# Multi-Array Design for Hydrokinetic Turbines in Hydropower Canals

^{*}

## Abstract

**:**

## 1. Introduction

_{∞}can be designated for performance and axial force computations. Alternatively, the use of the inflow speed U

_{0}instead of U

_{∞}at the denominator would complicate the assessment as multiple and contemporary flow speed measurements and data processing would be required at each cross-section. Indeed, the turbine thrust coefficient and the power coefficient are better defined as C

_{T,∞}= T/(½ρA

_{T}U

_{∞}

^{2}) and C

_{P,∞}= P/(½ρA

_{T}U

_{∞}

^{3}), with ρ denoting the water density, T denoting the turbine thrust force, P representing the mechanical power, A

_{T}denoting the rotor swept area, and U

_{∞}representing the far-field flow speed. The array thrust coefficient and power coefficient are given by C

_{TA,∞}= T

_{A,i}/(½ρA

_{array}U

_{∞}

^{2}) and C

_{PA,∞}= P

_{A,i}/(½ρA

_{array}U

_{∞}

^{3}), respectively, with T

_{A,i}and P

_{A,I}denoting the thrust force and power developed at the ith array, and A

_{array}denoting the array’s transversal area. Since U

_{∞}is constant in a steady flow and is uniquely determined before any assessment, it is preferable to relate the dimensionless coefficients to the undisturbed flow [2].

_{T,∞}and C

_{P,∞}of equal turbines in canals of a prismatic shape. With the help of tests performed at the Unitn Hydraulics Laboratory on plates immersed in a water channel, theoretical insights on the asymptotic trend of the maximum backwater from array to array are discussed.

## 2. Methods

_{∞}, h

_{∞}, and Fr

_{∞}). A Manning’s coefficient n of 0.031 at a certain cross-section was calibrated with a design flow rate Q

_{c}of 124 m

^{3}/s, depth h

_{∞}= 6.52 m, and local bed slope S of 3.84 × 10

^{−4}, along with a determination of the bulk flow speed U

_{∞}(1.439 m/s) and Froude number Fr

_{∞}(0.223). Geometric details of the cross-section depicted in Figure 2 were retrieved from Malini [25]. Technical data were drawn from local sources with authorization and are not disclosed in this work. The Froude number is calculated as Fr

_{∞}= U

_{∞}(gA

_{∞}/w

_{∞})

^{−}

^{1/2}, with U

_{∞}denoting the mean far-field flow speed, A

_{∞}representing the transversal wetted area, w

_{∞}denoting the free surface width, and g denoting gravity acceleration.

#### 2.1. Single-Array Scheme

_{f}. This justifies why the mean value of the entropy parameter at gauged sites remains approximately constant regardless of the level of discharge.

_{A,∞}is given by the sum of the turbines’ swept area divided by the wetted area (1).

_{A,∞}= N

_{T}A

_{T}/A

_{∞}

_{T}or from a centerline to another centerline, as further illustrated in Figure 4. According to Vogel et al. [14], the turbine blockage is given by Equation (2), with A

_{F,j}denoting the flow passage associated with the jth turbine, from left to right, and A

_{T,j}denoting the turbine’s swept area.

_{T,j}= A

_{T}/A

_{F,j}

_{0}, both DMS simulations would predict the same outputs. Otherwise, different values of B

_{T,j}would lead to an uneven distribution of thrust and power outputs for the turbines. Figure 4 shows examples of the subdivision of flow passages, in which higher blockage is assigned centrally, and lower blockage is assigned laterally. The inflow speed is assumed to be uniformly distributed upstream of the station {ith}. For a wide cross-section in navigable waterways, the array can be positioned on a side of the channel, resulting in an asymmetrical distribution of the flow through, but this case is not discussed in this work for the sake of brevity.

#### 2.1.1. Iterative Scheme for a Single-Array

_{0}/U

_{ref}is defined in a domain 0 < ϕ ≤ 1, while the ith array blockage ratio (3) and Froude number (4) are scaled from B

_{A,ref}and Fr

_{ref}to iteratively solve the hydraulic transition. Notably, for the first array from downstream U

_{ref}= U

_{∞}, Fr

_{ref}= Fr

_{∞}and B

_{A,ref}= B

_{A,∞}, but for the additional arrays, the downstream flow is represented by a reference station, which is simply indicated as “ref”. The array thrust coefficient C

_{TA,ref}, i.e., the ratio of the resultant thrust force generated over the entire structure to the reference dynamic pressure force (5), is used to solve the degree of momentum conservation between upstream and downstream stations {0, ref}, whose equation is given in (6).

_{A}= ϕ/B

_{A,ref}

^{3/2}(w

_{0}/w

_{ref})

^{1/2}Fr

_{ref}

_{TA,ref}= T

_{A}/(½ ρ U

_{ref}

^{2}N

_{T}A

_{T})

_{ref}

^{2}ϕ

^{3}− (C

_{TA,ref}B

_{A,ref}Fr

_{ref}

^{2}) ϕ

^{2}+ 1 = 0

_{A}= ∑T

_{j}. Notice that the array area N

_{T}× A

_{T}appears at the denominator in (5). Hence, the cubic polynomial (6) must be solved numerically for ϕ once C

_{TA,ref}is determined, as B

_{A,ref,}and Fr

_{ref}are known. This mathematical statement holds for a single turbine and multiple turbines as well, but in this case, array blockage and the resultant axial force are used instead. The DMS model [10] is applied repeatedly for each turbine so that C

_{TA,ref}is attained after the accumulation of all thrust forces. The updated solution of (6) is used to yield the new U

_{ref}, B

_{ref}, and Fr

_{ref}. According to the canal’s shape, a continuity equation must be solved to achieve w

_{0}and thus Fr, as illustrated in Table 1 of [2], whilst the whole cycle is repeated until the convergence of the inflow factor ϕ. To investigate the flow speed variation from the most upstream array to downstream, a global inflow factor ϕ

_{g}is also defined as U

_{0}/U

_{∞.}

#### 2.1.2. Power Optimization Schemes

_{Ω}, and for each prescribed Ω, the previous iterative method is executed. The array blockage ratio is then calculated based on the upstream cross-section, and the single-turbine blockage ratio B

_{T}(j) is determined through a getBlockage function based on the geometrical separation of the flow passages from the inflow cross-section of depth h

_{0}and area A

_{0}, whilst N

_{T}represents the number of turbines within the array. The getBlockage function is recalled at each inflow factor iteration, as the change in the cross-section area A

_{0}and depth h

_{0}are both functions of ϕ.

_{T}(j) have been established, the Double Multiple Streamtube function is carried out for all turbines, thus determining the turbine thrust force T, power P, and minimum depth h

_{w}(see the paragraph “Near and far wake modeling”), which are consecutively used in the outer process described in the flowchart of the multi-array scheme in Section 2.2. Thus, array thrust force T

_{A}and power P

_{A}are calculated by integrating all the contributions within the array, thereby yielding the thrust coefficient C

_{TA,ref}from (5). The inflow factor ϕ is updated by solving Equation (6). The process is repeated until ϕ converges for a single rotor speed Ω, and then for all rotor speeds. Finally, the array power is optimized and a single rotor speed λ

_{∞,opt}is achieved for all rotors.

_{T}between the turbines influences the performance through the locally generated blockage. Hence, if activated, the environmental control function is recalled before the determination of the array thrust T

_{A}and power P

_{A}.

#### 2.1.3. Environmental Constraint

_{i,}

**> Δh**

_{∞}_{lim}, a rotor speed modifier is introduced in the code. By default, this corresponds to a reduction of 1 rpm as a first step, but it can be modified by the user. This is applied first to the most downstream array so that the code re-executes the computations for all arrays with the same rotor speed modifier. The code verifies the environmental constraint once more at the end of the procedure. If the result is negative, the code implements a new step for speed reduction, inducing a drop in the rotor performance to respect the constraint.

#### 2.1.4. Wake Transition

_{D}and the transition to bluff body dynamics in a vertical-axis rotor to determine the wake recovery length. Assuming the ratio of time scales (t

_{v}/t

_{conv}), i.e., the percentage of convection time required by the blades to close the gaps in between them during rotation, and the geometric factor (R

_{T}/c), with R

_{T}denoting the turbine radius and c denoting the blade chord, the dynamic solidity is formulated in Equation (7).

_{D}= 1 − (R

_{T}/c) (t

_{v}/t

_{conv}) = 1 − 1/(2 π σ

_{A}λ)

_{A}represents the rotor solidity defined by Araya (N

_{b}× c/(2πR

_{T})), while λ is the tip speed ratio (ωR

_{T}/U

_{0}). Although both definitions are valid, σ

_{A}differs from our definition of solidity, i.e., σ = N

_{b}× c/(2R

_{T}), which is introduced later. The time scale t

_{v}represents the time required by the operating turbine to close the gaps between the blades t

_{v}= l/(N

_{b}U

_{0}λ), whilst the convective time scale is denoted as t

_{conv}, which is the time required for a free stream particle to travel the distance l = 2πR

_{T}(1 − σ

_{A}), i.e., t

_{conv}= l/U

_{0}, with l denoting the sum of the gaps in the circumference of the turbine rotorIf λ >> 1, the turbine’s wake mimics that of a full cylinder with with σ

_{D}≈ 1, but if λ > 1/(2πσ

_{A}), the dynamic solidity σ

_{D}increases monotonically with λ or σ

_{A}. If λ = 0, the degree of interaction of the incident flow with the blades is unknown as it depends on the orientation of the rotor and since σ

_{D}is indeterminate; otherwise, if λ < 1/(2πσ

_{A}), the dynamic solidity is negative σ

_{D}< 0 with a loss of physical meaning. Based on the spectrum of the Strouhal number St vs. the downstream length (d

_{t}/D

_{T}), Araya et al. demonstrated the existence of a streamwise point where blade vortices decay to the point where they no longer account for the most energetic fluctuations in the flow so that bluff body oscillations start to dominate the wake dynamics. The far wake exhibited low-frequency oscillations typical of bluff bodies near St = 0.26, which is consistent with HAWT wake studies, e.g., those conducted by Chamorro et al. [29] and Okulov et al. [30].

_{D}and the downstream transition location X

_{t}where a shear-layer instability occurs, thus achieving the near-wake length in turbine diameters D

_{T}. This relation was determined by performing a linear regression between the location of the wake transitions and dynamic solidities of different vertical-axis turbines. An increasing σ

_{D}gave weaker shed vortices with a faster transition into the wake.

_{t}= D

_{T}(4.78 − 4.93 σ

_{D})

#### 2.1.5. Wake Recovery

_{D}and the absolute minimum velocity was determined by Araya et al. [23], indicating that an increasing σ

_{D}yields a faster rate or recovery of the velocity deficit into the wake, i.e., the higher the solidity, the greater the initial velocity deficit. The velocity recovery profiles of vertical-axis rotors were aligned to that of a solid cylinder according to the predictions of Schlichting [31] and Iungo and Porté-Agel [32] for turbulent free-shear flows and horizontal-axis wind turbine wakes. Since the maximum Reynolds stress was found on the spanwise plane, it is reasonable to assume that the turbine aspect ratio AR plays a significant role in the wake distribution. Indeed, an increasing aspect ratio produces more significant spanwise fluctuations in the wake recovery. The downstream recovery of the wake velocity deficit is expressed by the power law illustrated by Ouro et al. [24], which can be reversed into Equation (9) to determine the far-wake length d

_{f}. In this equation, U

_{w}represents the known streamwise velocity in the near wake, i.e., (U

_{w}= α

_{w}U

_{0}), and a and b represent the power law coefficient and exponent according to Araya [23] for the rotor of a given aspect ratio. From the velocity deficit ΔU, one determines the local flow speed along the centerline U

_{#}= ΔU + U

_{0}. The length d

_{f}is found by substituting U

_{#}with the undisturbed flow speed U

_{∞}; thus, a reduction in velocity yields ΔU = U

_{∞}− U

_{0}.

_{f}= D

_{T}[(ΔU/U

_{0}+ U

_{w})/a]

^{(1/b)}

#### 2.1.6. Including Inflow and Wake Regions

_{0}was determined via the previous iterative procedure, whilst the depth h

_{∞}is known past the very-downstream array. The inflow region is the region where streamlines are sensitive to the rotor induction, which spans from {0} to the turbine inlet. Its length is assumed by default (d

_{0}/D

_{T}= 5.0), whereas the Standard [33] indicates (d

_{0}/D

_{T}= 2.5), according to the modeling of wind turbine inflow [34]. With 2-D modeling, Adamski [35] introduced blockage to include the bottom of a channel for an unbounded rotor. Plots of the dynamic pressure demonstrated that the wake persists further downstream for turbines at the free surface proximity, and the wake expands more uniformly for deeper installation. The minimum wake velocity at the turbine centerline at 20 D

_{T}downstream diminished for placement close to the free surface. Adamski demonstrated the enhancement of free surface fluctuations with turbine placement near the free surface, with stationary waves generated downstream and with amplitude decaying in space.

_{w}/D

_{T}≤ 2), as drawn approximately with a dashed line in Figure 7.

_{w}/D

_{t}≤ 5), the wake starts to vertically and laterally expand with a larger ambient turbulent flow entrainment that increases the turbulent fluxes and intensity, and momentum starts to recover faster [24]. In the far-wake region, the core momentum is recovered according to turbine dynamic solidity and aspect ratio such that Reynolds stresses decay to their minimum at (d

_{f}/D

_{T}= 14) and (d

_{f}/D

_{T}= 10), respectively. The code automatically determines a near-wake length according to Araya’s procedure once dynamic solidity σ

_{D}(7) has been derived.

_{w}, the free surface deformation over the rotor from downstream h

_{w}to upstream h

_{0}can be drawn by a cosine wave of a half amplitude, whose wavelength results in twice the sum (d

_{0}+ d

_{w}). The depth h

_{w}past each turbine is approximated from Equation (10) according to Whelan et al. [37], wherein α

_{w}denotes the wake interference factor from free surface actuator disc polynomial and where the effective turbine drag coefficient C

_{D}is known iteratively from the DMS code.

_{w}/h

_{0}= 1 − Fr

^{2}/2(C

_{D}+ α

_{w}

^{2}− 1)

_{d}

_{0}is added to the lengths of the near-wake region d

_{w}, transition region d

_{t}, and far-wake region d

_{f}. However, to account for uncertainty due to the mutual disturbance of multi-turbine wakes that carry turbulence further downstream, a correction factor is introduced. This factor has the same function of including an array-scale wake size as illustrated by Nishino and Willden for tidal applications [12]. Since a semi-empirical relation for the recovery length due to the superposition of multi-turbine wakes has not yet been standardized in the literature, a user-dependent parameter was introduced to enhance the far-wake length. A default value of f

_{c}= 1.1 is assigned in this work. The transition region’s wake length is also dependent on another default parameter that was set (d

_{t}/D

_{T}= 3.0) according to the research of Müller et al. [36]. The above procedure is valid for the very-downstream array. The assessment of the additional turbine arrays is described in the multi-array scheme paragraph.

#### 2.1.7. Backwater Region

_{0}are known, a direct step method is developed at this juncture to determine the backwater distribution of the gradually varied flow. This can be performed according to the energy or momentum principle according to Henderson [22]. At each step, the former reduces the water depth slightly from the previous value of a discrete dy so that the new wetted area A, the hydraulic radius R

_{h}, and mean flow speed U can be calculated. Since the Chezy coefficient C is given by the ratio (R

_{h}

^{1/6}/n) and the local friction slope S

_{f}is inversely proportional to the square of C according to Manning’s formula, S

_{f}is simply derived from (11). Once the mean values S

_{J,avg,}and Fr

_{avg}are determined from two steps, the discrete longitudinal specific energy variation (dE/dx)

_{avg}is derived from the 1-D Saint-Venant Equation (12). The amount (dE/dx)

_{avg}divided by dE (13) yields the distance dx, and the coordinates x and h are collected from the inflow depth h

_{0}to the far-field h

_{∞}.

_{f}= U

^{2}n

^{2}/R

_{h}

^{4/3}

_{f,avg}

_{avg}

^{2})

_{avg}is computed as a function of the friction slope from (15), whilst the longitudinal steps dx are found by averaging the discrete momentum variation dM to the mean step longitudinal area, as shown in (16), with p denoting the wetted perimeter.

_{avg}= ρg R

_{h}S

_{f,avg}

_{avg}/(ρg))

#### 2.2. Multi-Array Scheme

_{A}, with or without the environmental control. After the estimation of the wake recovery length dW, the first array is placed at the longitudinal coordinate X

_{array}(1) = X

_{ref}+ dW, thus establishing the initial downstream coordinate X

_{ref}. In our case, X

_{ref}= 0, whilst dW yields the distance from the determined zero to the first array’s installation. The latter array is placed at a distance d

_{A}from the former. By making use of the backwater estimate, depths and flow velocities are determined for the channel at each discrete step up to recovery.

_{ref}(2) yields the reference parameters, i.e., Fr

_{ref}, B

_{ref}, U

_{ref}, and h

_{ref}. After setting X

_{ref}(i) = X

_{array}(i), the code iterates the optimization tool and the wake function to allow a new X

_{ref}to be found at convergence. Indeed, this represents the first estimation, as the wake must develop entirely such that X

_{ref}(i) = X

_{array}(i) − dW. This implies that X

_{ref}for the second array is achieved at a gradually varied station immersed in the backwater. At the reference position, the depth from the still water level to the top of the turbines is again determined, allowing the free surface LMADT equations of the turbine model to account for the actual installation depth, which are further updated during the array’s iteration.

#### 2.3. Optimum Design Solution

_{max}= [1/2ρA

_{T}U

_{∞}× max(F), min(Δh), min(LCoE)]

## 3. Results and Discussion

#### 3.1. Sensitivity Analysis

_{g}—defined as U

_{i}/U

_{∞}, with the subscript (i) related to the actual array—starting from downstream. The same three-bladed geometry of the prototype [37] is used for the simulations, but lower solidity and a higher aspect ratio are chosen to reduce fatigue and torque ripple. In this work, solidity is defined as the chord-to-diameter ratio multiplied by the number of blades, i.e., σ = N

_{b}× c/(2R

_{T}). For the first assessment, a rotor size (D

_{T}× H

_{T}) of 2.0 × 3.0 m is selected, with a shaft diameter D

_{0}of 0.18 m, solidity σ = 0.36, and aspect ratio (AR) of 1.5. The blade profile is an asymmetric DU06W200, which is less prone to dynamic stall at low tip speed ratios, with a chord length c of 0.24 m. Since the computational time increases with the number of arrays, faster simulation is facilitated with an azimuthal displacement of 15 deg, whilst the starting turbine depth is 0.2 m from the still water level. The streamtube expansion is not activated to limit the computational cost of the simulation. For the same reason, no environmental constraint is switched on for the initial predictions. The sensitivity of array thrust and power based on turbine spacing d

_{T}and array spacing d

_{A}is investigated in the following sections.

#### 3.1.1. Turbine Spacing (d_{T})

_{TA}

_{,∞}vs. global inflow factor ϕ

_{g}for a plant of four arrays is shown on the left side of Figure 10 according to different turbine spacings d

_{T}(different colors). Each marker represents the array C

_{TA}

_{,∞}from the most upstream array (left side) to downstream (right side).

_{T}) are chosen and the array spacing d

_{A}is set to 50 m. The lower turbine spacing of 2.5 m guarantees a safety margin of 0.5 m between the blades of neighboring turbines, while the higher spacing of 3.5 m represents the limit for four turbines (2.0 × 3.0 m in size) in relation to the available wetted area and the selected clearance between the top of the turbine and the still water level as if ideally deployed in the undisturbed flow. On the right side of Figure 10, the array power coefficient C

_{PA}

_{,∞}is investigated. C

_{TA}

_{,∞}and C

_{PA}

_{,∞}are calculated as T

_{A,i}/(½ρA

_{array}U

_{∞}

^{2}) and P

_{A,i}/(½ρA

_{array}U

_{∞}

^{3}), respectively.

_{T}increases; however, for a single simulation of four arrays and proceeding upstream (from right to the left), the array power decreases linearly. The first step in d

_{T}of 0.5 m (from 2.5 to 3.0 m) implies the highest drop in C

_{TA}

_{,∞}and C

_{PA}

_{,∞}, while a second step induces a smaller difference. Since the turbine spacing d

_{T}defines each flow passage and the turbine blockage ratio B

_{T,j}, the higher the d

_{T,}the smaller the blockage for the core turbines, which is usually expected to convert more power in the array. Therefore, a reduction in thrust force and power is expected as d

_{T}increases. Maximum thrust and power are achieved by the very-downstream array depicted on the right. At each hydraulic transition, mechanical power and total thrust force drop with the reduction in the inflow factor ϕ

_{g}.

^{2}> 0.99) for all simulations. Moreover, linearity holds only for a small plant. As better illustrated in the “Large plants” paragraph, once the scale of the plant is expanded, the behavior is no longer linear.

#### 3.1.2. Array Spacing (d_{A})

_{T}is fixed, thrust and power sensitivity vs. global inflow factor ϕ

_{g}is tested by varying the array spacing d

_{A}from 50 m to 500 m, as illustrated in Figure 11. Since the rotor shape and turbine spacing d

_{T}are the same in these predictions, the first array downstream shares the same outputs regardless of d

_{A}. Array thrust and power output increase slightly from array to array as d

_{A}increases because the backwater reduces progressively with distance, thus determining a higher recovery of the water velocity.

_{A}is high. This concept extends to an ideal condition. If the spacing between two arrays exceeds the recovery length of the backwater, the power converted from both is the same. This proves the physical consistency of the model. Hence, from the perspective of pure energy conversion, the best choice would be to ensure that they are place far apart, thus affording as much energy as possible and reducing, albeit partially, the risk of upstream flooding.

_{TA}

_{,∞}and C

_{PA}

_{,∞}reduce progressively at the upstream arrays, suggesting that the upstream rotors are optimized at a lower rotor speed than that of the downstream rotors. In the image, the C

_{PA}

_{,∞}from the same plant is linearly distributed over a straight line regardless of array spacing and number. Hence, for a small-scale plant, linearity is again demonstrated with R

^{2}> 0.99. However, the deviation in C

_{TA}

_{,∞}and C

_{PA}

_{,∞}at the second array between all feasible d

_{A}is not large, and it becomes more consistent from the third array onward.

#### 3.2. Array Design

^{2}, thus generating a constant blockage ratio (B

_{A},

_{∞}) of 0.279 based on Equation (1). Turbines are scaled according to Table 1, assuming a fixed turbine solidity, thus excluding the dependency of

**C**on σ. However, the turbine spacing d

_{P}_{,∞}_{T}cannot be fixed due to the different turbine sizes within a limited canal cross-section. Hence, although B

_{A,}

_{∞}and σ are constant, this does not hold for d

_{T}, whose variations slightly affect hydrodynamic forces. Notice that the turbine spacing d

_{T}is chosen to be as small as possible to increase the turbine blockage and maximize array power extraction. Again, a minimum breadth of 0.5 m is left as a safety margin for all predictions.

_{T}/H

_{T}. Moreover, the chord is scaled according to the above definition of σ, yielding c = 0.24 × R

_{T}. The investigation is performed with arrays consisting of two larger turbines with an AR of 0.75, three turbines with an AR of 1.125, and four smaller turbines with an AR of 1.5. The longitudinal spacing is fixed, i.e., d

_{A}= 250 m. The array power coefficients C

_{PA}

_{,∞}and free surface variation Δh

_{i}

_{,∞}distributed over the arrays for each turbine aspect ratio are outlined in Figure 12.

_{PA}

_{,∞}is achieved by four closely spaced turbines with an AR = 1.5. Although C

_{PA}

_{,∞}does not vary noticeably between cases (b) and (c), the highest array performance is achieved by a row of four closely spaced turbines with an AR = 1.5, while a poor result is determined for a row of two larger turbines (AR = 0.75). According to the results of Section 3.1, the array power output decreases from the most downstream array toward the upstream direction. The limitation of higher power conversion is represented by the increased free surface variation Δh

_{i,}

_{∞}determined with respect to the undisturbed flow depth, i.e., (h

_{i}− h

_{∞}).

_{T,}

_{∞}and power coefficient C

_{P,}

_{∞}curves, respectively, vs. the tip speed ratio λ

_{∞}. Although not evident in these images, the depth in the canal increases from the first to the last array, and the curves of C

_{T,}

_{∞}and C

_{P,}

_{∞}slightly drop from downstream to upstream. The inflow speed of the arrays placed furthest upstream drops, and this reduces, albeit partially, the performance of the upstream rotors. Notice that the lateral turbines and core turbines are equal when aligned two by two, and thus show equal power coefficients. This is a drawback of the momentum model, which is incapable of catching differences between inward or outward rotation directions.

_{lim}based on specific regulations or channel manager decisions, one may predict different power outputs from each array. If Δh

_{lim}is smaller than the expected hydrostatic head variation generated by a certain array of turbines operating at maximum power, a sub-optimal response induced by the rotor speed limitation would be achieved. This would mean a lower power output from the turbines drawn for higher mechanical power, thus limiting the choice of the best array design solution between lower-capacity plants.

#### 3.3. Large Plants

_{p}54 mm × w

_{p}280 mm) built from a porous metallic frame deployed to occupy the rectangular channel span (w

_{c}= 305 mm) almost entirely. Depth and Froude numbers were allowed to vary according to a subcritical hydraulic regime (Fr < 1) by adjusting the level of downstream drain and thus automatically modifying the blockage ratio.

_{∞,opt}.

_{d}is fixed from the canal bed, with the submergence depth increasing with each successive array. In contrast, if the turbine depth is maintained constant at the free surface proximity, higher depths would be expected upstream.

_{i}reduces the mechanical power availability. The array (or turbine) power output P

_{A}scales linearly for a small number of arrays (≤5); then, it is approximated by a quadratic polynomial to about 20–25 arrays (which is still reasonable for large plants). For an extremely large number of arrays, a logarithmic curve fits the data, with R

^{2}> 0.99.

_{i}− h

_{i−}

_{1}) is approximated with a power law y = a × x

^{b}, in which x denotes the number of arrays, the coefficient a > 0, and the exponent b < 0. Ideally, Δ(h

_{i}− h

_{i−}

_{1}) would drop to zero asymptotically, suggesting a constant inflow depth for a large number of arrays. However, this can be approximated as the deviation in water depth Δ(h

_{i}− h

_{i−}

_{1}) approaches the numerical accuracy of the model (i.e., 1 mm) in Figure 16, implying that the inflow hydrostatic head is expected to remain constant after a certain number of arrays within the imposed physical tolerance. Moreover, a slight drop in depth between two arrays is still present due to the backwater. This trend can be generalized for any array layout of aligned turbines.

_{i}− h

_{i−}

_{1}). If a tolerance of 5 mm was allowed (which is more consistent with the typical sensitivity level of many probes), an asymptotic deviation would be reached after about 15 arrays (in this example). Thus, the expected far-field condition in a large plant of aligned rotors more closely resembles Figure 15, even though this distribution is valid for ideal canals.

## 4. Conclusions

- The exploitable capacity in a channel flow can be estimated with turbines operating at maximum performance and considering canals and rotors’ geometry, turbine spacing, and array spacing.
- The tool incorporates an environmental assessment of a canal, providing visualizations of a 1-D free surface distribution showing the backwater propagating from each successive array from a downstream cross-section.
- By integrating a Double Multiple Streamtube model operating in a confined environment with a variable Froude number and blockage ratio, the tool overcomes the limitations encountered in ideal analytical modeling with actuator discs. Indeed, the model accounts for the turbine’s fluid dynamic losses, turbulent mixing loss, and channel friction losses to calculate upstream free surface propagation.

- A sensitivity analysis demonstrated that closely spaced turbines in aligned arrays convert more power than widely spaced turbines. Indeed, an increase in turbine spacing shrinks the size of each flow passage and enhances the level of turbine blockage. As confirmed by the predictions, higher turbine blockage is achieved by the rotors deployed at the center of the row, which notoriously, harness more energy.
- If the hydrokinetic plant design involves a canal stretch of a given length, the sensitivity analysis shows that closely spaced arrays perform inadequately in subcritical flows. Wide array spacing allows for the partial recovery of superior flow conditions for the new array, thus implying higher power output.
- Assuming a scheduled number of rows with an equal number of turbines of a given solidity, the comparison of different array layouts for the investigated canal cross-section demonstrates that four small turbines of a high aspect ratio are more beneficial than three medium-size turbines or two large turbines of a small aspect ratio. While constraining the maximum expected free surface variation in the plant, a sub-optimal response induced by the rotor speed limitation may be achieved by the same array designed for higher capacity.
- For a small plant (number of arrays ≤ 5), a row’s maximum power output drops linearly with respect to the number of arrays, whereas for a reasonable number of arrays in a large plant, this measure can be approximated by a quadratic polynomial.
- Since the deviation in the inflow depth ideally tends to be zero for a large multi-array plant, the backwater depth far upstream is speculated to remain constant at each inflow station, even though a reduction in depth within each array spacing is recurrent.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | Cross-section area, m^{2} |

A_{F} | Turbine flow-passage area, m^{2} |

A_{T} | Turbine swept area, m^{2} |

AR | Turbine aspect ratio |

$\overline{Ay}$ | Moment area, m^{2} |

B_{A} | Array blockage ratio, N_{T}A_{T}/A |

B_{T} | Turbine blockage ratio, A_{T}/A_{F} |

C_{P,}_{∞} | Turbine power coefficient, P/(½ρAU_{∞}^{3}) |

C_{PA,}_{∞} | Array power coefficient, P_{A}/(½ρA_{rray}U_{∞}^{3}) |

C_{T,}_{∞} | Turbine thrust coefficient, T/(½ρAU_{∞}^{2}) |

C_{TA,∞} | Array thrust coefficient, T_{A}/(½ρA_{rray}U_{∞}^{2}) |

c | Blade chord, m |

D_{h} | Hydraulic depth, A/w |

D_{0} | Shaft diameter, m |

D_{T} | Turbine diameter, m |

d | Wake length, m (see subscripts) |

d_{A} | Array spacing, m |

d_{T} | Turbine spacing, m |

dE | Specific energy variation, m |

dW | Total wake length, m |

dx | Discrete longitudinal step, m |

dy | Discrete variation in water depth, m |

Fr | Froude number, U/(gD_{h})^{½} |

g | Gravity acceleration, m/s^{2} |

H_{T} | Turbine height, m |

h | Water depth, m |

M | Channel momentum, m^{2} |

N_{A} | Number of arrays |

N_{T} | Number of turbines |

N_{Ω} | Number of selected rotor speeds |

n | Manning’s coefficient, m^{−1/3}s |

P | Turbine power, W |

P_{A} | Array power, W |

Q_{c} | Channel discharge, m^{3}/s |

R_{h} | Hydraulic radius, m |

R_{T} | Turbine radius, m |

S | Bed slope, m/m |

S_{f} | Friction slope, m/m |

T | Axial thrust force, N |

T_{A} | Array axial thrust force, N |

U | Mean flow speed, m/s |

w | Free surface width, m |

X | Axial coordinate from the origin, m |

ΔH | Total head variation, m |

Δh | Free surface variation, m |

Δ(h_{i} − h_{i}_{−1}) | Deviation in water depth, m |

Δh_{lim} | Environmental constraint, m |

ΔU | Flow speed variation, m/s |

Ω | Rotor speed, rpm |

α_{w} | Near-wake interference factor, U/U_{0} |

λ | Tip speed ratio, ωR_{T}/U |

ρ | Fluid density kg/m^{3} |

σ | Rotor solidity, N_{b} c/(2R_{T}) |

σ_{A} | Rotor solidity from [23], N_{b} c/(2πR_{T}) |

σ_{D} | Dynamic solidity, 1 − 1/(2πσ_{A}λ) |

ϕ | Inflow factor, U_{0}/U_{ref} |

ϕ_{g} | Global inflow factor, U_{0}/U_{∞} |

ω | Angular velocity rad/s |

Subscripts | |

array | Array position |

f | Far-wake |

opt | Optimum |

ref | Reference |

t | Transition-wake |

w | Near-wake |

Short forms | |

DMS | Double Multiple Streamtube |

HK | Hydrokinetic |

LMADT | Linear Momentum Actuator Disc Theory |

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**Figure 1.**An array of aligned vertical-axis cross-flow hydrokinetic turbines deployed in a regular canal. Image from HE Powergreen S.r.l. [3].

**Figure 2.**Cross-section of interest in the Biffis canal from Malini [25], which was modified according to the design specifications of this work.

**Figure 5.**Flowchart of power optimization scheme (M1) for a single array (equal rotor speed for each turbine).

**Figure 6.**Flowchart of power optimization scheme (M2) for a single array (different optimum rotor speeds for each turbine).

**Figure 7.**Near-wake distribution of four cross-flow turbines from [3].

**Figure 9.**(

**Left**) Example of free surface variation based on three turbine arrays of four 6 m

^{2}turbines vs. distance with array spacing of 50 m. The local free surface variation past the turbines is visualized for the central rotors. (

**Right**) Effect of the asymptotic backwater past the three turbine arrays.

**Figure 10.**(

**Left**) Effect of the turbine spacing d

_{T}on the array thrust coefficient C

_{TA}

_{,∞}vs. global inflow factor ϕ

_{g}for a plant of four arrays. (

**Right**) Effect of the turbine spacing d

_{T}on the array power coefficient C

_{PA}

_{,∞}vs. global inflow factor ϕ

_{g}for a plant of four arrays. The three series of arrays from downstream to upstream are depicted with the same color from the right to left.

**Figure 11.**(

**Left**) Effect of the array spacing d

_{A}on the array thrust coefficient C

_{TA}

_{,∞}vs. global inflow factor ϕ

_{g}for a plant of four arrays. (

**Right**) Effect of the array spacing d

_{A}on the array power coefficient C

_{PA}

_{,∞}vs. global inflow factor ϕ

_{g}for a plant of four arrays. The most downstream array is indicated with a black marker. The series of arrays from downstream to upstream are depicted with the same color from right to left.

**Figure 12.**(

**Left**) Array power coefficient C

_{PA}

_{,∞}vs. number of arrays N

_{A}. (

**right**) Free surface variation Δh

_{i,∞}vs. number of arrays N

_{A}. Different colors indicate the various simulations (blue markers indicate 2-turbine arrays with AR = 0.75, red markers indicate 3-turbine arrays with AR = 1.125, and yellow markers indicate 4-turbine arrays with AR = 1.5).

**Figure 13.**Optimum design: 4 arrays of 4 turbines with AR = 1.5 operating at λ

_{∞,opt}. The left column indicates the local free surface deformation in space for each turbine with flow directed from right to left. The central and last columns represent turbine thrust coefficient C

_{T,}

_{∞}and power coefficient C

_{P,}

_{∞}, respectively, for each turbine of the array.

**Figure 14.**(

**Left**) Discharge Q

_{c}= 32.3 l/s, spacing between the plates d = 0.316 m, height from the bottom y

_{z}= 0.115 m, upstream depth h

_{0}= 0.20 m, upstream flow speed U

_{0}= 0.530 m/s, and measured thrust force over the first plate = 7.51 N. (

**Right**) Discharge Q

_{c}= 35.7 l/s, spacing between the plates d = 1.0 m, height from the bottom y

_{z}= 0.101 m, upstream depth h

_{0}= 0.201 m, upstream flow speed U

_{0}= 0.583 m/s, and measured thrust force over the first plate = 5.54 N.

**Figure 15.**The asymptotic trend of water depth variation Δh caused by increasing the number of arrays or turbines. Although not appreciable, backwater depth decreases upstream from each hydraulic transition. (Image is not to scale).

**Table 1.**Cases investigated with respect to array design. Nomenclature: N

_{A}number of arrays, N

_{T}number of turbines per array, d

_{A}array spacing, d

_{T}turbine spacing, A

_{T}turbine swept area, AR turbine aspect ratio, D

_{T}turbine diameter, H

_{T}turbine height, and c blade chord.

Case | N_{A} | N_{T} | d_{A} | dT | A_{T} | AR | D_{T} | H_{T} | c |
---|---|---|---|---|---|---|---|---|---|

[#] | [#] | [#] | [m] | [m] | [m^{2}] | [-] | [m] | [m] | [m] |

a | 4 | 2 | 250 | 4.5 | 12 | 0.75 | 4 | 3 | 0.48 |

b | 4 | 3 | 250 | 3.167 | 8 | 1.125 | 2.667 | 3 | 0.32 |

c | 4 | 4 | 250 | 2.5 | 6 | 1.5 | 2 | 3 | 0.24 |

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## Share and Cite

**MDPI and ACS Style**

Cacciali, L.; Battisti, L.; Dell’Anna, S.
Multi-Array Design for Hydrokinetic Turbines in Hydropower Canals. *Energies* **2023**, *16*, 2279.
https://doi.org/10.3390/en16052279

**AMA Style**

Cacciali L, Battisti L, Dell’Anna S.
Multi-Array Design for Hydrokinetic Turbines in Hydropower Canals. *Energies*. 2023; 16(5):2279.
https://doi.org/10.3390/en16052279

**Chicago/Turabian Style**

Cacciali, Luca, Lorenzo Battisti, and Sergio Dell’Anna.
2023. "Multi-Array Design for Hydrokinetic Turbines in Hydropower Canals" *Energies* 16, no. 5: 2279.
https://doi.org/10.3390/en16052279