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Cavitating response of passively controlled tidal turbines
Journal of Fluids and Structures 66 (2016) 462–475
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Cavitating response of passively controlled tidal turbines Ramona B. Barber n, Michael R. Motley Department of Civil and Environmental Engineering, University of Washington, Seattle, WA 98195, USA
a r t i c l e in f o
abstract
Article history: Received 17 September 2015 Received in revised form 24 June 2016 Accepted 25 August 2016
The use of passive pitch control in marine hydrokinetic turbine (MHK) blades has been shown to provide structural and performance improvements over similar systems with non-adaptive blades. Recent work has demonstrated such advantages as increased lifetime energy capture, reduced hydrodynamic instabilities, and improved efficiency, load shedding, fatigue behavior, and structural integrity. Additionally, passively adaptive blades have been shown to delay cavitation and decrease cavitation volume for marine propellers. For MHK turbines, a reduction in cavitation susceptibility could increase effectiveness of the turbine and reduce the rate of fatigue on the system. In this work, a previously validated, coupled 3-D boundary element method-finite element method solver is used to model two adaptive blades and a non-adaptive reference blade under both uniform and non-uniform inflow conditions. A numerical investigation of the onset of sheet cavitation on an MHK turbine blade and the structural response of a blade under cavitating conditions is presented. Results show that passive control can be used to delay cavitation and decrease cavitation volume under normal operating conditions. Cavitation is shown to increase the rate of fatigue on the system and has the potential to lead to unanticipated failures if proper cavitation analysis is not integral to the MHK turbine design process. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Tidal turbines Cavitation Passive pitch control Adaptive composites Fluid–structure interaction
1. Introduction Sustainably harvesting the kinetic energy of marine tidal currents is becoming an increasingly feasible alternative to the use of traditional power sources. The high energy density and reliability of these currents make them a very attractive option compared to more stochastic renewable sources such as wind and solar power. Theoretically, the energy contained in the world's oceans far exceeds global power demand; studies suggest that marine currents alone have the potential to supply a significant portion of future electricity needs (Ben Elghali et al., 2007; Blunden and Bahaj, 2007). Though there are many benefits to tidal current energy extraction, marine turbines face significant challenges. Blade design is a critical factor in the implementation of marine hydrokinetic (MHK) turbines, as the relatively slender blades must withstand the large, dynamic fluid forces inherent to the marine environment. An additional concern specific to marine turbines is fluid cavitation, which can cause vibrations, performance decay, and an increased rate of fatigue. Maintenance needs are considerably hard to address for underwater turbines purposefully placed in locations of extreme current, necessitating the development of a turbine uniquely suited to its harsh environment and able to operate for long periods of time without need of repair. n
Corresponding author. E-mail address: [email protected] (R.B. Barber).
http://dx.doi.org/10.1016/j.jfluidstructs.2016.08.006 0889-9746/& 2016 Elsevier Ltd. All rights reserved.
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Today, most marine turbine blades are constructed from fiber reinforced composite materials. These materials provide excellent strength-to-weight and stiffness-to-weight ratios, improved fatigue behavior, and are easier to manufacture. An additional benefit of composite materials is that their anisotropic nature allows turbine blades to be hydroelastically tailored during the design process to modify system performance over the expected range of operating conditions. By manipulating the fiber axis of the material, the relationship between bending deformations (blade tip deflection) and twisting deformations (change in blade pitch) can be controlled. Exploiting this intrinsic bend-twist deformation coupling behavior inherent to laminated composites can create a nearly instantaneous, passive mechanism to adapt blade pitch to the flow. Preliminary studies show that these passively adaptive blades can provide system performance improvements such as increased lifetime energy capture, reduced hydrodynamic instabilities, and improved efficiency, load shedding, fatigue behavior, and structural integrity (Motley and Barber, 2014a; Nicholls-Lee et al., 2013). An added benefit of adaptive blades that has not yet been fully explored is their reduced susceptibility to cavitation. Every marine structure that moves at a high velocity relative to the incident flow is at risk of cavitation. In accordance with Bernoulli's principle, an increase in the velocity of a fluid will cause a decrease in hydrostatic pressure. When the pressure in the flow falls below the vapor pressure of the fluid, vapor bubbles form and collapse at a high rate. Cavitation contributes to pitting, corrosion, vibration, and fatigue in many marine structures (Kumar and Saini, 2010); to avoid cavitation, limits are placed on MHK turbine depth, blade length, and rotational speed. Preliminary studies have indicated that the use of bendtwist coupled composite material could reduce cavitation on hydrofoils (Gowing et al., 1998). In research on marine propellers, numerical and experimental results have shown that adaptive composite propellers can provide improved cavitation performance and increased energy efficiency over their non-adaptive counterparts when operating at off-design conditions or in spatially varying flows (Young, 2007, 2008; Motley et al., 2009). Using a previously validated, coupled 3-D boundary element method–finite element method (BEM–FEM) solver, this work presents a numerical investigation of the effect of sheet cavitation on adaptive pitch MHK turbine blades. A set of adaptive blades and a non-adaptive reference blade are modeled in realistic, non-uniform inflow conditions. This study investigates the onset of sheet cavitation on an MHK turbine blade and the structural response of that blade and the full turbine system under cavitating conditions.
2. Turbine parameters The model used for a baseline reference in this work is a two-bladed turbine with a diameter of 20 m. The center of the hub is located 30 m above the sea floor and approximately 15 m below the surface. The reference blade geometry was taken from the optimized blade presented in Bir et al. (2011) and is shown in Fig. 1. The turbine geometry was designed for a variable speedvariable pitch system with a maximum rotational speed of 11.5 rpm; a tip speed ratio of 7 was found to be the optimum operating point until the maximum rotational limit is reached. As the limit on rotational speed was placed in part to avoid the chance of cavitation, that constraint has been removed for this work in order to evaluate the response of adaptive blades to cavitation.
Fig. 1. Schematic of the turbine geometry and wake considered in this work.
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Fig. 2. A schematic of passive pitch deformation, adapted from Barber (2015).
2.1. Passive control of turbine blades In order to avoid overloading the generator associated with an MHK turbine, each system must have a method of limiting power production in extreme flow conditions. Often, modern turbines are built with a motorized control mechanism in the hub of each blade which is programmed to adapt the blade pitch to the instantaneous velocity. This added mechanism relies on system feedback, thus it creates a lag between velocity change and pitch change. In contrast, a passively adaptive rotor blade exploits the anisotropic nature of fiber reinforced composites to create an inherent bend-twist coupled response control mechanism in the blade. In this way, passively controlled blades are able to independently adjust to changing flow conditions without the need for a mechanical driver. Additionally, passive adaptation creates a nearly instantaneous structural response that is difficult to achieve with an active mechanism because of the high flow variation and excitation frequencies in water. The bend-twist deformations are load-dependent and the magnitude and direction of the deformations are controlled by material design; thus it is possible to create an adaptive blade geometry for a more efficient system. To create a passive control mechanism, the material strong axis, defined by the anisotropic nature of the composite fibers, is rotationally offset from the longitudinal axis of the blade by a specified amount. This creates a load-dependent deformation mechanism in which an applied shear load, for example, will result in twisting as well as bending deformations even in the absence of a torsional load. An adaptive turbine blade can be designed to pitch to feather by decreasing the angle of attack or pitch to stall by increasing the angle of attack, depending on composite laminate layup sequence (Motley and Barber, 2014a). Fig. 2 shows a schematic of the passive geometry change. In practice, composite blades are made up of a multitude of layers of composite fibers. It has been shown, however, that a multi-layered structure can be modeled using an equivalent unidirectional fiber angle, θeq, which can be found such that the effective stiffness and degree of bend-twist coupling is approximately equal to that of the physical layup sequence (Liu and Young, 2009; Motley and Young, 2011a,b; Young, 2008). The equivalent unidirectional fiber angle is used to create seven blade models for this study. Three blades were designed to have varying biases to pitch towards stall (θeq < 0°), or increase their angle of attack and therefore decrease blade pitch, and three to pitch to feather (θeq > 0°), decreasing angle of attack and increasing blade pitch. The six adaptive blades were compared to a non-adaptive (θeq = 0°) reference blade. The composite material modeled is the carbon fiber reinforced polymer Hexcel IM7–8552, with material properties as defined in Table 1, where the 1-axis is defined as parallel to the fibers and the 2-axis as perpendicular to the fibers within each laminate layer (Camanho and Lambert, 2006).
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Table 1 Material properties of Hexcel IM7–8552 (Camanho and Lambert, 2006).
E1 = 171.42 GPa E2 = E3 = 9.08 GPa G12 = G13 = 5.29 GPa
G23 = 3.13 GPa ν12 = ν13 = 0.32 ν23 = 0.45
2.2. Effects of cavitation Cavitation susceptibility is a major limit state in modern marine turbine design. Research has shown that cavitation contributes to pitting, corrosion, vibration, and fatigue of marine structures (Bahaj and Myers, 2003; Wang et al., 2011). Additionally, the vibration caused by cavitation dramatically increases noise generation, a particular concern of turbine systems located in or near sensitive marine mammal habitats. The onset of cavitation also has a drastic effect on the power generation of the system. In general, the lift and drag coefficients of a hydrofoil do not vary as the ambient pressure decreases; the pressures on the upper and lower surfaces decrease in tandem. However, as cavitation develops along the top of the foil, the pressure on the top surface can no longer decrease because it is limited by the vapor pressure. The pressure on the bottom surface is not restricted, however, and continues to fall, progressively approaching the pressure on the top of the foil. As the pressure differential across the foil decreases, there is a corresponding decrease in the lift coefficient. Additionally, the resultant change in the shape of the pressure distribution causes an increase in drag. Thus, cavitation causes both a decrease in lift and an increase in drag, limiting the power generation as well as exacerbating the loads on the system. For these reasons, it is essential to include the study of cavitation in the analysis and development of MHK systems. Though the effects of cavitation have been studied on other marine structures, little research has been conducted on the possible impact of cavitation on an MHK turbine blade. Understanding the structural response of a cavitating blade is essential to predicting system life and the susceptibility of the blade to various failure modes.
3. Numerical analysis 3.1. Fluid–structure interaction In order to predict turbine response, accurate modeling of the fluid–structure interaction (FSI) effects that cause the loaddependent deformation of the blade is essential. To this end, turbine blades are modeled with a previously developed and validated, fully coupled boundary element method–finite element method (BEM–FEM) solver (Kinnas and Fine, 1993; Kinnas et al., 2003; Young and Kinnas, 2001, 2003; Young et al., 2006,2010; Young, 2007, 2008; Motley et al., 2009) that includes Froude number effects and is capable of considering spatially varying inflow and unsteady sheet cavitation on both sides of the blade surface. In the BEM code, the fluid behavior is assumed to be governed by the incompressible Euler equations solved in a bladefixed rotating coordinate system. The total velocity is expressed as the vector sum of the inflow velocity and the perturbation velocity potential, ∇Φ , corresponding to the turbine-induced flow field. The perturbation flow field Φ can be treated as incompressible, inviscid, and irrotational and satisfies the Laplace equation ∇2Φ = 0. To consider FSI effects, the perturbation potential Φ is linearly decomposed into two parts: a part due to large rigid blade rotation (ϕ) and one due to small elastic blade deformation (φ). Each part is solved separately using a 3-D BEM by applying appropriate kinematic and dynamic boundary conditions (see Young, 2007, 2008; Young et al., 2010). The decomposition of Φ allows the total pressure to be similarly separated into rigid blade (Pr) and elastic blade (Pv) components as follows:
⎡1 ⎤ ∂ϕ 1 Pr = ρ⎢ |Vin|2 − − |Vtr|2 ⎥ ⎣2 ⎦ ∂t 2
(1)
⎡ ∂φ ⎤ − Vin·∇φ⎥ Pv = ρ⎢ − ⎣ ∂t ⎦
(2)
where Vin is the inflow velocity and Vtr is the fluid velocity due to rigid blade rotation. Likewise, the total hydrodynamic force can be expressed as a sum of the forces due to rigid blade rotation and those due to elastic blade deformation. User-defined sub-routines in the commercial FEM solver ABAQUS/Standard are employed to obtain these forces from the decomposed pressure output of the BEM solver by applying the respective pressure terms to the nodes in the FEM model via the element shape functions, i.e,
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∫ ⎡⎣ N⎤⎦T {Pr}dS T {Fv } = ∫ ⎡⎣ N⎤⎦ {Pv }dS {Fr} =
(3) (4)
The hydrodynamic force due to elastic blade deformation can be expressed as functions of nodal velocity and acceleration and written
{Fv } = − [MH]{u¨ } − [CH]{u̇ }
(5)
where the [MH] and [CH] matrices represent the hydrodynamic added mass and damping inherent to a structure moving through a dense fluid. ABAQUS is then used to solve the final dynamic equation of motion,
( ⎡⎣ M⎤⎦ + ⎡⎣ MH⎤⎦){ u¨ } + ( ⎡⎣ C⎤⎦ + ⎡⎣ CH⎤⎦){ u̇ } + ⎡⎣ K⎤⎦{ u} = { Fce} + { Fco} + { Fr}
(6)
where {u¨ }, {u̇ }, and {u} are the structural nodal acceleration, velocity, and displacement vectors, respectively; [M], [C], and [K] are the structural mass, damping, and stiffness matrices; {Fce} is the centrifugal force and {Fco} is the Coriolis force. The blades are represented as fully fixed at the hub connection. The effects of large, nonlinear blade deformations are considered by iterating between the fluid and solid solvers until the solution converges. Detailed formulations of the solution algorithm, governing equations, and boundary conditions can be found in Young (2007, 2008), Motley and Barber (2014b) and Barber (2015). 3.2. Cavitation prediction For the purposes of cavitation prediction and analysis, a pressure coefficient is defined relative to the local flow,
−CP =
P0 − Pt 1 ρn2D2 2
(7)
where P0 is the freestream pressure, Pt is the total pressure at the point in question, ρ is the fluid density, n is the rotational speed, and D is the turbine diameter. The local velocity squared term n2D2 is used instead of the more common freestream velocity squared in order to facilitate a direct comparison with the cavitation number,
σn =
P0 − Pv 1 ρn2D2 2
(8)
where Pv is the saturated vapor pressure in the fluid. Thus, if at any point the pressure coefficient CP falls below the cavitation number sn, the potential for cavitation development exists. To determine the existence and extent of the potential cavity planform at each step, a Newton–Raphson iteration is used until the cavity closure condition is satisfied. The cavity closure scheme used in this method requires both the cavity height and the cavity thickness to equal zero on the trailing edge of the planform. Dynamic and kinematic boundary conditions are applied to the cavitating and wetted surfaces to uniquely define the perturbation potential. These conditions include requirements for flow tangency to both the turbine blades and cavity surfaces, constant pressure along the cavity equal to the vapor pressure of the fluid, and the cavity closure requirements noted above. This process is described in depth in Kinnas et al. (2003), Young and Kinnas (2001), and Young (2008). This method of sheet cavitation prediction has been extensively validated against experimental results for rigid and adaptive marine propellers and rigid turbine blades (Kinnas and Fine, 1993; Kinnas et al., 2003; Young and Kinnas, 2001; Young, 2007, 2008; Young et al., 2010). There are no experimental data for adaptive turbines available for validation at this time; this is a focus of future work for the authors. 3.3. Inflow velocity profile A uniform, steady inflow velocity profile is often adequate to obtain basic system performance metrics and observe the structural behavior of an MHK turbine. The analysis associated with such a profile is relatively simple and fast compared to the time-dependent problem of a non-uniform inflow profile. In actual marine conditions, however, an MHK turbine operates in a more complex field, and a realistic, non-uniform flow profile is required for a more robust understanding of the system dynamics. An example of the flow a turbine might realistically experience is shown in Fig. 3. This data was collected to inform the design, siting, and permitting of a pilot scale tidal energy array in Admiralty Inlet in Puget Sound, WA (Polagye and Thomson, 2013). Fig. 3 demonstrates the highly dynamic nature of the instantaneous velocity profile; though this profile is difficult and impractical to model, boundary layer effects can be approximated with a power-law estimate of the flow velocity across the water column to estimate dynamic effects. Here, a 1/7th-power law is applied, as shown in Fig. 4. The fluid velocity, V, at a specific height above the sea floor, h, is defined as
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Fig. 3. Top: Measured flow direction and depth-averaged velocity magnitude data from Admiralty Inlet, WA. Bottom: Measured velocity magnitude over depth from Admiralty Inlet, WA.
Fig. 4. Schematic of the non-uniform velocity profile applied in the unsteady analysis.
Fig. 5. Predicted change in pitch for six adaptive blades compared to a non-adaptive reference blade. The presence of cavitating conditions is indicated with larger, yellow-filled symbols and dotted lines. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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Fig. 6. Pressure distribution and presence of cavitation (shown in white) on the suction and pressure sides of the reference blade for one full rotation at V ¼ 3.0 m/s.
⎛ ⎞ ⎛ h ⎞1/7 V ⎜ h⎟ = Vavg ⎜ ⎟ ⎜ ⎟ ⎝ h0 ⎠ ⎝ ⎠
(9)
where Vavg is the mean inflow velocity in the turbine's plane of rotation and h0 is the distance from the center of the hub to the sea floor. This model does not produce a realistic representation of any instantaneous velocity field but is accurate as a time-averaged profile, which captures the essential velocity gradient of the flow. Including this gradient allows for the consideration of the periodic, unbalanced loading caused by boundary layer effects; in many cases, these forces have a significant impact on the performance and structural response of the turbine system.
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Fig. 7. Cavitation volume normalized by turbine radius R3 over one full rotation at increasing inflow velocities.
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Fig. 8. Predicted thrust coefficient on each blade over one full rotation at V ¼ 2.7 m/s and V¼ 2.9 m/s.
4. Results 4.1. Cavitation inception and growth To begin a study of cavitation, it is logical to start with an investigation into the conditions at which the onset of cavitation occurs. Cavitation inception in this case refers to the onset of sheet cavitation, in accordance with previous work by the authors. In this study, three pitch to stall blades (θeq < 0°) and three pitch to feather blades (θeq > 0°) with increasing levels of flexibility are compared to a non-adaptive (θeq = 0°) reference blade. The higher end of the velocity spectrum presented above is examined to predict the potential for the development of cavitation on each blade. The tip speed ratio for all following analyses is held constant at 7 in accordance with the optimal operating point find in Bir et al. (2011). Fig. 5 shows the change of pitch for each blade as a function of velocity, with indications marking cavitating conditions. The blades experienced pitch change as designed, though the reference blade exhibited a minor degree of feathering due to the inherent flexibility in the material and bias of the geometry. The effect of blade fiber angle on the onset of cavitating conditions is clear in this figure; the blades with a pitch to stall bias show cavitation at lower velocities than the reference blade, while the pitch to feather blades delay cavitation. This trend is strongest in the blades with the largest adaptive biases. After the onset of cavitating conditions, the blades tend to experience an increase in the rate of pitch change. This is likely due to the increased forces associated with cavitation. Fig. 6 shows the pressure distribution and presence of cavitation on the pressure and suction sides of a single blade as it rotates. The particular blade shown in this figure is the 0° reference blade at an inflow velocity V¼3.0 m/s, which is well representative of any moderately cavitating blade in this study. The onset of cavitation appears on the suction side of the
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Fig. 9. Predicted maximum displacement of each blade over one full rotation at V¼ 2.7 m/s and V ¼2.9 m/s.
blade as expected, initially at the point of the blade's rotation nearest the free surface, where the fluid flow is strongest and the surrounding pressure is lowest. As cavitation progresses, it begins to spread and appear throughout the upper half of the rotational sweep of the blade as shown in Fig. 6; however, due to the pressure and velocity differentials between the extreme blade positions, at no point is it possible for the cavitating volume to be constant around a full revolution. Thus, as the blade advances, cavitation forms and disappears in each rotation, causing a periodic and unbalanced loading. This periodic effect, as well as the significance of the increased rate of pitch change under cavitating conditions, can be seen clearly in Fig. 7. This figure shows the volume of cavitation on the reference and adaptive blades during a full revolution from V ¼2.7 m/s to V ¼3.0 m/s. Here the cavitation volume is normalized by the turbine radius R3. As the inflow velocity increases, cavitation forms and grows on each blade. The blades with a pitch to stall bias start to cavitate first, with the largest bias blades showing the most cavitation. This is consistent with the onset of cavitating conditions shown in Fig. 5. The cavitation volume on these blades grows dramatically with a small increase in velocity, due to the increasing rate of pitch change shown above. The pitch to feather blades, however, delay cavitation compared to the pitch to stall and reference blades. The feathering blades show both a volume reduction and slower cavitation growth compared to the other blades, as the increased rate of pitch change reduces the angle of attack significantly. Again, the benefit is greatest in the blades with the largest adaptive bias. It is also possible to compare the extent of the revolution of each blade for which cavitation is apparent. At the onset of cavitation, the cavity volume is visible only around the top of the blade's rotation. For blades operating at a more advanced state of cavitation, cavitation is apparent through more than half of the rotation. In all cases shown, the bottom of the revolution, nearest the sea floor, is free of cavitation.
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Fig. 10. Harmonic excitation in the blade due to axial forces, normalized by the 0th harmonic amplitude.
4.2. Structural response of a cavitating blade The onset of cavitation has a strong impact on blade forces and therefore deformation. For a more detailed analysis of structural response, this section focuses on two adaptive blades (θeq = ± 5°) and the reference blade (θeq = 0°). The fluctuation of forces on the system in cavitating conditions can be seen in the unsteady nature of the thrust coefficient (CT), shown in Fig. 8. At V¼2.7 m/s, the −5° blade has just entered cavitating conditions. There is very slight unsteadiness evidenced in the CT at that inflow velocity, but in general the CT curves for all three blades are smooth. At V¼2.9 m/s, however, there are large fluctuations in the CT for the −5° pitch to stall blade, and unsteadiness in all three blades. The +5° pitch to feather blade has the smoothest CT profile of the three, corresponding to the minimal cavitation volume shown in Fig. 7. The effect of these fluctuating forces can be seen in the maximum tip displacement of each blade, shown in Fig. 9. The magnitude of the displacement in both adaptive blades is higher than that of the reference blade, with the pitch to stall blade undergoing moderately larger deformations than the feathering blade. This result is consistent with earlier findings and is due to the fact that the pitch to stall blade experiences slightly higher forces than the feathering blade. The notable feature of these figures is the difference that the onset of cavitation creates. At V¼ 2.7 m/s, the 0° and +5° blades are fully wetted while the −5° blade has just entered cavitating conditions. The smooth lines in the wetted blade profiles show the gradual change in blade deformation brought on by the rotation though a non-uniform velocity profile, while the pitch to stall blade shows the effects of cavitation through very small oscillations in the displacement curve. As the fluid velocity is increased to V¼2.9 m/s, cavitation is apparent on all three blades. The feathering blade is only just beyond the onset of cavitation and shows small oscillations similar to the pitch to stall blade at V¼2.7 m/s. However, the other two blades show more significant cavitation and experience strongly unsteady displacements. These fluctuating effects in the blade displacements will increase the rate of fatigue on the system and could lead to unanticipated failures if proper cavitation analysis is not integral to the design process.
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Fig. 11. Harmonic excitation in the blade due to moment forces, normalized by the 0th harmonic amplitude.
The structural effects of fluttering caused by cavitation can also be seen in a harmonic analysis. Both the individual blade harmonics (defined with respect to blade-fixed coordinates) and the shaft, or full system, harmonics (in global coordinates) for the reference blade and the two adaptive pitch blades at V ¼2.7 m/s and 2.9 m/s are presented in this section. For ease of comparison, the magnitudes of the harmonic excitation are normalized by the amplitude of the 0th harmonic. The harmonics for a fully wetted analysis are compared to those found by the cavitating analysis in each figure for reference. The axial force blade harmonics are shown in Fig. 10. At the lower inflow velocity V¼2.7 m/s, low level excitation is found for the −5° blade in the higher harmonic modes, reflecting the onset of cavitating conditions. Very slight excitation is also found in the mid-level modes for the 0° reference blade; this likely reflects the imminent onset of cavitation for that blade. At V¼2.9 m/s, excitation is found in the higher modes of all three blades in the cavitation analysis that cannot be seen in the simple wetted analysis. The +5° feathering blade is shown to perform significantly better than both the reference and pitch to stall blades, in that it experiences much less excitation in the higher harmonic modes. This reflects the lower volume of cavitation activity on the blade (as shown in Fig. 7). The same pattern can be seen in the harmonics due to the moment on the blade, shown in Fig. 11. The high mode excitation is slightly stronger in this figure, due to the twisting nature of the blade. Again, solely the cavitating blades show high mode excitation. Similarly, Fig. 12 shows the harmonic response on the full system due to torque. The higher-mode excitation due to cavitation is significantly greater here than in the single blade analysis, due to the fact that cavitation is initiating and disappearing as each blade nears and passes the apex of its rotation, creating an uneven forcing on the system. In this figure, as in Figs. 10 and 11, the pitch to feather blade shows much weaker excitation in the higher modes for both the blade and shaft harmonics. It is essential to identify these high-frequency harmonic excitations induced by cavitation, as they contribute to blade and system fatigue over the lifetime of a turbine. Though the high-frequency harmonics are a concern for fatigue, they are not likely to cause resonance in the system.
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Fig. 12. Harmonic excitation in the system due to torque, normalized by the 0th harmonic amplitude. Table 2 Modal frequencies (Hz). Mode
θeq = − 5°
θeq = 0°
θeq =+ 5°
1 2 3 4 5
4.916 15.167 20.455 23.414 26.122
5.097 16.098 19.127 24.871 26.029
4.519 15.265 19.947 24.701 26.096
Table 2 shows the associated modal frequencies for the first five modes in each blade. These frequencies are much higher than the rate of rotation; at V¼3.0 m/s, the system is rotating at 0.334 cycles/s.
5. Conclusions and future work In this work, a numerical study of the effect of cavitation on adaptive pitch MHK turbine blades is presented. A previously validated, coupled 3-D BEM–FEM solver is used to model two sets of adaptive blades and a non-adaptive reference blade under realistic, non-uniform inflow conditions. Specifically, this study investigates the onset of cavitation on an MHK turbine blade and the structural response of that blade and the full turbine system under cavitating conditions. Numerical
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analysis suggests that passively adaptive pitch to feather blades can be used both to delay cavitation and to reduce cavitation volume over the blade surface. An analysis using a non-uniform inflow representative of the inflow in Admiralty Inlet, WA predicted cavitating conditions at well within the range of normal operating conditions and demonstrated the importance of cavitation analysis and design in MHK turbine development. After the onset of cavitation, all three blades in this study exhibited a similar structural response. A blade at the onset of cavitation will start to experience oscillations in thrust and therefore resulting deformations. As the cavitation volume increases, stronger fluctuations in the blade displacement are found. A harmonic analysis of each blade reflects this fluttering, showing high-frequency harmonic excitation in a cavitating blade that is not present in an equivalent fully wetted blade. The fluctuations in deformation combined with the higher mode excitation will increase the rate of fatigue on the system and could lead to unanticipated failures if proper cavitation analysis is not integral to the design process. The response to cavitation is identical regardless of material orientation; thus, to reduce the negative effects of cavitation it is necessary to design a system that will avoid cavitation over as much of the normal operating range as possible. The benefit found in the passively adaptive, pitch to feather blade in this study is in its ability to both delay the onset of cavitation and to maintain a reduced cavity volume after that point, allowing the system a greater amount of time spent operating smoothly and decreasing the exposure to fatigue-inducing oscillations. The results of this research suggest that an adaptive blade, designed carefully for site-specific parameters, could push the onset of cavitating conditions beyond the range of normal operation and effectively avoid the effects of cavitation in all but the most extreme conditions. Further study is needed to complete the knowledge necessary to design such a blade, however. While the work presented here provides an insight into cavitation response, it considers only sheet cavitation, and added complexities associated with other modes such as tip vortex cavitation or cavitation collapse will require supplementary research. Additionally, the analysis of a larger suite of blade profiles will add to the understanding of the potential benefits to be gained with the use of an adaptive pitch mechanism. Finally, a full system harmonic analysis is needed to adequately explore the possibility of resonant excitation. Acknowledgments This work was facilitated through the use of advanced computational, storage, and networking infrastructure provided by the Hyak supercomputer system, supported in part by the University of Washington eScience Institute.
References Bahaj, A., Myers, L., 2003. Fundamentals applicable to the utilisation of marine current turbines for energy production. Renew. Energy 28, 2205–2211. Barber, R., Motley, M., 2015. A numerical study of the effect of passive control on cavitation for marine hydrokinetic turbines. In: European Wave and Tidal Energy Conference, Nantes, France. Ben Elghali, S., Benbouzid, M., Charpentier, J., 2007. Marine tidal current electric power generation technology: state of the art and current status. In: Electric Machines & Drives Conference, pp. 1407–1412. Bir, G., Lawson, M., Li, Y., 2011. Structural design of a horizontal-axis tidal current turbine composite blade. In: International Conference on Ocean, Offshore and Arctic Engineering. Blunden, L., Bahaj, A., 2007. Tidal energy resource assessment for tidal stream generators. J. Power Energy 221 (2), 137–146. Camanho, P., Lambert, M., 2006. A design methodology for mechanically fastened joints in laminated composite materials. Compos. Sci. Technol. 66, 3004–3020. Gowing, S., Coffin, P., Dai, C., 1998. Hydrofoil cavitation improvements with elastically coupled composite materials. In: Proceedings of 25th American Towing Tank Conference, Iowa City, IA. Kinnas, S., Fine, N., 1993. A numerical nonlinear analysis of the flow around two- and three-dimensional partially cavitating hydrofoils. J. Fluid Mech. 254, 151–181. Kinnas, S., Lee, H., Young, Y., 2003. Modeling of unsteady sheet cavitation on marine propeller blades. Int. J. Rotat. Machin. 9 (4), 263–277. Kumar, P., Saini, R., 2010. Study of cavitation in hydro turbines – a review. Renew. Sustain. Energy Rev. 14, 374–383. Liu, Z., Young, Y., 2009. Utilization of bending-twisting coupling effects for performance enhancement of composite marine propellers. J. Fluids Struct. 25, 1102–1116. Motley, M., Young, Y., 2011. Influence of uncertainties on the response and reliability of self-adaptive composite rotors. Compos. Struct. 94 (1), 114–120. Motley, M., Young, Y., 2011. Performance-based design and analysis of flexible composite propulsors. J. Fluids Struct. 27 (8), 1310–1325. Motley, M., Barber, R., 2014. Passive control of marine hydrokinetic turbine blades. Compos. Struct. 110, 133–139. Motley, M., Barber, R., 2014b. Passive pitch control of horizontal axis marine hydrokinetic turbine blades. In: 33rd International Conference on Ocean, Offshore and Arctic Engineering. Motley, M., Liu, Z., Young, Y., 2009. Utilizing fluid–structure interactions to improve energy efficiency of composite marine propellers in specially varying wake. Compos. Struct. 90 (3), 304–313. Nicholls-Lee, R., Turnock, S., Boyd, S., 2013. Application of bend-twist coupled blades for horizontal axis tidal turbines. Renew. Energy 50, 541–550. Polagye, B., Thomson, J., 2013. Tidal energy resource characterization: methodology and field study in admiralty inlet, puget sound, US. Proc. Inst. Mech. Eng. Part A J. Power Energy 227 (3), 352–367. Wang, J., Piechna, J., Müller, N., 2011. A novel design and preliminary investigation of composite material marine current turbine. Arch. Mech. Eng. 58 (4), 355–366. Young, Y., 2007. Time-dependent hydroelastic analysis of cavitating propulsors. J. Fluids Struct. 23, 269–295. Young, Y., 2008. Fluid–structure interaction analysis of flexible composite marine propellers. J. Fluids Struct. 24, 799–818. Young, Y., Kinnas, S., 2001. A bem for the prediction of unsteady midchord face and/or back propeller cavitation. J. Fluids Eng. 123, 311–319. Young, Y., Kinnas, S., 2003. Analysis of supercavitating and surface-piercing propeller flows via BEM. Comput. Mech. 32, 269–280. Young, Y., Michael, T., Seaver, M., Trickey, S., 2006. Numerical and experimental investigations of composite marine propellers. In: 26th Symposium on Naval Hydrodynamics, Rome, Italy. Young, Y., Motley, M., Yeung, R., 2010. Three-dimensional numerical modeling of the transient fluid–structure interaction response of tidal turbines. J. Offshore Mech. Arctic Eng. 132, 011101.
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Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs
Cavitating response of passively controlled tidal turbines Ramona B. Barber n, Michael R. Motley Department of Civil and Environmental Engineering, University of Washington, Seattle, WA 98195, USA
a r t i c l e in f o
abstract
Article history: Received 17 September 2015 Received in revised form 24 June 2016 Accepted 25 August 2016
The use of passive pitch control in marine hydrokinetic turbine (MHK) blades has been shown to provide structural and performance improvements over similar systems with non-adaptive blades. Recent work has demonstrated such advantages as increased lifetime energy capture, reduced hydrodynamic instabilities, and improved efficiency, load shedding, fatigue behavior, and structural integrity. Additionally, passively adaptive blades have been shown to delay cavitation and decrease cavitation volume for marine propellers. For MHK turbines, a reduction in cavitation susceptibility could increase effectiveness of the turbine and reduce the rate of fatigue on the system. In this work, a previously validated, coupled 3-D boundary element method-finite element method solver is used to model two adaptive blades and a non-adaptive reference blade under both uniform and non-uniform inflow conditions. A numerical investigation of the onset of sheet cavitation on an MHK turbine blade and the structural response of a blade under cavitating conditions is presented. Results show that passive control can be used to delay cavitation and decrease cavitation volume under normal operating conditions. Cavitation is shown to increase the rate of fatigue on the system and has the potential to lead to unanticipated failures if proper cavitation analysis is not integral to the MHK turbine design process. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Tidal turbines Cavitation Passive pitch control Adaptive composites Fluid–structure interaction
1. Introduction Sustainably harvesting the kinetic energy of marine tidal currents is becoming an increasingly feasible alternative to the use of traditional power sources. The high energy density and reliability of these currents make them a very attractive option compared to more stochastic renewable sources such as wind and solar power. Theoretically, the energy contained in the world's oceans far exceeds global power demand; studies suggest that marine currents alone have the potential to supply a significant portion of future electricity needs (Ben Elghali et al., 2007; Blunden and Bahaj, 2007). Though there are many benefits to tidal current energy extraction, marine turbines face significant challenges. Blade design is a critical factor in the implementation of marine hydrokinetic (MHK) turbines, as the relatively slender blades must withstand the large, dynamic fluid forces inherent to the marine environment. An additional concern specific to marine turbines is fluid cavitation, which can cause vibrations, performance decay, and an increased rate of fatigue. Maintenance needs are considerably hard to address for underwater turbines purposefully placed in locations of extreme current, necessitating the development of a turbine uniquely suited to its harsh environment and able to operate for long periods of time without need of repair. n
Corresponding author. E-mail address: [email protected] (R.B. Barber).
http://dx.doi.org/10.1016/j.jfluidstructs.2016.08.006 0889-9746/& 2016 Elsevier Ltd. All rights reserved.
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Today, most marine turbine blades are constructed from fiber reinforced composite materials. These materials provide excellent strength-to-weight and stiffness-to-weight ratios, improved fatigue behavior, and are easier to manufacture. An additional benefit of composite materials is that their anisotropic nature allows turbine blades to be hydroelastically tailored during the design process to modify system performance over the expected range of operating conditions. By manipulating the fiber axis of the material, the relationship between bending deformations (blade tip deflection) and twisting deformations (change in blade pitch) can be controlled. Exploiting this intrinsic bend-twist deformation coupling behavior inherent to laminated composites can create a nearly instantaneous, passive mechanism to adapt blade pitch to the flow. Preliminary studies show that these passively adaptive blades can provide system performance improvements such as increased lifetime energy capture, reduced hydrodynamic instabilities, and improved efficiency, load shedding, fatigue behavior, and structural integrity (Motley and Barber, 2014a; Nicholls-Lee et al., 2013). An added benefit of adaptive blades that has not yet been fully explored is their reduced susceptibility to cavitation. Every marine structure that moves at a high velocity relative to the incident flow is at risk of cavitation. In accordance with Bernoulli's principle, an increase in the velocity of a fluid will cause a decrease in hydrostatic pressure. When the pressure in the flow falls below the vapor pressure of the fluid, vapor bubbles form and collapse at a high rate. Cavitation contributes to pitting, corrosion, vibration, and fatigue in many marine structures (Kumar and Saini, 2010); to avoid cavitation, limits are placed on MHK turbine depth, blade length, and rotational speed. Preliminary studies have indicated that the use of bendtwist coupled composite material could reduce cavitation on hydrofoils (Gowing et al., 1998). In research on marine propellers, numerical and experimental results have shown that adaptive composite propellers can provide improved cavitation performance and increased energy efficiency over their non-adaptive counterparts when operating at off-design conditions or in spatially varying flows (Young, 2007, 2008; Motley et al., 2009). Using a previously validated, coupled 3-D boundary element method–finite element method (BEM–FEM) solver, this work presents a numerical investigation of the effect of sheet cavitation on adaptive pitch MHK turbine blades. A set of adaptive blades and a non-adaptive reference blade are modeled in realistic, non-uniform inflow conditions. This study investigates the onset of sheet cavitation on an MHK turbine blade and the structural response of that blade and the full turbine system under cavitating conditions.
2. Turbine parameters The model used for a baseline reference in this work is a two-bladed turbine with a diameter of 20 m. The center of the hub is located 30 m above the sea floor and approximately 15 m below the surface. The reference blade geometry was taken from the optimized blade presented in Bir et al. (2011) and is shown in Fig. 1. The turbine geometry was designed for a variable speedvariable pitch system with a maximum rotational speed of 11.5 rpm; a tip speed ratio of 7 was found to be the optimum operating point until the maximum rotational limit is reached. As the limit on rotational speed was placed in part to avoid the chance of cavitation, that constraint has been removed for this work in order to evaluate the response of adaptive blades to cavitation.
Fig. 1. Schematic of the turbine geometry and wake considered in this work.
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Fig. 2. A schematic of passive pitch deformation, adapted from Barber (2015).
2.1. Passive control of turbine blades In order to avoid overloading the generator associated with an MHK turbine, each system must have a method of limiting power production in extreme flow conditions. Often, modern turbines are built with a motorized control mechanism in the hub of each blade which is programmed to adapt the blade pitch to the instantaneous velocity. This added mechanism relies on system feedback, thus it creates a lag between velocity change and pitch change. In contrast, a passively adaptive rotor blade exploits the anisotropic nature of fiber reinforced composites to create an inherent bend-twist coupled response control mechanism in the blade. In this way, passively controlled blades are able to independently adjust to changing flow conditions without the need for a mechanical driver. Additionally, passive adaptation creates a nearly instantaneous structural response that is difficult to achieve with an active mechanism because of the high flow variation and excitation frequencies in water. The bend-twist deformations are load-dependent and the magnitude and direction of the deformations are controlled by material design; thus it is possible to create an adaptive blade geometry for a more efficient system. To create a passive control mechanism, the material strong axis, defined by the anisotropic nature of the composite fibers, is rotationally offset from the longitudinal axis of the blade by a specified amount. This creates a load-dependent deformation mechanism in which an applied shear load, for example, will result in twisting as well as bending deformations even in the absence of a torsional load. An adaptive turbine blade can be designed to pitch to feather by decreasing the angle of attack or pitch to stall by increasing the angle of attack, depending on composite laminate layup sequence (Motley and Barber, 2014a). Fig. 2 shows a schematic of the passive geometry change. In practice, composite blades are made up of a multitude of layers of composite fibers. It has been shown, however, that a multi-layered structure can be modeled using an equivalent unidirectional fiber angle, θeq, which can be found such that the effective stiffness and degree of bend-twist coupling is approximately equal to that of the physical layup sequence (Liu and Young, 2009; Motley and Young, 2011a,b; Young, 2008). The equivalent unidirectional fiber angle is used to create seven blade models for this study. Three blades were designed to have varying biases to pitch towards stall (θeq < 0°), or increase their angle of attack and therefore decrease blade pitch, and three to pitch to feather (θeq > 0°), decreasing angle of attack and increasing blade pitch. The six adaptive blades were compared to a non-adaptive (θeq = 0°) reference blade. The composite material modeled is the carbon fiber reinforced polymer Hexcel IM7–8552, with material properties as defined in Table 1, where the 1-axis is defined as parallel to the fibers and the 2-axis as perpendicular to the fibers within each laminate layer (Camanho and Lambert, 2006).
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Table 1 Material properties of Hexcel IM7–8552 (Camanho and Lambert, 2006).
E1 = 171.42 GPa E2 = E3 = 9.08 GPa G12 = G13 = 5.29 GPa
G23 = 3.13 GPa ν12 = ν13 = 0.32 ν23 = 0.45
2.2. Effects of cavitation Cavitation susceptibility is a major limit state in modern marine turbine design. Research has shown that cavitation contributes to pitting, corrosion, vibration, and fatigue of marine structures (Bahaj and Myers, 2003; Wang et al., 2011). Additionally, the vibration caused by cavitation dramatically increases noise generation, a particular concern of turbine systems located in or near sensitive marine mammal habitats. The onset of cavitation also has a drastic effect on the power generation of the system. In general, the lift and drag coefficients of a hydrofoil do not vary as the ambient pressure decreases; the pressures on the upper and lower surfaces decrease in tandem. However, as cavitation develops along the top of the foil, the pressure on the top surface can no longer decrease because it is limited by the vapor pressure. The pressure on the bottom surface is not restricted, however, and continues to fall, progressively approaching the pressure on the top of the foil. As the pressure differential across the foil decreases, there is a corresponding decrease in the lift coefficient. Additionally, the resultant change in the shape of the pressure distribution causes an increase in drag. Thus, cavitation causes both a decrease in lift and an increase in drag, limiting the power generation as well as exacerbating the loads on the system. For these reasons, it is essential to include the study of cavitation in the analysis and development of MHK systems. Though the effects of cavitation have been studied on other marine structures, little research has been conducted on the possible impact of cavitation on an MHK turbine blade. Understanding the structural response of a cavitating blade is essential to predicting system life and the susceptibility of the blade to various failure modes.
3. Numerical analysis 3.1. Fluid–structure interaction In order to predict turbine response, accurate modeling of the fluid–structure interaction (FSI) effects that cause the loaddependent deformation of the blade is essential. To this end, turbine blades are modeled with a previously developed and validated, fully coupled boundary element method–finite element method (BEM–FEM) solver (Kinnas and Fine, 1993; Kinnas et al., 2003; Young and Kinnas, 2001, 2003; Young et al., 2006,2010; Young, 2007, 2008; Motley et al., 2009) that includes Froude number effects and is capable of considering spatially varying inflow and unsteady sheet cavitation on both sides of the blade surface. In the BEM code, the fluid behavior is assumed to be governed by the incompressible Euler equations solved in a bladefixed rotating coordinate system. The total velocity is expressed as the vector sum of the inflow velocity and the perturbation velocity potential, ∇Φ , corresponding to the turbine-induced flow field. The perturbation flow field Φ can be treated as incompressible, inviscid, and irrotational and satisfies the Laplace equation ∇2Φ = 0. To consider FSI effects, the perturbation potential Φ is linearly decomposed into two parts: a part due to large rigid blade rotation (ϕ) and one due to small elastic blade deformation (φ). Each part is solved separately using a 3-D BEM by applying appropriate kinematic and dynamic boundary conditions (see Young, 2007, 2008; Young et al., 2010). The decomposition of Φ allows the total pressure to be similarly separated into rigid blade (Pr) and elastic blade (Pv) components as follows:
⎡1 ⎤ ∂ϕ 1 Pr = ρ⎢ |Vin|2 − − |Vtr|2 ⎥ ⎣2 ⎦ ∂t 2
(1)
⎡ ∂φ ⎤ − Vin·∇φ⎥ Pv = ρ⎢ − ⎣ ∂t ⎦
(2)
where Vin is the inflow velocity and Vtr is the fluid velocity due to rigid blade rotation. Likewise, the total hydrodynamic force can be expressed as a sum of the forces due to rigid blade rotation and those due to elastic blade deformation. User-defined sub-routines in the commercial FEM solver ABAQUS/Standard are employed to obtain these forces from the decomposed pressure output of the BEM solver by applying the respective pressure terms to the nodes in the FEM model via the element shape functions, i.e,
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∫ ⎡⎣ N⎤⎦T {Pr}dS T {Fv } = ∫ ⎡⎣ N⎤⎦ {Pv }dS {Fr} =
(3) (4)
The hydrodynamic force due to elastic blade deformation can be expressed as functions of nodal velocity and acceleration and written
{Fv } = − [MH]{u¨ } − [CH]{u̇ }
(5)
where the [MH] and [CH] matrices represent the hydrodynamic added mass and damping inherent to a structure moving through a dense fluid. ABAQUS is then used to solve the final dynamic equation of motion,
( ⎡⎣ M⎤⎦ + ⎡⎣ MH⎤⎦){ u¨ } + ( ⎡⎣ C⎤⎦ + ⎡⎣ CH⎤⎦){ u̇ } + ⎡⎣ K⎤⎦{ u} = { Fce} + { Fco} + { Fr}
(6)
where {u¨ }, {u̇ }, and {u} are the structural nodal acceleration, velocity, and displacement vectors, respectively; [M], [C], and [K] are the structural mass, damping, and stiffness matrices; {Fce} is the centrifugal force and {Fco} is the Coriolis force. The blades are represented as fully fixed at the hub connection. The effects of large, nonlinear blade deformations are considered by iterating between the fluid and solid solvers until the solution converges. Detailed formulations of the solution algorithm, governing equations, and boundary conditions can be found in Young (2007, 2008), Motley and Barber (2014b) and Barber (2015). 3.2. Cavitation prediction For the purposes of cavitation prediction and analysis, a pressure coefficient is defined relative to the local flow,
−CP =
P0 − Pt 1 ρn2D2 2
(7)
where P0 is the freestream pressure, Pt is the total pressure at the point in question, ρ is the fluid density, n is the rotational speed, and D is the turbine diameter. The local velocity squared term n2D2 is used instead of the more common freestream velocity squared in order to facilitate a direct comparison with the cavitation number,
σn =
P0 − Pv 1 ρn2D2 2
(8)
where Pv is the saturated vapor pressure in the fluid. Thus, if at any point the pressure coefficient CP falls below the cavitation number sn, the potential for cavitation development exists. To determine the existence and extent of the potential cavity planform at each step, a Newton–Raphson iteration is used until the cavity closure condition is satisfied. The cavity closure scheme used in this method requires both the cavity height and the cavity thickness to equal zero on the trailing edge of the planform. Dynamic and kinematic boundary conditions are applied to the cavitating and wetted surfaces to uniquely define the perturbation potential. These conditions include requirements for flow tangency to both the turbine blades and cavity surfaces, constant pressure along the cavity equal to the vapor pressure of the fluid, and the cavity closure requirements noted above. This process is described in depth in Kinnas et al. (2003), Young and Kinnas (2001), and Young (2008). This method of sheet cavitation prediction has been extensively validated against experimental results for rigid and adaptive marine propellers and rigid turbine blades (Kinnas and Fine, 1993; Kinnas et al., 2003; Young and Kinnas, 2001; Young, 2007, 2008; Young et al., 2010). There are no experimental data for adaptive turbines available for validation at this time; this is a focus of future work for the authors. 3.3. Inflow velocity profile A uniform, steady inflow velocity profile is often adequate to obtain basic system performance metrics and observe the structural behavior of an MHK turbine. The analysis associated with such a profile is relatively simple and fast compared to the time-dependent problem of a non-uniform inflow profile. In actual marine conditions, however, an MHK turbine operates in a more complex field, and a realistic, non-uniform flow profile is required for a more robust understanding of the system dynamics. An example of the flow a turbine might realistically experience is shown in Fig. 3. This data was collected to inform the design, siting, and permitting of a pilot scale tidal energy array in Admiralty Inlet in Puget Sound, WA (Polagye and Thomson, 2013). Fig. 3 demonstrates the highly dynamic nature of the instantaneous velocity profile; though this profile is difficult and impractical to model, boundary layer effects can be approximated with a power-law estimate of the flow velocity across the water column to estimate dynamic effects. Here, a 1/7th-power law is applied, as shown in Fig. 4. The fluid velocity, V, at a specific height above the sea floor, h, is defined as
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Fig. 3. Top: Measured flow direction and depth-averaged velocity magnitude data from Admiralty Inlet, WA. Bottom: Measured velocity magnitude over depth from Admiralty Inlet, WA.
Fig. 4. Schematic of the non-uniform velocity profile applied in the unsteady analysis.
Fig. 5. Predicted change in pitch for six adaptive blades compared to a non-adaptive reference blade. The presence of cavitating conditions is indicated with larger, yellow-filled symbols and dotted lines. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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Fig. 6. Pressure distribution and presence of cavitation (shown in white) on the suction and pressure sides of the reference blade for one full rotation at V ¼ 3.0 m/s.
⎛ ⎞ ⎛ h ⎞1/7 V ⎜ h⎟ = Vavg ⎜ ⎟ ⎜ ⎟ ⎝ h0 ⎠ ⎝ ⎠
(9)
where Vavg is the mean inflow velocity in the turbine's plane of rotation and h0 is the distance from the center of the hub to the sea floor. This model does not produce a realistic representation of any instantaneous velocity field but is accurate as a time-averaged profile, which captures the essential velocity gradient of the flow. Including this gradient allows for the consideration of the periodic, unbalanced loading caused by boundary layer effects; in many cases, these forces have a significant impact on the performance and structural response of the turbine system.
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Fig. 7. Cavitation volume normalized by turbine radius R3 over one full rotation at increasing inflow velocities.
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Fig. 8. Predicted thrust coefficient on each blade over one full rotation at V ¼ 2.7 m/s and V¼ 2.9 m/s.
4. Results 4.1. Cavitation inception and growth To begin a study of cavitation, it is logical to start with an investigation into the conditions at which the onset of cavitation occurs. Cavitation inception in this case refers to the onset of sheet cavitation, in accordance with previous work by the authors. In this study, three pitch to stall blades (θeq < 0°) and three pitch to feather blades (θeq > 0°) with increasing levels of flexibility are compared to a non-adaptive (θeq = 0°) reference blade. The higher end of the velocity spectrum presented above is examined to predict the potential for the development of cavitation on each blade. The tip speed ratio for all following analyses is held constant at 7 in accordance with the optimal operating point find in Bir et al. (2011). Fig. 5 shows the change of pitch for each blade as a function of velocity, with indications marking cavitating conditions. The blades experienced pitch change as designed, though the reference blade exhibited a minor degree of feathering due to the inherent flexibility in the material and bias of the geometry. The effect of blade fiber angle on the onset of cavitating conditions is clear in this figure; the blades with a pitch to stall bias show cavitation at lower velocities than the reference blade, while the pitch to feather blades delay cavitation. This trend is strongest in the blades with the largest adaptive biases. After the onset of cavitating conditions, the blades tend to experience an increase in the rate of pitch change. This is likely due to the increased forces associated with cavitation. Fig. 6 shows the pressure distribution and presence of cavitation on the pressure and suction sides of a single blade as it rotates. The particular blade shown in this figure is the 0° reference blade at an inflow velocity V¼3.0 m/s, which is well representative of any moderately cavitating blade in this study. The onset of cavitation appears on the suction side of the
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Fig. 9. Predicted maximum displacement of each blade over one full rotation at V¼ 2.7 m/s and V ¼2.9 m/s.
blade as expected, initially at the point of the blade's rotation nearest the free surface, where the fluid flow is strongest and the surrounding pressure is lowest. As cavitation progresses, it begins to spread and appear throughout the upper half of the rotational sweep of the blade as shown in Fig. 6; however, due to the pressure and velocity differentials between the extreme blade positions, at no point is it possible for the cavitating volume to be constant around a full revolution. Thus, as the blade advances, cavitation forms and disappears in each rotation, causing a periodic and unbalanced loading. This periodic effect, as well as the significance of the increased rate of pitch change under cavitating conditions, can be seen clearly in Fig. 7. This figure shows the volume of cavitation on the reference and adaptive blades during a full revolution from V ¼2.7 m/s to V ¼3.0 m/s. Here the cavitation volume is normalized by the turbine radius R3. As the inflow velocity increases, cavitation forms and grows on each blade. The blades with a pitch to stall bias start to cavitate first, with the largest bias blades showing the most cavitation. This is consistent with the onset of cavitating conditions shown in Fig. 5. The cavitation volume on these blades grows dramatically with a small increase in velocity, due to the increasing rate of pitch change shown above. The pitch to feather blades, however, delay cavitation compared to the pitch to stall and reference blades. The feathering blades show both a volume reduction and slower cavitation growth compared to the other blades, as the increased rate of pitch change reduces the angle of attack significantly. Again, the benefit is greatest in the blades with the largest adaptive bias. It is also possible to compare the extent of the revolution of each blade for which cavitation is apparent. At the onset of cavitation, the cavity volume is visible only around the top of the blade's rotation. For blades operating at a more advanced state of cavitation, cavitation is apparent through more than half of the rotation. In all cases shown, the bottom of the revolution, nearest the sea floor, is free of cavitation.
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Fig. 10. Harmonic excitation in the blade due to axial forces, normalized by the 0th harmonic amplitude.
4.2. Structural response of a cavitating blade The onset of cavitation has a strong impact on blade forces and therefore deformation. For a more detailed analysis of structural response, this section focuses on two adaptive blades (θeq = ± 5°) and the reference blade (θeq = 0°). The fluctuation of forces on the system in cavitating conditions can be seen in the unsteady nature of the thrust coefficient (CT), shown in Fig. 8. At V¼2.7 m/s, the −5° blade has just entered cavitating conditions. There is very slight unsteadiness evidenced in the CT at that inflow velocity, but in general the CT curves for all three blades are smooth. At V¼2.9 m/s, however, there are large fluctuations in the CT for the −5° pitch to stall blade, and unsteadiness in all three blades. The +5° pitch to feather blade has the smoothest CT profile of the three, corresponding to the minimal cavitation volume shown in Fig. 7. The effect of these fluctuating forces can be seen in the maximum tip displacement of each blade, shown in Fig. 9. The magnitude of the displacement in both adaptive blades is higher than that of the reference blade, with the pitch to stall blade undergoing moderately larger deformations than the feathering blade. This result is consistent with earlier findings and is due to the fact that the pitch to stall blade experiences slightly higher forces than the feathering blade. The notable feature of these figures is the difference that the onset of cavitation creates. At V¼ 2.7 m/s, the 0° and +5° blades are fully wetted while the −5° blade has just entered cavitating conditions. The smooth lines in the wetted blade profiles show the gradual change in blade deformation brought on by the rotation though a non-uniform velocity profile, while the pitch to stall blade shows the effects of cavitation through very small oscillations in the displacement curve. As the fluid velocity is increased to V¼2.9 m/s, cavitation is apparent on all three blades. The feathering blade is only just beyond the onset of cavitation and shows small oscillations similar to the pitch to stall blade at V¼2.7 m/s. However, the other two blades show more significant cavitation and experience strongly unsteady displacements. These fluctuating effects in the blade displacements will increase the rate of fatigue on the system and could lead to unanticipated failures if proper cavitation analysis is not integral to the design process.
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Fig. 11. Harmonic excitation in the blade due to moment forces, normalized by the 0th harmonic amplitude.
The structural effects of fluttering caused by cavitation can also be seen in a harmonic analysis. Both the individual blade harmonics (defined with respect to blade-fixed coordinates) and the shaft, or full system, harmonics (in global coordinates) for the reference blade and the two adaptive pitch blades at V ¼2.7 m/s and 2.9 m/s are presented in this section. For ease of comparison, the magnitudes of the harmonic excitation are normalized by the amplitude of the 0th harmonic. The harmonics for a fully wetted analysis are compared to those found by the cavitating analysis in each figure for reference. The axial force blade harmonics are shown in Fig. 10. At the lower inflow velocity V¼2.7 m/s, low level excitation is found for the −5° blade in the higher harmonic modes, reflecting the onset of cavitating conditions. Very slight excitation is also found in the mid-level modes for the 0° reference blade; this likely reflects the imminent onset of cavitation for that blade. At V¼2.9 m/s, excitation is found in the higher modes of all three blades in the cavitation analysis that cannot be seen in the simple wetted analysis. The +5° feathering blade is shown to perform significantly better than both the reference and pitch to stall blades, in that it experiences much less excitation in the higher harmonic modes. This reflects the lower volume of cavitation activity on the blade (as shown in Fig. 7). The same pattern can be seen in the harmonics due to the moment on the blade, shown in Fig. 11. The high mode excitation is slightly stronger in this figure, due to the twisting nature of the blade. Again, solely the cavitating blades show high mode excitation. Similarly, Fig. 12 shows the harmonic response on the full system due to torque. The higher-mode excitation due to cavitation is significantly greater here than in the single blade analysis, due to the fact that cavitation is initiating and disappearing as each blade nears and passes the apex of its rotation, creating an uneven forcing on the system. In this figure, as in Figs. 10 and 11, the pitch to feather blade shows much weaker excitation in the higher modes for both the blade and shaft harmonics. It is essential to identify these high-frequency harmonic excitations induced by cavitation, as they contribute to blade and system fatigue over the lifetime of a turbine. Though the high-frequency harmonics are a concern for fatigue, they are not likely to cause resonance in the system.
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Fig. 12. Harmonic excitation in the system due to torque, normalized by the 0th harmonic amplitude. Table 2 Modal frequencies (Hz). Mode
θeq = − 5°
θeq = 0°
θeq =+ 5°
1 2 3 4 5
4.916 15.167 20.455 23.414 26.122
5.097 16.098 19.127 24.871 26.029
4.519 15.265 19.947 24.701 26.096
Table 2 shows the associated modal frequencies for the first five modes in each blade. These frequencies are much higher than the rate of rotation; at V¼3.0 m/s, the system is rotating at 0.334 cycles/s.
5. Conclusions and future work In this work, a numerical study of the effect of cavitation on adaptive pitch MHK turbine blades is presented. A previously validated, coupled 3-D BEM–FEM solver is used to model two sets of adaptive blades and a non-adaptive reference blade under realistic, non-uniform inflow conditions. Specifically, this study investigates the onset of cavitation on an MHK turbine blade and the structural response of that blade and the full turbine system under cavitating conditions. Numerical
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analysis suggests that passively adaptive pitch to feather blades can be used both to delay cavitation and to reduce cavitation volume over the blade surface. An analysis using a non-uniform inflow representative of the inflow in Admiralty Inlet, WA predicted cavitating conditions at well within the range of normal operating conditions and demonstrated the importance of cavitation analysis and design in MHK turbine development. After the onset of cavitation, all three blades in this study exhibited a similar structural response. A blade at the onset of cavitation will start to experience oscillations in thrust and therefore resulting deformations. As the cavitation volume increases, stronger fluctuations in the blade displacement are found. A harmonic analysis of each blade reflects this fluttering, showing high-frequency harmonic excitation in a cavitating blade that is not present in an equivalent fully wetted blade. The fluctuations in deformation combined with the higher mode excitation will increase the rate of fatigue on the system and could lead to unanticipated failures if proper cavitation analysis is not integral to the design process. The response to cavitation is identical regardless of material orientation; thus, to reduce the negative effects of cavitation it is necessary to design a system that will avoid cavitation over as much of the normal operating range as possible. The benefit found in the passively adaptive, pitch to feather blade in this study is in its ability to both delay the onset of cavitation and to maintain a reduced cavity volume after that point, allowing the system a greater amount of time spent operating smoothly and decreasing the exposure to fatigue-inducing oscillations. The results of this research suggest that an adaptive blade, designed carefully for site-specific parameters, could push the onset of cavitating conditions beyond the range of normal operation and effectively avoid the effects of cavitation in all but the most extreme conditions. Further study is needed to complete the knowledge necessary to design such a blade, however. While the work presented here provides an insight into cavitation response, it considers only sheet cavitation, and added complexities associated with other modes such as tip vortex cavitation or cavitation collapse will require supplementary research. Additionally, the analysis of a larger suite of blade profiles will add to the understanding of the potential benefits to be gained with the use of an adaptive pitch mechanism. Finally, a full system harmonic analysis is needed to adequately explore the possibility of resonant excitation. Acknowledgments This work was facilitated through the use of advanced computational, storage, and networking infrastructure provided by the Hyak supercomputer system, supported in part by the University of Washington eScience Institute.
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