The idea of integrating existing hydropower plants with hydrokinetic turbines was suggested by Liu and Packey [1
] for the development of Combined-Cycle Hydropower Systems, for which the objective was to harvest additional energy from the tailwater of the flows exiting dams.
The energy recovery of hydropower canals was then discussed by Cacciali et al. [2
] to attempt the installation of cross-flow HK devices in inlet hydropower canals, i.e., artificial waterways connecting basins with a subcritical flow, with lateral banks that conveniently accommodate structural works. This application is suitable in some parts of northern Italy, which is a mountainous area with several hydroelectric plants built more than a hundred years ago. An example of a hydro–hydro application is illustrated in Figure 1
concerning the rotors being tested by HE Powergreen S.r.l. [3
] in the Biffis canal, i.e., a waterway used to supply the hydropower plant in Bussolengo (VR), Italy.
The deployment of multiple arrays represents a trade-off between power optimization and environmental control. Indeed, besides the energy harvested, an adverse effect is posed by the backwater generated from the operating turbines, thus implying the requirement for automated depth measurement. This forces the designer to strictly design the array such that it limits the backwater induced by the plant. On the other hand, the basin operator cannot allow flooding hazards to occur due to faulty design.
The most relevant works intended for canals and fluvial applications are presented in the following material; some of these are dedicated to array power optimization in tidal flows, including assessments of turbine spacing and array spacing.
Turbine arrays were investigated by Gauvin-Tremblay and Dumas [4
] in aligned and staggered configurations; the investigation showed that the local blockage and turbine–wake interaction affect the performance of an array, which, in turn, dominates the downstream row. The wake deflection originating from the upstream vertical-axis turbines highlighted a negative impact on the staggered configuration and clarified that an aligned layout with inward rotations of the turbines should be selected. Their work demonstrated no advantages in terms of power extraction for the aligned configuration over a staggered layout in a wide channel, whilst higher power output was shown by the aligned configuration in a narrower environment. The authors also observed higher mean power extraction per turbine by increasing the longitudinal spacing between the turbine arrays; however, they also affirmed that the larger the lateral spacing, the higher the total power. Almost all the aforementioned results were confirmed by the model presented in our work, except for those regarding lateral spacing, as will be clarified later.
Gonzalez-Gorbeña et al. [5
] conducted an interesting optimization of turbine arrays with a surrogate-based method for regular and irregular channels in both uniform and non-uniform flow scenarios. Inline and staggered arrays were investigated, leading to the determination of capacity factors and profitability. Polynomial and radial basis function surrogate methods were used to fit an objective function based on two variables: the lateral and the longitudinal spacings. This method is promising, although the downstream flow does not seem to affect the upstream flow, wherein horizontal rotors were simplified as actuator discs.
Okulov et al. [6
] tested longitudinally aligned horizontal axis hydrokinetic turbines with particle image velocimetry to measure the flow properties upstream and downstream of the devices including up to four elements, highlighting the interference of the turbine wake on the downstream turbine, which was closely deployed. The authors remarked on a continuous drop in turbine performance due to the upstream wake reducing the inflow speed with higher turbulence. Unlike Okulov, we clarify that optimal rotor placement in a narrow canal or riverbed should enable the rotor’s detachment from reduced longitudinal spacings due to turbulent phenomena. This was clarified through the on-site experience of H.E. Powergreen S.r.l. [3
], which demonstrated evident backwater phenomena due to the variation in rotor speed and lateral spacing.
Septyaningrum et al. [7
] investigated co-rotating and counter-rotating turbines and conducted an analysis of turbulence intensity in the interaction zone, thereby demonstrating that flow superposition is not the only reason for flow acceleration, as an additional influence is contributed by a closer spacing between the turbines, which confers a jet-type flow effect for the bypass flow. The authors proved that a faster velocity superposition and lower turbulence intensity are achieved with inward counter-rotating rotors. Although not discussing the backwater phenomenon, the authors indicated that the performance of downstream turbines in a multi-array layout would be sub-optimal if operating in the wake of upstream turbines, concluding that downstream turbines should be placed beyond that area.
Riglin et al. [8
] simulated different array layouts with a CFD solver based on RANS equations mimicking a hydro-farm. A substantial drop in power was observed from the upstream to downstream units, showing that the operation of a turbine in the wake of the upstream turbines significantly reduces the energy available downstream.
Vennell et al. [9
] reviewed the main parameters affecting the large turbine arrays in tidal flows, highlighting that the enhanced drag due to power extraction reduces the mean flow speed as the flow approaches the rotors. Moreover, the authors specified that optimally tuned turbines in large arrays perform differently from tuned isolated turbines. Some of the effects of a large array discussed by Vennell can be applied to on-land subcritical canals, e.g., in efforts to boost a turbine’s output by adjusting its relative position within an array. In this work, this effect is obtained by modifying the spacing between the turbines. Vennell indicated that higher capacities can be obtained by adding supplementary turbines to an array, but the behavior of the power curve differs among wide and narrow channels.
A major contribution to array-based analytical models can be attributed to Nishino, Willden, and Draper, whereas the confined actuator disc theory was extended in its free surface double-disc version in our previous work [10
], which was subsequently updated with a method accounting for the backwater in subcritical flows [2
Although staggered turbine layouts have been assessed in relation to tidal flows [11
], their actual application in hydropower canals is questionable, as closely spaced rows would cause an increase in turbulent effects and hydrostatic depth, which is only tolerable to a small extent, and which will be explained further later. With aligned rotors, the hydraulic transition to be solved at each turbine array resembles the solution introduced by Cacciali et al. [2
]. Once the equilibrium in hydrostatic pressure is reached upstream of each turbine array, the computation of the average flow speed in the cross-section is iteratively feasible. Since the maximum power output can be reached by respecting the environmental constraints, i.e., avoiding overbank flows due to backwater, only aligned rotors are investigated.
Nishino and Willden [11
] described the flow disturbance due to multiple turbines involving a large portion of the channel. The flow proximity was ideally assessed on a large scale with actuator discs due to power extraction in a tidal environment, showing that the turbine-scale wake expansion occurs faster than the expansion around the turbine array. Draper and Nishino [13
] developed a model for centered and staggered actuator discs and concluded that two arrays should be closely spaced only if staggered; otherwise, centered arrangements should be kept as far apart as possible, which is in agreement with our findings. Subsequently, a theoretical model accounting for both the turbine and array scales was introduced by Vogel et al. [14
] for a finite-width array of tidal turbines. However, these analytical approaches are controversial if applied to narrow canals of an arbitrary cross-sectional geometry when a subcritical flow is subject to downstream control.
Since undisturbed subcritical flow occurs downstream of any hydraulic resistance [15
], a reference flow speed U∞
can be designated for performance and axial force computations. Alternatively, the use of the inflow speed U0
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 CT,∞
) and CP,∞
), with ρ
denoting the water density, T
denoting the turbine thrust force, P
representing the mechanical power, AT
denoting the rotor swept area, and U∞
representing the far-field flow speed. The array thrust coefficient and power coefficient are given by CTA,∞
) and CPA,∞
), respectively, with TA,i
denoting the thrust force and power developed at the i
th array, and Aarray
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
The hydraulic transition due to the turbine array entails free surface deformation in canals, waterways, and rivers, with a cross-section almost entirely occupied by cross-flow or axial-flow hydrokinetic turbines. In a previous study, Myers and Bahaj illustrated the free surface effects achieved experimentally in water channels at various speeds [16
]. Moreover, the variation in the clearance between the top of the turbine and the still water level can also create undesirable phenomena, as clarified further on.
The potential for hydrokinetic energy exploitation has been determined in several works [17
], with most of them estimating the total head variation in a certain section of the channel without introducing a turbine model. These assessments offer an undoubtable advantage in terms of simplicity and allow for direct access to data when designing hydrokinetic systems.
However, a rigorous estimation of this potential that accounts for the free surface development in the investigated canal stretch has not yet been provided. Due to the complexity and computational cost of the numerical works conducted with CFD simulations on turbine arrays, estimates of the available capacity and effective power output in a simple, quick, and effective manner are rare. In addition, analytical models only provide a macroscopic picture of the energy extraction in water flows that do not differentiate each turbine in the array [4
]. We agree that analytical models concerning ideal actuator discs draw interesting conclusions regarding the layouts of the arrays, but they do not include important aspects of turbine operation and turbine–wake interaction. This fact prompted us to consider designing a model that overcomes the limitations of the actuator discs, i.e., including the fluid-dynamic losses of each turbine, and avoiding the excessive computational costs necessitated by CFD models.
Therefore, a tool that quantitatively estimates the exploitable resources in a canal to pre-size the hydrokinetic plant was developed, which incorporated the complete prediction of the free surface variation induced by the rotors from a uniform station (downstream) to the exhaustion of the backwater (far upstream). This has not been presented in other works. Moreover, different prismatic channel geometries can be considered, thus adding flexibility to the simulation.
Real rotors are predicted by this tool, thus overcoming the limitations concerning the approximation of ideal rotors with actuator discs. The fluid flow distant from each rotor is then simplified as a 1-D channel flow based on a subcritical hydraulic regime, i.e., respecting the physics of open channel flows.
However, based on momentum models, this tool does not allow for the direction of rotation of the rotors to be distinguished, nor does it allow for the mutual influence of the wakes to be evaluated. The treatment of the development of wakes is executed by exploiting regressions, and this simplifies the modeling process because power dissipation is treated quantitatively. It is emphasized that the method requires that the wake of an upstream array does not affect the fluid dynamics of the downstream array. Indeed, on an industrial scale, the array problem should be treated over a consistent longitudinal stretch. Unlike some of the cited works, the deployment of rotating elements in the wake of other rotors for systems designed over long, longitudinal stretches seems illogical. For design purposes, it is critical to ensure sufficient longitudinal spacing to recover, albeit partially, the velocity and depth of the flow.
The integrated use of a DMS model for vertical-axis turbines [10
] with a hydraulic model for solving the hydraulic transition for single rotors [2
], the computation of the level of backwater generation [22
], and a simplified investigation of the recovery length downstream of the array from the research in [23
] enable the rapid prediction of the available and exploitable capacity.
By incorporating these methods, this work aims to model multiple arrays with a numerical procedure with a low computational cost, thus obtaining the expected CT,∞ and CP,∞ 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.
The proposed method is introduced in a single-array and multi-array scheme, which are presented in Section 2.1
and Section 2.2
, respectively. Section 2.1.1
includes the definitions of turbine and array blockage ratios, whereas Section 2.1.2
presents an optimization scheme with which to build the power curve of a single turbine and select the best performance at the optimum tip speed ratio. This procedure is repeated for all turbines in the array. When the turbine’s thrust and power are determined, the total thrust force of the array relaxes the inflow blockage and Froude number. Once converged, the code updates each rotor’s power and thrust force. Hence, a scheme between M1 and M2 must be selected. According to scheme M1, an equal rotor speed is assumed for all turbines in the array; according to scheme M2, each turbine is optimized at a certain rotor speed, implying higher array power.
Eventually, an environmental boundary condition can be activated (Section 2.1.3
) to limit the backwater to a prescribed depth variation. Section 2.1.4
, Section 2.1.5
, and Section 2.1.6
are dedicated to the approximation of the near-wake, transition-wake, and far-wake past the turbines. Hence, Section 2.1.7
shows the methods for estimating the reduction in the backwater upstream according to energy or momentum principles.
The results discussed in Section 3
focus on sensitivity analysis (Section 3.1
), array design (Section 3.2
), and simulations of large plants (Section 3.3
). First, a sensitivity analysis is performed concerning turbine spacing and then regarding array spacing for a plant built with four arrays and four turbines per array. The simulations dedicated to the array design in Section 3.2
show the best prediction within a selected number of cases based on a constant blockage ratio and a fixed level of rotor solidity, thereby enabling a reduction in the number of variables. The inclusion of a limit in the backwater would alter the output of this prediction, thus directing the design toward less-expensive and lower-power solutions. Section 3.3
shows the expected free surface variation from a simulation performed for a plant of many arrays. Finally, conclusions and prospects are discussed in Section 4
3. Results and Discussion
The momentum method was tested to investigate the effect of turbine spacing and array spacing on axial forces and power, which indicated an increase in thrust reasonably proportional to the rise in backwater depth. The predictions were carried out based on the concession discharge of the hydropower canal depicted in Figure 2
, which was subject to ongoing hydrokinetic turbine testing.
An optimization procedure was thus conducted on some design cases based on different array features. Array blockage and turbine solidity were kept constant to assess the most suitable design solutions, and the turbine aspect ratio was allowed to vary between alternatives to meet the requirements of the fixed array blockage ratio.
A final computation is performed based on the research shown in [38
] to predict the distribution of the free surface variation and maximum power output for a large plant incorporating many arrays.
3.1. Sensitivity Analysis
Predictions of the array thrust force and power are collected vs. the global inflow factor ϕg—
defined as Ui
, 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
). For the first assessment, a rotor size (DT
) of 2.0 × 3.0 m is selected, with a shaft diameter D0
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 dT
and array spacing dA
is investigated in the following sections.
3.1.1. Turbine Spacing (dT)
The array thrust coefficient CTA,∞
vs. global inflow factor ϕg
for a plant of four arrays is shown on the left side of Figure 10
according to different turbine spacings dT
(different colors). Each marker represents the array CTA,∞
from the most upstream array (left side) to downstream (right side).
Spacings of 2.5 m, 3.0 m, and 3.5 m between each rotor vertical-axis (corresponding to 1¼, 1½, and 1¾ equivalent lengths based on turbine diameters DT
) are chosen and the array spacing dA
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 CPA,∞
is investigated. CTA,∞
are calculated as TA,i
) and PA,i
Both array thrust and power drop as the turbine spacing dT 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 dT of 0.5 m (from 2.5 to 3.0 m) implies the highest drop in CTA,∞ and CPA,∞, while a second step induces a smaller difference. Since the turbine spacing dT defines each flow passage and the turbine blockage ratio BT,j, the higher the dT, 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 dT 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 ϕg.
Linearity is demonstrated for the variations in power with a very high correlation (R2 > 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 (dA)
is fixed, thrust and power sensitivity vs. global inflow factor ϕg
is tested by varying the array spacing dA
from 50 m to 500 m, as illustrated in Figure 11
. Since the rotor shape and turbine spacing dT
are the same in these predictions, the first array downstream shares the same outputs regardless of dA
. Array thrust and power output increase slightly from array to array as dA
increases because the backwater reduces progressively with distance, thus determining a higher recovery of the water velocity.
Hence, a greater amount of energy can be harnessed at the farthest array if dA 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.
Again, CTA,∞ and CPA,∞ 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 CPA,∞ 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 R2 > 0.99. However, the deviation in CTA,∞ and CPA,∞ at the second array between all feasible dA is not large, and it becomes more consistent from the third array onward.
3.2. Array Design
The previous sensitivity analysis demonstrates the improvement in the array power output with a reduced turbine spacing, even though higher axial forces and depths are afforded. Another finding is that highly spaced arrays harness more energy than closer arrays. However, the choice of the array spacing is based on the available length of the channel stretch and the expected investment, and this implies knowledge of the number of arrays. It is expected that a cash flow assessment of the best predictions will be incorporated in future works to achieve the most convenient design solution.
In the ideal scenario considered herein, 1 km of the canal is dedicated to simulations with four turbine arrays, with the target of maintaining a swept area of 24 m2
, thus generating a constant blockage ratio (BA,∞
) of 0.279 based on Equation (1). Turbines are scaled according to Table 1
, assuming a fixed turbine solidity, thus excluding the dependency of CP,∞
. However, the turbine spacing dT
cannot be fixed due to the different turbine sizes within a limited canal cross-section. Hence, although BA,∞
are constant, this does not hold for dT
, whose variations slightly affect hydrodynamic forces. Notice that the turbine spacing dT
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.
By fixing the height of the turbines to 3.0 m and maintaining a level of solidity of σ
= 0.36, the diameter is determined from the ratio AT
. Moreover, the chord is scaled according to the above definition of σ
, yielding c
= 0.24 × RT
. 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., dA
= 250 m. The array power coefficients CPA,∞
and free surface variation Δhi,∞
distributed over the arrays for each turbine aspect ratio are outlined in Figure 12
The prediction demonstrates that a higher CPA,∞
is achieved by four closely spaced turbines with an AR
= 1.5. Although CPA,∞
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 Δhi,∞
determined with respect to the undisturbed flow depth, i.e., (hi
The outputs of the optimum solution of this assessment are shown in Figure 13
from the most downstream array (#1) to the most upstream (#4) array. The schemes on the left column represent the local free surface distribution generated by each turbine over its predicted longitudinal extension. The direction of the flow is from right (inflow) to left (flowing past the area of turbulent dissipation). Since the turbines deployed at the center of the canal convert more power, they generate higher free surface variation, whilst a mild variation occurs for the lateral turbines. The flow past array (#1) represents the undisturbed flow achieved downstream of the hydrokinetic plant.
The central and right schemes represent the thrust coefficient CT,∞ and power coefficient CP,∞ 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 CT,∞ and CP,∞ 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.
By activating the environmental constraint to limit the backwater to a value of Δhlim based on specific regulations or channel manager decisions, one may predict different power outputs from each array. If Δhlim 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
Measurements of the drag forces were carried out at the DICAM Unitn Hydraulic Laboratory with plates (hp 54 mm × wp 280 mm) built from a porous metallic frame deployed to occupy the rectangular channel span (wc = 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.
The images in Figure 14
were obtained by placing two of the perforated plates at a constant spacing and fixed elevation from the channel bed. Notice that the flow is directed from right to left. Considering the limited channel length available in the facility, other plates were allowed to be “added” upstream of the last plate only by repeating the test with a new regulation of the downstream flow through the drain. The previous backwater was recreated downstream for a new test, for which there were no changes in the spacing between the plates. Depths were accurately measured at a precise spacing from the upstream plate to allow for reliable matching with the new regulated depths.
The aforementioned phenomenon occurs if one approximates the channel as an ideal water body free of obstacles, whose geometric cross-section is uniformly distributed over the length of interest, and with a constant elevation of the plate from the channel bottom. Hence, the steady distribution of depth, flow speed, thrust, and power may be predicted over a certain distance to improve the modeling of combined hydroelectric–hydrokinetic systems.
Since the turbines are expected to operate at maximum performance, the resultant backwater depth is not as severe as that generated at higher rotor speeds. The highest turbine thrust force occurs at a rotor speed at which almost no mechanical power is collected at the shaft and the energy dissipation in the channel flow is maximum. To render a test more consistent with real turbines, the porosity of the plates should be adjusted progressively upstream to approximate the regulation of the optimum turbine tip speed ratio λ∞,opt.
According to the measurements of the drag forces and depth of each plate, an asymptotic trend of the deviation in the inflow depth was speculated for a high number of arrays of optimized turbines.
shows the expected reduction in the water depth variation at the i
th array, even though bed slope and exact free surface deformation are neglected in the image. As mentioned, the turbine elevation zd
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.
Therefore, a progressive drop in flow speed upstream is inevitable unless the new row is placed after the backwater exhaustion, even when many kilometers away.
A single-turbine test was carried out to most accurately visualize this phenomenon and reduce the inaccuracy resulting from the numerical procedure. For this purpose, a prediction is made wherein a series of equal turbines of the type shown in [38
] are positioned with an array spacing of 10 m in the canal illustrated in the same work.
The water depth increases after each hydraulic transition, and the total head variation ΔHi reduces the mechanical power availability. The array (or turbine) power output PA 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 R2 > 0.99.
For the case investigated, the deviation in the inflow depth Δ(hi
) is approximated with a power law y
, in which x
denotes the number of arrays, the coefficient a
> 0, and the exponent b
< 0. Ideally, Δ(hi
) 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(hi
) 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.
Therefore, the imposed physical tolerance to the model determines the length scale to ascertain the asymptotic deviation Δ(hi
). 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.
The assumptions of an infinitely long channel with a homogeneous, unobstructed geometry and stationary flow do not realistically represent the characteristics of large plants. Indeed, a transient variation in the flow rate would imply that the discharge cannot be assumed to be constant over the entire stretch of a channel at the same time. Turbulence, local head losses, and finite lengths of artificial canals further complicate the real flow distribution.
A similar problem was illustrated by Karteznikova and Ravens [39
] with respect to a homogeneous distribution of closely spaced turbines in a regular channel. The authors modeled an almost flat free surface far upstream, speculating that a uniform flow with a change in Manning’s roughness coefficient had been achieved, even though a non-realistic power extraction scheme based on Garrett and Cummins’ equation [40
] was included.