Two-Way Fluid–Solid Interaction Analysis for a Horizontal Axis Marine Current Turbine with LES

Article Two-Way Fluid–Solid Interaction Analysis for a Horizontal Axis Marine Current Turbine with LES

Jintong Gu 1,2, Fulin Cai 1,*, Norbert Müller 2, Yuquan Zhang 3 and Huixiang Chen 2,4

1 College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China; [email protected] 2 Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA; [email protected] (N.M.); [email protected] (H.C.) 3 College of Energy and Electrical Engineering, Hohai University, Nanjing 210098, China; [email protected] 4 College of Agricultural Engineering, Hohai University, Nanjing 210098, China * Correspondence: ﬂ[email protected]; Tel.: +86-139-5195-1792

Received: 2 September 2019; Accepted: 23 December 2019; Published: 27 December 2019

Abstract: Operating in the harsh marine environment, fluctuating loads due to the surrounding turbulence are important for fatigue analysis of marine current turbines (MCTs). The large eddy simulation (LES) method was implemented to analyze the two-way fluid–solid interaction (FSI) for an MCT. The objective was to afford insights into the hydrodynamics near the rotor and in the wake, the deformation of rotor blades, and the interaction between the solid and fluid field. The numerical fluid simulation results showed good agreement with the experimental data and the influence of the support on the power coefficient and blade vibration. The impact of the blade displacement on the MCT performance was quantitatively analyzed. Besides the root, the highest stress was located near the middle of the blade. The findings can inform the design of MCTs for enhancing robustness and survivability.

Keywords: marine current turbine; large eddy simulation; two-way ﬂuid–solid interaction; vibration; stress; vortex

1. Introduction As a kind of clean energy with huge reserves, marine current can be predicted and utilized accurately [1–3]. The marine current turbine (MCT) was adapted first from wind turbines, and considerable part of the technology can be directly applied. However, the density of water is about 800-times higher than that of air. Compared with wind turbines, water loads on MCTs are much larger. Besides, there are more challenges, such as biofouling, marine debris, and cavitation [4]. In the early stage, the experimental research on MCTs mainly focused on fluid dynamics and performance. Coiro et al. [5] obtained the characteristics of horizontal axis tidal current turbine power and thrust for a range of tip speed ratio (TSR) and hub pitch angle through towing tank tests. Bahaj et al.[6] investigated the effect of flow speed, rotation speed and hub pitch angle on MCT performance in a towing tank and a cavitation tunnel. They also studied the secondary effects of rotor depth, yaw and dual rotor interference. Song et al. [7] focused on the tip vortex generated by different raked blade tip shapes to optimize the performance of MCTs and verified their results by towing tank experiments with models. Gaurier et al. [8] studied the influence of wave–current interactions on the MCT performance and measured strains of blades in a flume tank. An increasing number of researchers are interested in developing numerical methods that could simulate complex flows near MCTs. Those are also useful for field or laboratory tests because they

Water 2020, 12, 98; doi:10.3390/w12010098 www.mdpi.com/journal/water Water 2020, 12, 98 2 of 16 could afford more detailed hydrodynamic information about the region of interest and accurately predict the performance of the turbine. The early famously employed methods are the actuator disc theory, which ignores the shape of rotor, and blade element momentum method, which ignores the span-wise flow. These simplifications greatly reduce the computational cost. The former one can well-predict the wake field far away from the turbine [9–11], and the latter provides a satisfactory results of turbine performance [12,13]. However, the flow structure around the rotor, particularly the tip vortices, and the load on the blade cannot be fully described [14,15]. Complex and strong turbulence near the MCT is caused by fluid–solid interaction, including wake vortex generation, dynamic stall, or instability of hydrodynamic loads. Recently, computational fluid dynamics (CFD) methods were widely employed that use 3D cells to represent explicitly the MCT’s geometry and motion. The fluid field is computed using Reynolds-averaged Navier–Stokes (RANS) or large eddy simulation (LES) as a turbulence model. RANS models are justifiably popular, as the computational requirements are affordable though the time-averaging of the velocity field does not realistically represent the time-varying fluctuations associated with blade-generated turbulence or instantaneous fluid–blade interaction [14,16]. By contrast, LES can resolve large-scale flow structures present in the velocity field [17,18]. Furthermore, increasing computational technology efficiently compensates the known drawbacks of LES, namely its high computational cost, especially for highly turbulent flows. This makes it more possible to have highly resolved dense meshes and obtain detailed flow information. Consequently, some researchers have used LES to investigate the fluid dynamics and performance of MCTs. McNaughton et al. [19] applied sliding mesh LES method to model the flow and tip vortices of MCT, which were influenced by the support. Kang et al. [20] utilized curvilinear immersed boundary LES method to investigate the wake of MCT. Most of previous studies have investigated the performance of MCTs and flow field with little consideration of structural response. To reduce the maintenance cost and risk, Singh et al. [21] transferred the blade surface pressure from the steady flow field simulation to the structural analysis code. Then, the stress, strain, and deformation were calculated to verify the safety of the blades. Due to the unstable current and induced alternating loads, MCTs face the possibility of fatigue failure. Also, it is beneficial to consider the interaction between the structure and the flow field in evaluating the performance and the details of the structure response of MCTs. Badshah et al. [22] compared the one-way and two-way FSI method applied to an MCT modeled as a structural-steel solid. The maximum blade displacement was only 1/2000 of its radius. In addition, the RANS model was used, which cannot reflect the influence of the blade variation on the flow. Bai et al. [23] presented an immersed boundary method and two free surface methods to simulate the MCT with FSI in a relatively small computational domain without consideration of the support or discussion of the blade deformation. The object of this paper was to investigate the interaction between the fluid field and MCT with support in a two-way FSI, using LES. LES can more accurately solve large energetic scales in the flow field better than RANS [24], thus providing more details for the FSI simulation of the MCT rotor that is typically subjected to considerable turbulence, and in reverse for the impact on the performance of MCT and the flow field. The model was validated by comparing the predicted thrust and power coefficients with experimental measurements for a 0.8-m MCT presented in the work of Bahaj et al. [6].

2. Computational Equations and Method

2.1. LES Governing Equations In a turbulent ﬂow, eddies of various sizes are generated. Direct numerical simulation (DNS) could, theoretically, directly solve the whole spectrum of turbulent length scale. However, its cost is proportional to the cube of Reynolds number. For most realistic engineering problems concerning high Reynolds number ﬂows, DNS is still too expensive. Water 2020, 12, 98 3 of 16

Mass, momentum, and energy are transported mainly by large eddies, which are dictated by boundary conditions and geometries within the involved flow. Small eddies depend less on the geometries. With LES, larger eddies that can be captured by cells are resolved directly, whereas the effects of smaller eddies that cannot be captured by cells are accounted for with modeling, indirectly, using a sub grid model. This involves a filtering process, which makes it possible to simulate the transient flow with high-performance computers, although it still needs sufficient space and time resolution. A filtered variable (denoted by an overbar, such as Φ) at a point x is defined by the following convolution: Z Φ(x) = Φ(x0)G(x, x0)dx0, (1) D where D is the fluid domain, and G is the filter function determining the scale of the resolved eddies. The finite-volume discretization itself implicitly affords the filtering procedure: Z 1 ϕ(x) = ϕ(x0)dx0, x0 V, (2) V ∈ V where V is the volume of a computational cell. The filter function, G(x, x0), implied here is then

( 1 V , x0 V G(x, x0) = ∈ . (3) 0, x0 otherwise

For incompressible ﬂows, ﬁltering the continuity and momentum equations, one obtains

∂u i = 0, (4) ∂xi and ∂ u u ∂ui i j ∂τij 1 ∂p ∂σij + + = + f i + , (5) ∂t ∂xj ∂xj −ρ ∂xi ∂xj where u is the velocity vector, i and j range from 1 to 3 to denote three directions and velocity components respectively, ρ is the density, f is a volume forcing term, and σij is the stress tensor due to molecular viscosity expressed by ! ∂ui ∂uj 2 ∂ui σij υ + δij , (6) ≡ ∂xj ∂xi − 3 ∂xi and τij is the subgrid-scale stress expressed by

τ u u u u , (7) ij ≡ i j − i j where v is the kinematic viscosity, δij is Kronecker delta, which is 1 if i = j, and 0 otherwise. The subgrid-scale stress is unknown and requires modeling. Here, the Boussinesq hypothesis is adopted to compute the subgrid-scale stress 1 τ τ δ = 2vtS , (8) ij − 3 kk ij − ij where τkk is the isotropic part of the subgrid-scale stresses, which is not modeled but added to the ﬁltered static pressure term. Sij is the rate-of-strain tensor for the resolved scale expressed by ! 1 ∂ui ∂uj Sij + , (9) ≡ 2 ∂xj ∂xi Water 2020, 12, 98 4 of 16

and vt is the subgrid-scale turbulent viscosity. Here, the Wall-Adapting Local Eddy Viscosity (WALE) Model is adopted, it is modeled by

3/2 d d SijSij 2 υt = LS , (10) 5/2 5/4 + d d SijSij SijSij

d where Ls and Sij are deﬁned, respectively, as

1/3 Ls = min κd, CwV , (11)

d 1 2 2 1 2 ∂ui Sij = gij + gij δij gkk, gij = , (12) 2 − 3 ∂xj where κ is the von Kármán constant, d is the distance to the nearest wall, and Cw, the WALE constant, is 0.325, which has produced consistently superior results under intensive validation [25].

2.2. Solid Governing Equation The conservation equation mainly used in solid solution is as follows:

.. ρsds = fs + σs, (13) ∇· .. where, ρs is the solid density, ds is the local acceleration vector, fs is the body force, including the centrifugal force and ﬂuid ﬁeld pressure, σs is the Cauchy’s stress tensor expressed by

σs = Hsεs, (14) where, Hs is the tensor of the elastic coeﬃcients, εs is the Cauchy’s strain tensor expressed by

1 T εs = ds + ( ds) , (15) 2 ∇i ∇i where, ds is the solid displacement [26].

2.3. Two-Way FSI Iterative Computation In two-way FSI analysis, the fluid pressure is applied to the solid, where, in turn, the deformation of the solid affects the flow field. The computational domain are generally divided into fluid field and solid field, both of which are defined by materials, boundaries, and initial conditions. The interaction occurs at the interface between two fields. Two-way FSI iterative computation is also called partition method because the fluid equation and the solid equation are solved separately and continuously, and always update the latest information provided by the other part of the coupling system. This iteration continues until the solution of the coupling equation converges. The calculation process could be summarized as: getting the solutions at t + ∆t by iterating between the fluid model and the solid model. Assume the initial solution

1 0 t ds− = ds = ds, (16)

τ0 = τt f f , (17) where, τ f is the ﬂuid stress. With iteration k = 1, 2, ..., the equilibrium solution for each time step Xt+∆t is obtained. Water 2020, 12, 98 5 of 16

k (1) The fluid solution vector X f is obtained from k k 1 k 2 F X , λ d − + (1 λ )d − = 0, (18) f f d s − d s where, F f represents the correlative fluid formulas of the terms in the brackets. In fluid analysis, the solution is obtained by specifying the solid displacement. Since the fluid and the solid are not solved in the same matrix, the solid displacement was solved using the relaxation factor 0 < λd < 1, which is conducive to iterative convergence. (2) The stress residuals are calculated and compared with the tolerance due to the need to meet the stress criteria. When the criteria are met, steps 3–5 are skipped. k (3) The solid solution vector Xs is obtained from k k k 1 Fs X , λττ + (1 λτ)τ − = 0, (19) s f − f where, Fs represents the correlative solid formulas of the terms in the brackets. Likewise, stress relaxation factor λτ (0 < λτ < 1) is used in fluid stress. (4) The fluid node displacement is determined by the specified boundary conditions

k k k 1 d = λ d + 1 λ d − . (20) f d s − d s (5) Similarly, the displacement standard needs to be met, and the displacement residual is calculated and compared with the tolerance. If there is no convergence, the process returns to step 1 and iterates again. (6) The ﬂuid and solid solutions is saved.

3. Computational Domain and Veriﬁcation

3.1. Test Case Description Experimental data is available for turbine performance characteristics in terms of power and thrust coefficients for ranges of TSR and blade pitch settings. In this case, the data from the towing tank tests of Bahaj0s group at the University of Southampton [6] were modeled because the geometry of the MCT was provided along with data for model verification. Although there was a lack of experimental data for the velocity distribution in the wake or for the pressure distribution along the blades, additional CFD simulations were conducted to study the sensitivity of these parameters to TSR and inflow turbulence. Calculations were carried out with both RANS [15,27,28] and LES models [19,29]. The MCT tested in [6] and modeled within this research was equipped with three blades and a rotor with diameter of 0.8 m. The diameter of the nacelle and of the support was 0.1 m. The blades were developed from NACA 63-8xx profiles, with chord, thickness, and pitch distributions given at 17 locations along the blade. The towing tank had a depth and width of 1.8 m and 3.7 m, respectively, resulting in a blockage ratio of 7.5%. The rotor was centered 0.96 m above the bed. Among the results of the power coefficient (CP) and thrust coefficient (CT) from the tank test, the MCT with 20◦ hub pitch angle at zero yaw angle has the best performance. The 20◦ hub pitch angle refers to the nose-tail angle of the blade at a radius of 0.2R (R is the rotor radius), while the pitch angle at the blade tip is 5◦.

3.2. Computational Domain Figure1 illustrates the computation domain as it extends a distance of 5 D upstream and 15D downstream from the point of origin in the rotor, where the rotational axes for the blade pitch cross. The domain was divided into an external domain and an internal domain of the rotor with sliding mesh [16]. The internal domain had a diameter of 1.5D (where D is the diameter of the rotor) and extended to a distance of 1/4D upstream and 1/8D downstream from the point of origin in the rotor. Water 2020, 12, 98 6 of 16

For visualization purposes, only in Figure1, the hexahedral mesh is given as discretized with 3 × 106 cells. Figure2a shows the surface of the coarse 3 106 cells mesh on the rotor, hub and support. × Figure2b gives an enlargement of a part of the blade surface mesh around r = 0.4R. Figure2c shows the blade surface mesh around r = 0.4R with a mesh three-times denser in all three directions, resulting in 81 106 cells. Because LES performance is improved with a ﬁner mesh for resolving even smaller × eddies, the actual simulation was performed with a mesh of 100 106 cells as shown in Figure2d,e, × where the wall-normal and wall-parallel (streamwise and spanwise) spacing was decreased in all three directions as a wall was approached (Figure2e). The adaptive mesh method was used via HyperMesh to avoid high near-wall aspect ratio, which remained between 1 and 6. Finally, the blade surface mesh consisted of 145 nodes in the chordwise and 756 nodes in the spanwise directions. The y plus was about 0.4 to 3 at r = 0.4R. WaterWater 2019 2019, ,11 11, ,x x FOR FOR PEER PEER REVIEW REVIEW 66 ofof 1616

6 FigureFigure 1.1. Computation domain discretized withwith 33 × 106 hexahedralhexahedral cells. cells. Figure 1. Computation domain discretized with 3 ×× 10 hexahedral cells.

FigureFigure 2. 2.2. ( (a(aa)) )The TheThe surface surface meshmesh mesh ofof of the the the rotor, rotor, rotor, hub hub hub and and and support support support in coarsein in coarse coarse mesh; mesh; mesh; (b) The( b(b) )The bladeThe blade blade surface surface surface mesh = = mesharoundmesh aroundaroundr 0.4 𝑟R𝑟;(c =) = The0.4𝑅 0.4𝑅; ; blade (c(c) ) TheThe surface bladeblade mesh surfacesurface around meshmeshr around0.4aroundR densiﬁed 𝑟𝑟 = = 30.4𝑅 0.4𝑅 times densifieddensified in all 3 directions; 33 timestimes inin ( d all)all The 33 = directions;bladedirections; surface ( d(d) ) meshThe The blade blade acturally surface surface used mesh mesh in the acturally acturally simulation; used used (e in )in The the the sectionsimulation; simulation; at r ( e(0.4e) )The RThewith section section adapted at at 𝑟=0.4𝑅𝑟=0.4𝑅 mesh. withwith adapted adapted mesh. mesh.

3.3.3.3. Verification Verification of of the the Model Model BeforeBefore the the FSI FSI simulation, simulation, the the fl fluiduid model model was was verified. verified. Fluent Fluent 19.1 19.1 in in ANSYS ANSYS workbench workbench was was usedused for for the the CFD CFD simulation. simulation. The The above-described above-described LE LESS turbulence turbulence model model was was applied. applied. The The top top of of the the computationalcomputational domain domain was was set set as as a a slip slip wall, wall, and and the the inlet inlet velocity velocity to to the the computational computational domain domain was was uniformly 𝑈 =1.4 m/s. The outflow boundary condition was used TSR = 𝜔𝑅/𝑈 was varied in the uniformly 𝑈=1.4 m/s. The outflow boundary condition was used TSR = 𝜔𝑅/𝑈 was varied in the rangerange between between 4 4 and and 12 12 by by changing changing rotational rotational speed speed 𝜔𝜔. .The The Reynolds Reynolds number, number, based based on on the the inlet inlet 6 −5 velocityvelocity and and diameter, diameter, was was 1.12 1.12 × × 10 106. .Convergence Convergence was was specified specified as as RMS RMS residual residual of of 1 1 × × 10 10−5. .The The mainmain nondimensional nondimensional load load parameters parameters were were 𝑄𝜔𝑄𝜔 powerpower coefficient: coefficient: 𝐶 𝐶== , , (21) 11 (21) 𝜌𝜋𝑅𝜌𝜋𝑅𝑈𝑈 22

Water 2020, 12, 98 7 of 16

3.3. Verification of the Model Before the FSI simulation, the fluid model was verified. Fluent 19.1 in ANSYS workbench was used for the CFD simulation. The above-described LES turbulence model was applied. The top of the computational domain was set as a slip wall, and the inlet velocity to the computational domain was uniformly U0 = 1.4 m/s. The outflow boundary condition was used TSR = ωR/U0 was varied in the range between 4 and 12 by changing rotational speed ω. The Reynolds number, based on the inlet velocity and diameter, was 1.12 106. Convergence was specified as RMS residual of 1 10 5. × × − The main nondimensional load parameters were

Qω power coefficient : CP = , (21) 1 2 3 2 ρπR U0 T thrust coefficient : CT = . (22) 1 2 2 2 ρπR U0 where, Q is the torque, T is the thrust. Bahaj et al. [6] reported experimental values of CP, CT, and TSR with a blockage correction applied to identify them as ‘equivalent open-water’ measurements. The same correction was applied to the numerical results here, with the thrust coefficient calculated from numerical results. With the blockage ratio 7.5%, a typical CT = 0.8 resulted in multiplicative blockage correction factors of 92.2%, 94.7%, and 97.3% for the power coefficient, thrust coefficient, and TSR, respectively. A complete mesh sensitivity study was performed with different mesh resolutions (see Table1). The LES was grid-dependent, i.e., the finer the mesh, the more length-scales of turbulence were resolved explicitly. Hence, the fine mesh was chosen and time step size was set as 1 10 5 s. × − Table 1. Mesh details and sensitivity study for mean blade loading at TSR = 5.67.

Mesh Number of Cells Time Step (s) Y Plus CP CT Coarse 3 106 1 10 4 1.2–9 0.393 0.707 × × − Medium 28 106 1 10 4 0.8–6 0.42 0.761 × × − Fine 100 106 1 10 4 0.4–3 0.431 0.763 × × − Fine 100 106 1 10 5 0.4–3 0.431 0.763 × × −

Figure3 compares time averaged blockage-corrected power and thrust coe fficients from pure CFD and FSI against the experimental data [6]. At each TSR, the simulation results were averaged over four rotations after the power and thrust coefficients achieved a repeatable periodic behavior. The numerical results both agree well with the experimental data. The only exception is the highest TSR, where the pure CFD and FSI underpredicted the CP by about 40% and 30%, respectively. The predicted CP results show the similar trend with the fitting curve of experimental data. With the increase of TSR, the CP increased to a maximum value and then decreased. Hence, the fluid model was deemed to be acceptable for further use in the numerical FSI simulation analysis. The CP and CT results at three different TSRs from FSI simulation are presented here in advance. It can be seen that the CP and CT at TSR = 4.9, 8.3 and 11.3 predicted by FSI are closer to the experimental data, while the CP and CT at TSR = 5.67 predicted by pure CFD are closer to the experimental data. Water 2019, 11, x FOR PEER REVIEW 7 of 16

𝑇 thrust coefficient: 𝐶 = . 1 (22) 𝜌𝜋𝑅𝑈 2 where, 𝑄 is the torque, 𝑇 is the thrust.

Bahaj et al. [6] reported experimental values of 𝐶 , 𝐶, and TSR with a blockage correction applied to identify them as ‘equivalent open-water’ measurements. The same correction was applied to the numerical results here, with the thrust coefficient calculated from numerical results. With the blockage ratio 7.5%, a typical 𝐶 = 0.8 resulted in multiplicative blockage correction factors of 92.2%, 94.7%, and 97.3% for the power coefficient, thrust coefficient, and TSR, respectively. A complete mesh sensitivity study was performed with different mesh resolutions (see Table 1). The LES was grid-dependent, i.e., the finer the mesh, the more length-scales of turbulence were resolved explicitly. Hence, the fine mesh was chosen and time step size was set as 1 × 10−5 s.

Table 1. Mesh details and sensitivity study for mean blade loading at TSR = 5.67.

Mesh Number of Cells Time Step (s) Y Plus CP CT Coarse 3 × 106 1 × 10−4 1.2–9 0.393 0.707 Medium 28 × 106 1 × 10−4 0.8–6 0.42 0.761 Fine 100 × 106 1 × 10−4 0.4–3 0.431 0.763 Fine 100 × 106 1 × 10−5 0.4–3 0.431 0.763

Figure 3 compares time averaged blockage-corrected power and thrust coefficients from pure CFD and FSI against the experimental data [6]. At each TSR, the simulation results were averaged over four rotations after the power and thrust coefficients achieved a repeatable periodic behavior. The numerical results both agree well with the experimental data. The only exception is the highest

TSR, where the pure CFD and FSI underpredicted the 𝐶 by about 40% and 30%, respectively. The predicted 𝐶 results show the similar trend with the fitting curve of experimental data. With the increase of TSR, the 𝐶 increased to a maximum value and then decreased. Hence, the fluid model was deemed to be acceptable for further use in the numerical FSI simulation analysis. The 𝐶 and 𝐶 results at three different TSRs from FSI simulation are presented here in advance. It can be seen Waterthat 2020the , 𝐶12, 98and 𝐶 at TSR = 4.9, 8.3 and 11.3 predicted by FSI are closer to the experimental 8data, of 16 while the 𝐶 and 𝐶 at TSR = 5.67 predicted by pure CFD are closer to the experimental data.

0.6 1.2 Exp data Exp fit 0.5 1 CFD FSI 0.4 0.8 P 0.3 T 0.6 C C Exp data Exp fit 0.2 0.4 CFD FSI 0.1 0.2

0 0 24681012 24681012 TSR TSR (a) (b)

FigureFigure 3.3. ComparisonsComparisons between between numerical numerical and and experimental experimental data data from from Bahaj Bahaj et al. et [6al.] at[6] di ffaterent different TSR. (TSR.a) power (a) power coeffi cientcoefficient and (b and) thrust (b) thrust coefficient coefficient

4. Two-way FSI Numerical Simulation

4.1. Two-Way FSI Model

The two-way FSI numerical simulation was also carried out in ANSYS workbench, where the fluid domain and the solid domain were established separately. The fluid domain was the computational domain described above, with the rotor surface set as the FSI interface to transmit the pressure information and receive the deformation information of the rotor. Because the blades will always be deformed, dynamic mesh method was adopted to avoid cells with negative volumes. To facilitate the analysis of interaction details, only the rotor deformation was simulated, whereas the deformation of the support was left for further studies in the future. Due to its long-time operation in seawater, the MCT is generally manufactured from composite materials. However, the blades in the tank test are solid bodies manufactured from T6082-T6 aluminum alloy [6]. To be consistent with the experiment, aluminum alloy was chosen as the rotor material in this study. The properties of T6082-T6 aluminum alloy are provided in Table2. The trailing edge of blade was very thin, which was difficult to discretize into hexahedral cells. Therefore, and to reduce the error caused by the information exchange and interpolation of nodes on the FSI interface, the quadrilateral cells on the rotor surface were extracted from the fluid domain and split into triangular cells with a diagonal line. Then, the tetrahedral cells in the rotor interior were generated freely, obtaining the solid domain with 2.5 105 cells. For visualization purpose, the cell size in Figure4 was the same as in Figure2a. Like the × surface of the fluid domain on the rotor, the rotor surface of the solid domain was set to FSI interface. Fluent and Transient Structure are mutually iterative computations through a two-way coupled solver. TSR = 5.67 was chosen in the FSI simulation, i.e., ω = 20.43 rad/s before the blockage correction. These simulations were performed on high-performance computers using 128 processors, with a single FSI simulation taking three months and a single LES taking one month.

Table 2. Properties of T6082-T6 aluminum alloy.

Parameters Values Density 2700 kg/m3 Young’s Modulus 70 GPa Poisson’s Ratio 0.33 Tensile yield strength 270 MPa Tensile ultimate strength 330 MPa Water 2019, 11, x FOR PEER REVIEW 8 of 16

4. Two-way FSI Numerical Simulation

4.1. Two-Way FSI Model The two-way FSI numerical simulation was also carried out in ANSYS workbench, where the fluid domain and the solid domain were established separately. The fluid domain was the computational domain described above, with the rotor surface set as the FSI interface to transmit the pressure information and receive the deformation information of the rotor. Because the blades will always be deformed, dynamic mesh method was adopted to avoid cells with negative volumes. To facilitate the analysis of interaction details, only the rotor deformation was simulated, whereas the deformation of the support was left for further studies in the future. Due to its long-time operation in seawater, the MCT is generally manufactured from composite materials. However, the blades in the tank test are solid bodies manufactured from T6082-T6 aluminum alloy [6]. To be consistent with the experiment, aluminum alloy was chosen as the rotor material in this study. The properties of T6082-T6 aluminum alloy are provided in Table 2. The trailing edge of blade was very thin, which was difficult to discretize into hexahedral cells. Therefore, and to reduce the error caused by the information exchange and interpolation of nodes on the FSI interface, the quadrilateral cells on the rotor surface were extracted from the fluid domain and split into triangular cells with a diagonal line. Then, the tetrahedral cells in the rotor interior were generated freely, obtaining the solid domain with 2.5 × 105 cells. For visualization purpose, the cell size in Figure 4 was the same as in Figure 2a. Like the surface of the fluid domain on the rotor, the rotor surface of the solid domain was set to FSI interface. Fluent and Transient Structure are mutually iterative computations through a two-way coupled solver. TSR = 5.67 was chosen in the FSI simulation, i.e., 𝜔 = 20.43 rad/s before the blockage correction. These simulations were performed on high-performance computers using 128 processors, with a single FSI simulation taking three months and a single LES taking one month.

Table 2. Properties of T6082-T6 aluminum alloy. Parameters Values Density 2700 kg/m3 Young’s Modulus 70 GPa Poisson’s Ratio 0.33 Water 2020, 12, 98 9 of 16 Tensile yield strength 270 MPa Tensile ultimate strength 330 MPa

Water 2019, 11, x FOR PEER REVIEW 9 of 16 FigureFigure 4. 4.The The surfacesurface mesh of the solid solid domain domain of of the the rotor. rotor. 4.2. Results and Discussion 4.2. Results and Discussion Figure 5 presents the power coefficient and thrust coefficient versus the angle of rotation during Figure5 presents the power coe fficient and thrust coefficient versus the angle of rotation during the the fifth to seventh rotation obtained from CFD simulation and FSI simulation at the same TSR, fifth to seventh rotation obtained from CFD simulation and FSI simulation at the same TSR, respectively. respectively. Both plots reveal that the 𝐶 and 𝐶 obtained from pure CFD decayed to a minimum C C Bothvalue plots every reveal 120° that corresponding the P and toT aobtained blade passing from pureby the CFD support. decayed The tosupport a minimum affected value the everyflow field 120◦ corresponding to a blade passing by the support. The support affected the flow field of the wake and of the wake and reduced 𝐶 and 𝐶 by about 1.5% and 1.7% from the maximum value every time a reducedblade passedCP and by.C T by about 1.5% and 1.7% from the maximum value every time a blade passed by.

(a) (b)

FigureFigure 5. 5.( a()a Power) Power coe coefficientﬃcient and and (b ()b thrust) thrust coe coefficientﬃcient versus versus angle angle of rotation of rotation obtained obtained from from CFD onlyCFD and only FSI. and FSI.

TheTheC 𝐶Pand andC T𝐶values values obtained obtained from from FSI FSI were were about about 2.4% 2.4% andand 1.6%1.6% lower than those those from from CFD CFD duedue to to the the displacement displacement ofof thethe blade.blade. However,However, as mentioned above, in in Figure Figure 33,, it is worth noting thatthat the the FSI FSI did did not not alwaysalways underpredictunderpredict the C𝐶P compared compared toto thethe pure CFD. Figure 66 illustratesillustrates the bladeblade total total displacement displacement atat 207207°.◦. TheThe maximum total displacement is is 1/27 1/27𝑅R, ,at at the the tip tip of of the the blade blade (Point(Point 3). 3). For For each each section section of of the the blade, blade, the the total total displacement displacement of of the the nose nose and and tail tail (e.g., (e.g., Points Points 1 1 and and 2) relative2) relative to each to each other other was was less less than than 1 110 × 105 m,−5 m, indicating indicating that that the the twist twist of of the the blade blade can can be be viewed viewed as × − negligiblyas negligibly small. small.

Figure 6. Total displacement of the blade at 207°.

Figure 7 shows the displacement in flow direction at the tip of each blade versus the angle of rotation. Points 4 and 5 were at tip of the other two blades at same location on the profile as Point 3 in Figure 6. Point 3 was on the tip of the blade, which was in front of the support at 0° in Figure 7 like shown in Figures 1 and 2a. Point 5 was on the tip of the blade, which was marked by the ellipse in Figure 2a. Point 4 was on the tip of the blade left in Figure 2a. There was a delay of about 15° after each blade passes in front of the support until the displacement of the blade tip decayed to a

Water 2019, 11, x FOR PEER REVIEW 9 of 16

4.2. Results and Discussion Figure 5 presents the power coefficient and thrust coefficient versus the angle of rotation during the fifth to seventh rotation obtained from CFD simulation and FSI simulation at the same TSR, respectively. Both plots reveal that the 𝐶 and 𝐶 obtained from pure CFD decayed to a minimum value every 120° corresponding to a blade passing by the support. The support affected the flow field of the wake and reduced 𝐶 and 𝐶 by about 1.5% and 1.7% from the maximum value every time a blade passed by.

(a) (b)

Figure 5. (a) Power coefficient and (b) thrust coefficient versus angle of rotation obtained from CFD only and FSI.

The 𝐶 and 𝐶 values obtained from FSI were about 2.4% and 1.6% lower than those from CFD due to the displacement of the blade. However, as mentioned above, in Figure 3, it is worth noting that the FSI did not always underpredict the 𝐶 compared to the pure CFD. Figure 6 illustrates the blade total displacement at 207°. The maximum total displacement is 1/27𝑅, at the tip of the blade (Point 3). For each section of the blade, the total displacement of the nose and tail (e.g., Points 1 and 2)Water relative2020, 12 to, 98 each other was less than 1 × 10−5 m, indicating that the twist of the blade can be viewed10 of 16 as negligibly small.

Figure 6. Total displacement of the blade at 207◦. Figure 6. Total displacement of the blade at 207°. Figure7 shows the displacement in ﬂow direction at the tip of each blade versus the angle of rotation.Figure Points 7 shows 4 and the 5 were displacement at tip of the in otherflow twodirection blades at at the same tip locationof each onblade the versus proﬁle the as Point angle 3 of in rotation.Figure6. Points Point 3 4 was and on5 were the tip at oftip the of the blade, other which two wasblades in frontat same of thelocation support on the at 0 profile◦ in Figure as Point7 like 3 inshown Figure in 6. Figures Point 31 wasand 2ona. the Point tip5 of was the onblade, the tipwhich of the was blade, in front which of the was support marked at 0° by in the Figure ellipse 7 like in shownFigure2 ina. PointFigures 4 was1 and on 2a. the Point tip of 5 the was blade on leftthe intip Figure of the2 a.blade, There which was a was delay marked of about by 15 the◦ afterellipse each in FigurebladeWater 2019 passes 2a., 11Point, inx FOR front 4 wasPEER of the onREVIEW the support tip of until the theblade displacement left in Figure of the2a. bladeThere tip was decayed a delay to of a minimumabout 15°10 value.after of 16 eachThe plottedblade passes points showin front a periodic of the support trend with unt ail periodthe displacement of 360 and theof the scatter blade indicates tip decayed additional to a minimum value. The plotted points show a periodic trend with◦ a period of 360° and the scatter higher-frequency oscillations of the blades throughout the rotation. indicates additional higher-frequency oscillations of the blades throughout the rotation.

Figure 7. Displacement in flow direction versus the angle of rotation for the tip of each blade (Points 3–5). Figure 7. Displacement in flow direction versus the angle of rotation for the tip of each blade (Points Figure3–5). 8 shows the displacement velocity in flow direction of Point 3 against the angle of rotation. The majority of points were concentrated between 0.2 m/s to 0.2 m/s. The blade oscillating in flow − directionFigure changed 8 shows the the relative displacement velocity approaching velocity in flow the blade direction and henceof Point the 3 local against velocity the triangle.angle of Figurerotation.9 illustrates The majority a local of velocitypoints were triangle concentrated at the blade between tip. It is −0.2 well-known m/s to 0.2 that m/s. the The approaching blade oscillating flow in flow direction changed the relative velocity approaching the blade and hence the local velocity velocity u is less than the initial free-stream velocity U0 In Figure 10, u = U0 was applied, and the tangentialtriangle. Figure and radial 9 illustrates components a local were velocity not considered. triangle Forat the the rigidblade non-displaced tip. It is well-known blade, the that inflow the approaching flow velocityu 𝑢 is less than the initial free-stream velocity 𝑈 In Figure 10, 𝑢=𝑈 was velocity angle θ = arctan = 9.72◦, the pitch angle β = 5◦, and the angle of attack α = 4.72◦. To applied, and the tangentialωr and radial components were not considered. For the rigid non-displaced blade, the inflow velocity angle 𝜃 = arctan = 9.72°, the pitch angle 𝛽= 5°, and the angle of attack (∆) 𝛼= 4.72°. To obtain the flow angle 𝜃 versus the angle of rotation in Figure 10, 𝜃 = arctan was used, where ∆𝑢 represents the displacement velocity in flow direction of Point 3 given in Figure 8. The majority of the points in Figure 10 fell in the range of 8° to 11°, indicating the high-frequency change of inflow velocity angle causing the higher-frequency fluctuations of 𝐶 predicted by FSI simulation in Figure 5. Cyclic behavior cannot be observed in both Figures 8 and 10. It is suspected that the flow separation was responsible for this phenomenon, which require further investigation.

Water 2020, 12, 98 11 of 16

(u+∆u) obtain the flow angle θ versus the angle of rotation in Figure 10, θ = arctan ωr was used, where ∆u represents the displacement velocity in flow direction of Point 3 given in Figure8. The majority of the points in Figure 10 fell in the range of 8◦ to 11◦, indicating the high-frequency change of inflow velocity angle causing the higher-frequency fluctuations of CP predicted by FSI simulation in Figure5. Cyclic behavior cannot be observed in both Figures8 and 10. It is suspected that the flow separation WaterwasWater responsible2019 2019, ,11 11, ,x x FOR FOR for PEER PEER this REVIEW REVIEW phenomenon, which require further investigation. 1111 ofof 1616

FigureFigure 8. 8. Displacement Displacement velocity velocity in in flow ﬂowflow direction direction of of the the Point Point 3 3 versus versusversus the thethe angle angleangle of ofof rotation. rotation.rotation.

Figure 9. Local velocity triangle of blade tip section. FigureFigure 9. 9. Local Local velocity velocity triangle triangle of of blade blade tip tip section. section.

FigureFigure 1111 illustratesillustrates thethe contourcontour ofof equivalentequivalent (v(vonon Mises)Mises) stressstress onon bladeblade suctionsuction side,side, wherewhere thethe stressesstresses werewere higherhigher thanthan onon thethe pressurepressure side. side. AtAt 207°, 207°, thethe maximummaximum valuevalue ofof 130 130 MPa MPa occurredoccurred onon thethe rootroot inin thethe PointPoint 6.6. ItIt waswas lessless thanthan thethe tensiletensile yieldyield strengthstrength ofof 270270 MPaMPa andand tensiletensile ultimateultimate strengthstrength of of 330 330 MPa. MPa. Beyond Beyond the the root, root, a a maximum maximum value value of of 81 81 MPa MPa occurred occurred near near 0.5𝑅.0.5𝑅. The The root root can can bebe thickened thickened to to avoid avoid damage. damage. However, However, it it is is wort worthh noting noting that that the the maximum maximum stress stress occurred occurred around around 0.5𝑅0.5𝑅 beyond beyond thethe rootroot atat everyevery moment,moment, inin thethe rangerange wherewhere bladesblades werewere brokenbroken [1][1]..

Water 2020, 12, 98 12 of 16 Water 2019, 11, x FOR PEER REVIEW 12 of 16

Water 2019, 11, x FOR PEER REVIEW 12 of 16

Figure 10. FlowFlow angle angle θ𝜃 of of thethe PointPoint 33 versusversus thethe angleangleof ofrotation. rotation.

Figure 11 illustrates the contour of equivalent (von Mises) stress on blade suction side, where the stresses were higher than on the pressure side. At 207◦, the maximum value of 130 MPa occurred on the root in the Point 6. It was less than the tensile yield strength of 270 MPa and tensile ultimate strength of 330 MPa. Beyond the root, a maximum value of 81 MPa occurred near 0.5R. The root can be thickened to avoid damage. However, it is worth noting that the maximum stress occurred around 0.5R beyond the root at every moment, in the range where blades were broken [1]. Figure 10. Flow angle 𝜃 of the Point 3 versus the angle of rotation.

Figure 11. Contour of equivalent (von Mises) stress on blade suction side at 207°.

Figure 12 shows the equivalent (von Mises) stress normalized by 130 MPa at the root of each blade versus the angle of rotation, where Points 7 and 8 were at the same location as Point 6 and on the same blade as Points 4 and 5, respectively. All blades experienced a maximum stress variation of 5.8% during one rotation due to the support. Bahaj et al. considered the influence of the support on MCT performance and placed it as far as at least one rotor radius downstream of the rotor. The variation Figure 11. would be largerFigure a velocity 11. Contour profile ofof of equivalentequivalent 1/7th power (von(von lawMises)Mises) was stressstress used onon instead bladeblade suctionsuction of uniform sideside atat velocity 207207°.◦. [22]. Relative to the time average, LES can not only predict the effect of the support on the Figure 12 shows the equivalent (von Mises) stress normalized by 130 MPa at the root of each performanceFigure 12 of shows the MCT, the equivalent but also (voncapture Mises) the stresstip vortices normalized disturbed by 130 by MPa the at support, the root ofas eachshown blade in blade versus the angle of rotation, where Points 7 and 8 were at the same location as Point 6 and on the Figureversus the13. angleQ-criterion of rotation, was used:where 𝑄=0.5(𝛺Points 7 and𝛺 8 were−𝑆𝑆 at )the, where same 𝛺location is the asvorticity Point 6 andtensor on andthe same 𝑆 is same blade as Points 4 and 5, respectively. All blades experienced a maximum stress variation of 5.8% theblade rate-of-strain as Points 4 and tensor. 5, respectively. The helicoidal All blades tip vortices experienced were a generated maximum bystress the variation fast-moving of 5.8% blade during tip during one rotation due to the support. Bahaj et al. considered the influence of the support on MCT surfaceone rotation and then due transported to the support. downstream. Bahaj et These al. co helicnsideredal structures the influence closely ofattached the support the shear on layer,MCT performance and placed it as far as at least one rotor radius downstream of the rotor. The variation rollingperformance up between and placed the fast it as flow far rightas at leastabove one the ro tiptor of radius each bladedownstream and the of low-velocity the rotor. The rotor variation wake. would be larger a velocity profile of 1/7th power law was used instead of uniform velocity [22]. Thewould radius, be larger at which a velocity the tip profile vortices of 1/7th traveled power downst law wasream used in theinstead wake, of expandeduniform velocity with the [22] radius. of Relative to the time average, LES can not only predict the effect of the support on the performance the sheerRelative layer to and the increased time average, beyond LESthe radius can not of thone rotor.ly predict Figure the 13 alsoeffect shows of the an supportadditional on three the of the MCT, but also capture the tip vortices disturbed by the support, as shown in Figure 13. Q-criterion vortices,performance which of originatedthe MCT, frombut also the rootcapture of the the bl tipades. vortices They flowed disturbed downstream by the support, around asthe shown hub and in was used: Q = 0.5 ΩijΩij SijSij , where Ω is the vorticity tensor and S is the rate-of-strain tensor. wereFigure maintained 13. Q-criterion by the was sheer− used: layer 𝑄=0.5(𝛺 roll-up between𝛺 −𝑆 𝑆the) , low-velocitywhere 𝛺 is thehub vorticity boundary tensor layer and and 𝑆the is The helicoidal tip vortices were generated by the fast-moving blade tip surface and then transported higher-velocitythe rate-of-strain wake tensor. of the The rotor. helicoidal The root tip vortices vortices rotated were in generated the direction by oppositethe fast-moving to the tip blade vortices. tip Figuresurface 14 and shows then the transported eddy viscosity downstream. in the central These cut helic plane.al structures The high closelyvalues correspondattached the to shear vortices layer, in Figurerolling 13.up between the fast flow right above the tip of each blade and the low-velocity rotor wake. The radius, at which the tip vortices traveled downstream in the wake, expanded with the radius of the sheer layer and increased beyond the radius of the rotor. Figure 13 also shows an additional three vortices, which originated from the root of the blades. They flowed downstream around the hub and were maintained by the sheer layer roll-up between the low-velocity hub boundary layer and the higher-velocity wake of the rotor. The root vortices rotated in the direction opposite to the tip vortices. Figure 14 shows the eddy viscosity in the central cut plane. The high values correspond to vortices in Figure 13.

Water 2020, 12, 98 13 of 16

downstream. These helical structures closely attached the shear layer, rolling up between the fast ﬂow right above the tip of each blade and the low-velocity rotor wake. The radius, at which the tip vortices traveled downstream in the wake, expanded with the radius of the sheer layer and increased beyond the radius of the rotor. Figure 13 also shows an additional three vortices, which originated from the root of the blades. They ﬂowed downstream around the hub and were maintained by the sheer layer roll-up between the low-velocity hub boundary layer and the higher-velocity wake of the rotor. The root vortices rotated in the direction opposite to the tip vortices. Figure 14 shows the eddy Water 2019, 11, x FOR PEER REVIEW 13 of 16 Waterviscosity 2019, 11 in, x the FOR central PEER REVIEW cut plane. The high values correspond to vortices in Figure 13. 13 of 16

Figure 12. Equivalent (von Mises) stress on the root of the three blades normalized by 130 MPa versus FigureFigure 12. 12. EquivalentEquivalent (von (von Mises) Mises) stress stress on on the the root root of of the the three three blades blades normalized normalized by by 130 130 MPa MPa versus versus the angle of rotation. thethe angle angle of of rotation. rotation.

Figure 13. Iso-Q surfaces colored by velocity in ﬂow direction. Figure 13. Iso-Q surfaces colored by velocity in flow direction. Figure 13. Iso-Q surfaces colored by velocity in flow direction.

Water 2020, 12, 98 14 of 16 Water 2019, 11, x FOR PEER REVIEW 14 of 16

FigureFigure 14. 14. TheThe eddy eddy viscosity viscosity in in the the central central cut cut plane. plane.

5. Conclusions 5. Conclusions As a form of ocean energy conversion, MCT has been widely accepted by the industry and As a form of ocean energy conversion, MCT has been widely accepted by the industry and researchers. The most concern is reducing design, installation, and maintenance costs. Based on this researchers. The most concern is reducing design, installation, and maintenance costs. Based on this purpose, a two-way FSI method, suitable for MCTs, was described and implemented. LES was chosen purpose, a two-way FSI method, suitable for MCTs, was described and implemented. LES was chosen instead of the time-averaged model to capture accurate pressure information from the flow field at any instead of the time-averaged model to capture accurate pressure information from the flow field at time, blade vibration, passing by the support and the tip vortices, which expand outward. any time, blade vibration, passing by the support and the tip vortices, which expand outward. Bahaj’s group expected the reductions of the CP and CT of the MCT every time a blade passed by Bahaj’s group expected the reductions of the 𝐶 and 𝐶 of the MCT every time a blade passed the support to be very small. The results of this numerical simulation quantified these reductions at by the support to be very small. The results of this numerical simulation quantified these reductions 1.5% and 1.7%, respectively. The oscillation and vibration of the blade reduced the CP and CT by an at 1.5% and 1.7%, respectively. The oscillation and vibration of the blade reduced the 𝐶 and 𝐶 by additional 2.4% and 1.6%, respectively. The maximum total displacement was 1/27R at the tip of the an additional 2.4% and 1.6%, respectively. The maximum total displacement was 1/27𝑅 at the tip of blade. With the blade displacement, the displacement velocity of the blade tip changed the velocity the blade. With the blade displacement, the displacement velocity of the blade tip changed the triangle continuously and periodically. velocity triangle continuously and periodically. The maximum equivalent (von Mises) stress occurred on the root on the suction side, and a high The maximum equivalent (von Mises) stress occurred on the root on the suction side, and a high R valuevalue occurred occurred near near 0.5 0.5𝑅 at at everyevery moment.moment. DuringDuring every every rotation, rotation, the the stresses stresses varied varied around around 5.8% 5.8% for foreach each blade, blade, which which would would be largerbe larger if the if profiledthe profiled inflow inflow was considered.was considered. This canThis considerably can considerably impact the fatigue loading with an MCT, typically experiencing in the order of 1 108 rotational cycles over a impact the fatigue loading with an MCT, typically experiencing in the order× of 1 × 108 rotational cycles over20-year a 20-year life span life [span30]. [30]. ComparedCompared with with pure pure CFD CFD simulation simulation without without considering considering solid solid deformation deformation and and one-way one-way fluid–solidfluid–solid interaction interaction simulation, simulation, two-way two-way fluid–solid fluid–solid interaction interaction simulation simulation can can well-predict well-predict the the stressstress variation variation experienced experienced by by blades blades under under more more complex complex loads. loads. InIn the the future, future, we we expect expect to to explore explore the impact ofof thethe deformationdeformation of of the the support support and and the the influence influence of ofa freea free surface surface and and waves waves on the on performance, the performance, and to simulateand to simulate the MCT inthe a velocityMCT in sheara velocity environment. shear environment.Author Contributions: Conceptualization, J.G. and F.C.; Data curation, J.G.; Formal analysis, J.G. and Y.Z.; Funding acquisition, J.G. and Y.Z.; Investigation, J.G.; Methodology, J.G. and H.C.; Project administration, F.C. Authorand N.M.; Contributions: Resources, N.M.;Conceptualization, Software, J.G. J.G. and and H.C.; F.C.; Supervision, Data curation, F.C.; J.G.; Validation, Formal analysis, J.G.; Visualization, J.G. and Y.Z.; J.G.; FundingWriting—original acquisition, draft, J.G. J.G.;and Y.Z.; Writing—review Investigation, & J.G.; editing, Methodology, F.C. and N.M. J.G. All and authors H.C.; Project have read administration, and agreed toF.C. the andpublished N.M.; Resources, version of theN.M.; manuscript. Software, J.G. and H.C.; Supervision, F.C.; Validation, J.G.; Visualization, J.G.; Writing—originalFunding: This research draft, J.G.; was supportedWriting—review by Fundamental & editing, Research F.C. and FundsN.M. All for authors the Central have Universities read and agreed (2016B41514), to the publishedPriority Academic version of Program the manuscript. Development of Jiangsu Higher Education Institutions (YS11001), National Natural Science Foundation of China (51809083), Natural Science Foundation of Jiangsu Province (BK20180504) and China Funding:Postdoctoral This Science research Foundation was supported (2019M651678), by Fundamental China Scholarship Research Council Funds (201706710022). for the Central Universities (2016B41514),Acknowledgments: PriorityThis Academic work has Program been performed Development at theTurbomachinery of Jiangsu Higher Laboratory Education at Michigan Institutions State (YS11001), University Nationalwith the Natural support fromScience Research Foundation Institute of ofChina Hydraulic (51809083) and Hydropower, Natural Science Engineering Foundation at Hohai of Jiangsu University. Province (BK20180504) and China Postdoctoral Science Foundation (2019M651678), China Scholarship Council Conflicts of Interest: The authors declare no conflict of interest. (201706710022).

Acknowledgments: This work has been performed at the Turbomachinery Laboratory at Michigan State University with the support from Research Institute of Hydraulic and Hydropower Engineering at Hohai University.

Conflicts of Interest: The authors declare no conflict of interest.

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