Impact of Environmental Turbulence on the Performance and Loadings of a Tidal Stream Turbine

The method of Large-Eddy Simulation (LES) is employed to compute the flow around a horizontal axis tidal stream turbine using the in-house code Hydro3D, which has been successfully applied to the analysis of complex problems such as flow over vegetation [41, 42] or flow over dunes [43]. Hydro3D resolves the spatially-filtered three-dimensional incompressible Navier-Stokes equations which read, where u and p are the fluid velocities and pressure, ν and ν_t are the viscosity of the fluid and that added by the sub-grid scale model, and f is the source term from the IB method that enforces the no-slip condition at the location of the IB markers [44]. First and second order velocity derivatives are approximated using fourth-order central differences. The fractional-step method is used to advance in time these equations [45] using a low-storage third-order Runge-Kutta algorithm e.g. [46]. The Poisson-type pressure equation is solved using a multigrid approach which provides a scalar field onto which the predicted velocities are projected in order to fulfill the divergence-free condition. The sub-grid scale (sgs) stress tensor is approximated using the WALE model from Nicoud and Ducros [47] which does not explicitly calculate the sgs-viscosities near the wall and hence adapts well to its use in conjunction with moving IB bodies [48].

A direct forcing Immersed Boundary (IB) method originally developed by Uhlmann [44] is implemented in Hydro3D, in which the representation of solid boundaries within the computational domain is accomplished by the discretisation of the geometry in a set of regularly spaced Lagrangian markers constituting the solid domain [11, 48]. This Lagrangian grid is detached from the fixed fluid mesh and the interpolation of quantities between fluid and solid domains is accomplished via delta functions [49]. The immersed boundary method is a compromise between numerical accuracy and stability, as moving boundaries in highly turbulent flows pose a challenge to any numerical method. And whilst this method is unable to resolve explicitly the viscous part of the laminar boundary layer developing over the hydrofoil, and hence is unable to resolve the laminar-to-turbulent boundary layer transition, it nevertheless proved to be accurate enough when employing adequate grid spacings and time steps. This LES-IB method has been validated in complex fluid-structure interaction applications such as vortex-induced vibrations [50], dynamic stall in pitching airfoils [51], or flow around vertical axis tidal stream turbines [48]. Most relevant to this study is the validation of the current method for the horizontal axis tidal turbine operating over a flat bed by Ouro et al. [11], which is exactly the same turbine as the one to be studied herein.

Hydro3D uses a domain decomposition technique to divide the computational domain into rectangular sub-domains on which the equations are solved using multiple processors and these communicate via Message Passing Interface (MPI) protocol. Cevheri et al. [52] implemented a local mesh refinement method that allows to refine the fluid mesh in certain areas of interest, e.g. near an immersed solid, while coarser grid resolution can be used far from these thereby reducing the computational load of the LES.

The simulated horizontal axis tidal turbine is the discrete model of a 1/30th-scale prototype comprises three blades and is presented in Fig. 2. The rotor radius R is 0.2 m measured from the centre of the rotor and the radius of the hub r is 0.05 m. A model of the turbine was tested in Cardiff University’s hydraulic flume of rectangular cross-section and flat bottom. The Reynolds number based on the chord length, c and freestream velocity, U₀, is Re_c ≈ 60,00, which is the same as in Ouro et al. [11].

The extracted power of a tidal stream turbine is often obtained for a wide range of rotational speeds or tip speed ratios respectively calculated as λ = ΩR/U₀, where Ω is the turbine’s rotational speed. The optimum rotational speed of a tidal turbine is achieved when power extraction is maximum and in the simulations presented here the tip speed ratio is λ ≈ 3, which is very similar to the expected tip speed ratio of the prototype. The dimensionless performance of the device is computed as the ratio of the power generated by the turbine, P_T to that available in the water, P_W, represented by the power coefficient C_P and defined here as, where Q is the torque generated by all blades, ρ is the fluid density and A = πR² is the area swept by the rotor.

The flap-wise bending moment is the main contributor to the structural stresses at the blade-hub juncture as its values are generally 3–4 times higher than those from edge-wise bending moments [11]. The forces in the streamwise direction are denoted thrust forces (T) and are responsible for the flap-wise bending moment \({M_{\theta }}^{T}\) whilst the tangential force Q generated by the blades is responsible for the edge-wide bending moment \({M_{\theta }}^{Q}\). Following [11], the coefficients of these two structural moments are defined as,

Even though the chord Reynolds number of the full-scale prototype is approximately 1–2 magnitudes greater, the lift and drag coefficients (and hence the dimensionless coefficient of power and bending moments) of the simulated turbine are expected to be very similar to the prototype due to the fact that the approach flow is highly turbulent and thus the effects of turbulent boundary layer transition are deemed marginal.

Figure 6 presents iso-surfaces of instantaneous negative pressure fluctuations (\(p^{\prime }=p-P\), where p and P are the instantaneous and time-averaged pressures,respectively) and these are coloured with time-averaged streamwise velocity. Minima of the pressure fluctuation are reliable indicators for vortical motion (e.g. [43, 53, 56]), and hence allows visualising large-scale turbulence structures of this flow. The flow separates at the crest of the dune and generates a shear layer which, followed by Kelvin-Helmholtz (KH) instabilities, leads to the formation of elongated spanwise vortices denoted as KH rollers. These can grow up to the size of the dune’s height and due to the high streamwise velocity are subsequently convected downstream [56]. Along the boundary layer recovery over the stoss side of the dune, Hair-Pin (HP) vortices are similar to those observed by Stoesser et al. [43], and these move upwards interacting with the overlying high-momentum flow and eventually forming Vertical Rollers (VR) that approach the turbine as observed in Fig. 6. Other secondary flow structures, such as streamwise-oriented (S) vortices, are also developed.

The instantaneous flow field behind the turbine is depicted in Fig. 7 in a longitudinal plane at y = y_hub, at half width of the domain, with contours of u- and v-velocities and instantaneous turbulent kinetic energy, \(k^{\prime }\). Figure 7 allows to identify the main turbulent structures generated by the turbine and the interaction between the wake of the turbine and the near bed flow structures in the trough downstream of the turbine. A recirculation region behind the hub is identified by the streamwise velocity contours of Fig. 7a. Two low-momentum areas are marked in Fig. 7a and these correspond to the recirculation in the dune’s trough and in the hub’s wake in the region x/D = 1.5 − 4. There is an area of high-momentum flow between the turbine’s lower tip and the dune crest. This high-velocity area extends until x/D = 2 − 3 which is where the interaction between the turbine’s and dune’s wakes takes place and which is marked as an oval in Fig. 7b. Between the turbine’s top tip and the free-surface there is another region of high-velocities that interacts with the turbine’s wake.

Top and bottom tip vortices are recognised from the instantaneous spanwise velocity (v/U₀) contours of Fig. 7b. These energetic flow structures are generated by the shear induced by the blades’ tip into the incident flow. The blades rotate with positive v-velocity when moving above the hub’s centre (z > z_hub) while with negative v-velocity for z < z_hub. This motion induces the fluid flow to react with an opposite distribution of transversal velocities with negative values over the hub’s centre and positive below it. Contours of v-velocity also show the interaction between the wakes generated by the turbine and the second dune. The mixing of the wakes starts approximately after x/D = 2, and this is more noticeable between x/D = 2.5 − 4.0. Noteworthy is the fact that the turbine wake does not exhibit the same characteristics as the wake downstream of a tidal stream turbine placed on a flat bed, where the far wake starts to exhibit a spiraling motion e.g. [12] and [11].

Contours of the instantaneous turbulent kinetic energy (\(k^{\prime }\)) are presented in Fig. 7c and these depict secondary turbulent structures derived from the interaction of the wakes. The area of greatest \(k^{\prime }\) is found in the dune’s trough as flow reattachment triggers turbulence structures of high turbulence intensity levels (see Fig. 5). The operating turbine introduces significant turbulent kinetic energy by the blades, their tips and the hub.

A three-dimensional view of the turbine-generated turbulence structures is presented in Fig. 8 using iso-surfaces of λ₂ to educe coherent turbulence together with contours of relative pressure, p, in a longitudinal plane in the channel centre. λ₂ is the second eigenvalue of the sum of the symmetric and antisymmetric parts of the velocity gradient tensor and a vortex core is identified by an isosurface of a negative λ₂. The signature of the tip vortices is clearly visible in the form of low-pressure areas (marked as 1c, 2c etc.) as well as the hub’s wake, which starts meandering downstream of the turbine as identified from isosurfaces of a chosen negative pressure fluctuation (very similar to the one shown in [11] and not shown for brevity). The tip vortex denoted as “1a” corresponds to the latest vortex generated by blade 1, or tip vortex denoted “2c” is one that has been shed by blade 2. Shortly downstream of the turbine, bottom tip vortices break down and loose coherence due to their interaction with the ambient flow as just vortices 1a, 2a and 3a are visible from the pressure contours, and vortex 1b is also appreciated but appears to be segregated into 2 cores. Tip vortices near the water surface remain coherent during their advection downstream although the spacing between consecutive vortices is irregular due to their interaction with the turbulent ambient flow, e.g. separation between vortices 2b–3b is larger than between 1c and 2c. The λ₂ iso-surfaces show that the characteristic roller shape of the tip vortices [11] is lost in vortex 1b whilst vortex 3a (which is closer to the rotor) is smoother and more compact. Such a loss of coherence is due to their interaction with the turbulent ambient flow. Chamorro et al. [57] visualised experimentally the wake of a HATT using PIV records and also observed that tip vortices feature irregularly spaced patterns in highly turbulent flow.

The time-averaged flow field in a longitudinal plane is presented in Fig. 9 with contours and vectors of normalised streamwise velocity (a) and turbulent kinetic energy k (b). Streamwise velocities increase on the stoss side of the upstream dune with the highest velocities found at the dune crest just upstream of the turbine. The turbine extracts energy contained in the approach flow inducing an area of low momentum immediately downstream of it. Outside of the turbine’s swept area the flow is accelerated generating high-momentum zones above and below the turbine’s blade tips, the locations of which are indicated with dotted lines in Fig. 9. There is a low-pressure area in the dune’s trough and the downwards pointing velocity vectors between 1 ≤ z/k ≤ 2 at 1 ≤ x/D ≤ 2 in Fig. 9a indicate that the fast-moving flow found below the turbine’s swept area starts to fill this area.

Three wake regions are discerned in Fig. 9: The near-wake extends between 0 < x/D < 1.5 featuring a short recirculation behind the hub. The low-velocity region covering 1.5 < x/D < 4 (blue areas in Fig. 9a) is the characteristic signature of the mid-wake region. Compared to the flat-bed channel results of Ouro et al. [11], see Fig. 10, the presence of an irregular seabed causes the wake to be asymmetric in the vertical direction. The flow decelerates above the dune’s trough and the boundary layer deficit downstream of the dune’s recirculation zone begins to be filled with high-momentum fluid and hence the turbine’s wake is skewed towards the dune trough. Figure 10 plots streamwise velocity contours of both flat-bed (top) and dune-bed (bottom) cases showing that the low-momentum, mid-wake behind the turbine is somewhat wider (due to the flow expanding in the trough) and slightly further back in the dune-case in comparison with the flat-bed case. The far-wake region of the dune-bed case begins at x/D ≈ 4 and is characterised by fast wake recovery, i.e. the velocity deficit is recovered quickly due to the acceleration of the flow over the downstream dune. In contrast the flow field of the flat bed case recovers much slower in the far wake and the presence of the turbine is still visible at x/D = 8.

Contours and vectors of the mean turbulent kinetic energy (k) are presented in Fig. 9b. Shortly upstream of the turbine at x/D = − 0.5, the flow approaching the turbine features larger k values in the lower half of the water-depth, the signature of the turbulence coming from the upstream dune. In the near-wake region, the main regions of high k (red areas in Fig. 9b) coincide with the rotor’s swept area as the blades introduce additional turbulence into the flow, in the form of trailing edge and tip vortices. Also, high areas of k are found in the recirculation area of the rotor hub. At 1D downstream of the turbine, the tip vortices and the shear layer above the dune’s recirculation zone are responsible for significant levels of k as evidenced by the long k vectors. In the mid-wake the k levels are relatively low and tip vortices diffuse by action of the shear of the free-stream flow. Near the bottom of the channel the zone of high k in the dune’s trough merges with the one due to the energetic tip vortices of the near bed blades. At the beginning of the far-wake at x/D ≈ 4, the interaction of the tip vortices with the ambient flow increases again the magnitude of k as observed from the vectors above z/k > 3, while there are also high levels of turbulent kinetic energy at 1 < z/k > 2 due to the interaction between the dune’s and turbine’s wakes. The two peaks of turbulence kinetic energy at x/D = 4 merge to a large area of high levels of k after x/D ≈ 6. Figure 11 plots contours of the mean turbulent kinetic energy of both flat-bed (top) and dune-bed (bottom) cases. The most noteworthy difference is the significantly higher turblence behind the turbine operating over the dune bed and its extension into the far wake.

Figure 12a presents profiles of the normalised time-averaged streamwise velocity of the flat-bed and dune-bed cases at selected locations in the centre-line of the channel. This allows more a quantitative comparisons between the two flows. The flow upstream of the turbine, at x/D = − 1 is almost identical, however the dune-case features a slightly higher streamwise velocity and a more uniform distribution than the flat-bed case. The first profile downstream of the turbine (x/D = 1) is also quite similar, however slightly higher streamwise velocity is found underneath the rotor in the dune-bed case, the result of the slowly rising channel bed just upstream of the turbine (inducing an acceleration near the bed). The widening of the cross-section in the dune-bed case allows the flow to expand and streamwise velocities in the mid-wake of the dune-case are lower than in the flat-bed case x/D = 4. In the dune bed case the velocity deficit behind the turbine is almost recovered by x/D = 6, whereas the flow over the flat-bed has not fully recovered by x/D = 8. In contrast the streamwise velocity over the dune appears fully recovered by x/D = 8.

Figure 12b presents profiles of the normalised streamwise velocity fluctuation of the flat-bed and dune-bed cases at selected locations in the centre-line of the channel. The streamwise turbulence upstream of the turbine, at x/D = − 1 is significantly different: the dune-case approach flow having considerably higher streamwise velocity fluctuations than the flow over the flat bed, due to its bedform-induced (or rough-bed) turbulence. In the first profile downstream of the turbine (x/D = 1) the streamwise turbulence is quite similar between the two flows, except for the significant peak in the dune-bed case around z/k = 1, which signifies the location of the shear-layer between the dune’s recirculation zone and the fast moving fluid above it. Streamwise turbulence levels remain high near the bed in the dune-bed case and low in the mid-wake x/D = 2, whilst in the flat-bed case the onset of wake meandering signified by high levels of \(rms(u^{\prime })\) is observed from x/D = 2 onwards. Further downstream of the turbine streamwise turbulence levels are similar (at x/D = 6) before the flow over the dune-bed features aforementioned higher levels of turbulence, and at x/D = 8 this is a remarkable 25% higher streamwise turbulence in the dune-bed flow compared to the flat-bed flow.

Figure 13 depicts the distribution of the time-averaged streamwise velocity (left column), mean (k, centre column), and instantaneous (\(k^{\prime }\), right column) turbulent kinetic energy in selected cross-sections located at x/D = 0.5,1.5,3.5 and 7.5 downstream of the turbine. In the plane closest to the turbine, the characteristic signature of the turbine near-wake is well appreciated with the circular low-momentum region with negative U/U₀ values behind the hub, and high levels of turbulent kinetic energy behind the hub and along the perimeter swept by the blade’s tips. The contours of \(k^{\prime }\) reveal the presence of individual tip vortices triggered by the three blades and the dynamics of the hub’s wake.

Shortly downstream, at x/D = 1.5, the circular shape of the turbine’s wake is still visible from the U/U₀ contours that also show the turbulence originating from the dune’s trough at z/D < 0.4 in the form of high- and low-speed streaks, previously identified by [43]. The time-averaged k contours suggest that the highest levels of k are found in the dune’s trough meanwhile the circular turbulence of the tip vortices is more diffused than at x/D = 0.5 due the loss of coherence of the tip vortices as they interact with (and get sheared by) the high-momentum free-stream flow. This is better illustrated by the instantaneous turbulent kinetic energy contours on the left. The cross-section at x/D = 3.5 is located at the stoss side of the dune where the boundary layer recovery enhances the mixing of the turbine’s wake with the highly turbulent ambient flow. The low-momentum region still coincides with the turbine’s swept area (dashed lines in Fig. 13) and it has merged with the low-momentum area near the bed. Similar behaviour is observed for the area of highest values of mean turbulent kinetic energy, i.e. merging of turbine- and bed-generated turbulence. The contours of instantaneous k exhibit pockets of high k as a result of the turbine’s wake interacting with the turbulent flow generated in the dune’s trough. Further downstream of the turbine, at x/D = 7.5, the streamwise velocities appear to be more evenly distributed over the whole cross-section with only minimal signs of the presence of a turbine upstream of this location. The largest time-averaged k levels are found within the turbine’s swept area a signature of wake meandering and this is also visible from the instantaneous turbulent kinetic energy contours.

Time-averaged quantities of the normalised streamwise velocity (U/U₀) and turbulence intensity (\(I(\%)=(1/3(u^{\prime }+v^{\prime }+w^{\prime })^{0.5}*100\)) along the centreline of the channel at the hub’s height, z = z_hub are presented in Fig. 14a) and mean velocity fluctuations (\(rms(u^{\prime })\), \(rms(v^{\prime })\), \(rms(w^{\prime })\)) and turbulent kinetic energy (k) are presented in Fig. 14b). Upstream of the turbine (x/D < 0), the values of U/U₀ are fairly uniform with U ≈ 1.2U₀, i.e. approximately 20% higher than the bulk velocity which is owing to the use of sidewalls, hence velocities in the centre of the channel are higher than the cross-sectional average. In the approach flow the turbulence intensity is highest at x/D = − 8 with a value of I = 20% and its magnitude diminishes progressively until a minimum of I ≈ 8% is attained at x/D = − 2. The fluctuation of vertical velocity (\(w^{\prime }/U_{0}\)) is the main contributor to k, especially at − 2 < x/D < 0, while velocity fluctuations in x − and y −direction are similar upstream of the turbine. The flow conditions upstream of the turbine at x/D = − 0.5 are often adopted as reference values to predict/assess turbine performance. Here, these are U_hub ≈ 1.3U₀ and I_hub ≈ 14.5%.

The two dashed lines in Fig. 14a and b mark the boundaries of the three characteristic wake regions, as proposed from Fig. 9, and the profiles provide evidence of the chosen extent of each of these regions. The near-wake is characterised by the sudden drop in velocity and spike in turbulence followed by a significant drop-off in turbulence intensity/fluctuations. In the mid-wake, extending between 1.5 < x/D < 4, low streamwise velocities plateau and turbulence levels in this low-momentum zone are relatively constant. In the far-wake, the velocity recovers and the turbulence intensity levels increase again due to high-momentum fluid filling the wake which results in wake meandering and thus high levels of \(rms(u^{\prime })/U_{0}\) [35] are observed. The boundary layer recovers along the stoss side of the dune which results in increasing the values of \(rms(w^{\prime })/U_{0}\).

The identification of large-scale flow structures can be accomplished via temporal autocorrelations of velocity fluctuations (ρ_i) which allows to determine the integral time and length scales of the turbulent structures present in the flow [28, 58], and it is defined as: where i (= 1,2,3) refers to the velocity component, t denotes time, and τ is the time lag adopted to measure the phase correlation. The integral time scale, \(T_{u_{i}}\) is calculated by integrating the autocorrelation function from τ = 0 until when ρ_i = 0, meanwhile the spatial length scale is calculated as \(L_{u_{i}}=T_{u_{i}} {U}\), where U is the local time-averaged streamwise velocity. Figure 15a shows the autocorrelation function of the three velocity fluctuations obtained at location P1, which is just upstream of the turbine. The largest integral time scale is obtained for T_u ≈ 0.17 s indicating that the correlation of the flow in the streamwise direction is more coherent than in the spanwise or transversal directions for which T_v ≈ 0.084 s and T_w ≈ 0.083 s. The integral length scale value is here L_u ≈ 0.15 m and it provides an estimation of the dimension of the large-scale structures in the flow [31]. This value gives L_u ≈ 0.75R in terms of rotor radius suggesting that the large-scale flow structures shown in Fig. 6 impact the blades unevenly and hence the instantaneous structural loading on each blade varies significantly (as is demonstrated in Fig. 17 below). Chamorro et al. [59] obtained a similar value of L_u ≈ 0.7R indicating that the current case adopting the dune-like bathymetry represents realistic environmental turbulent flow conditions. In order to justify the assumptions made to compute \(T_{u_{i}}\) a second location is chosen, P3, to estimate integral time scales. The correlations of Fig. 15b show peaks from which the blade frequency can be inferred, evidence that temporal autocorrelations for these types of (shear) flows are reasonably reliable in estimating large-scale turbulence frequencies and length-scales. The correlation of Fig. 15a shows a secondary peak at τ ≈ 2.1 s suggesting that a dune generated turbulence structure occurs approximately every two seconds.

The anisotropy ratio \(rms(u^{\prime })\):\(rms(v^{\prime })\):\(rms(w^{\prime })\) at location P1, i.e. shortly upstream of the turbine, is found to be 1:0.64:0.88. These values are somewhat dissimilar to the 1:0.75:0.56 found at the Sound of Islay (Scotland) tidal site [28] or to two-dimensional channel flows [60] in which the ratio is 1:0.71:0.55. The present values reflect the influence of the dune-induced turbulence with the vertical velocity fluctuations dominating over those in the transversal direction. According to Mycek et al. [20], ratios of mean turbulence intensities are site-dependent and vary according to the dominating turbulence structures at a given deployment site, e.g. such as the presence of large obstacles in the bathymetry such as Horse Rock in Ramsey Sound [9].

The frequency associated with periodic events due to large-scale flow structures in the velocity time series in Fig. 17a is revealed via Power Spectral Density (PSD) from u − and w-velocities series obtained at four points, denoted P1, P2, P3 and P4 (see Fig. 8), and these are presented in Fig. 16. P1 is shortly upstream of the turbine, P2 and P3 are located at x_hub + 0.3D and at x_hub + 0.8D, respectively, at a distance of D/2 from the hub’s centre, i.e. at the projected turbine swept perimeter that coincide with the path described by the tip vortices, and P4 is aligned with the hub’s centre located at a distance of 1.8R downstream. At P1 a low-frequency peak in the u-velocity spectrum is appreciated, labelled as f_L and \(f_{L}= 0.45~\mathrm {s}^{-1}\), as a result of the periodic turbulent events generated by the dune upstream of the turbine, such as the vertical rollers depicted in Fig. 6. In the w-velocity spectrum there is also an energy peak at frequency f_L although there is another peak with higher associated energy at a frequency of f_s = 0.89 Hz, which is linked to bed-induced turbulent events occurring at lower frequency in time than those contributing for f_L. Spectra at P1 feature a decay slope of -1 in the production range prior to the inertial sub-range that follows the −5/3 slope expected from homogenous turbulence, which is also commonly found in boundary layer flows [61]. The −5/3-region spans approximately one decade of frequencies and given the relatively high Re this is sufficient for an LES. After the inertial sub-range the slope is significantly steeper and this is due to the sgs-model dissipating energy efficiently.

The turbine introduces turbulent kinetic energy into the flow as result of its operation and the flow structures induced by the turbine blades, e.g. tip vortices, are responsible for increasing the spectral energy associated with frequencies in the energy production range. P2 and P3 are located in the near-wake downstream of the turbine at x = 0.6R and x = 1.8R, respectively. The dominating large-scale structures at these locations are identified by the energy peaks occurring at a frequency coincident with the blade passing f_b = Ω/2π ⋅ N_b = 5.75 Hz, and thus these peaks are associated with the tip vortices [62]. Note that the latter energy peaks exhibit higher spectral energy than that found at P1 linked to the low-frequency turbulent structures from the turbulent flow approaching the turbine. The turbine-induced turbulence changes the energy decay rate in the production subrange at P2 and P3 with a flatter slope when compared to the −1 slope found at P1, i.e. in the free-stream flow. It is also observed that comparing the u-velocity spectrum at P2 and P3, the slopes at the latter location is steeper compared to that in the former which is closer to the turbine rotor. Jin et al. [58] observed a similar trend from flatter to steeper slopes in the energy cascade when moving from the near-wake to the far-wake of a horizontal axis wind turbine. Energy peaks at blade passing frequency harmonics, i.e. 2f_b and 3f_b, are observed at P2 especially for the w-velocity spectrum. However, as the tip vortices are convected downstream they interact with the ambient turbulent flow and lose coherence, thus at P3 the spectral energy associated with f_b is smaller than that at P2 and energy peaks at f_b harmonics are no longer observed. The breakdown of the energetic tip vortices due to the turbulent ambient flow was highlighted in Fig. 8. Turbulence properties in the middle of the near-wake are revealed via spectra at P4, which show an energy peak at f_b most noticeably in the w-velocity signal. This is the result of the recurrent shedding of blade root vortices which are generated by the sharp end of the blades near the hub [11].

Survivability of tidal turbines relies on the ability of their material to sustain large load fluctuations due to sudden velocity and pressure variations in the flow field. Figure 17 presents the time history of the streamwise velocity upstream of the turbine obtained at P1 (located at x_hub − 0.5D, y_hub, z_hub, see Fig. 8) normalised by the mean streamwise velocity at the turbine hub’s height U_hub, the power and thrust coefficients Cp_dune and Ct_dune normalised with their respective mean value obtained in the flat bed simulation Cp_flat and Ct_flat of [11], and the coefficients of flap-wise (\(C_{M_{\theta }}^{T}\)) and edge-wise (\(C_{M_{\theta }}^{Q}\)) bending moments.

The time series of u at P1 exhibits severable noticeable drops during recurrent turbulent events, e.g. at t/T ≈ 0.2, 4.8, 8.8, 15.2 and 18.6 considering T = 2π/Ω = 0.5236 s as the time of a full rotor revolution. These events are a reflection of the periodic energetic turbulent structures generated at the upstream dune as visualised in Fig. 6. The instantaneous power and thrust coefficients drop markedly when these events pass through as depicted in Fig. 17b, as a result of the sudden drop in speed of the approaching flow. After each event, the power generation of the device is quickly recovered as a result of the following increase of streamwise velocity, suggesting significant load fluctuations on all turbine components. Noteworthy is the fact that a drop in streamwise velocities is associated with a sudden increase in vertical velocity (not shown for brevity), the result of turbulent ejection events which travelling on a upwards path towars the water surface [43].

Figure 17c and d present the bending moment coefficients for each of the blades comprising the rotor and highlights that each blade is affected differently during the turbulent events. For instance, at time t/T ≈ 4.8, blade 1 is almost in vertical position and thus its tip is near to the water surface, whilst blades 2 and 3 are below half water depth and hence their tips are closer to the channel bed. The bathymetry-induced turbulence structures are more energetic near the bed and thus induce on blades 2 and 3 a larger impact than on blade 1. The results confirm that the size of the turbulent structures is significantly smaller than the turbine diameter as it was verified from the integral spatial length scale.

Table 1 presents averaged/maximum values of structural bending moments for the three blades together with the results of Ouro et al. [11] for the flat-bed channel simulations. The “range” quantity, here calculated as the difference between maximum and minimum values, is a key parameter in the prediction of fatigue loads [7].

The computed time-averaged structural coefficients of the present simulation are very similar to those found by [11] for the cases of the turbine operating above a flat-bed channel subjected to different levels of turbulence intensity. The mean flapwise bending moment coefficient \(C_{M_{\theta }}^{T} = 0.1012 \) is only approx. 0.101 which is similar to the 0.0964 and 0.0975 values obtained for the I = 10% and 20% cases, respectively. Such small variation under different turbulence conditions follow the findings of Blackmore et al. [22] who stated that even though turbulence intensities in the flow vary considerably, mean bending moments stay within certain bounds, i.e. no great differences are expected irrespective of the turbulence intensity in the flow. An analogous trend is found for \(C_{M_{\theta }}^{Q}\) with a value of \(C_{M_{\theta }}^{Q}= 0.0259\) for the present case with a bed of dunes whereas it is \(C_{M_{\theta }}^{Q}= 0.0230\) or \(C_{M_{\theta }}^{Q}= 0.0235\), respectively for two artificial turbulent inflow cases [11].