Experimental setup and characterization of incoming flows
All experiments were carried out at the University of Edinburgh’s FloWave Ocean Energy Research Centre, a circular combined wave-current tank designed for large-scale testing of offshore renewable energy devices. The reservoir has a diameter of 25 m and, for the current campaign, was operated in current mode only with a working water depth of 1.92 m. A schematic of the installation is shown in Figure 6a of the extended data. The experiments were carried out at six inflow velocities in the range You∞ = 0.444 to 0.923 ms−1corresponding to a Reynolds number based on the diameter,
$$\mathrm{Re}\equiv \frac{{u}_{\infty }d}{\nu },$$
(2)
from 0.49 × 106 at 1.16 × 106Or ν is the kinematic viscosity and d is the diameter of the turbine. See Supplementary Note 4 for a discussion of Reynolds number scaling.
FloWave turbulence is between 8% and 9% and integral length scales approximately 25% of water depth28. These turbulent features are consistent with energetic tidal sites such as, for example, Warness Falls.29. There, the turbulence intensity is between 5% and 11%, with integral length scales of approximately 30–50% of the channel depth. The average vertical shear in FloWave is approximately a power law with exponent 1/12 (Extended Data, Fig. 6b), which is only slightly weaker than the 1/10 power law profile commonly used to represent tidal flows.30. Extended Data Table 1 summarizes the comparison between FloWave and representative tide sites.
Turbine and blades
A three-blade horizontal axis turbine with a rotor diameter d = 1.26 m was installed in the center of the tank, at the height of the hub zhub = 0.876 m. The turbine is equipped with a brushless DC motor and a 25:1 planetary gearbox. The main components of the turbine are depicted in Figure 7 of the extended data.
The results presented in the main text refer to four sets of blades: one set of rigid blades used as a reference and three sets of rubber blades. The base rigid blades are CNC machined (SS316) using a modified NACA 63-415 hydrofoil profile along the span. The profile has been modified to ensure a blunt trailing edge, as specified in ref.31. These blades had negligible deformations at the flaps and edges. Notably, the blades of large-scale tidal turbines exhibit some structural conformity. However, here the model-scale rigid blade has been deliberately made rigid to provide defined geometry.
The main text presents results for rigid blades and a set of rubber blades with optimal stiffness. Conversely, the extended data figures also include results for the other two sets of rubber blades, with both higher and lower stiffness than optimal. The flexible blades were manufactured by injecting a proprietary rubber blend into a mold and around a steel spar. All rubber blades have the same mold geometry. See extended data,Fig. 8, for diagrams of the rubber blade. The aeroelastic properties of the blade are adapted along its span by varying the ratio between the width of the rigid spar and the length of the local chord, which increases from tip to root with an average of 45%.
Instrumentation, data acquisition and uncertainty
Thrust and torque are measured by a force transducer installed between the turbine rotor and the motor shaft (Extended Data Fig. 7). The transducer is custom-made, manufactured by Applied Measurements, rated for 1,300 N axial load and 100 N·m torque, and resolution of 0.005 N and 0.0004 N·m, respectively. In this study, thrust is the net force measured by the hub transducer in the flow direction, and torque is the net moment in the edge direction around the hub axis. Additionally, each blade root was fitted with a custom root bending moment sensor in the flap direction. Calibration matrices for each sensor were obtained in situ prior to testing by applying known weights at four points along the cantilever beam attached to the sensor. All force signals were transmitted to a National Instruments NI 9203 acquisition system via a slip ring. The load cells were reset to zero without flow before each series of measurements. For each measurement, loads were recorded at a sampling rate of 128 Hz for 300 s. The angular speed of the rotor was measured using an inductive proximity sensor which emitted one pulse per blade passage (three pulses per revolution), making it possible to determine the period of revolution and calculate the angular speed.
The main quantities measured were the torque (Q), thrust (T), bending moment of the root in the direction of the flap (M) and the angular velocity (ω). The measured loads and power are not dimensioned using standard turbine coefficients:
$${C}_{P}\equiv \frac{Q\omega }{\frac{1}{8}\rho {u}_{\infty }^{3}\pi {d}^{2}},$$
(3)
$${C}_{T}\equiv \frac{T}{\frac{1}{8}\rho {u}_{\infty }^{2}\pi {d}^{2}},$$
(4)
$${C}_{M}\equiv \frac{M}{\frac{1}{8}\rho {u}_{\infty }^{2}\pi {d}^{2}}.$$
(5)
The top speed ratio is defined as
$$\lambda \equiv \frac{\omega d}{2{u}_{\infty }},$$
(6)
Or ω is the angular speed of the turbine in radians per second.
Sensor accuracy and calibration uncertainties were propagated to non-dimensional coefficients using standard bias limit uncertainty analysis.32including thrust torque transducer manufacturer’s specifications (0.1% of full scale capacity) and key uncertainties of You∞, ω and geometric parameters. For the current configuration, the maximum total bias limits are around 10.−2 For CP. and of order 10−2 For CT (representative values). Supplementary Note 1 provides the complete derivations and tables.
DIC and blade shape reconstruction
Blade deformation was measured using a phase-locked DIC with underwater cameras and LED illumination (LaVision). The camera system had a fixed focal length of 1.1 m and a field of view of approximately 0.87 m × 0.56 m, sufficient to capture the full span of the blade (0.56 m). Phase locking was achieved by using the proximity sensor signal to trigger image acquisition when a blade passed a fixed azimuthal position, with horizontal span and downward tangential velocity. For each condition, between 89 and 447 instantaneous strain fields were recorded over a period of 300 s, depending on You∞ And λ.
Images were processed in LaVision DaVis 11 using a subset size of 21 × 21 pixels and a grid pitch of 2 × 2 pixels, resulting in a spatial resolution of approximately 0.68 mm near the blade tip (approximately 2% of the tip chord). A reference image in still water was acquired before each series of measurements. In the current configuration, the DIC was used to measure the deformation of the surface on the extrados side of the blade. The complete blade profile was reconstructed by combining the measured chord deformation with the known undeformed thickness distribution.
Fluid-structure interaction simulations
A tailor-made fluid-structure interaction (FSI) solver was developed for the present study. The solver combines in-house code based on blade element moment theory (BEMT), a panel method with an integral boundary layer formulation (XFOIL) and commercial finite element analysis software (Abaqus). The code uses a two-level iterative process, as shown schematically in Figure 9 of the extended data. The process begins with an initial geometry of the blade. In the outer loop, the BEMT solver is used to calculate axial and tangential induction factors over a finite number of blade sections at different radial coordinates. Then, the flow velocity and angle of attack experienced by each blade section are calculated. These values are used as initial conditions for the FSI inner loop. First, the pressure distributions around each blade section are calculated with XFOIL. Then, the pressure distributions of each blade section are combined to obtain a three-dimensional pressure distribution on the blade, which is used as input to Abaqus, in which a new deformed shape is calculated. The Abaqus model includes the steel spar. This new shape is then discretized into sections and sent back to XFOIL to recalculate the pressure field. This iterative process continues until the blade shape converges. This equilibrium deformed blade geometry is then fed back into the external BEMT loop to update the induction factors. This entire two-level process is repeated until convergence. The power and thrust coefficients of the entire turbine are then calculated by integrating the loads calculated with XFOIL for each blade section and multiplying by three to account for the three-blade turbine.
This code was used to design the unconstrained mold shape for each blade stiffness, ensuring that the equilibrium blade shape matched the reference rigid blade at the design flow velocity and tip speed ratio. The model-scale simulations in Extended Data Figs. Figures 4 and 5 are initialized with the mold geometry and use the actual stiffness of the metal spar and rubber. In the large-scale simulations (extended data, Fig. 5), the stiffness of the spar and rubber is 10.6 times higher than that at the model scale to ensure that the model and large-scale Cauchy number for the spar and rubber are the same, where
$$\mathrm{Ca}\equiv \frac{\rho {u}_{\infty }^{2}}{E},$$
(7)
Or ρ is the density of the fluid and E is Young’s modulus.
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