2.1. Test facility

The tests were conducted at FloWave (https://www.flowave.eng.ed.ac.uk/) at the University of Edinburgh, a 25 m diameter tank with pumps and wave paddles positioned around its circumference, able to generate current and waves in any direction. Sutherland et al. (Reference Sutherland, Noble, Steynor, Davey and Bruce2017) have quantified the velocity shear and turbulence at a range of conditions at the facility, providing depth profiles and turbulence quantities at a range of nominal flow speeds. Following their work, a flow speed of $0.8\ {\rm m}\ {\rm s}^{-1}$ was chosen for the present test campaign so that comparisons could be made in the analysis. This relatively high flow speed was chosen so as to achieve as high a blade chord Reynolds number as possible, in the range $1.4 \unicode{x2013} 2 \times 10^5$ at an inflow speed of $0.8\ {\rm m} {\rm s}^{-1}$, to promote post-transitional blade flow. Details on the Reynolds number independence of the results that confirm $0.8\ {\rm m}\ {\rm s}^{-1}$ as a suitable flow speed are provided in Appendix A. At this tank flow speed the streamwise turbulence intensity is in the range of 6–8 % and the mean shear profile is well described by a power law with an exponent of one-sixth to one-ninth, varying spatially throughout the tank.

2.2. Turbine model

The tests used two identical three-bladed horizontal-axis tidal turbines with rotor diameter $d = 1.2$ m, shown in figure 1. The rotors were designed specifically for the twin turbine configuration when deployed in this test facility, using a methodology that accounts for local blockage effects and provides superior rotor performance by exploiting constructive interference effects (Cao, Willden & Vogel Reference Cao, Willden and Vogel2018; Cao Reference Cao2020). A two-stage hydrodynamic rotor design process was performed; first using RANS-BE in which boundary and turbine proximity effects are accounted for through the computational stencil, which was then followed by blade resolved simulation using a moving reference frame method. The second stage is necessary to adjust the design for complex flows at the root and tip where lower-order RANS-BE solutions are less accurate. A full description of the hydrodynamic design method is given in Cao (Reference Cao2020).

Figure 1.

Photograph of the twin tidal turbines during operation.

The rotors were mounted to existing drive-trains and nacelles provided by the University of Edinburgh. These drive-trains made use of a thrust and torque transducer upstream of the shaft bearing and seal arrangement, allowing loads to be measured without any mechanical losses. Further details of the nacelle and drive-train are given in Payne, Stallard & Martinez (Reference Payne, Stallard and Martinez2017), whereas details of the rotor manufacture and integration for these tests may be found in McNaughton et al. (Reference McNaughton, Cao, Vogel and Willden2019). The two turbines operated independently of each other and thus there was no phase locking between the blade azimuth positions. Further, owing to small fluctuations in the motor speeds the relative phase difference between the two turbines varied in time.

2.3. Test matrix

Tests were performed to characterise the flow in the absence of turbines, as well as two turbine configurations, single and twin. (Owing to project timescales the flow measurements without turbines were performed in May 2021 whereas the turbine tests were completed two years earlier in February 2019. For both experimental campaigns tank settings were the same and a comparison of data points performed during both campaigns is provided in Appendix B.) The turbine layouts are shown in figure 2. For clarity, the turbines are referred to as the North and South turbines, with the North turbine being the rotor used in both single and twin configurations, and the South turbine the additional rotor used only in the twin configuration.

Figure 2.

(a) Schematic of FloWave layout (adapted from Noble et al. Reference Noble, Davey, Smith, Kaklis, Robinson and Bruce2015; Sutherland et al. Reference Sutherland, Noble, Steynor, Davey and Bruce2017) indicating layout of turbines, note that streamlines are illustrative only. (b) Turbine positions and naming conventions used in this study. Not to scale.

The turbines were positioned with their rotor plane 1.3$d$ downstream of the mid plane of the tank, based on experience of previous tests such as Sutherland et al. (Reference Sutherland, Noble, Steynor, Davey and Bruce2017) to minimise spatial changes in flow conditions. The rotors were located at mid-water depth and for the twin configuration the tip-to-tip distance between the rotors was $s = d/4$ with the rotors symmetrical placed either side of the tank centreline. For practical reasons, the North turbine position did not change between single and twin configurations meaning that there was a minor asymmetry during the single turbine experiments. An additional asymmetry from both turbines rotating in the same direction was also present for the twin turbine case. This was due to the model design and ease of machining six identical blades rather than two sets of three. Although it is unlikely that this had noticeable effect on the mean interactional effects of the turbines, this asymmetry may have resulted in uneven loading or flow effects. Using counter-rotating turbines would have prevented this and provided an opportunity to better understand some of the blade loading effects discussed later in § 5.3.

The global blockage ratio (ratio of total frontal area of turbines to channel cross-sectional area), $B_G$, and the local blockage ratio (ratio of a turbine's frontal area to the local flow passage cross-sectional area), $B_L$, are defined as

(2.1a,b)\begin{equation} B_G = \frac{n A}{w h}, \quad B_L = \frac{A}{(s+d)h}, \end{equation}

with $A$ the swept area of a single turbine's rotor, $w$ the channel width, $h$ the depth and $n$ the number of turbines. We adopt the definition of local blockage from Nishino & Willden (Reference Nishino and Willden2012) for consistency with previous works. Note that this definition takes no account of the aspect ratio of the local flow passage nor of anisotropy at turbine fence ends. Noble et al. (Reference Noble, Draycott, Nambiar, Sellar, Steynor and Kiprakis2020) presented details on the current generation properties of the test facility, demonstrating that the circumferentially placed pumps cause the mean flow path lines to contract with increasing curvature towards the tank sides as illustrated in figure 2. This has the effect of narrowing the effective width, $w$, of the flow passage through the rotor plane, which has been estimated previously by Gaurier et al. (Reference Gaurier2020) as 15 m. The effective width was evaluated computationally by Cao (Reference Cao2020) through comparison of simulated loads in rectangular cross-section channels of varying width to the present experiments, indicating that the effective rectangular channel width of the tank was 12 m for these tests. Using this value, the global blockage for the twin case (9.4 %) is twice that of the single rotor (4.7 %). The local blockage for the twin case with a 1/4 diameter spacing is 38 %. This relatively high level of local blockage was chosen to be close to 40 %, as the partial fence model of Nishino & Willden (Reference Nishino and Willden2012) shows this to deliver optimum performance for very long turbine fences in low levels of global blockage. For shorter fences, Nishino & Willden (Reference Nishino and Willden2013) showed increasing the number of turbines has a significant effect on potential performance benefits, with a very long fence being beyond the 40 turbines simulated. For lower numbers of turbines, they suggested a lower optimal local blockage is required to achieve maximum power. The quarter diameter spacing used in these experiments matches the optimal spacing of Nishino & Willden (Reference Nishino and Willden2013) for a twin turbine fence, although their numerical simulations used a diameter to depth ratio of 0.5 rather than our value of 0.6 and, hence, our resulting local blockage is higher.

2.4. Flow measurements

Flow measurements at the facility were made using Nortek Vectrino acoustic Doppler velocimeters (ADVs), which use four beams to sample three velocity components over a control volume of around $1\ {\rm cm}^3$ at 100 Hz. Details of the ADV processing are given in Appendix B. Five minutes of data were found to be sufficient to achieve stationarity of the mean signals, with further details provided in Appendix A.

For flow mapping without the turbines, two ADVs were used to measure the flow across three streamwise planes located at the rotor plane and at 2$d$ and 3$d$ upstream of it. The ADVs were mounted on a frame with a fixed lateral separation of 600 mm (${=}d/2 = 2s$). The frame was mounted to the carriage gantry and movable in the vertical and lateral directions, whereas the gantry's streamwise movement allowed for positioning between the planes. Measurements were taken on a grid of 12 cross-stream points and four vertical points, with measurement spacing equal to the turbine separation $s$ and aligned with the turbine centrelines as shown in figure 3. Although lateral motion was unrestricted, the maximum depth that the ADVs could reach was 1.3 m below the water surface, which corresponds to 75 % of the depth across the rotors (the rotors were placed mid-depth giving a tip submersion of $d/3=0.4$ m).

Figure 3.

Measurement points (crosses) taken in streamwise planes in the absence of turbines at the rotor plane and $2d$ and $3d$ upstream of it. The solid circles indicate the turbine outlines when installed. Dashed lines indicate division of a rotor area into sub-areas for calculation of the power-weighted rotor average speed.

With the turbines installed, flow measurements were only possible with one ADV (corresponding to ADV 1 in Appendix B), with a set-up similar to above for lateral and streamwise positioning. All flow measurements with turbines installed were at turbine hub height (also the mid water depth). For the majority of the flow mappings a fixed speed of 77.5 rpm was used, which for a reference flow speed of $0.8\ {\rm m}\ {\rm s}^{-1}$ leads to a nominal tip-speed ratio (TSR) of $\lambda ^*=6.1$. A selection of flow measurements were also made upstream of the single turbine with no resistance applied by the generator, allowing the rotors to free-wheel with only rotor drag and mechanical resistance from the drive-train components.

3.1. Undisturbed flow

Time-averaged contours of velocity magnitude and turbulence intensity are shown in figure 4 for the three planes considered without the turbines present. Although velocity shear associated with the vertical boundary layer in the tank is expected, a substantial variation in the lateral direction also occurs. The flow accelerates between the 3$d$ upstream and rotor planes, caused by the narrowing of the effective width of the flow as depicted in figure 2. Recalling that both the 2$d$ and 3$d$ planes are upstream of the tank's mid-plane, this acceleration suggests the effective width of the channel narrows until at least this location.

Figure 4.

Contour plots of (a) velocity magnitude (${\rm m}\ {\rm s}^{-1}$) and (b) turbulence intensity (%) on the rotor plane and planes 2$d$ and 3$d$ upstream of it. The dashed circles indicate the projected rotor locations. Measurements were made without turbines installed. Note inverted $x$-axis of plots so that the left-hand circle is the South turbine to maintain consistency with figure 2(b). Flow direction is then into the page.

To normalise the performance metrics, a reference flow speed is required that represents the flow that will travel through the rotor plane. Once installed, turbines influence the approach flow and so the reference flow speed should be measured far enough upstream of the rotor plane that this effect is minimal. IEC (2013) recommends that the reference flow speed is measured between 2$d$ and 5$d$ upstream of the rotor plane and is a rotor-equivalent flow speed. This accounts for the power available in the flow through the cube-root-mean-cube of the velocity magnitude, and shear over the rotor plane through an area-weighted average of multiple measurements. As the undisturbed flow at FloWave is shown to vary both laterally and vertically, we compute a rotor-equivalent flow speed to account for this, following an approach close to that of IEC (2013). With reference to figure 3 and letting $\boldsymbol {u}_{\boldsymbol {i}}$ be the instantaneous velocity vector at grid point $i$, the rotor-equivalent flow speed is calculated as

(3.2)\begin{equation} U_j = \frac{1}{A} \sum_{i \in \mathcal{P}_j} A_i \overline{\lVert\boldsymbol{u}_{\boldsymbol{i}}\rVert^3}^{{1}/{3}}, \end{equation}

the subscript $j = N$ or $S$ is the North or South rotor, respectively, and $\mathcal {P}_j$ is the envelope of the projected rotor $j$. Although the difficulty in measuring the rotor-equivalent speed in tidal environments is well documented (see, e.g., McNaughton et al. Reference McNaughton, Harper, Sinclair and Sellar2015; Starzmann et al. Reference Starzmann, Jeffcoate, Scholl, Bischof and Elsaesser2015; Harrold, Ouro & O'Doherty Reference Harrold, Ouro and O'Doherty2020), controlled laboratory environments provide an opportunity to measure this with high confidence due to stationarity of the flow. Although this choice of reference flow speed maintains consistency with IEC (2013) it is recognised that a root-mean-square may be more appropriate for normalising of forces. We observe that for our measurements the ratio between $U_N$ and $U_S$ varies by less than 0.1 % if they are computed by either of these methods and the physical values change by less than 0.5 %. As a result we choose to use the cube-root-mean-cube for calculating the equivalent flow speed (3.2) for normalising of both power and forces. We also define the array's reference flow speed $U_0 = (U_N + U_S)/2$, which is used in the following sections to normalise the flow field plots. Values for these reference speeds are provided in table 1.

Table 1.

Values of rotor equivalent flow speed and vertical and horizontal velocity variations on the projected rotor areas 3$d$ upstream of the rotor plane.

The flow at the rotor plane is necessarily affected once the turbines are installed. Undisturbed rotor plane measurements are useful in understanding the background flow at the test facility, although once installed, the turbines modify the flow through the rotor plane from that shown in figure 4. For experiments using high blockage ratios or streamwise flow variation, such as those we present in this paper, we believe a suitable reference flow speed must therefore be defined upstream of the rotors at a distance such that rotor interference is minimised. Flow measurements made with the turbines installed demonstrated that the flow at 2$d$ is much more affected by the turbines operation than at 3$d$ upstream; this is discussed in the following section and in Appendix B. For this reason, the corresponding turbine's value for $U_j$ from the 3$d$ upstream rotor plane is used.

The shear over each of the rotors is considered in terms of the relative vertical and horizontal velocity shear over the rotor, respectively:

(3.3a,b)\begin{equation} \gamma_z = \frac{d}{U_j} \left\lvert\frac{\partial u_x}{\partial z}\right\rvert \approx \frac{d}{U_j} \left\lvert\frac{\Delta {u_x}|_z }{\Delta z}\right\rvert \quad\mathrm{and}\quad \gamma_y = \frac{d}{U_j} \left\lvert\frac{\partial u_x}{\partial y}\right\rvert \approx \frac{d}{U_j} \left\lvert\frac{\Delta {u_x}|_y }{\Delta y}\right\rvert \end{equation}

where $\Delta y = d$ and $\Delta z = 3d/4$ following the range of measurement points and $\Delta {u_x}|_i$ is the difference of the streamwise velocity component evaluated at the extremities of $\Delta i$. Values for these velocity shears on the plane 3$d$ upstream of the rotor plane are provided in table 1, where we note the horizontal flow is greater towards the tank edges. In addition to higher overall rotor equivalent speed, the area that the North rotor will occupy experiences 22 % (horizontal) and 30 % (vertical) greater velocity shear than that of the South turbine. This infers that the North turbine's blades will experience greater flow variations and likely lead to higher load fluctuations as discussed in § 5.3.

The turbulence intensity is lower and more uniform over the South turbine's rotor plane. As with the velocity, we observe streamwise variation in both the magnitude and spatial uniformity of turbulence intensity between planes. This demonstrates a complex spatially varying flow that should be accounted for when analysing turbine performance and loading in this tank.

Variations in the approach flow may be associated with the curvature in flow through the tank as shown in figure 2 and were observed by Noble et al. (Reference Noble, Davey, Smith, Kaklis, Robinson and Bruce2015) to reduce with increasing flow speed. It is unclear why these differences in approach speed occur but they may be due to non-uniformity of flow through the tank's pumps or mixing between these flows. For our tests it can be considered that the flow is primarily formed from the output of three (around 2 m diameter) flow drive units. These use low-solidity propellers (with associated reduction in swirl) and rotate in the same direction, potentially giving localised mean flow variation (as observed by Germain Reference Germain2008).

3.2. Turbine flow

Figure 5 demonstrates how the streamwise velocity component, turbulence intensity and streamwise integral length scales change along the axes corresponding to the turbine centrelines without the turbines installed and when installed at a range of operating points. As discussed previously, flow acceleration is visible when there are no turbines, although once installed, the flow speed reduces as it approaches the rotor plane. Although the background acceleration is still observed between 3$d$ and 2$d$ upstream of the North and South rotors, the actual flow speeds are reduced due to the turbines. After 2$d$ upstream the turbine resistance effect dominates over the background flow and the flow decelerates as it approaches the rotor plane. For the single turbine, the free-wheel (high thrust, high rotational speed and low torque) leads to a far more significant reduction in flow speed than the design point (low thrust, low speed and high torque) demonstrating the importance of thrust on upstream flow.

Figure 5.

Assessment of streamwise velocity, turbulence intensity and streamwise length scale at distance $x$ upstream of rotor plane along turbine centreline.

There is a general streamwise decrease in background turbulence intensity without turbines present. When the turbines are operating there is a slight uplift in turbulence intensity in front of the rotor plane, which is amplified in the case of freewheeling where thrust and flow deceleration are maximised. Note that no measurement was taken at $1d$ upstream of the single rotor at the design point so that no conclusion can be drawn about whether this uplift occurs in this case. We associate the high turbulence intensity upstream of the North rotor with increased cross-stream shear in the approach flow speed at all distances upstream. As turbine thrust increases, so does the velocity difference and therefore shear between the array bypass and through-turbine flows. Increased shear yields an increase in turbulence intensity when turbines are present, with this effect further amplified at maximum thrust.

The streamwise integral length scales exhibit a generally decreasing streamwise trend with and without the turbines present, although there is a large degree of variation in the data. However, we consistently observe that the length scales upstream of the South rotor are two to three times smaller than those approaching the North turbine, which may affect the resulting loads. The upstream length scales are relatively large as a proportion of turbine diameter, although not unrealistic for tidal turbines in the field, where length scales are highly site dependent. Here we observe a maximum $L_{11}/h \approx 0.35$ whereas Milne et al. (Reference Milne, Sharma, Flay and Bickerton2013) report a maximum $L_{11}/h \approx 0.23$ for field measurements of a tidal channel with relevant flow characteristics for power generation, although their measurements were lower in the water column at about one-third of the depth above seabed. Sutherland et al. (Reference Sutherland, Noble, Steynor, Davey and Bruce2017) reported variations in the streamwise integral length scales at various locations at FloWave, up to a maximum of $L_{11}/h \approx 0.25$. The reason for this value being lower than we observe is a consequence of spatial variations in flow.

To understand the spatial variation of the flow through the turbine array, flow measurements were taken at various locations for the single and twin cases operating at the design point. Flow vectors indicating the flow speed and direction are shown in figure 6. The acceleration around the sides of the array is observed for both single and twin configurations, with differences in the single and twin array bypass flows appearing slight (view the overlaid vectors on the north side of the array); differences are discussed quantitatively later through the transects plotted in figure 7. Significant acceleration is observed between the two rotors, demonstrating the flow straightening and jetting effect that is expected between turbines in a closely spaced multi-turbine array (Nishino & Willden Reference Nishino and Willden2013). Streamwise flow 2$d$ upstream of the rotors is also presented in figure 6, demonstrating the lateral variation in the approach flow as well as the variation in this from single to twin configurations. The South turbine experiences a 3.5 % lower rotor-equivalent flow speed than the North, leading to the unintended consequence that the turbines will not experience the same TSR when controlled to operate at the same rotational speed. This demonstrates the importance of performing detailed flow measurements upstream of turbines in such test configurations.

Figure 6.

Flow vectors for the single and twin turbine configurations operating at the design speed. Vectors are scaled so that each grid spacing corresponds to $U/U_0 = 1$. Solid and dashed lines are used to show the cross-stream variation of the flow speed $2d$ upstream of the rotor plane for the single and twin turbine configurations, respectively, with numerical values indicated by the $U/U_0$ scale on the upper axis.

Figure 7.

Streamwise transects of flow velocity (a) and turbulence intensity (b) at different cross-stream positions. (c) and (d) Transect positions for the single and twin configurations, respectively, with markers indicating measurement locations and labels describing distance between transect and turbine/array outboard edge.

A more detailed comparison of the streamwise flow speed and turbulence intensity is provided in figure 7, where streamwise transects of the variables are presented at different cross-stream locations running through the array. Where transects were taken off both the north and south sides of the fence the values were not dissimilar and have been averaged without affecting the analysis. The traverse at 1/8$d$ for the single turbine intersects with the rotor's wake and shows a clear flow deceleration downstream of the turbine, whereas the centreline traverse in the case of the twin turbines (also at 1/8$d$ from both turbines) shows a significant initial acceleration of the flow as it passes between the turbines, which is followed by a more modest deceleration of the flow indicating a lack of internal expansion of the turbine wakes. The change in bypass flow at 1/4$d$ and 1/2$d$ off the sides of the array is moderately affected by the change from a single to a twin turbine fence, with the twin fence maintaining a stronger bypass flow downstream of the turbines, which is consistent with the greater overall resistance presented by two turbines. It is also noted that whereas the turbulence intensity is different upstream of the rotors for the different configurations, downstream of the rotor plane there is little difference in the values for single and twin arrays at the $1/4d$ and $1/2d$ transects.

4.1. Normalised quantities

Data analysis is performed using integrated quantities for the TSR, thrust coefficient and power coefficient. These are defined respectively as

(4.1a–c)\begin{equation} \lambda = \frac{\omega d}{2 U_j}, \quad C_{T} = \frac{T}{\tfrac{1}{2} \rho U_j^2 A}, \quad C_{P} = \frac{\omega Q}{\tfrac{1}{2} \rho U_j^3 A}, \end{equation}

with $T$, $Q$ and $\omega$ the time averaged rotor thrust, torque and rotational speed, respectively, and $\rho$ is the fluid density and subscript $j$ indicates the North or South rotor following (3.2). Similarly, the flapwise (FW) and edgewise (EW) root bending moments (RBMs) are non-dimensionalised as

(4.2a,b)\begin{equation} C_{FW} = \frac{M_{FW}}{\tfrac{1}{2} \rho U_j^2 A {(d / 2)} }, \quad C_{EW} = \frac{M_{EW}}{\tfrac{1}{2} \rho U_j^2 A {(d / 2}) }. \end{equation}

Bending moment coefficients are presented in terms of phase averages, which have been calculated by binning the loads according to azimuth angle in 5$^\circ$ increments. The loads from all three blades are combined by applying an angular offset and then averaging (note the South turbine's EW RBM contains data from only two blades owing to an error in the amplifier saturating the signal on the third blade). For the EW bending moment the self-weight of the blade is subtracted according to azimuth position using the centre of mass and its value from the CAD file of the blade. Depending on the TSR there are 300–500 revolutions of data in each phase average, with three times this amount of blade data due to the combining of each blade's data.

Prior to phase averaging, an additional pre-processing calibration of the blade RBM transducers was performed to remove sensitivities from bolt tightening and drift described in Appendix C.

4.2. Blockage considerations

This study seeks to investigate the performance benefits that can be obtained through constructive interference effects, i.e. local blockage. A consequence of arraying an additional turbine laterally in the tank is to simultaneously increase the global blockage for the twin turbine configuration. Ideally, a global blockage correction would be applied to all data so that the effects of the finite width of the tank can be eliminated in isolation. However, only single-scale blockage corrections have thus far been developed and the application of such corrections would attempt to remove all blockage effects, including the sought constructive interference effects, from the experimental results.

To account for global blockage, we follow the correction of Bahaj et al. (Reference Bahaj, Molland, Chaplin and Batten2007) which is briefly summarised in Appendix D. Prior studies comparing blockage corrections have shown it to perform well both experimentally (Ross & Polagye Reference Ross and Polagye2020) (note Ross & Polagye (Reference Ross and Polagye2020) reference this correction to Barnsley & Wellicome (Reference Barnsley and Wellicome1990) who applied the correction to wind turbine data) and numerically (Zilic de Arcos, Tampier & Vogel Reference Zilic de Arcos, Tampier and Vogel2020) for unblocking tidal turbine datasets. The premise of the blockage correction is that there exists an equivalent unblocked domain in which an alternative upstream flow, $U_F$, exists that results in the same flow speed through the turbine, $U_1$, and resulting thrust, $T$, as occur in the blocked domain in which the approach flow is $U_A$. The objective of the blockage correction is to develop the velocity ratio $U_F/U_A$ so that performance data can be corrected and related to the equivalent unblocked domain. Replacing the system of (D1)–(D4) and (D5) (see Appendix D) by the two functions respectively:

(4.3a,b)\begin{equation} \frac{U_1}{U_A} = f_B\left( B_A , C_{T,A} \right), \quad \frac{U_F}{U_A} = f_F\left( \frac{U_1}{U_A} , C_{T,A} \right) \end{equation}

where given a blockage ratio $B_A$ and thrust coefficient $C_{T,A}$, $f_B$ provides the ratio $U_1/U_A$ between the flow speed through the turbine and the undisturbed upstream flow for the blocked domain and $f_F$ provides the ratio $U_F/U_A$ between the undisturbed upstream flow for the unblocked and blocked domains. The performance coefficients for the blocked domain $A$ are then scaled to their unblocked equivalents using the following relations:

(4.4a–c)\begin{equation} \lambda_F = \lambda_A \frac{U_A}{U_F} , \quad C_{T,F} = C_{T,A} \left( \frac{U_A}{U_F} \right)^2 , \quad C_{P,F} = C_{P,A} \left( \frac{U_A}{U_F} \right)^3 , \end{equation}

with subscripts $F$ denoting the unblocked freestream equivalent coefficients. We note that such a blockage correction is based on the idealised flow assumption of uniform flow which is not the case in our experiments and that further uncertainty arises from the inhomogeneous undisturbed flow field in FloWave, as detailed in § 3.1.

Such a correction cannot be used to just remove the global blockage effects in the twin turbine case as local and global effects are nonlinearly interwoven. Rather we choose to invert the blockage correction for the single turbine dataset to find what the performance would be if the single turbine was operating in domain $B$ with blockage $2B_G$ and approach flow $U_B$ by finding the upstream velocity ratio between these two blocked domains:

(4.5a,b)\begin{equation} \frac{U_B}{U_A} = \frac{ f_B\left( B_G , C_{T,A} \right) } { f_B\left( 2 B_G , C_{T,B} \right) }, \quad C_{T_B} = C_{T,A} \left( \frac{U_A}{U_B} \right)^2 \end{equation}

which may be solved by iteration. The correction of the performance coefficients to the blockage-increased domain is the same as in (4.4a–c) with the updated velocity ratio found above.