- Email: [email protected]
The flow regime and hydrodynamic performance for a pitching hydrofoil
Renewable Energy 150 (2020) 412e427
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
The flow regime and hydrodynamic performance for a pitching hydrofoil Mengjie Zhang a, Qin Wu a, *, Guoyu Wang a, Biao Huang a, **, Xiaoying Fu b, Jie Chen a a b
School of Mechanical Engineering, Beijing Institute of Technology, Beijing, China State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 May 2019 Received in revised form 31 December 2019 Accepted 2 January 2020 Available online 7 January 2020
The objective of this paper is to study the flow regime and hydrodynamic performance for a pitching Clark-Y hydrofoil. The aims are to (1) improve the understanding of the interplay between the hydrodynamic performance, unsteady flow structures and dynamic motion of the hydrofoil, (2) study the influence of the pitching rate on the transition of different flow regimes. The experimental investigations were conducted in the looped cavitation tunnel, and the dynamic moment measurement system was applied to obtain the hydrodynamic forces. The pitching hydrofoil is controlled to rotate from aþ ¼ 10 to aþ ¼ 15 firstly, then goes from aþ ¼ 15 to a ¼ 5 , finally goes back to aþ ¼ 10 from a ¼ 5 . The pitching rate is varying with the Reynolds number Re ¼ 4.4 105. The numerical investigations were performed by solving the incompressible URANS equations using the coupled k-u SST turbulence model and g-Req transition model. The numerical results agree well with the experimental measurements. The pitching motion affects the turbulence kinetic energy distribution around the hydrofoil, leading to the delay or acceleration of the transition between different flow patterns. During the pitching process, higher level of turbulence kinetic energy distribution causes earlier transition from laminar to turbulence. Moreover, hysteresis effect of the hydrodynamic force is observed. For the upstroke stage, the higher pitching rate promotes the laminar separation slightly and intensifies the delay of turbulence separation. For the downstroke stage, the higher pitching rate promotes the turbulence separation extensively. The first leading edge vortex (LEV) and anticlockwise trailing edge vortex (TEV) are delayed with the increase of pitching rate, which is responsible to the delay of dynamic stall. The lower pitching rate shrinks the hysteresis loops and intensifies the fluctuation of the dynamic force. © 2020 Published by Elsevier Ltd.
Keywords: Hydrodynamic performance Flow pattern Pitching hydrofoil Lagrangian coherent structures
1. Introduction Because of the high cost of non-renewable fossil fuels, the development of renewable energy sources has been promoted [1e3]. Among which, hydroenergy has attracted much attention because it is resource-rich, widely distributed and clean. As one of the important hydraulic machinery, the water turbines have been widely applied [4e6]. However, the effective angle of attack of the blades often changes due to varying inflow and active/passive body motions in the water turbine system [7e9]. The dynamic motion of boundary caused by the varying angle of attack is often accompanied with different flow regimes, including laminar, transition,
* Corresponding author. ** Corresponding author. E-mail address: [email protected] (Q. Wu). https://doi.org/10.1016/j.renene.2020.01.006 0960-1481/© 2020 Published by Elsevier Ltd.
turbulence and stall, leading to complex and unsteady hydrodynamic load. Especially, laminar-turbulent transition and dynamic stall always attract much attention, as they play an important role in the performance of engineering system. Therefore, it is very important to understand the dynamic interaction between the pitching motion, unsteady flow structures and hydrodynamic performance. It does not only help improve the overall design of hydraulic and marine structures, but also promote better performance for harvesting energy [10]. Many studies of flow regimes and hydrodynamic performance around the static foils have been conducted [11e13], with focus on the flow separation mechanism. Yang et al. [14] investigated the transient behavior of the flow separation on a NASA low-speed GA (W)-1 airfoil at the Reynolds number Re ¼ 6.8 104 by a highresolution PIV system. The results showed that the increased adverse pressure gradient caused the laminar boundary layer separation on the airfoil, and the separated laminar boundary layer
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
413
Fig. 1. Schematic of the cavitation tunnel.
transited to turbulent flow rapidly by generating unsteady vortex structures due to Kelvin-Helmholtz instabilities, which reattached to the upper surface of the airfoil as a turbulent boundary layer. Wei et al. [15] experimentally studied the flow separation control for hydrofoils with leading-edge tubercles in a water tunnel at the Reynolds number Re ¼ 1.4 104. The cross-stream flow measurements indicated that the streamwise counter-rotating vortex pairs were generated over the tubercles and mitigated flow separation. Compared with the experimental research, many researchers have also conducted numerical study to investigate the flow structure around hydrofoils. The flow separation and transition around a NACA0012 airfoil with a ¼ 4 and Re ¼ 1.0 105 were investigated by Liu et al. [16]. They found that the disturbances in the near wake propagated upstream in the form of acoustic waves and induced disturbances to the separated shear layer over the upper surface of the airfoil. Yang et al. [17] studied separated boundary-layer transition on a flat plate with a semicircular leading edge, with focus on the physics of separated boundary-layer transition induced by a change of surface curvature. The results showed that there are some vortical structures at various stages of the transition process. In order to consider the impact of the dynamic boundary on the flow structure of the hydrofoil, many studies about the pitching foil have also been investigated. Franc et al. [18] experimentally studied the impact of unsteadiness on the attached cavitation around an oscillating hydrofoil. Results presented that a cavity detaches behind a laminar separation of the boundary layer under steady conditions, and the unsteadiness affects both the cavity and the boundary layer, essentially through a convection effect and a delay effect. Anderson et al. [19] studied propulsive performance of a harmonically oscillating foil through force and power measurements, as well as visualization data. Results showed that highefficiency conditions are associated with the formation of a moderately strong leading-edge vortex, which is convected downstream and interacts with the trailing-edge vortex, eventually resulting in the formation of a reverse Karman street. Pedro et al. [20] simulated the dynamics of a flapping hydrofoil and found that the thrust and efficiency are significantly affected by the Strouhal number, maximum pitch angle and phase angle. Akhtar et al. [21] numerically examined the performance of a foil undergoing
Fig. 2. Schematic of the pitching hydrofoil system.
oscillating motion in the wake of another oscillating foil. It was found that the shedding vortex of upstream hydrofoil increases the effective angle of the downstream hydrofoil, resulting to the formation of a stronger leading edge vortex on the downstream hydrofoil, thus improves the performance (thrust and efficiency) of the downstream foil. Much effort has been conducted on the flow regimes around static and pitching hydrofoil. However, the effect of dynamic boundary on the flow regime and hydrodynamic performance is still implicit, the transition of different flow regimes around the pitching foil have not been explained clearly yet. The objective of this paper is to study flow regimes and hydrodynamic performance around the pitching hydrofoil. The aims are to (1) improve the understanding of the interplay between unsteady flow structure, hydrofoil motion and hydrodynamic performance, (2) investigate the impact of the pitching rate on the flow structure and hydrodynamic performance, especially for the evolution of flow regime. The experimental setup and the numerical methods are presented in Section 2 and 3, respectively. In Section 4, the numerical setup and verification are given. The detailed flow regimes, flow structure and hydrodynamic performance for fixed cases and pitching cases with three pitching rates are discussed in Section 5. Finally, the major findings and future work are summarized in Section 6. 2. Experimental setup Experiments were conducted in the looped cavitation tunnel [22] with a test section of 190 mm height, 70 mm width and 700 mm length, as shown in Fig. 1. Two main control parameters, upstream pressure and velocity, are measured respectively by the vacuum gauge (with the uncertainty 0.25% of the maximum range, 0.1Mpa) and the electromagnetic flowmeter (with the uncertainty 0.5% of the maximum range, 1550 m3/h). The detailed performance parameters of Cavitation tunnel is given in Table 1. Fig. 2 shows the schematic of the pitching hydrofoil system, which mainly includes PC(Personal Computer), controller, motor, hydrofoil, rotating transformer, moment sensor and Data
Table 1 The performance parameters of Cavitation tunnel. Minimum cavitation number
Maximum velocity
Steady speed
Coefficient of stability
Reduced pressure maintenance.
Vacuum range
0.30
20 m/s
4.68e12.78 m/s
0.71e1.93%
1.11e12.83%
0e0.1 Mpa
414
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 3. Variation of the angle of attack (AOA) a with time (a_ * ¼ 0:086; 0:0215; 0:172).
Fig. 4. Fluid domain and boundary conditions.
Acquisition Card. It is mounted horizontally in the tunnel test section. The controller and the motor (150 ST-M23020) realize the reciprocating oscillating motion of the hydrofoil. The oscillating motion of the hydrofoil is closed-loop controlled by a rotating transformer (TS2640N321E64), with the precision of the angle of attack a of the hydrofoil is ±0.1, as shown in Fig. 2. The IC chip (AD2S1210) of the rotating transformer is applied to convert analog output signals into digital position (angle) signals. The moment of hydrofoil M is measured by the moment sensor (SPI-M2210A), which is mounted on a rotating shaft, as shown in Fig. 2. When the
Fig. 6. Comparisons of the experimental moment coefficient and the numerical moment coefficient for fixed case and pitching case (a_ * ¼ 0:086).
shaft deforms, the change of the bridge resistance results in the change of the electrical signal. Synchronously, the torque is collected by data acquisition card (NI 6356). The sampling frequency of the moment sensor is 20 kHz, and the precision is 0.02N.m. Table 2 Grid convergence study based on Richardson extrapolation.
Fig. 5. Mesh distribution.
Mesh I Mesh II Mesh III
Grid nodes
CL-mean
GCI(%)
396000 792000 1584000
1.02362 1.13617 1.18815
GCI21 ¼ 0.11795 GCI32 ¼ 0.04908
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 7. The Contours of z-vorticity superimposed on the instantaneous streamlines and the velocity vector at different angles of attack.
415
416
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 8. The skin friction distribution along the hydrofoil at different attack angle as noted in Fig. 7.
In this paper, the nominal free stream velocity U∞ is fixed at 6.3 m/s, corresponding to a Reynolds number Re]U∞c/ n ¼ 4.4 105, where c is the chord of the hydrofoil and n is the kinematic viscosity of the liquid. The pressure in the tunnel test section is set to 1atm. The Clark-Y hydrofoil, with c ¼ 70 mm chord and s ¼ 69 mm span, pitches about an axis located at 50% from the leading edge. The pitching hydrofoil is controlled to rotate from aþ ¼ 10 to aþ ¼ 15 firstly, then goes from aþ ¼ 15 to a ¼ 5 , finally goes back to aþ ¼ 10 from a ¼ 5 , with the oscillation amplitude Da ¼ 5 and the mean angle of attack adjusted to a0 ¼ 10 , as shown in Fig. 3. In the pitching process, the angular velocity is at a constant speed. Multiple periodic motions are measured and simulated. We mainly focused on one typical cycle to investigate the flow regimes and hydrodynamic performance around the pitching hydrofoil in this work. According to the direction of motion, the stage of the increasing attack angle is called the upstroke phase, which is denoted with aþ. And the stage of the declining attack angle is called the downstroke phase, which is denoted with a. Hence, the pitching period can be divided into four stages–upstroke I, downstroke I, downstroke II, upstroke II, as shown in Fig. 3. The angular velocity is defined as a_ ¼ af =Tf , where af is defined as af ¼ 4 Da and Tf is the total duration of the transient pitching motion. The non-dimensional angular velocity based on the chord length c and the upstream velocity U∞ is defined as a_ * ¼ ða_ cÞ =ðU∞ DaÞ. In this paper, for the analysis of typical flow regimes around the pitching hydrofoil, the pitching rate is set to 0.086. As for the influence of the pitching rate on the flow structures and corresponding hydrodynamics, the cases with different pitching rate, a_ * ¼ 0:0215, a_ * ¼ 0:086 and a_ * ¼ 0:172, have been compared, as shown in Fig. 3.
v uj ¼0 vxj
r
! vðui Þ v ui uj vp v þ þ ¼ vt vxi vxj vxj
(1)
m
vui vxj
! (2)
where, u denotes the velocity, x denotes the coordinate and subscripts i and j denote the directions of the Cartesian coordinates. r denotes the fluid density, t denotes the time, p denotes the pressure, m denotes the fluid viscosity.
3.2. Turbulence model Many previous studies have highlighted the importance to take into account the transition in the hydrodynamic prediction [23e26]. The g-Req transition model consists of the intermittency
3. Numerical methods 3.1. Basic governing equations The two-dimensional Reynolds-Averaged Navier-Stokes (RANS) equations for unsteady incompressible turbulent flow with the continuity and momentum equation are expressed as follows:
Fig. 9. The evolution of flow regimes around fixed hydrofoil and pitching hydrofoil for Re ¼ 4.4 105, U∞ ¼ 6.3 m/s, a_ * ¼ 0:086 (black solid lines: fixed case; discrete red points: pitching case). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
417
Fig. 10. The non-dimensional turbulence kinetic energy distribution around hydrofoil for pitching case and fixed case.
transport equation which associates transition to the local variable to turn on the production term of the turbulent kinetic energy downstream of the transition point and trigger the transition, and the momentum thickness transport equation which is solved to capture the non-local influence of the turbulence intensity, as it ties the empirical correlation to the onset criteria in the intermittency equation. It can be described in detail as follows:
" ! # vðrgÞ v rUj g v m t vg þ ¼ Pg Eg þ mþ vt vxj sf vxj vxj " # vðrReqt Þ v rUj Reqt v vReqt þ ¼ Pqt þ s ðm þ m t Þ vt vxj qt vxj vxj
(3)
(4)
where, g is intermittency, Pg and Pqt are transition sources, Eg is destruction/relaminarization, Reqt is local transition Reynolds number. In this study, the revised k-u SST turbulence model [27], which couples the k-u SST turbulence model [28] and the g-Req transition model, are used to solve the URANS equations.
" # vðrkÞ v ruj k v vk ~ ~ þ ¼ P k Dk þ ðm þ sk mt Þ vt vxj vxj vxj
original production terms and destruction term respectively. 3.3. Lagrangian coherent structures (LCSs) From the Lagrangian viewpoint, the fluid flow is a dynamic system of fluid particles. The Lagrangian Coherent Structures (LCSs) can be regarded as a trajectory-based approach [29]. The LCSs are extracted from the Finite-Time Lyapunov Exponent (FTLE), which characterizes the separation rate of neighborhood trajectories during a finite time. To characterize the separation rate of the infinitely close trajectories, the Lyapunov exponent is defined as:
s ¼ lim
t/∞ jdx0 j/0
vðruÞ v rUj u v m vu þ ¼ Cu Pu bu ru2 þ mþ t vxj vt vxi sk vxi 1 vk vu þ 2rð1 F1 Þsu2 u vxi vxi
DTt0LE ðx0 Þ ¼
where, k and u denote the turbulent kinetic energy and the specific turbulent dissipation, respectively, F1 denotes blending functions. ~ and destruction term D ~ are And the revised production terms P k k defined as:
~ ¼g P P k eff k
(7)
a_ * ¼ 0:086
(8)
where, geff denotes the revised coefficient, Pk and Dk denote the
vxðt0 þ TLE ; t0 ; x0 Þ T vxðt þ TLE ; t0 ; x0 Þ vx0 vx0
(10)
where, TLE denotes the time interval, x (t0þTLE; t0, x0) denotes the new position of point x0 after TLE. Then the finite-time Lyapunov exponent (FTLE) during the time interval TLE is defined as:
sTt0LE ðx0 Þ ¼ (6)
(9)
where, x0 denotes an arbitrary point in the dynamical system. Based on the Cauchy-Green deformation tensor, the separation rate can be acquired as:
(5)
1 jdxðx0 ; tÞj ln t jdx0 j
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 1 ln lmax DTt0LE ðx0 Þ jTLE j
(11)
where, lmax ðDTt0LE ðx0 ÞÞ denotes the max eigenvalue of the CauchyGreen deformation tensor. Based on the FTLE field, the LCSs can be obtained with ridges of the FTLE field and has been proven helpful to capture the vortex boundary [30e33]. 4. Numerical setup and verification 4.1. Numerical setup According to the experimental setup, the computational domain and boundary conditions are presented in Fig. 4, including two domains connected with a sliding interface. The rectangular static
418
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 11. The skin friction distribution along the hydrofoil and the contour of turbulence kinetic energy superimposed on velocity profile at different separation position with a ¼ 8 and 13 .
domain has a length of 10c and a height of 2.7c, and the circular dynamic domain has a diameter of 2c. The interaction between the static and dynamic domains is controlled by the CFX expression language (CEL) subroutine. The pitching motion of the foil is simulated by setting the pitching motion of the dynamic domain, which is the cylinder region around the mid-chord of foil. The inlet velocity and the outlet pressure are set according to the experimental condition. A no-slip boundary condition is imposed on the hydrofoil surface, and symmetry conditions are imposed on the top and bottom tunnel boundaries. We used a structured grid strategy in computational mesh. Fig. 5 shows the mesh distribution and the refined grids around the leading edge and the trailing edge. There
are 500 nodes placed in the boundary layer, which is selected to meet yþ ¼ yut/n z 1. Three meshes, which give a constant refinement ratio r ¼ Nfine/ Ncourse ¼ 2, are used in the grid convergence study, where N is the number of grid nodes, as shown in Table 2. The GCI index is estimated based on Richardson extrapolation [34e36]. The mean lift coefficient of one cycle CL-mean is considered integral quantities, which is defined as CL-mean ¼ SCL/TS (TS is the time step, CL ¼ L/ (0.5rU2∞sc), L is the lift). As the GCI are less than 5%, the results for mesh III are considered as mesh independent, and the subsequent simulations are performed with the meshing scheme of mesh III. In addition, the finite volume method is applied for the space
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
discretization. The high order resolution scheme was used for the convective term, with the second order central difference scheme used for the diffusion term in the governing equations. The second order backward euler scheme was used for the transient term. And the time step was set to 5 105 s (Tref/200, where Tref ¼ c/U∞). This treatment is an effective way to investigate the fundamental vortex structure, as reported recently by many researchers [37,38].
419
the moment coefficient curve of pitching cases is prominent. Detailed analysis of the flow structures and corresponding hydrodynamics will be presented as following.
5. Results and discussion 5.1. Flow structures and corresponding hydrodynamics for the fixed and pitching hydrofoil
4.2. Numerical validation Fig. 6 compares the measured and predicted moment coefficients for the fixed cases and pitching case of a_ * ¼ 0:086. The xcoordinate is angle of attack (AOA) a, and the y-coordinate is the moment coefficient CM (CM ¼ M/(0.5rU2∞sc2)), with the negative moment indicated the clockwise direction. We can see that the predicted results agree well with the experimental results. For the upstroke I, the moment coefficient increases with fluctuation gradually. When the angle of attack begins to decline, the moment also declines gradually. Until a ¼ 5 , the moment reaches the lowest value in the whole process. With the change of pitching direction, the angle of attack and moment again increases. Compared to the fixed cases, it can be noted that the fluctuation of
5.1.1. Typical flow regimes around the fixed hydrofoil Fig. 7 shows the contours of z-vorticity superimposed on the instantaneous streamlines and the velocity vector at different attack angle. The distributions of skin friction (Cf ¼ f/(0.5rU2∞, where f is the wall shear stress) along the hydrofoil at different attack angle are presented in Fig. 8. At a ¼ 2 , the flow is quasi-laminar, as shown in Fig. 7(a). According to the Prandtl separation criterion [31], it defines the laminar separation point at (vux/vy)y ¼ 0. We can see that the skin friction of the hydrofoil with a ¼ 2 is negative from x/ c ¼ 0.47e0.66, which is responsible to the transition from laminar to turbulent flow, as shown in Fig. 8. The location of x/c ¼ 0.47 is defined as laminar separation point, and the location of x/c ¼ 0.66 is
Fig. 12. The evolution of the predicted lift coefficient (CL), drag coefficient (CD) for Re ¼ 4.4 105, U∞ ¼ 6.3 m/s.
Fig. 13. FTLE field, the corresponding LCS and the trajectory of the specific points at various geometric angle of attack.
420
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
defined as turbulence reattachment point. In transition area, it can be noted in Fig. 7(a) that there are reversed velocity vector around the suction side of the hydrofoil. At a ¼ 5 , the skin friction of the hydrofoil is negative from x/c ¼ 0.26e0.40. It can be found that the locations of the laminar separation point and turbulence reattachment point all are advanced. While for the hydrofoil with a ¼ 8 , the laminar separation point is closer to the leading edge than that of a ¼ 2 and a ¼ 5 , as shown in Figs. 7(aec) and Fig. 8. For a ¼ 12 , the laminar flow transits to turbulent flow on the leading edge from x/c ¼ 0.1e0.2, which is corresponding to the reversed velocity vector on the leading edge, as shown in Figs. 7(d) and 8. It can be concluded that the transition from laminar to turbulent flow happens on closer and closer to the leading edge and the transition area gets smaller and smaller. Meanwhile the turbulent boundary layer separates from the trailing edge because of the adverse pressure gradient, with the skin friction being negative from x/c ¼ 0.71 to x/c ¼ 1, as shown in Figs. 7(d) and 8. The location of x/c ¼ 0.71 is defined as turbulence separation point, after that the flow couldn’t attach on the surface of hydrofoil and the light static stall happens. For a ¼ 19 , the turbulence separation point has occurred on the leading edge of hydrofoil, thus leading to the deep static stall, with the hydrodynamic performance declines gradually, as shown in Figs. 7(e) and 8.
5.1.2. Typical flow regimes around the pitching hydrofoil Fig. 9 shows the evolution of flow regimes around fixed hydrofoil and pitching hydrofoil of a_ * ¼ 0:086. The x-coordinate is angle of attack (AOA) a, and the y-coordinate is the nondimensional location x/c. According to the location of laminar separation, turbulence reattachment, and turbulence separation at different fixed angle of attack, the flow regime is divided into four types-Laminar, Transition, Turbulence and Stall, as shown with black solid lines in Fig. 9. In which, the discrete red points mark the location of laminar separation, turbulence reattachment, and turbulence separation during the pitching process, with hollow points represented the upstroke stages and the solid points represented the downstroke stage. It’s noted that the frontiers between different flow patterns of pitching cases is different with that of fixed cases. Fig. 10 presents the non-dimensional turbulence kinetic energy K* (K * ¼ K=U 2∞ , where K is the turbulence kinetic energy) distribution around hydrofoil for pitching case and fixed case. During the upstroke I(aþ ¼ 10 ~aþ ¼ 15 ), the laminar separation point moves toward the leading edge gradually, which is corresponding to the high turbulence kinetic energy on leading edge, as shown in Figs. 9 and 10(a). Because of the motion of the boundary, the turbulence kinetic energy around pitching hydrofoil declines slightly, thus leading to the delay of transition from
Fig. 14. The FTLE field, the corresponding LCS and the trajectory of the specific points at various geometric angle of attack a for a_ * ¼ 0:086, Re ¼ 4.4 105, U∞ ¼ 6.3 m/s.
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
421
Fig. 15. The evolution of flow regime around fixed cases and pitching cases of three pitching rates for Re ¼ 4.4 105, U∞ ¼ 6.3 m/s.
laminar to turbulence. Besides, the turbulent reattachment points and the turbulent separation points are also delayed, as showed in Figs. 9 and 10. The skin friction distribution along the hydrofoil and the contour of turbulence kinetic energy superimposed on velocity profile at the separation position with a ¼ 8 and 13 are presented in Fig. 11. Compared to the fixed cases of a ¼ 13 , the location of laminar separation point is delayed slightly at aþ ¼ 13 , as shown in Fig. 11(b). It is also found in Fig. 11(b) that the turbulence kinetic energy distribution around the trailing edge at aþ ¼ 13 are weaker than that of fixed case (a ¼ 13 ). For a ¼ 13 , the location of the turbulent separation point is x/c ¼ 0.64; for aþ ¼ 13 , the location of the turbulent separation point is x/c ¼ 0.70. At x/c ¼ 0.70, the reverse velocity has occurred on the hydrofoil boundary layer for fixed case (a ¼ 13 ), as showed in Fig. 11(b).
During the downstroke I(aþ ¼ 15 ~a ¼ 10 ), the laminar separation points are similar to that of the upstroke I. The turbulence kinetic energy declines with the decreasing attack angle, thus leading to the delay of turbulence reattachment, as showed in Fig. 10. However, we can find that the turbulence separation is expedited, thus leading to the obvious hysteresis in the downstroke I, as shown in Fig. 9. It’s noted in Fig. 11(b) that the location of the turbulent separation points for a ¼ 13 is x/c ¼ 0.57, which is corresponding to the increase of turbulence around the trailing edge. For velocity profiles at x/c ¼ 0.57 of the hydrofoil, as shown in Fig. 11(b), the velocity of a ¼ 13 is higher than that of aþ ¼ 13 and a ¼ 13 . During the downstroke II(a ¼ 10 ~a ¼ 5 ), the turbulence kinetic energy distribution around hydrofoil is similar to fixed
422
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 16. The non-dimensional turbulence kinetic energy distribution around hydrofoil for pitching cases of two pitching rates.
Fig. 17. The turbulence kinetic energy distribution contour around hydrofoil at different attack angle for different pitching rates.
cases, which is responsible to the similar laminar separation points, turbulence reattachment points and turbulent separation points, as showed in Figs. 9 and 10. It’s noted in Fig. 11(a) that the location of the turbulent separation points for a ¼ 8 and a ¼ 8 is uniform (x/ c ¼ 0.91), and the difference of velocity profiles is very small. During the upstroke II(a ¼ 5 ~aþ ¼ 10 ), the location of the laminar separation point is similar to that during the downstroke II(a ¼ 10 ~a ¼ 5 ). However, the turbulence reattachment points and turbulent separation points are delayed due to the dynamic boundary, as showed in Figs. 9 and 10. It is also found in Fig. 11(a) that the turbulence kinetic energy distribution around the trailing edge are weaker than that of fixed case (a ¼ 8 ), and the location of the turbulent separation points for aþ ¼ 8 is x/c ¼ 0.94. Fig. 12 shows the evolution of the predicted lift coefficient and drag coefficient (CD ¼ D/(0.5rU2∞sc), where D is the drag) for the fixed and the pitching case. In the upstroke I, the lift coefficient and drag coefficient increase linearly with the increase of the angle of attack. The lift coefficient of the pitching case is higher than that of the fixed cases. This is because the flow separation is delayed. Then the lift coefficient declines sharply when the angle of attack declines in the downstroke I. However, the values are lower than that
of the fixed cases, thus leading to the obvious hysteresis. This may due to the acceleration of the flow separation in the downstroke I. Besides, compared with the fixed case, it can be found that fluctuation of the lift coefficient curve of pitching case is very evident, which will be discussed in next section. For the downstroke II and upstroke II processes, the lift coefficients of the fixed cases and the pitching case are almost the same, and the trend of the drag coefficients of the fixed cases and the pitching case is basically the same. In order to further study the flow structure and hydrodynamic performance of pitching hydrofoil during the upstroke I and downstroke I stages, the FTLE field, the corresponding LCS and the trajectory of the specific points at typical angle of attack a are presented in Fig. 13. Fig. 13(a) shows the trajectory of the specific points (Points A/B/ C) in the adjacent vortices from t1:aþ ¼ 12.8 to t2:aþ ¼ 12.91 during the time interval TLE ¼ 0.00275s (55 numerical steps). The LCS A represents the boundary of vortex near the trailing edge of hydrofoil. As time goes on, the trajectory of the points C that are inside LCS A show a clockwise rotation, which is responsible for the formation of the trailing edge vortex (TEV). As for the trajectory of the Points A/B near the leading edge of hydrofoil, the points A that are outside LCS A move downstream, while the points B that are inside LCS A move toward the trailing edge and rotate clockwise, thus inducing a leading edge vortex (LEV). It is noted in Fig. 12 that the formed vortex structures lead to the slight fluctuation of hydrodynamic forces. The evolution of the LEV and the clockwise TEV in the upstroke I is presented in Fig. 14(a). It is noted that the group A represents the LEV and the group B represents this clockwise trailing edge vortex (TEV). With the increase of the angle of attack, the LEV moves toward the trailing edge and merges with the TEV, which is responsible to the increase of hydrodynamic curve, as showed in Fig. 12. The trajectory of the specific points (Points D/E/F) in the adjacent vortices from t6:aþ ¼ 14.42 to t7:aþ ¼ 14.49 during the time interval TLE ¼ 0.00175s (35 numerical steps) are presented in Fig. 13(b). It is found that the points E rotate anticlockwise on the trailing edge with the increasing angle. These trajectories suggest an anticlockwise vortex structure on the trailing edge of hydrofoil, which is responsible of the formation of the LCS E. Meanwhile, the dynamic curves start to decline gradually due to the interaction between clockwise and anticlockwise vortex, thus leading to the
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
slight dynamic stall. It’s found that the anticlockwise TEV plays an important role in the hydrodynamic performance around the pitching hydrofoil. Fig. 14(b) shows that the evolution of the clockwise and anticlockwise TEV from the upstroke I to the downstroke I. At t8:aþ ¼ 14.84 , the clockwise vortex (group E and LCS C) and anticlockwise vortex (group F and LCS F) are formed. Due to the interaction between the anticlockwise TEV (group F) and the clockwise TEV (group E), the LCS C and the anticlockwise TEV (group F) vanishes gradually, thus leading to the decline of hydrodynamic curve, as shown in Figs. 12 and 14(b). When the angle of attack reaches 15 , the hydrofoil begins to rotate anticlockwise. At t10:a ¼ 14.57, a new anticlockwise vortex (group G and LCS G) are formed again and interact with the clockwise TEV. In addition, Fig. 14(c) shows the evolution of the LEV, clockwise and anticlockwise TEV in the downstroke I. As shown from t11:a ¼ 13.54 to t13:a ¼ 13.2 in Fig. 14(c), we can find that the LEV, clockwise and anticlockwise TEV still are formed. However, the area of vortex wakens gradually with the decline of the angle of attack. 5.2. Influence of the pitching rate on the flow structures and corresponding hydrodynamics In order to further study the influence of the pitching rate, the evolution of flow regime around fixed cases and pitching cases of three pitching rates are presented in Fig. 15. Fig. 16 shows the nondimensional turbulence kinetic energy distribution around the pitching hydrofoil with different pitching rates. The turbulence kinetic energy distribution contour around the hydrofoil at typical attack of angles for different pitching rates are shown in Fig. 17. Fig. 18 shows that comparisons of the predicted pressure coefficients distributions at various geometric angle of attack a for three pitching rates. It is noted in Fig. 15 that the laminar separation points move
423
upward slightly with the increase of the pitching rate from aþ ¼ 10 to aþ ¼ 15 . However, the turbulence reattachment points and turbulent separation points are further delayed. This may result from the decline of turbulence kinetic energy around hydrofoil, as showed in Fig. 16. As shown in Fig. 17(c), we can see that the turbulence kinetic energy near the trailing edge at aþ ¼ 13 declined with the increase of the pitching rate. It is also noted in Fig. 18(c) that the adverse pressure gradient at aþ ¼ 13 declines with the increasing of pitching rate. During the downstroke I(aþ ¼ 15 ~a ¼ 10 ), the location of laminar separation points, the turbulence reattachment points and the turbulent separation points move upward with the increase of the pitching rate, as shown in Fig. 15. We can find that the turbulence kinetic energy and the adverse pressure gradient around the trailing edge of hydrofoil at a ¼ 13 with a_ * ¼ 0:172 are higher than a ¼ 13 with a_ * ¼ 0:0215 and 0.086, as shown in Figs. 17(d) and 18(d). During the downstroke II(a ¼ 10 ~a ¼ 5 ), the discrepancy between the cases with three pitching rates declined, as showed in Figs. 15 and 16. It’s noted in Figs. 17(b) and 18(b) that the turbulence kinetic energy and the adverse pressure gradient around the trailing edge of hydrofoil at a ¼ 8 is very similar. During the upstroke II(a ¼ 5 ~aþ ¼ 10 ), the laminar points and turbulence separation points don’t change much for different pitching rates. However, turbulence reattachment points are further delayed at high pitching rate, as shown in Fig. 15. We can see that the discrepancy of turbulence kinetic energy and adverse pressure gradient at aþ ¼ 8 decline, as shown in Figs. 17(a) and 18(a). It’s also noted in Fig. 17(a) that the turbulent distribution around the leading edge of hydrofoil at a_ * ¼ 0:172 are lower than that of others. To further analyze the effect of pitching rate on the hydrodynamic performance, the evolution of the predicted lift coefficient (CL) for three pitching rates are presented in Fig. 19. As shown in
Fig. 18. Comparisons of the predicted pressure coefficients distributions at various geometric angle of attack a for three pitching rates.
424
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 19. The evolution of the predicted lift coefficient (CL) for three pitching rates.
Fig. 19, the maximum lift coefficient increases with the increase of the pitching rate. Moreover, the higher pitching rate broadens the hysteresis loops and the lower pitching rate intensify fluctuation of the dynamic forces, as shown in Fig. 19. It also reveals that the pitching rate, where there are consecutive vortical patterns in the lift coefficient curves, has considerable influence on the strength of these vortices and their initiation location during the pitching motion. Figs. 20 and 21 show the FTLE field, the corresponding LCS and the trajectory of the specific points at various geometric angle of attack a for a_ * ¼ 0:0215 and a_ * ¼ 0:172 respectively. Fig. 20(a) shows the trajectory of the specific points (Points A’/ B0 ) in the adjacent vortices from t’1:aþ ¼ 12.21 to t’2:aþ ¼ 12.27 during the time interval TLE ¼ 0.0015s (30 numerical steps) with a_ * ¼ 0:0215. It is found that the first LEV (LCS A0 ) occurs on t’2:aþ ¼ 12.27, which is earlier than that of a_ * ¼ 0:086. Moreover, as shown in Fig. 19, the numbers of effective LEV lead to the highfrequency fluctuations of lift coefficient. It can be concluded that the formation of the LEV effects the lift evolution. At t’3: aþ ¼ 13.89 , the first anticlockwise TEV (group D0 and LCS C0 ) is formed, which is earlier than that with a_ * ¼ 0:086 (t8:aþ ¼ 14.49 ). Similarly, the dynamic performance curves start to decline gradually, which is responsible to dynamic stall in advance. As shown in Fig. 20(c), when the angle of attack reaches to t’4:aþ ¼ 14.96 , the anticlockwise TEV again is formed. From t’4:aþ ¼ 14.96 to
t’6:a ¼ 14.97, the evolution of vortices is similar to the case of a_ * ¼ 0:086. The trajectory of the specific points (Points A’’/B00 ) in the adjacent vortices from t’’1:aþ ¼ 14.29 to t’’2:aþ ¼ 14.78 during the time interval TLE ¼ 0.01225s (245 numerical steps) with a_ * ¼ 0:172 is presented in Fig. 21(a). As the pitching rate is increased to a_ * ¼ 0:172, the attack angle at which the first LEV (group A00 and LCS A00 ) forms is changed from t3:aþ ¼ 13.05 at a_ * ¼ 0:086 to t’’2:aþ ¼ 14.78 at a_ * ¼ 0:172. From t’’2:aþ ¼ 14.78 to t’’4:a ¼ 14.89 , the interaction between the LEV and clockwise TEV is similar to the case of a_ * ¼ 0:086, as shown in Fig. 21(b). Moreover, there aren’t anticlockwise TEV in the upstroke I process. The dynamic stall didn’t occur in this process. Until the angle of attack reaches t’’5:a ¼ 13.25 in the downstroke I, the first anticlockwise TEV (group F00 and LCS C00 ) is formed, as shown in Fig. 21(c). Fig. 22 shows the location evolution of the first LEV and first anticlockwise TEV with three pitching rates. It is found that the first LEV and anticlockwise TEV are delayed with the increase of the pitching rate. The location of the first LEV and the numbers of LEV affect the dynamic performance around the pitching hydrofoil. Moreover, the interaction between the clockwise and anticlockwise TEV results in the decline of hydrodynamic performance. As shown from the stall line in Fig. 22, it also reveals that the anticlockwise TEV plays an important role in the dynamic stall.
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
425
Fig. 20. The FTLE field, the corresponding LCS and the trajectory of the specific points at various geometric angle of attack a for a_ * ¼ 0:0215, Re ¼ 4.4 105, U∞ ¼ 6.3 m/s.
6. Conclusions In this work, different flow regimes and hydrodynamic performance around the fixed and pitching hydrofoil with different pitching rates are investigated at Re ¼ 4.4 105. In general, the pitching motion and the pitching rate have significant effect on the transition between the different flow patterns (especially laminarturbulent transition and turbulence separation) and hydrodynamic performance. The primary findings include: (1) For the fixed cases, with increase of the angle of attack, the laminar separation point is closer to the leading edge. The turbulence kinetic energy distribution around the hydrofoil has an important influence on both the hydrofoil boundary layer and the separated shear layer. (2) For the pitching hydrofoil, the dynamic boundary affects the turbulence kinetic energy distribution around the hydrofoil. Higher levels of turbulence kinetic energy cause earlier transition compared to the lower turbulence levels. From aþ ¼ 10 to a ¼ 10 , the location of laminar separation points, turbulence reattachment points and turbulence separation points are delayed or advanced due to the evolution of turbulence kinetic energy distribution. However, these locations don’t vary much from a ¼ 10 to aþ ¼ 10 because the differences of turbulence kinetic energy distribution
around hydrofoil between fixed cases and pitching case decrease. In addition, the interaction between the clockwise and anticlockwise TEV results in the dynamic stall. (3) The pitching rate plays an important role in flow regime and hydrodynamic performance. From aþ ¼ 10 to aþ ¼ 15 , the increase of pitching rate causes the acceleration of the transition from laminar to turbulence. However, the increase of pitching rate enlargers the delayed location of the turbulence reattachment points and turbulence separation points. From aþ ¼ 15 to a ¼ 10 , the high pitching rate promotes the turbulence separation immensely. From a ¼ 10 to aþ ¼ 10 , the effect of pitching rate on laminar separation and turbulence separation is limited. But the high pitching rate affects the turbulence reattachment partly. In addition, the higher pitching rate broadens the hysteresis loops. The first LEV and anticlockwise TEV are delayed with the increase of pitching rate, which is responsible to the delay of dynamic stall. Moreover, the lower pitching rate intensifies fluctuation of the dynamic force due to the strength and number of LEV. Although reasonable agreement between the experimental and numerical results has been obtained, some discrepancies are still existed. In order to acquire the spatial and temporal variation of the flow structures more accurately, Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES) will be further studied in
426
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 21. FTLE field, the corresponding LCS and the trajectory of the specific points at various geometric angle of attack a for a_ * ¼ 0:172, Re ¼ 4.4 105, U∞ ¼ 6.3 m/s.
flow regime and dynamic performance will be also further discussed in the future.
Author contribution statement Zhang Mengjie: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - Original Draft. Wu Qin: Software, Validation, Writing - Review & Editing, Supervision, Funding acquisition. Wang Guoyu: Resources, Supervision, Funding acquisition. Huang Biao: Conceptualization, Software, Validation, Writing Review & Editing, Supervision, Funding acquisition. Fu Xiaoying: Resources, Funding acquisition. Chen Jie: Project administration.
Fig. 22. The location evolution of the first LEV and anticlockwise TEV with the pitching rate.
the future work. In addition, the three dimensional effect of the flow due to the gap flow will be discussed to better illuminate the underlying physics. Especially, the impact of the cavitation on the
Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: As we have too similar research topics and we have conducted investigation based on the too similar research objects, there is a potential conflict of interest existing between us.
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant Nos: 51839001, 51909002 and 91752105), the National Natural Science Foundation of Beijing (Grant No: 3172029), and the Fundamental Research Funds for the Central Universities. References ~o, Transient analysis of variable-speed [1] R. Melício, V.M.F. Mendes, J.P.S. Catala wind turbines at wind speed disturbances and a pitch control malfunction, Appl. Energy 88 (4) (2011) 1322e1330. lez, E. Figueres, G. Garcer [2] L.G. Gonza a, O. Carranza, Maximum-power-point tracking with reduced mechanical stress applied to wind-energy-conversionsystems, Appl. Energy 87 (7) (2010) 2304e2312. [3] H.R. Karbasian, J.A. Esfahani, E. Barati, The power extraction by flapping foil hydrokinetic turbine in swing arm mode, Renew. Energy 88 (2016) 130e142. [4] S. Park, S. Park, S.H. Rhee, Influence of blade deformation and yawed inflow on performance of a horizontal axis tidal stream turbine, Renew. Energy 92 (2016) 321e332. [5] J. Zanette, D. Imbault, A. Tourabi, A design methodology for cross flow water turbines, Renew. Energy 35 (5) (2010) 997e1009. [6] D. Li, H. Wang, Y. Qin, Z. Li, X. Wei, D. Qin, Mechanism of high amplitude low frequency fluctuations in a pump-turbine in pump mode, Renew. Energy 126 (2018) 668e680. [7] H. Zhang, Q. Wu, Y. Li, B. Huang, G. Wang, Numerical investigation of the deformation characteristics of a composite hydrofoil with different ply angles, Ocean Eng. 163 (2018) 348e357. [8] B. Huang, A. Ducoin, L.Y. Young, Physical and numerical investigation of cavitating flows around a pitching hydrofoil, Phys. Fluids 25 (10) (2013) 102e109. [9] B. Huang, Q. Wu, G.Y. Wang, Numerical simulation of unsteady cavitating flows around a transient pitching hydrofoil, Sci. China Technol. Sci. 57 (1) (2014) 101e116. [10] D.T. Akcabay, Y.L. Young, Hydroelastic response and energy harvesting potential of flexible piezoelectric beams in viscous flow, Phys. Fluids 24 (2012), 054106. [11] Q. Wu, C.C. Wang, B. Huang, et al., Measurement and prediction of cavitating flow-induced vibrations, J. Hydrodyna. Ser. B. 30 (6) (2018) 1064e1071. [12] Q. Wu, B. Huang, G.Y. Wang, et al., Experimental and numerical investigation of hydroelastic response of a flexible hydrofoil in cavitating flow, Int. J. Multiph. Flow 74 (2015) 19e33. [13] M. Liu, L. Tan, Y. Liu, Y. Xu, S. Cao, Large eddy simulation of cavitation vortex interaction and pressure fluctuation around hydrofoil ALE15, Ocean Eng. 163 (2018) 264e274. [14] Z.F. Yang, F. Haan, H. Hu, H.W. Ma, An Experimental Investigation on the Flow Separation on a Low-Reynolds-Number Airfoil, 45th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings, Reno, Nevada, 2007. [15] Z. Wei, T.H. New, Y.D. Cui, An experimental study on flow separation control of hydrofoils with leading-edge tubercles at low Reynolds number, Ocean Eng. 108 (2015) 336e349. [16] H. Shan, L. Jiang, C. Liu, Direct numerical simulation of flow separation around a NACA 0012 airfoil, Comput. Fluids 34 (9) (2005) 1096e1114. [17] Z. Yang, P.R. Voke, Large-Eddy Simulation of boundary layer separation and
[18] [19] [20] [21]
[22]
[23] [24]
[25]
[26]
[27]
[28] [29] [30]
[31]
[32]
[33]
[34] [35] [36]
[37]
[38]
427
transition at a change of surface curvature, J. Fluid Mech. 439 (439) (2001) 305e333. J.P. Franc, J.M. Michel, Unsteady attached cavitation on an oscillating hydrofoil, J. Fluid Mech. 193 (1988) 171e189. J.M. Anderson, K. Streitlien, D.S. Barrett, et al., Oscillating foils of high propulsive efficiency, J. Fluid Mech. 360 (1998) 41e72. G. Pedro, A. Suleman, N. Djilali, A numerical study of the propulsive efficiency of a flapping hydrofoil, Int. J. Numer. Methods Fluids 42 (2003) 493. I. Akhtar, R. Mittal, G.V. Lauder, et al., Hydrodynamics of a biologically inspired tandem flapping foil configuration, Theor. Comput. Fluid Dyn. 21 (3) (2007) 155e170. Q. Wu, B. Huang, G.Y. Wang, et al., The transient characteristics of cloud cavitating flow over a flexible hydrofoil, Int. J. Multiph. Flow 99 (2018) 162e173. R.B. Langtry, F.R. Menter, Transition Modeling for General CFD Applications in Aeronautics, AIAA Paper, 2005, p. 522. D. Li, H. Wang, Z. Li, T.K. Nielsen, R. Goyal, X. Wei, D. Qin, Transient characteristics during the closure of guide vanes in a pump-turbine in pump mode, Renew. Energy 118 (2018) 973e983. D. Li, Y. Qin, Z. Zuo, H. Wang, S. Liu, X. Wei, Numerical simulation on pump transient characteristic in a model pump turbine, J. Fluids Eng. 141 (11) (2019) 111101. M.J. Zhang, Q. Wu, B. Huang, G.Y. Wang, Lagrangian-based numerical investigation of aerodynamic performance of an oscillating foil, Acta Mech. Sin. 34 (5) (2018) 839e854. R.B. Langtry, F.R. Menter, S.R. Likki, Y.B. Suzen, et al., A correlation-based transition model using local variables-Part I: model formulation, J. Turbomach. 128 (2006) 413e422. F.R. Menter, Improved two-equation k-u turbulence models for aerodynamic flows, NASA Tech. Memo. 34 (1992) 103975. G. Haller, G. Yuan, Lagrangian coherent structures and mixing in two dimensional turbulence, Phys. D Nonlinear Phenom. 147 (2000) 352e370. J.N. Tang, C.C. Tseng, N.F. Wang, Lagrangian-based investigation of multiphase flows by finite-time Lyapunov exponents, Acta Mech. Sin. 28 (3) (2012) 612e624. C.C. Tseng, P.B. Liu, Dynamic behaviors of the turbulent cavitating flows based on the Eulerian and Lagrangian viewpoints, Int. J. Heat Mass Transf. 102 (2016) 479e500. Q. Wu, B. Huang, G.Y. Wang, Lagrangian-based investigation of the transient flow structures around a pitching hydrofoil, Acta Mech. Sin. 32 (1) (2016) 64e74. M.J. Zhang, H. Chen, Q. Wu, et al., Experimental and numerical investigation of cavitating vortical patterns around a tulin hydrofoil, Ocean Eng. 173 (2019) 298e307. L.F. Richardson, J.A. Gaunt, The deferred approach to the limit, Philos. Trans. R. Soc., A 226 (636e646) (1927) 299e349. E. Jak, P.C. Hayes, Procedure of estimation and reporting of uncertainty due to discretization in CFD applications, J. Fluids Eng. 130 (7) (2008), 078001. Y. Long, X.P. Long, B. Ji, T. Xing, Verification and validation of Large Eddy Simulation of attached cavitating flow around a Clark-Y hydrofoil, J. Multiph. Flow 115 (2019) 93e107. O. Coutier-Delgosha, J.L. Reboud, Y. Delannoy, Numerical simulation of the unsteady behaviour of cavitating flows, Int. J. Numer. Methods Fluids 42 (2003) 527e548. B. Ji, X.W. Luo, R.E.A. Arndt, et al., Large Eddy Simulation and theoretical investigations of the transient cavitating vortical flow structure around a NACA66 hydrofoil, Int. J. Multiph. Flow 68 (2015) 121e134.
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
The flow regime and hydrodynamic performance for a pitching hydrofoil Mengjie Zhang a, Qin Wu a, *, Guoyu Wang a, Biao Huang a, **, Xiaoying Fu b, Jie Chen a a b
School of Mechanical Engineering, Beijing Institute of Technology, Beijing, China State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 May 2019 Received in revised form 31 December 2019 Accepted 2 January 2020 Available online 7 January 2020
The objective of this paper is to study the flow regime and hydrodynamic performance for a pitching Clark-Y hydrofoil. The aims are to (1) improve the understanding of the interplay between the hydrodynamic performance, unsteady flow structures and dynamic motion of the hydrofoil, (2) study the influence of the pitching rate on the transition of different flow regimes. The experimental investigations were conducted in the looped cavitation tunnel, and the dynamic moment measurement system was applied to obtain the hydrodynamic forces. The pitching hydrofoil is controlled to rotate from aþ ¼ 10 to aþ ¼ 15 firstly, then goes from aþ ¼ 15 to a ¼ 5 , finally goes back to aþ ¼ 10 from a ¼ 5 . The pitching rate is varying with the Reynolds number Re ¼ 4.4 105. The numerical investigations were performed by solving the incompressible URANS equations using the coupled k-u SST turbulence model and g-Req transition model. The numerical results agree well with the experimental measurements. The pitching motion affects the turbulence kinetic energy distribution around the hydrofoil, leading to the delay or acceleration of the transition between different flow patterns. During the pitching process, higher level of turbulence kinetic energy distribution causes earlier transition from laminar to turbulence. Moreover, hysteresis effect of the hydrodynamic force is observed. For the upstroke stage, the higher pitching rate promotes the laminar separation slightly and intensifies the delay of turbulence separation. For the downstroke stage, the higher pitching rate promotes the turbulence separation extensively. The first leading edge vortex (LEV) and anticlockwise trailing edge vortex (TEV) are delayed with the increase of pitching rate, which is responsible to the delay of dynamic stall. The lower pitching rate shrinks the hysteresis loops and intensifies the fluctuation of the dynamic force. © 2020 Published by Elsevier Ltd.
Keywords: Hydrodynamic performance Flow pattern Pitching hydrofoil Lagrangian coherent structures
1. Introduction Because of the high cost of non-renewable fossil fuels, the development of renewable energy sources has been promoted [1e3]. Among which, hydroenergy has attracted much attention because it is resource-rich, widely distributed and clean. As one of the important hydraulic machinery, the water turbines have been widely applied [4e6]. However, the effective angle of attack of the blades often changes due to varying inflow and active/passive body motions in the water turbine system [7e9]. The dynamic motion of boundary caused by the varying angle of attack is often accompanied with different flow regimes, including laminar, transition,
* Corresponding author. ** Corresponding author. E-mail address: [email protected] (Q. Wu). https://doi.org/10.1016/j.renene.2020.01.006 0960-1481/© 2020 Published by Elsevier Ltd.
turbulence and stall, leading to complex and unsteady hydrodynamic load. Especially, laminar-turbulent transition and dynamic stall always attract much attention, as they play an important role in the performance of engineering system. Therefore, it is very important to understand the dynamic interaction between the pitching motion, unsteady flow structures and hydrodynamic performance. It does not only help improve the overall design of hydraulic and marine structures, but also promote better performance for harvesting energy [10]. Many studies of flow regimes and hydrodynamic performance around the static foils have been conducted [11e13], with focus on the flow separation mechanism. Yang et al. [14] investigated the transient behavior of the flow separation on a NASA low-speed GA (W)-1 airfoil at the Reynolds number Re ¼ 6.8 104 by a highresolution PIV system. The results showed that the increased adverse pressure gradient caused the laminar boundary layer separation on the airfoil, and the separated laminar boundary layer
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
413
Fig. 1. Schematic of the cavitation tunnel.
transited to turbulent flow rapidly by generating unsteady vortex structures due to Kelvin-Helmholtz instabilities, which reattached to the upper surface of the airfoil as a turbulent boundary layer. Wei et al. [15] experimentally studied the flow separation control for hydrofoils with leading-edge tubercles in a water tunnel at the Reynolds number Re ¼ 1.4 104. The cross-stream flow measurements indicated that the streamwise counter-rotating vortex pairs were generated over the tubercles and mitigated flow separation. Compared with the experimental research, many researchers have also conducted numerical study to investigate the flow structure around hydrofoils. The flow separation and transition around a NACA0012 airfoil with a ¼ 4 and Re ¼ 1.0 105 were investigated by Liu et al. [16]. They found that the disturbances in the near wake propagated upstream in the form of acoustic waves and induced disturbances to the separated shear layer over the upper surface of the airfoil. Yang et al. [17] studied separated boundary-layer transition on a flat plate with a semicircular leading edge, with focus on the physics of separated boundary-layer transition induced by a change of surface curvature. The results showed that there are some vortical structures at various stages of the transition process. In order to consider the impact of the dynamic boundary on the flow structure of the hydrofoil, many studies about the pitching foil have also been investigated. Franc et al. [18] experimentally studied the impact of unsteadiness on the attached cavitation around an oscillating hydrofoil. Results presented that a cavity detaches behind a laminar separation of the boundary layer under steady conditions, and the unsteadiness affects both the cavity and the boundary layer, essentially through a convection effect and a delay effect. Anderson et al. [19] studied propulsive performance of a harmonically oscillating foil through force and power measurements, as well as visualization data. Results showed that highefficiency conditions are associated with the formation of a moderately strong leading-edge vortex, which is convected downstream and interacts with the trailing-edge vortex, eventually resulting in the formation of a reverse Karman street. Pedro et al. [20] simulated the dynamics of a flapping hydrofoil and found that the thrust and efficiency are significantly affected by the Strouhal number, maximum pitch angle and phase angle. Akhtar et al. [21] numerically examined the performance of a foil undergoing
Fig. 2. Schematic of the pitching hydrofoil system.
oscillating motion in the wake of another oscillating foil. It was found that the shedding vortex of upstream hydrofoil increases the effective angle of the downstream hydrofoil, resulting to the formation of a stronger leading edge vortex on the downstream hydrofoil, thus improves the performance (thrust and efficiency) of the downstream foil. Much effort has been conducted on the flow regimes around static and pitching hydrofoil. However, the effect of dynamic boundary on the flow regime and hydrodynamic performance is still implicit, the transition of different flow regimes around the pitching foil have not been explained clearly yet. The objective of this paper is to study flow regimes and hydrodynamic performance around the pitching hydrofoil. The aims are to (1) improve the understanding of the interplay between unsteady flow structure, hydrofoil motion and hydrodynamic performance, (2) investigate the impact of the pitching rate on the flow structure and hydrodynamic performance, especially for the evolution of flow regime. The experimental setup and the numerical methods are presented in Section 2 and 3, respectively. In Section 4, the numerical setup and verification are given. The detailed flow regimes, flow structure and hydrodynamic performance for fixed cases and pitching cases with three pitching rates are discussed in Section 5. Finally, the major findings and future work are summarized in Section 6. 2. Experimental setup Experiments were conducted in the looped cavitation tunnel [22] with a test section of 190 mm height, 70 mm width and 700 mm length, as shown in Fig. 1. Two main control parameters, upstream pressure and velocity, are measured respectively by the vacuum gauge (with the uncertainty 0.25% of the maximum range, 0.1Mpa) and the electromagnetic flowmeter (with the uncertainty 0.5% of the maximum range, 1550 m3/h). The detailed performance parameters of Cavitation tunnel is given in Table 1. Fig. 2 shows the schematic of the pitching hydrofoil system, which mainly includes PC(Personal Computer), controller, motor, hydrofoil, rotating transformer, moment sensor and Data
Table 1 The performance parameters of Cavitation tunnel. Minimum cavitation number
Maximum velocity
Steady speed
Coefficient of stability
Reduced pressure maintenance.
Vacuum range
0.30
20 m/s
4.68e12.78 m/s
0.71e1.93%
1.11e12.83%
0e0.1 Mpa
414
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 3. Variation of the angle of attack (AOA) a with time (a_ * ¼ 0:086; 0:0215; 0:172).
Fig. 4. Fluid domain and boundary conditions.
Acquisition Card. It is mounted horizontally in the tunnel test section. The controller and the motor (150 ST-M23020) realize the reciprocating oscillating motion of the hydrofoil. The oscillating motion of the hydrofoil is closed-loop controlled by a rotating transformer (TS2640N321E64), with the precision of the angle of attack a of the hydrofoil is ±0.1, as shown in Fig. 2. The IC chip (AD2S1210) of the rotating transformer is applied to convert analog output signals into digital position (angle) signals. The moment of hydrofoil M is measured by the moment sensor (SPI-M2210A), which is mounted on a rotating shaft, as shown in Fig. 2. When the
Fig. 6. Comparisons of the experimental moment coefficient and the numerical moment coefficient for fixed case and pitching case (a_ * ¼ 0:086).
shaft deforms, the change of the bridge resistance results in the change of the electrical signal. Synchronously, the torque is collected by data acquisition card (NI 6356). The sampling frequency of the moment sensor is 20 kHz, and the precision is 0.02N.m. Table 2 Grid convergence study based on Richardson extrapolation.
Fig. 5. Mesh distribution.
Mesh I Mesh II Mesh III
Grid nodes
CL-mean
GCI(%)
396000 792000 1584000
1.02362 1.13617 1.18815
GCI21 ¼ 0.11795 GCI32 ¼ 0.04908
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 7. The Contours of z-vorticity superimposed on the instantaneous streamlines and the velocity vector at different angles of attack.
415
416
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 8. The skin friction distribution along the hydrofoil at different attack angle as noted in Fig. 7.
In this paper, the nominal free stream velocity U∞ is fixed at 6.3 m/s, corresponding to a Reynolds number Re]U∞c/ n ¼ 4.4 105, where c is the chord of the hydrofoil and n is the kinematic viscosity of the liquid. The pressure in the tunnel test section is set to 1atm. The Clark-Y hydrofoil, with c ¼ 70 mm chord and s ¼ 69 mm span, pitches about an axis located at 50% from the leading edge. The pitching hydrofoil is controlled to rotate from aþ ¼ 10 to aþ ¼ 15 firstly, then goes from aþ ¼ 15 to a ¼ 5 , finally goes back to aþ ¼ 10 from a ¼ 5 , with the oscillation amplitude Da ¼ 5 and the mean angle of attack adjusted to a0 ¼ 10 , as shown in Fig. 3. In the pitching process, the angular velocity is at a constant speed. Multiple periodic motions are measured and simulated. We mainly focused on one typical cycle to investigate the flow regimes and hydrodynamic performance around the pitching hydrofoil in this work. According to the direction of motion, the stage of the increasing attack angle is called the upstroke phase, which is denoted with aþ. And the stage of the declining attack angle is called the downstroke phase, which is denoted with a. Hence, the pitching period can be divided into four stages–upstroke I, downstroke I, downstroke II, upstroke II, as shown in Fig. 3. The angular velocity is defined as a_ ¼ af =Tf , where af is defined as af ¼ 4 Da and Tf is the total duration of the transient pitching motion. The non-dimensional angular velocity based on the chord length c and the upstream velocity U∞ is defined as a_ * ¼ ða_ cÞ =ðU∞ DaÞ. In this paper, for the analysis of typical flow regimes around the pitching hydrofoil, the pitching rate is set to 0.086. As for the influence of the pitching rate on the flow structures and corresponding hydrodynamics, the cases with different pitching rate, a_ * ¼ 0:0215, a_ * ¼ 0:086 and a_ * ¼ 0:172, have been compared, as shown in Fig. 3.
v uj ¼0 vxj
r
! vðui Þ v ui uj vp v þ þ ¼ vt vxi vxj vxj
(1)
m
vui vxj
! (2)
where, u denotes the velocity, x denotes the coordinate and subscripts i and j denote the directions of the Cartesian coordinates. r denotes the fluid density, t denotes the time, p denotes the pressure, m denotes the fluid viscosity.
3.2. Turbulence model Many previous studies have highlighted the importance to take into account the transition in the hydrodynamic prediction [23e26]. The g-Req transition model consists of the intermittency
3. Numerical methods 3.1. Basic governing equations The two-dimensional Reynolds-Averaged Navier-Stokes (RANS) equations for unsteady incompressible turbulent flow with the continuity and momentum equation are expressed as follows:
Fig. 9. The evolution of flow regimes around fixed hydrofoil and pitching hydrofoil for Re ¼ 4.4 105, U∞ ¼ 6.3 m/s, a_ * ¼ 0:086 (black solid lines: fixed case; discrete red points: pitching case). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
417
Fig. 10. The non-dimensional turbulence kinetic energy distribution around hydrofoil for pitching case and fixed case.
transport equation which associates transition to the local variable to turn on the production term of the turbulent kinetic energy downstream of the transition point and trigger the transition, and the momentum thickness transport equation which is solved to capture the non-local influence of the turbulence intensity, as it ties the empirical correlation to the onset criteria in the intermittency equation. It can be described in detail as follows:
" ! # vðrgÞ v rUj g v m t vg þ ¼ Pg Eg þ mþ vt vxj sf vxj vxj " # vðrReqt Þ v rUj Reqt v vReqt þ ¼ Pqt þ s ðm þ m t Þ vt vxj qt vxj vxj
(3)
(4)
where, g is intermittency, Pg and Pqt are transition sources, Eg is destruction/relaminarization, Reqt is local transition Reynolds number. In this study, the revised k-u SST turbulence model [27], which couples the k-u SST turbulence model [28] and the g-Req transition model, are used to solve the URANS equations.
" # vðrkÞ v ruj k v vk ~ ~ þ ¼ P k Dk þ ðm þ sk mt Þ vt vxj vxj vxj
original production terms and destruction term respectively. 3.3. Lagrangian coherent structures (LCSs) From the Lagrangian viewpoint, the fluid flow is a dynamic system of fluid particles. The Lagrangian Coherent Structures (LCSs) can be regarded as a trajectory-based approach [29]. The LCSs are extracted from the Finite-Time Lyapunov Exponent (FTLE), which characterizes the separation rate of neighborhood trajectories during a finite time. To characterize the separation rate of the infinitely close trajectories, the Lyapunov exponent is defined as:
s ¼ lim
t/∞ jdx0 j/0
vðruÞ v rUj u v m vu þ ¼ Cu Pu bu ru2 þ mþ t vxj vt vxi sk vxi 1 vk vu þ 2rð1 F1 Þsu2 u vxi vxi
DTt0LE ðx0 Þ ¼
where, k and u denote the turbulent kinetic energy and the specific turbulent dissipation, respectively, F1 denotes blending functions. ~ and destruction term D ~ are And the revised production terms P k k defined as:
~ ¼g P P k eff k
(7)
a_ * ¼ 0:086
(8)
where, geff denotes the revised coefficient, Pk and Dk denote the
vxðt0 þ TLE ; t0 ; x0 Þ T vxðt þ TLE ; t0 ; x0 Þ vx0 vx0
(10)
where, TLE denotes the time interval, x (t0þTLE; t0, x0) denotes the new position of point x0 after TLE. Then the finite-time Lyapunov exponent (FTLE) during the time interval TLE is defined as:
sTt0LE ðx0 Þ ¼ (6)
(9)
where, x0 denotes an arbitrary point in the dynamical system. Based on the Cauchy-Green deformation tensor, the separation rate can be acquired as:
(5)
1 jdxðx0 ; tÞj ln t jdx0 j
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 1 ln lmax DTt0LE ðx0 Þ jTLE j
(11)
where, lmax ðDTt0LE ðx0 ÞÞ denotes the max eigenvalue of the CauchyGreen deformation tensor. Based on the FTLE field, the LCSs can be obtained with ridges of the FTLE field and has been proven helpful to capture the vortex boundary [30e33]. 4. Numerical setup and verification 4.1. Numerical setup According to the experimental setup, the computational domain and boundary conditions are presented in Fig. 4, including two domains connected with a sliding interface. The rectangular static
418
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 11. The skin friction distribution along the hydrofoil and the contour of turbulence kinetic energy superimposed on velocity profile at different separation position with a ¼ 8 and 13 .
domain has a length of 10c and a height of 2.7c, and the circular dynamic domain has a diameter of 2c. The interaction between the static and dynamic domains is controlled by the CFX expression language (CEL) subroutine. The pitching motion of the foil is simulated by setting the pitching motion of the dynamic domain, which is the cylinder region around the mid-chord of foil. The inlet velocity and the outlet pressure are set according to the experimental condition. A no-slip boundary condition is imposed on the hydrofoil surface, and symmetry conditions are imposed on the top and bottom tunnel boundaries. We used a structured grid strategy in computational mesh. Fig. 5 shows the mesh distribution and the refined grids around the leading edge and the trailing edge. There
are 500 nodes placed in the boundary layer, which is selected to meet yþ ¼ yut/n z 1. Three meshes, which give a constant refinement ratio r ¼ Nfine/ Ncourse ¼ 2, are used in the grid convergence study, where N is the number of grid nodes, as shown in Table 2. The GCI index is estimated based on Richardson extrapolation [34e36]. The mean lift coefficient of one cycle CL-mean is considered integral quantities, which is defined as CL-mean ¼ SCL/TS (TS is the time step, CL ¼ L/ (0.5rU2∞sc), L is the lift). As the GCI are less than 5%, the results for mesh III are considered as mesh independent, and the subsequent simulations are performed with the meshing scheme of mesh III. In addition, the finite volume method is applied for the space
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
discretization. The high order resolution scheme was used for the convective term, with the second order central difference scheme used for the diffusion term in the governing equations. The second order backward euler scheme was used for the transient term. And the time step was set to 5 105 s (Tref/200, where Tref ¼ c/U∞). This treatment is an effective way to investigate the fundamental vortex structure, as reported recently by many researchers [37,38].
419
the moment coefficient curve of pitching cases is prominent. Detailed analysis of the flow structures and corresponding hydrodynamics will be presented as following.
5. Results and discussion 5.1. Flow structures and corresponding hydrodynamics for the fixed and pitching hydrofoil
4.2. Numerical validation Fig. 6 compares the measured and predicted moment coefficients for the fixed cases and pitching case of a_ * ¼ 0:086. The xcoordinate is angle of attack (AOA) a, and the y-coordinate is the moment coefficient CM (CM ¼ M/(0.5rU2∞sc2)), with the negative moment indicated the clockwise direction. We can see that the predicted results agree well with the experimental results. For the upstroke I, the moment coefficient increases with fluctuation gradually. When the angle of attack begins to decline, the moment also declines gradually. Until a ¼ 5 , the moment reaches the lowest value in the whole process. With the change of pitching direction, the angle of attack and moment again increases. Compared to the fixed cases, it can be noted that the fluctuation of
5.1.1. Typical flow regimes around the fixed hydrofoil Fig. 7 shows the contours of z-vorticity superimposed on the instantaneous streamlines and the velocity vector at different attack angle. The distributions of skin friction (Cf ¼ f/(0.5rU2∞, where f is the wall shear stress) along the hydrofoil at different attack angle are presented in Fig. 8. At a ¼ 2 , the flow is quasi-laminar, as shown in Fig. 7(a). According to the Prandtl separation criterion [31], it defines the laminar separation point at (vux/vy)y ¼ 0. We can see that the skin friction of the hydrofoil with a ¼ 2 is negative from x/ c ¼ 0.47e0.66, which is responsible to the transition from laminar to turbulent flow, as shown in Fig. 8. The location of x/c ¼ 0.47 is defined as laminar separation point, and the location of x/c ¼ 0.66 is
Fig. 12. The evolution of the predicted lift coefficient (CL), drag coefficient (CD) for Re ¼ 4.4 105, U∞ ¼ 6.3 m/s.
Fig. 13. FTLE field, the corresponding LCS and the trajectory of the specific points at various geometric angle of attack.
420
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
defined as turbulence reattachment point. In transition area, it can be noted in Fig. 7(a) that there are reversed velocity vector around the suction side of the hydrofoil. At a ¼ 5 , the skin friction of the hydrofoil is negative from x/c ¼ 0.26e0.40. It can be found that the locations of the laminar separation point and turbulence reattachment point all are advanced. While for the hydrofoil with a ¼ 8 , the laminar separation point is closer to the leading edge than that of a ¼ 2 and a ¼ 5 , as shown in Figs. 7(aec) and Fig. 8. For a ¼ 12 , the laminar flow transits to turbulent flow on the leading edge from x/c ¼ 0.1e0.2, which is corresponding to the reversed velocity vector on the leading edge, as shown in Figs. 7(d) and 8. It can be concluded that the transition from laminar to turbulent flow happens on closer and closer to the leading edge and the transition area gets smaller and smaller. Meanwhile the turbulent boundary layer separates from the trailing edge because of the adverse pressure gradient, with the skin friction being negative from x/c ¼ 0.71 to x/c ¼ 1, as shown in Figs. 7(d) and 8. The location of x/c ¼ 0.71 is defined as turbulence separation point, after that the flow couldn’t attach on the surface of hydrofoil and the light static stall happens. For a ¼ 19 , the turbulence separation point has occurred on the leading edge of hydrofoil, thus leading to the deep static stall, with the hydrodynamic performance declines gradually, as shown in Figs. 7(e) and 8.
5.1.2. Typical flow regimes around the pitching hydrofoil Fig. 9 shows the evolution of flow regimes around fixed hydrofoil and pitching hydrofoil of a_ * ¼ 0:086. The x-coordinate is angle of attack (AOA) a, and the y-coordinate is the nondimensional location x/c. According to the location of laminar separation, turbulence reattachment, and turbulence separation at different fixed angle of attack, the flow regime is divided into four types-Laminar, Transition, Turbulence and Stall, as shown with black solid lines in Fig. 9. In which, the discrete red points mark the location of laminar separation, turbulence reattachment, and turbulence separation during the pitching process, with hollow points represented the upstroke stages and the solid points represented the downstroke stage. It’s noted that the frontiers between different flow patterns of pitching cases is different with that of fixed cases. Fig. 10 presents the non-dimensional turbulence kinetic energy K* (K * ¼ K=U 2∞ , where K is the turbulence kinetic energy) distribution around hydrofoil for pitching case and fixed case. During the upstroke I(aþ ¼ 10 ~aþ ¼ 15 ), the laminar separation point moves toward the leading edge gradually, which is corresponding to the high turbulence kinetic energy on leading edge, as shown in Figs. 9 and 10(a). Because of the motion of the boundary, the turbulence kinetic energy around pitching hydrofoil declines slightly, thus leading to the delay of transition from
Fig. 14. The FTLE field, the corresponding LCS and the trajectory of the specific points at various geometric angle of attack a for a_ * ¼ 0:086, Re ¼ 4.4 105, U∞ ¼ 6.3 m/s.
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
421
Fig. 15. The evolution of flow regime around fixed cases and pitching cases of three pitching rates for Re ¼ 4.4 105, U∞ ¼ 6.3 m/s.
laminar to turbulence. Besides, the turbulent reattachment points and the turbulent separation points are also delayed, as showed in Figs. 9 and 10. The skin friction distribution along the hydrofoil and the contour of turbulence kinetic energy superimposed on velocity profile at the separation position with a ¼ 8 and 13 are presented in Fig. 11. Compared to the fixed cases of a ¼ 13 , the location of laminar separation point is delayed slightly at aþ ¼ 13 , as shown in Fig. 11(b). It is also found in Fig. 11(b) that the turbulence kinetic energy distribution around the trailing edge at aþ ¼ 13 are weaker than that of fixed case (a ¼ 13 ). For a ¼ 13 , the location of the turbulent separation point is x/c ¼ 0.64; for aþ ¼ 13 , the location of the turbulent separation point is x/c ¼ 0.70. At x/c ¼ 0.70, the reverse velocity has occurred on the hydrofoil boundary layer for fixed case (a ¼ 13 ), as showed in Fig. 11(b).
During the downstroke I(aþ ¼ 15 ~a ¼ 10 ), the laminar separation points are similar to that of the upstroke I. The turbulence kinetic energy declines with the decreasing attack angle, thus leading to the delay of turbulence reattachment, as showed in Fig. 10. However, we can find that the turbulence separation is expedited, thus leading to the obvious hysteresis in the downstroke I, as shown in Fig. 9. It’s noted in Fig. 11(b) that the location of the turbulent separation points for a ¼ 13 is x/c ¼ 0.57, which is corresponding to the increase of turbulence around the trailing edge. For velocity profiles at x/c ¼ 0.57 of the hydrofoil, as shown in Fig. 11(b), the velocity of a ¼ 13 is higher than that of aþ ¼ 13 and a ¼ 13 . During the downstroke II(a ¼ 10 ~a ¼ 5 ), the turbulence kinetic energy distribution around hydrofoil is similar to fixed
422
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 16. The non-dimensional turbulence kinetic energy distribution around hydrofoil for pitching cases of two pitching rates.
Fig. 17. The turbulence kinetic energy distribution contour around hydrofoil at different attack angle for different pitching rates.
cases, which is responsible to the similar laminar separation points, turbulence reattachment points and turbulent separation points, as showed in Figs. 9 and 10. It’s noted in Fig. 11(a) that the location of the turbulent separation points for a ¼ 8 and a ¼ 8 is uniform (x/ c ¼ 0.91), and the difference of velocity profiles is very small. During the upstroke II(a ¼ 5 ~aþ ¼ 10 ), the location of the laminar separation point is similar to that during the downstroke II(a ¼ 10 ~a ¼ 5 ). However, the turbulence reattachment points and turbulent separation points are delayed due to the dynamic boundary, as showed in Figs. 9 and 10. It is also found in Fig. 11(a) that the turbulence kinetic energy distribution around the trailing edge are weaker than that of fixed case (a ¼ 8 ), and the location of the turbulent separation points for aþ ¼ 8 is x/c ¼ 0.94. Fig. 12 shows the evolution of the predicted lift coefficient and drag coefficient (CD ¼ D/(0.5rU2∞sc), where D is the drag) for the fixed and the pitching case. In the upstroke I, the lift coefficient and drag coefficient increase linearly with the increase of the angle of attack. The lift coefficient of the pitching case is higher than that of the fixed cases. This is because the flow separation is delayed. Then the lift coefficient declines sharply when the angle of attack declines in the downstroke I. However, the values are lower than that
of the fixed cases, thus leading to the obvious hysteresis. This may due to the acceleration of the flow separation in the downstroke I. Besides, compared with the fixed case, it can be found that fluctuation of the lift coefficient curve of pitching case is very evident, which will be discussed in next section. For the downstroke II and upstroke II processes, the lift coefficients of the fixed cases and the pitching case are almost the same, and the trend of the drag coefficients of the fixed cases and the pitching case is basically the same. In order to further study the flow structure and hydrodynamic performance of pitching hydrofoil during the upstroke I and downstroke I stages, the FTLE field, the corresponding LCS and the trajectory of the specific points at typical angle of attack a are presented in Fig. 13. Fig. 13(a) shows the trajectory of the specific points (Points A/B/ C) in the adjacent vortices from t1:aþ ¼ 12.8 to t2:aþ ¼ 12.91 during the time interval TLE ¼ 0.00275s (55 numerical steps). The LCS A represents the boundary of vortex near the trailing edge of hydrofoil. As time goes on, the trajectory of the points C that are inside LCS A show a clockwise rotation, which is responsible for the formation of the trailing edge vortex (TEV). As for the trajectory of the Points A/B near the leading edge of hydrofoil, the points A that are outside LCS A move downstream, while the points B that are inside LCS A move toward the trailing edge and rotate clockwise, thus inducing a leading edge vortex (LEV). It is noted in Fig. 12 that the formed vortex structures lead to the slight fluctuation of hydrodynamic forces. The evolution of the LEV and the clockwise TEV in the upstroke I is presented in Fig. 14(a). It is noted that the group A represents the LEV and the group B represents this clockwise trailing edge vortex (TEV). With the increase of the angle of attack, the LEV moves toward the trailing edge and merges with the TEV, which is responsible to the increase of hydrodynamic curve, as showed in Fig. 12. The trajectory of the specific points (Points D/E/F) in the adjacent vortices from t6:aþ ¼ 14.42 to t7:aþ ¼ 14.49 during the time interval TLE ¼ 0.00175s (35 numerical steps) are presented in Fig. 13(b). It is found that the points E rotate anticlockwise on the trailing edge with the increasing angle. These trajectories suggest an anticlockwise vortex structure on the trailing edge of hydrofoil, which is responsible of the formation of the LCS E. Meanwhile, the dynamic curves start to decline gradually due to the interaction between clockwise and anticlockwise vortex, thus leading to the
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
slight dynamic stall. It’s found that the anticlockwise TEV plays an important role in the hydrodynamic performance around the pitching hydrofoil. Fig. 14(b) shows that the evolution of the clockwise and anticlockwise TEV from the upstroke I to the downstroke I. At t8:aþ ¼ 14.84 , the clockwise vortex (group E and LCS C) and anticlockwise vortex (group F and LCS F) are formed. Due to the interaction between the anticlockwise TEV (group F) and the clockwise TEV (group E), the LCS C and the anticlockwise TEV (group F) vanishes gradually, thus leading to the decline of hydrodynamic curve, as shown in Figs. 12 and 14(b). When the angle of attack reaches 15 , the hydrofoil begins to rotate anticlockwise. At t10:a ¼ 14.57, a new anticlockwise vortex (group G and LCS G) are formed again and interact with the clockwise TEV. In addition, Fig. 14(c) shows the evolution of the LEV, clockwise and anticlockwise TEV in the downstroke I. As shown from t11:a ¼ 13.54 to t13:a ¼ 13.2 in Fig. 14(c), we can find that the LEV, clockwise and anticlockwise TEV still are formed. However, the area of vortex wakens gradually with the decline of the angle of attack. 5.2. Influence of the pitching rate on the flow structures and corresponding hydrodynamics In order to further study the influence of the pitching rate, the evolution of flow regime around fixed cases and pitching cases of three pitching rates are presented in Fig. 15. Fig. 16 shows the nondimensional turbulence kinetic energy distribution around the pitching hydrofoil with different pitching rates. The turbulence kinetic energy distribution contour around the hydrofoil at typical attack of angles for different pitching rates are shown in Fig. 17. Fig. 18 shows that comparisons of the predicted pressure coefficients distributions at various geometric angle of attack a for three pitching rates. It is noted in Fig. 15 that the laminar separation points move
423
upward slightly with the increase of the pitching rate from aþ ¼ 10 to aþ ¼ 15 . However, the turbulence reattachment points and turbulent separation points are further delayed. This may result from the decline of turbulence kinetic energy around hydrofoil, as showed in Fig. 16. As shown in Fig. 17(c), we can see that the turbulence kinetic energy near the trailing edge at aþ ¼ 13 declined with the increase of the pitching rate. It is also noted in Fig. 18(c) that the adverse pressure gradient at aþ ¼ 13 declines with the increasing of pitching rate. During the downstroke I(aþ ¼ 15 ~a ¼ 10 ), the location of laminar separation points, the turbulence reattachment points and the turbulent separation points move upward with the increase of the pitching rate, as shown in Fig. 15. We can find that the turbulence kinetic energy and the adverse pressure gradient around the trailing edge of hydrofoil at a ¼ 13 with a_ * ¼ 0:172 are higher than a ¼ 13 with a_ * ¼ 0:0215 and 0.086, as shown in Figs. 17(d) and 18(d). During the downstroke II(a ¼ 10 ~a ¼ 5 ), the discrepancy between the cases with three pitching rates declined, as showed in Figs. 15 and 16. It’s noted in Figs. 17(b) and 18(b) that the turbulence kinetic energy and the adverse pressure gradient around the trailing edge of hydrofoil at a ¼ 8 is very similar. During the upstroke II(a ¼ 5 ~aþ ¼ 10 ), the laminar points and turbulence separation points don’t change much for different pitching rates. However, turbulence reattachment points are further delayed at high pitching rate, as shown in Fig. 15. We can see that the discrepancy of turbulence kinetic energy and adverse pressure gradient at aþ ¼ 8 decline, as shown in Figs. 17(a) and 18(a). It’s also noted in Fig. 17(a) that the turbulent distribution around the leading edge of hydrofoil at a_ * ¼ 0:172 are lower than that of others. To further analyze the effect of pitching rate on the hydrodynamic performance, the evolution of the predicted lift coefficient (CL) for three pitching rates are presented in Fig. 19. As shown in
Fig. 18. Comparisons of the predicted pressure coefficients distributions at various geometric angle of attack a for three pitching rates.
424
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 19. The evolution of the predicted lift coefficient (CL) for three pitching rates.
Fig. 19, the maximum lift coefficient increases with the increase of the pitching rate. Moreover, the higher pitching rate broadens the hysteresis loops and the lower pitching rate intensify fluctuation of the dynamic forces, as shown in Fig. 19. It also reveals that the pitching rate, where there are consecutive vortical patterns in the lift coefficient curves, has considerable influence on the strength of these vortices and their initiation location during the pitching motion. Figs. 20 and 21 show the FTLE field, the corresponding LCS and the trajectory of the specific points at various geometric angle of attack a for a_ * ¼ 0:0215 and a_ * ¼ 0:172 respectively. Fig. 20(a) shows the trajectory of the specific points (Points A’/ B0 ) in the adjacent vortices from t’1:aþ ¼ 12.21 to t’2:aþ ¼ 12.27 during the time interval TLE ¼ 0.0015s (30 numerical steps) with a_ * ¼ 0:0215. It is found that the first LEV (LCS A0 ) occurs on t’2:aþ ¼ 12.27, which is earlier than that of a_ * ¼ 0:086. Moreover, as shown in Fig. 19, the numbers of effective LEV lead to the highfrequency fluctuations of lift coefficient. It can be concluded that the formation of the LEV effects the lift evolution. At t’3: aþ ¼ 13.89 , the first anticlockwise TEV (group D0 and LCS C0 ) is formed, which is earlier than that with a_ * ¼ 0:086 (t8:aþ ¼ 14.49 ). Similarly, the dynamic performance curves start to decline gradually, which is responsible to dynamic stall in advance. As shown in Fig. 20(c), when the angle of attack reaches to t’4:aþ ¼ 14.96 , the anticlockwise TEV again is formed. From t’4:aþ ¼ 14.96 to
t’6:a ¼ 14.97, the evolution of vortices is similar to the case of a_ * ¼ 0:086. The trajectory of the specific points (Points A’’/B00 ) in the adjacent vortices from t’’1:aþ ¼ 14.29 to t’’2:aþ ¼ 14.78 during the time interval TLE ¼ 0.01225s (245 numerical steps) with a_ * ¼ 0:172 is presented in Fig. 21(a). As the pitching rate is increased to a_ * ¼ 0:172, the attack angle at which the first LEV (group A00 and LCS A00 ) forms is changed from t3:aþ ¼ 13.05 at a_ * ¼ 0:086 to t’’2:aþ ¼ 14.78 at a_ * ¼ 0:172. From t’’2:aþ ¼ 14.78 to t’’4:a ¼ 14.89 , the interaction between the LEV and clockwise TEV is similar to the case of a_ * ¼ 0:086, as shown in Fig. 21(b). Moreover, there aren’t anticlockwise TEV in the upstroke I process. The dynamic stall didn’t occur in this process. Until the angle of attack reaches t’’5:a ¼ 13.25 in the downstroke I, the first anticlockwise TEV (group F00 and LCS C00 ) is formed, as shown in Fig. 21(c). Fig. 22 shows the location evolution of the first LEV and first anticlockwise TEV with three pitching rates. It is found that the first LEV and anticlockwise TEV are delayed with the increase of the pitching rate. The location of the first LEV and the numbers of LEV affect the dynamic performance around the pitching hydrofoil. Moreover, the interaction between the clockwise and anticlockwise TEV results in the decline of hydrodynamic performance. As shown from the stall line in Fig. 22, it also reveals that the anticlockwise TEV plays an important role in the dynamic stall.
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
425
Fig. 20. The FTLE field, the corresponding LCS and the trajectory of the specific points at various geometric angle of attack a for a_ * ¼ 0:0215, Re ¼ 4.4 105, U∞ ¼ 6.3 m/s.
6. Conclusions In this work, different flow regimes and hydrodynamic performance around the fixed and pitching hydrofoil with different pitching rates are investigated at Re ¼ 4.4 105. In general, the pitching motion and the pitching rate have significant effect on the transition between the different flow patterns (especially laminarturbulent transition and turbulence separation) and hydrodynamic performance. The primary findings include: (1) For the fixed cases, with increase of the angle of attack, the laminar separation point is closer to the leading edge. The turbulence kinetic energy distribution around the hydrofoil has an important influence on both the hydrofoil boundary layer and the separated shear layer. (2) For the pitching hydrofoil, the dynamic boundary affects the turbulence kinetic energy distribution around the hydrofoil. Higher levels of turbulence kinetic energy cause earlier transition compared to the lower turbulence levels. From aþ ¼ 10 to a ¼ 10 , the location of laminar separation points, turbulence reattachment points and turbulence separation points are delayed or advanced due to the evolution of turbulence kinetic energy distribution. However, these locations don’t vary much from a ¼ 10 to aþ ¼ 10 because the differences of turbulence kinetic energy distribution
around hydrofoil between fixed cases and pitching case decrease. In addition, the interaction between the clockwise and anticlockwise TEV results in the dynamic stall. (3) The pitching rate plays an important role in flow regime and hydrodynamic performance. From aþ ¼ 10 to aþ ¼ 15 , the increase of pitching rate causes the acceleration of the transition from laminar to turbulence. However, the increase of pitching rate enlargers the delayed location of the turbulence reattachment points and turbulence separation points. From aþ ¼ 15 to a ¼ 10 , the high pitching rate promotes the turbulence separation immensely. From a ¼ 10 to aþ ¼ 10 , the effect of pitching rate on laminar separation and turbulence separation is limited. But the high pitching rate affects the turbulence reattachment partly. In addition, the higher pitching rate broadens the hysteresis loops. The first LEV and anticlockwise TEV are delayed with the increase of pitching rate, which is responsible to the delay of dynamic stall. Moreover, the lower pitching rate intensifies fluctuation of the dynamic force due to the strength and number of LEV. Although reasonable agreement between the experimental and numerical results has been obtained, some discrepancies are still existed. In order to acquire the spatial and temporal variation of the flow structures more accurately, Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES) will be further studied in
426
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Fig. 21. FTLE field, the corresponding LCS and the trajectory of the specific points at various geometric angle of attack a for a_ * ¼ 0:172, Re ¼ 4.4 105, U∞ ¼ 6.3 m/s.
flow regime and dynamic performance will be also further discussed in the future.
Author contribution statement Zhang Mengjie: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - Original Draft. Wu Qin: Software, Validation, Writing - Review & Editing, Supervision, Funding acquisition. Wang Guoyu: Resources, Supervision, Funding acquisition. Huang Biao: Conceptualization, Software, Validation, Writing Review & Editing, Supervision, Funding acquisition. Fu Xiaoying: Resources, Funding acquisition. Chen Jie: Project administration.
Fig. 22. The location evolution of the first LEV and anticlockwise TEV with the pitching rate.
the future work. In addition, the three dimensional effect of the flow due to the gap flow will be discussed to better illuminate the underlying physics. Especially, the impact of the cavitation on the
Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: As we have too similar research topics and we have conducted investigation based on the too similar research objects, there is a potential conflict of interest existing between us.
M. Zhang et al. / Renewable Energy 150 (2020) 412e427
Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant Nos: 51839001, 51909002 and 91752105), the National Natural Science Foundation of Beijing (Grant No: 3172029), and the Fundamental Research Funds for the Central Universities. References ~o, Transient analysis of variable-speed [1] R. Melício, V.M.F. Mendes, J.P.S. Catala wind turbines at wind speed disturbances and a pitch control malfunction, Appl. Energy 88 (4) (2011) 1322e1330. lez, E. Figueres, G. Garcer [2] L.G. Gonza a, O. Carranza, Maximum-power-point tracking with reduced mechanical stress applied to wind-energy-conversionsystems, Appl. Energy 87 (7) (2010) 2304e2312. [3] H.R. Karbasian, J.A. Esfahani, E. Barati, The power extraction by flapping foil hydrokinetic turbine in swing arm mode, Renew. Energy 88 (2016) 130e142. [4] S. Park, S. Park, S.H. Rhee, Influence of blade deformation and yawed inflow on performance of a horizontal axis tidal stream turbine, Renew. Energy 92 (2016) 321e332. [5] J. Zanette, D. Imbault, A. Tourabi, A design methodology for cross flow water turbines, Renew. Energy 35 (5) (2010) 997e1009. [6] D. Li, H. Wang, Y. Qin, Z. Li, X. Wei, D. Qin, Mechanism of high amplitude low frequency fluctuations in a pump-turbine in pump mode, Renew. Energy 126 (2018) 668e680. [7] H. Zhang, Q. Wu, Y. Li, B. Huang, G. Wang, Numerical investigation of the deformation characteristics of a composite hydrofoil with different ply angles, Ocean Eng. 163 (2018) 348e357. [8] B. Huang, A. Ducoin, L.Y. Young, Physical and numerical investigation of cavitating flows around a pitching hydrofoil, Phys. Fluids 25 (10) (2013) 102e109. [9] B. Huang, Q. Wu, G.Y. Wang, Numerical simulation of unsteady cavitating flows around a transient pitching hydrofoil, Sci. China Technol. Sci. 57 (1) (2014) 101e116. [10] D.T. Akcabay, Y.L. Young, Hydroelastic response and energy harvesting potential of flexible piezoelectric beams in viscous flow, Phys. Fluids 24 (2012), 054106. [11] Q. Wu, C.C. Wang, B. Huang, et al., Measurement and prediction of cavitating flow-induced vibrations, J. Hydrodyna. Ser. B. 30 (6) (2018) 1064e1071. [12] Q. Wu, B. Huang, G.Y. Wang, et al., Experimental and numerical investigation of hydroelastic response of a flexible hydrofoil in cavitating flow, Int. J. Multiph. Flow 74 (2015) 19e33. [13] M. Liu, L. Tan, Y. Liu, Y. Xu, S. Cao, Large eddy simulation of cavitation vortex interaction and pressure fluctuation around hydrofoil ALE15, Ocean Eng. 163 (2018) 264e274. [14] Z.F. Yang, F. Haan, H. Hu, H.W. Ma, An Experimental Investigation on the Flow Separation on a Low-Reynolds-Number Airfoil, 45th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings, Reno, Nevada, 2007. [15] Z. Wei, T.H. New, Y.D. Cui, An experimental study on flow separation control of hydrofoils with leading-edge tubercles at low Reynolds number, Ocean Eng. 108 (2015) 336e349. [16] H. Shan, L. Jiang, C. Liu, Direct numerical simulation of flow separation around a NACA 0012 airfoil, Comput. Fluids 34 (9) (2005) 1096e1114. [17] Z. Yang, P.R. Voke, Large-Eddy Simulation of boundary layer separation and
[18] [19] [20] [21]
[22]
[23] [24]
[25]
[26]
[27]
[28] [29] [30]
[31]
[32]
[33]
[34] [35] [36]
[37]
[38]
427
transition at a change of surface curvature, J. Fluid Mech. 439 (439) (2001) 305e333. J.P. Franc, J.M. Michel, Unsteady attached cavitation on an oscillating hydrofoil, J. Fluid Mech. 193 (1988) 171e189. J.M. Anderson, K. Streitlien, D.S. Barrett, et al., Oscillating foils of high propulsive efficiency, J. Fluid Mech. 360 (1998) 41e72. G. Pedro, A. Suleman, N. Djilali, A numerical study of the propulsive efficiency of a flapping hydrofoil, Int. J. Numer. Methods Fluids 42 (2003) 493. I. Akhtar, R. Mittal, G.V. Lauder, et al., Hydrodynamics of a biologically inspired tandem flapping foil configuration, Theor. Comput. Fluid Dyn. 21 (3) (2007) 155e170. Q. Wu, B. Huang, G.Y. Wang, et al., The transient characteristics of cloud cavitating flow over a flexible hydrofoil, Int. J. Multiph. Flow 99 (2018) 162e173. R.B. Langtry, F.R. Menter, Transition Modeling for General CFD Applications in Aeronautics, AIAA Paper, 2005, p. 522. D. Li, H. Wang, Z. Li, T.K. Nielsen, R. Goyal, X. Wei, D. Qin, Transient characteristics during the closure of guide vanes in a pump-turbine in pump mode, Renew. Energy 118 (2018) 973e983. D. Li, Y. Qin, Z. Zuo, H. Wang, S. Liu, X. Wei, Numerical simulation on pump transient characteristic in a model pump turbine, J. Fluids Eng. 141 (11) (2019) 111101. M.J. Zhang, Q. Wu, B. Huang, G.Y. Wang, Lagrangian-based numerical investigation of aerodynamic performance of an oscillating foil, Acta Mech. Sin. 34 (5) (2018) 839e854. R.B. Langtry, F.R. Menter, S.R. Likki, Y.B. Suzen, et al., A correlation-based transition model using local variables-Part I: model formulation, J. Turbomach. 128 (2006) 413e422. F.R. Menter, Improved two-equation k-u turbulence models for aerodynamic flows, NASA Tech. Memo. 34 (1992) 103975. G. Haller, G. Yuan, Lagrangian coherent structures and mixing in two dimensional turbulence, Phys. D Nonlinear Phenom. 147 (2000) 352e370. J.N. Tang, C.C. Tseng, N.F. Wang, Lagrangian-based investigation of multiphase flows by finite-time Lyapunov exponents, Acta Mech. Sin. 28 (3) (2012) 612e624. C.C. Tseng, P.B. Liu, Dynamic behaviors of the turbulent cavitating flows based on the Eulerian and Lagrangian viewpoints, Int. J. Heat Mass Transf. 102 (2016) 479e500. Q. Wu, B. Huang, G.Y. Wang, Lagrangian-based investigation of the transient flow structures around a pitching hydrofoil, Acta Mech. Sin. 32 (1) (2016) 64e74. M.J. Zhang, H. Chen, Q. Wu, et al., Experimental and numerical investigation of cavitating vortical patterns around a tulin hydrofoil, Ocean Eng. 173 (2019) 298e307. L.F. Richardson, J.A. Gaunt, The deferred approach to the limit, Philos. Trans. R. Soc., A 226 (636e646) (1927) 299e349. E. Jak, P.C. Hayes, Procedure of estimation and reporting of uncertainty due to discretization in CFD applications, J. Fluids Eng. 130 (7) (2008), 078001. Y. Long, X.P. Long, B. Ji, T. Xing, Verification and validation of Large Eddy Simulation of attached cavitating flow around a Clark-Y hydrofoil, J. Multiph. Flow 115 (2019) 93e107. O. Coutier-Delgosha, J.L. Reboud, Y. Delannoy, Numerical simulation of the unsteady behaviour of cavitating flows, Int. J. Numer. Methods Fluids 42 (2003) 527e548. B. Ji, X.W. Luo, R.E.A. Arndt, et al., Large Eddy Simulation and theoretical investigations of the transient cavitating vortical flow structure around a NACA66 hydrofoil, Int. J. Multiph. Flow 68 (2015) 121e134.