangle shaftin substantially same efficiency of the turbine deflection the caseshowed of six different shaft the spacings showed substantially the same information. When comparing the six curves Figures and 16 for ◦ in ◦15 information. When comparing the six curves in Figures 15 and 16 for Ψ = 0 to Ψ = 30 , the relative showed substantially the same information. When comparing the six curves in Figures 15 and 16 for Ψ = 0° to Ψ = 30°,values the relative power coefficient values on the different curves corresponding to the power on thepower different curves corresponding to the same deflection angle areto quite Ψ = 0° coefficient to Ψ = 30°, the relative coefficient values on the different curves corresponding the same deflection angle are quite different. From Ψ = 30° to Ψ = 90°, the curves tend to be the same ◦ ◦ different. From Ψ = 30are toquite Ψ = 90 , the curves two to case = 3R same deflection angle different. Fromtend Ψ = to 30°betothe Ψ same = 90°,exception the curvesoftend be of theL same exception of two case of L = 3R and L = 2.5R at Ψ = 30°; however, from Ψ = 45° to Ψ = 90°, the curves ◦ ◦ ◦ and L = 2.5R at Ψcase = 30 from Ψ =at45 Ψ however, = 90 , thefrom curves to Ψ be=quite thecurves same. exception of two of ;Lhowever, = 3R and L = 2.5R Ψ =to 30°; Ψ =tend 45° to 90°, the tend to be quite the same. when According to Figures 14–16, when considering the arrangement of the According Figures considering the arrangement the turbine, increasing the deflection tend to be to quite the 14–16, same. According to Figures 14–16, whenofconsidering the arrangement of the turbine, increasing the deflection angle between the turbine axis connecting line and the flow angle between the turbine axis connecting and the direction wasand conducive to turbine, increasing the deflection angle line between theflow turbine axis appropriately connecting line the flow direction appropriately was conducive to the improvement of the overall power coefficient of the the improvement of the overall power coefficient of the turbine direction appropriately was conducive to the improvement ofarray. the overall power coefficient of the turbine array. turbine array.
Figure 14. Influence Influence of deflection deflection angle on on hydrodynamic characteristics characteristics of hydraulic hydraulic turbine where where Figure Figure 14. 14. Influence of of deflection angle angle on hydrodynamic hydrodynamic characteristics of of hydraulic turbine turbine where Z = 33 and L = 4R. Z and L L == 4R. Z= = 3 and 4R.
Figure 15. Influence of deflection angle on hydrodynamic characteristics of hydraulic turbine where 15. Influence Influenceofofdeflection deflectionangle angle hydrodynamic characteristics of hydraulic turbine Figure 15. onon hydrodynamic characteristics of hydraulic turbine wherewhere Z = 2. Z = 2. Z = 2.
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Figure 16. 16. Influence Influenceofofdeflection deflection angle hydrodynamic characteristics of hydraulic turbine Figure angle onon hydrodynamic characteristics of hydraulic turbine wherewhere Z = 2. Z = 2.
4. Conclusions 4. Conclusions This paper discussed the law that shows the influence of the turbine wake, spacing, and the deflection angle discussed between the connecting line and theturbine flow direction to the and overall This paper the turbine law thataxis shows the influence of the wake, spacing, the performance of the hydraulic turbine. At the same time, the blade and the rotor force characteristics deflection angle between the turbine axis connecting line and the flow direction to the overall were also analysed. The following conclusions can be drawn from the performance of the hydraulic turbine. At the same time, the blade andstudy. the rotor force characteristics of theThe vertical axis turbine has a great influence on the performance of the downstream wereThe alsowake analysed. following conclusions can be drawn from the study. turbine, in series it will reduceaxis the power of theinfluence downstream greatly. When Theand wake of the vertical turbinecoefficient has a great on turbine the performance of the the turbine array turbine, is arranged turbine incoefficient exactly theofupstream turbine wake it downstream and placing in seriesthe it downstream will reduce the power the downstream turbine should be avoided as much as possible. When the turbines are connected in parallel, reducing the greatly. When the turbine array is arranged placing the downstream turbine in exactly the axis spacing will increase overall power coefficient of possible. the turbine array. axis upstream turbine wake it the should be avoided as much as When theHowever, turbines the are small connected spacing makes installation and routinewill maintenance more difficult, and the force of the turbine blade and the in parallel, reducing the axis spacing increase the overall power coefficient array. rotor will increase. However, the small axis spacing makes installation and routine maintenance more difficult, and the (Ψ =and 0◦ ),the therotor relative forceIn ofseries the blade willefficiency increase. of the turbine is the smallest, and in parallel (Ψ = 90◦ ), ◦ –30◦ interval, the turbine power coefficient the power coefficient highest. In theofΨthe = 90 In series (Ψ = 0°), is thethe relative efficiency turbine is the smallest, and in parallel (Ψ = 90°), ◦ slowly decreased from maximum, and the Ψ =interval, 30 –0◦ the interval, thepower turbine array’s power the power coefficient is the highest. In the Ψ in = 90°–30° turbine coefficient slowly coefficient dropped to a minimum. ininterval, the arrangement of the turbine array,coefficient the angle decreased quickly from the maximum, and in theTherefore, Ψ = 30°–0° the turbine array's power between the axis oftothe turbine and the direction of the incoming flow shouldarray, try to the avoid the quickly dropped a double minimum. Therefore, in the arrangement of the turbine angle ◦ ◦ Ψ = −30 the –30 axis interval. between of the double turbine and the direction of the incoming flow should try to avoid the Ψ = −30°–30° interval.
Author Contributions: Conceptualization, G.L.; Methodology, G.L. and Q.C.; Software, G.L. and Q.C.; Formal Analysis, G.L.; Investigation, G.L.; Writing-Original Draft Preparation, G.L.; Writing-Review & Editing, G.L., Q.C. Author Contributions: Conceptualization, G.L.; Methodology, G.L. and Q.C.; Software, G.L. and Q.C.; Formal and H.G.; Funding Acquisition, G.L. Analysis, G.L.; Investigation, G.L.; Writing-Original Draft Preparation, G.L.; Writing-Review & Editing, G.L., This research funded byG.L. National Natural Science Foundation Program; grant number T (51679216). Funding: Q.C. and H.G.; Fundingwas Acquisition, Acknowledgments: The study is supported by National Natural Science Foundation Program. Funding: This research was funded by National Natural Science Foundation Program; grant number T Conflicts of Interest: The authors declare no conflict of interest. (51679216).
Acknowledgments: The study is supported by National Natural Science Foundation Program. Nomenclature Conflicts of Interest: The Momentum authors declare no conflict of interest. BEMT Blade Element Theory CFD Computational Fluid Dynamics Nomenclature LES Large Eddy Simulation RANS Average Momentum Navier-Stokes BEMT Reynolds Blade Element Theory BEMT Blade Element Momentum Theory CFD Computational Fluid Dynamics Xd, Yd coordinate spacing of the rotation centers of turbines in twin-turbine system LES Large Eddy Simulation ψ RANS angle betweenAverage the connection of the rotation centers of turbines and the X axis Reynolds Navier-Stokes L BEMT distance the rotation centers Theory of turbines BladeofElement Momentum t Xd, Yd timecoordinate spacing of the rotation centers of turbines in twin-turbine system angle between the connection of the rotation centers of turbines and the X axis t0ψ initial time L distance of the rotation centers turbines Ω rotating angular velocity about the of origin o t time initial time t0 Ω rotating angular velocity about the origin o
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translational velocity of the origin o velocity of uniform incoming flow at infinity azimuthal angle rotating angular velocity about the rotation center of turbine velocity potential fluid domain perturbation velocity potential pressure on the blade surface force on the blade moment of the blade moment coefficient of a single blade on turbine hydrodynamic force acting on the turbine hydrodynamic torque acting on the turbine rotor shaft resultant force coefficient of the turbine average torque coefficient of the turbine rotor shaft power coefficient of the turbine diameter of turbine radius of turbine wingspan length of blade chord length of blade blade number of a single turbine tip speed ratios relative power coefficient of double turbines array
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