applied sciences

Article Propulsion Performance of the Full-Active Flapping Foil in Time-Varying Freestream

Zhanfeng Qi 1,2, Lishuang Jia 1, Yufeng Qin 1, Jian Shi 1 and Jingsheng Zhai 2,*

1 National Ocean Technology Center, Tianjin 300112, China; [email protected] (Z.Q.); [email protected] (L.J.); [email protected] (Y.Q.); [email protected] (J.S.) 2 School of Marine Science and Technology, Tianjin University, Tianjin 300072, China * Correspondence: [email protected]



Received: 17 July 2020; Accepted: 3 September 2020; Published: 8 September 2020


Abstract: A numerical investigation of the propulsion performance and hydrodynamic characters of the full-active flapping foil under time-varying freestream is conducted. The finite volume method is used to calculate the unsteady Reynolds averaged Navier–Stokes by commercial Computational Fluid Dynamics (CFD) software Fluent. A mesh of two-dimensional (2D) NACA0012 foil with the Reynolds number Re = 42,000 is used in all simulations. We first investigate the propulsion performance of the flapping foil in the parameter space of reduced frequency and pitching amplitude at a uniform flow velocity. We define the time-varying freestream as a superposition of steady flow and sinusoidal pulsating flow. Then, we study the influence of time-varying flow velocity on the propulsion performance of flapping foil and note that the influence of the time-varying flow is time dependent. For one period, we find that the oscillating amplitude and the oscillating frequency coefficient of the time-varying flow have a significant influence on the propulsion performance of the flapping foil. The influence of the time-varying flow is related to the motion parameters (reduced frequency and pitching amplitude) of the flapping foil. The larger the motion parameters, the more significant the impact of propulsion performance of the flapping foil. For multiple periods, we note that the time-varying freestream has little effect on the propulsion performance of the full-active flapping foil at different pitching amplitudes and reduced frequency. In summary, we conclude that the time-varying incoming flow has little effect on the flapping propulsion performance for multiple periods. We can simplify the time-varying flow to a steady flow field to a certain extent for numerical simulation.

Keywords: full-active flapping foil; time-varying freestream; propulsion performance; hydrodynamic character

1. Introduction Exploring the ocean requires a variety of marine observation equipment. In particular, the marine mobile vehicle provides new means for the ocean’s development and utilization, which significantly expands the human ability to explore, understand, and exploit the ocean. Underwater propulsion technology is one of the critical technologies of the marine mobile vehicle, which directly affects the maneuverability, operability, and endurance of vehicles. Compared with the traditional propeller, the propulsion system based on flapping foil has high stability, high efficiency, excellent maneuverability, and environmental friendliness [1–3]. As a result of so many advantages, it is necessary to understand the hydrodynamic performance of the propulsion system of flapping foil fully and deeply. Analyzing the impact of different parameters on the flapping foil’s propulsion performance and trying to improve its propulsion performance is of great significance for revealing the propulsion mechanism and popularizing applications.

Appl. Sci. 2020, 10, 6226; doi:10.3390/app10186226 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 6226 2 of 18

The fluid–structure interaction between the flapping foil and flow field is complicated and unsteady [4,5]. In recent decades, many researchers have explored the propulsion mechanism of flapping foil and the effects of multiple factors on the propulsion performance through natural observation, numerical simulation, and experimental study [6–8]. Previous studies have found that factors such as the motion amplitude and reduced frequency [9], motion trajectory [10], St (Strouhal number) [11], pivot location [12], phase difference between heaving and pitching motions [13], shape and flexibility of the foil [14,15] can directly affect the hydrodynamic performance of the flapping foil. The reduced frequency and the amplitude of heaving and pitching motion are the main factors affecting the flapping flow field structure [9]. Koochesfahani and Manoochehr [16] study the hydrodynamic performance and wake vortex of the pure pitching flapping foil at a small amplitude through the tank experiment. The results show that the wake vortex structure can be effectively controlled by controlling the reduced frequency, pitching amplitude, and motion trajectory, which can significantly affect the flapping foil’s thrust. Ashraf [17] and Dewey [18] analyze the effect of the reduced frequency on the propulsion performance of full-active flapping foil. They find that there is a critical value for the reduced frequency. When the reduced frequency is less than the critical value, the mean thrust coefficient increases monotonously. However, when the reduced frequency is higher than the critical value, the propulsion performance of flapping foil decreases significantly. The influencing mechanism of heaving and pitching motion amplitude is similar to the reduced frequency [7,17]. The thickness, chord length, curvature, and flexibility also affect the foil’s hydrodynamic performance [11,19,20]. Besides, some researchers study the influence of non-sinusoidal motion trajectories on the propulsion performance of flapping foil and intend to improve it [21–25]. According to the above research, most researchers assume the freestream as uniform and steady when studying the hydrodynamic performance of flapping foil. In the marine environment, especially when approaching the surface and floor of the sea, the flow field is usually non-uniform and time-varying due to the seabed topography or the wind and waves. The conclusions based on the above assumption may not reflect the hydrodynamic performance of flapping foil under real conditions. The numerical simulation results obtained by simplifying the flow to a uniform flow may not be accurate enough. Therefore, it is of great practical significance to analyze the influence of non-uniform and time-varying freestream velocity on the propulsion performance of the flapping foil. We noticed a few studies on the influence of non-uniform and time-varying flow velocity on the propulsion performance of the flapping foil. In contrast, the influence of non-uniform flow velocity on the flapping foil’s energy harvesting performance has been investigated systematically [26–28]. Kinsey and Dumas [29] study the effect of small amplitude disturbances of freestream on the hydrodynamic performance of flapping foil and point out that the influence on the energy harvesting performance of the flapping foil is almost negligible. Zhu [26] and Cho [27] analyze the effect of linear shear flow on the energy harvesting performance of two-dimensional flapping foil. They find that the flow field under a low shear rate can slightly improve the flapping foil’s performance, while the energy harvesting efficiency of flapping foil reduces sharply when the shear rate is large enough. Ma [30] examines the effect of time-varying flow velocity on the energy harvesting performance of two-dimensional flapping foil. The study also points out that flapping foil’s hydrodynamic performance is greatly affected by the time-varying flow rate in one cycle. When calculating the mean performance in eight periods, the time-varying freestream velocity has little effect on the flapping foil’s energy harvesting performance. In addition, other researchers have found that the free surface and waves also have an important impact on the hydrodynamic performance of the flapping foil. Silva [31] investigates the possibility of extracting energy from gravity waves by a two-dimensional flapping foil. He points out that the marine propulsion of the flapping foil can absorb energy form sea waves and improve the propulsion performance when the design parameters are correctly maintained. Filippas and Belibassakis [32] analyze the hydrodynamic performance of the flapping foil under free surface and in waves by unsteady boundary element method. They note that significant efficiency of the flapping foil can be obtained under optimal wave conditions. Xu [33] investigates the propulsion of a flapping foil in Appl. Sci. 2020, 10, x 3 of 18 Appl. Sci. 2020, 10, 6226 3 of 18 frequency difference between the foil and waves has a significant effect on the hydrodynamic performance of the flapping foil. headingReferring wave throughto the non-uniform the velocity potentialflow on theorythe energy and indicatesharvesting that performance the frequency of di flappingfference betweenfoil, we noticethe foil that and both waves the has small a significant amplitude e ffshearect on flow the an hydrodynamicd the time-varying performance flow velocity of the flappinghave little foil. effect. Therefore,Referring we tocan the infer non-uniform that the influence flow on the of energy the non-uniform harvesting performanceflow with a ofsmall flapping amplitude foil, we on notice the propulsionthat both the performance small amplitude of the shear flapping flow andfoil theshould time-varying also be weak. flow velocityTo verify have the littleabove eff ect.inference, Therefore, we analyzewe can inferthe effect that the of influencetime-varying of the flow non-uniform velocity on flow the with flapping a small foil’s amplitude propulsion on the performance propulsion throughperformance numerical of the simulation. flapping foil In should this paper, also bethe weak. kinematic To verify model the of above full-active inference, flapping we analyze foil under the time-varyingeffect of time-varying flow velocity flow velocityis established. on the The flapping finite foil’s volume propulsion method performance (FVM) is used through to calculate numerical the unsteadysimulation. Reynolds In this paper,averaged the Navier–Stokes kinematic model (URANS of full-active) by commercial flapping CFD foil under software time-varying Fluent version flow 19.1.velocity We isfirst established. investigate The the finite propulsion volume methodperformance (FVM) of is flapping used to calculatefoil in the the parameter unsteady Reynoldsspace of reducedaveraged frequency Navier–Stokes and pitching (URANS) amplitude by commercial at a unif CFDorm software flow velocity. Fluent version Then, 19.1.we study We first the investigate influence ofthe time-varying propulsion performance flow velocity of on flapping the propulsion foil in the performance parameter space of fla ofpping reduced foil. frequency Moreover, and the pitching effect mechanismamplitude atof a uniformtime-varying flow velocity.flow velocity Then, on we the study hydrodynamic the influence performance of time-varying of flowflapping velocity foil onis explainedthe propulsion by analyzing performance the evolutions of flapping of foil.vortex Moreover, topology, the flow effect velocity, mechanism and pressure of time-varying distribution flow aroundvelocity the on foil. the hydrodynamic performance of flapping foil is explained by analyzing the evolutions of vortex topology, flow velocity, and pressure distribution around the foil. 2. Problem Description and Methodology 2. Problem Description and Methodology 2.1. Problem Description 2.1. Problem Description Referring to Zhu [26] and Cho [27], we note that the shear flow is mainly affected by the seafloor Referring to Zhu [26] and Cho [27], we note that the shear flow is mainly affected by the seafloor topography, while the marine vehicles propelled by flapping foil are mostly used in the vast ocean, topography, while the marine vehicles propelled by flapping foil are mostly used in the vast ocean, the influence of the terrain on the flow field can be ignored. Therefore, we only discuss the effect of the influence of the terrain on the flow field can be ignored. Therefore, we only discuss the effect time-varying flow velocity on the propulsion performance of active flapping wings in this paper. The of time-varying flow velocity on the propulsion performance of active flapping wings in this paper. sketch of full-active flapping foil under time-varying freestream velocity is illustrated in Figure 1. The sketch of full-active flapping foil under time-varying freestream velocity is illustrated in Figure1. The foil has two degrees of freedom, which can only translate in the vertical direction and rotate The foil has two degrees of freedom, which can only translate in the vertical direction and rotate around around the pivot, other degrees of freedom are strictly prohibited. The heaving and pitching motions the pivot, other degrees of freedom are strictly prohibited. The heaving and pitching motions of the of the flapping foil are fully prescribed and defined as a sinusoidal motion. A two-dimensional flapping foil are fully prescribed and defined as a sinusoidal motion. A two-dimensional NACA0012 NACA0012 foil is used in this paper, 𝑐 and 𝑑 are defined as the chord length and the thickness of foil is used in this paper, c and d are defined as the chord length and the thickness of the foil, the foil’s the foil, the foil’s pivot is located at one third of chord. pivot is located at one third of chord.

Figure 1. Figure 1. SketchSketch of of full-active full-active flapping flapping foil foil under under time-varying time-varying freestream freestream velocity. velocity. The heaving and pitching motion of the full-active flapping foil are defined as: The heaving and pitching motion of the full-active flapping foil are defined as: y(t) = y sin(2π f t) (1) yt() y00 sin(2 ft ) (1)

θθθ(t)=+Φ = θ sin(2 ππ f t + Φ) (2) (tft )00 sin(2 ) (2) Appl. Sci. 2020, 10, 6226 4 of 18

where y0 and θ0 are defined as the heaving and pitching amplitudes, respectively. f is the motion frequency, f ∗ is the non-dimensional coefficient of motion frequency, named reduced frequency, f ∗ = f c/U . Φ is the phase different between the heaving and pitching motions, Φ = π/2. Referring to ∞ the assumptions of time-varying flow velocity made by Gharali [34] and Chen [35], the time-varying freestream in this paper is considered as a superposition of steady flow and sinusoidal pulsating flow. Therefore, the time-varying flow U(t) is defined as:

U(t) = U + γ sin(2π fvt)U (3) ∞ ∞ where U is the constant term of flow, γ sin(2π fvt)U is the pulsating term of flow. γ and fv are ∞ ∞ defined as the oscillating amplitude and oscillating frequency of the time-varying flow. τ is defined as the oscillating frequency coefficient of the time-varying flow, fv = 2τ f . The instantaneous thrust coefficient, instantaneous lift coefficient and instantaneous moment coefficient of full-active flapping foil are expressed as: 1 C (t) = F (t)/ ρU2c (4) X X 2 1 C (t) = F (t)/ ρU2c (5) Y Y 2 1 C (t) = M(t)/ ρU2c2 (6) M 2 where FX, FY and M are defined as the thrust, lift and moment. ρ is the fluid density. The instantaneous input power PI(t) and propulsion power PI(t) are defined as:

PI(t) = PY(t) + Pθ(t) = FY(t)Vy(t) + M(t)Ω(t) (7)

PO(t) = FX(t)U(t) (8) where Vy is the heaving velocity, Ω is the pitching velocity. The mean input power coefficient and mean propulsion power coefficient are expressed as:

Z T 1 1 3 CPI = (PI(t)/ ρU c)dt (9) T 0 2

Z T 1 1 3 CPO = (PO(t)/ ρU c)dt (10) T 0 2 where T is defined as the period of the heaving and pitching motion of the flapping foil, T = 1/ f . The propulsion efficiency of the full-active flapping foil is defined as:

η = CPO/CPI (11)

2.2. Numerical Methods and Validation Based on the sliding mesh technology, the computational domain and boundary conditions for 2D flapping foil is illustrated in Figure2. We use PointWise software to mesh the 2D flapping foil. The entire computation domain is 120c 120c, the inner grid diameter is 35c. The inlet, outlet, upper wall and low × wall are divided into 25 equal parts, the sliding interface is divided into 120 equal parts, and the foil 5 wall is divided into 600 equal parts. The height of the first layer near the foil is set to 10e− , and grows 30 layers at a rate of 1.1, which ensures y+ < 1. The flow distribution in the viscous sub-layer and transition layer of the foil surface is solved by using the Navier-Stokes (NS) equation discretely. We use a sliding interface to divide the simulation domain into the inner grid and outer grid. The rigid movement of the inner grid can ensure that the inner grid is undeformed and improves the calculation accuracy near the foil. Referring to Equations (1) and (2), we program the user defined functions Appl. Sci. 2020, 10, x 5 of 18 andAppl. improves Sci. 2020, 10 ,the 6226 calculation accuracy near the foil. Referring to Equations (1) and (2), we program5 of 18 the user defined functions (UDFs) to control the heaving and pitching motions of the inner grid. With the(UDFs) motion to control of the theinner heaving grid, the and outer pitching grid motions adjusts of adaptively. the inner grid. The Withinlet thecondition motion for of thetime-varying inner grid, flowthe outer velocity grid is adjusts controlled adaptively. by the TheUDFs inlet according condition to for Equation time-varying (3). The flow outlet velocity condition is controlled is a pressure by the outlet,UDFs accordingand the upper to Equation and lower (3). wall The outletis symmetry. condition The is aReynolds pressure number outlet, and is 𝑅𝑒 the upper = 42,000 and, the lower Sparat– wall Allmarasis symmetry. turbulence The Reynolds model number is used, is Reand= the42,000, SIMPLE the Sparat–Allmaras algorithm is selected turbulence for modelpressure–velocity is used, and couplingthe SIMPLE calculation. algorithm The is selecteddiscrete formethods pressure–velocity of gradient and coupling pressure calculation. are least The squares discrete cell-based methods and of second-ordergradient and pressurediscrete, arerespectively. least squares Momentum, cell-based turbulent and second-order flow energy discrete, and turbulent respectively. dissipation Momentum, rate areturbulent all selected flow in energy second-order and turbulent upwind. dissipation Time disc rateretization are all selectedis set second-order in second-order implicit upwind. format. Time The convergencediscretization condition is set second-order of velocity implicit and continuity format. The is convergence1𝑒 . The judgment condition condition of velocity for and the continuity periodic simulation5 is that the change rate of the mean value of the instantaneous thrust coefficient is less than is 1e− . The judgment condition for the periodic simulation is that the change rate of the mean value of 1%.the instantaneousAccording to different thrust coe simulationfficient is lessparameters than 1%. of According flapping foil, to di aff erentstable simulation periodic solution parameters of the of flappingflapping foil foil, can a stable be obtained periodic after solution 10 cycles. of the flapping foil can be obtained after 10 cycles.

Figure 2. Computational domain and boundary conditions for two-dimensional (2D) flapping foil. Figure 2. Computational domain and boundary conditions for two-dimensional (2D) flapping foil. To verify the simulation strategy, the grid, time independence and validation are studied in our To verify the simulation strategy, the grid, time independence and validation are studied in our previous research [10]. We simulated four grid and time-steps levels. The comparison of the mean previous research [10]. We simulated four grid and time-steps levels. The comparison of the mean value of thrust coefficient, lift coefficient and propulsion efficiency corresponding to the different grid value of thrust coefficient, lift coefficient and propulsion efficiency corresponding to the different and time-steps levels is shown in Table1. In order to ensure the accuracy and calculation e fficiency, grid and time-steps levels is shown in Table 1. In order to ensure the accuracy and calculation we chose the mesh and time step as 80,000 cells and 800 time-steps for all simulation cases. We also efficiency, we chose the mesh and time step as 80,000 cells and 800 time-steps for all simulation cases. validated the current strategy against the classic literature [36] under the same conditions, as shown We also validated the current strategy against the classic literature [36] under the same conditions, in Figure3, and found that the result agrees well with the Kinsey and Dumas, which confirm the as shown in Figure 3, and found that the result agrees well with the Kinsey and Dumas, which simulation strategy of this paper is credible and accurate. confirm the simulation strategy of this paper is credible and accurate. Table 1. Grid and time independence. Table 1. Grid and time independence.

Grid Time Step CPO CPI η Grid Time Step 𝑪𝑷𝑶 𝑪𝑷𝑰 𝜼 800 ts 6.264 21.573 29.038% 80,000 cells 800 ts 6.264 21.573 29.038% 80,000 cells 1600 ts 6.293 21.752 28.930% 1600 ts 6.293 21.752 28.930% 800 ts 6.327 21.893 28.9% 160,000 cells 800 ts 6.327 21.893 28.9% 160,000 cells 1600 ts 6.407 22.210 28.846% 1600 ts 6.407 22.210 28.846%

3. Results and Discussions In this paper, we first analyze the propulsion performance of the full-active flapping foil in the parameter space of reduced frequency and pitching amplitude at a uniform freestream. Then, we study the effects of the oscillating amplitude, oscillating frequency coefficient, and simulation time Appl. Sci. 2020, 10, x 6 of 18 of the time-varying freestream on the propulsion performance of the full-active flapping foil. Finally, based on the evolution of the instantaneous thrust coefficient, wake vortex, velocity and pressure near the foil, we reveal the mechanism of the time-varying freestream on the hydrodynamic performance of the full-active flapping foil. The range of the oscillating amplitude and frequency of the time-varying freestream is selected based on the literature [30]. The other physical parameters of the flapping foil refer to the physical parameters of wave glider [10]. The main parameters are shown in Table 2. Appl. Sci. 2020, 10, 6226 6 of 18

(a) (b)

o FigureFigure 3.3. Comparison of results by the present study and Kinsey and Dumas for y𝑦0 ==1𝑐1c,, θ𝜃0 =70= 70 , ∗ 𝑓f ∗ ==0.140.14,, ((aa)) thrustthrust coecoefficientfficient and and ( b(b)) lift lift coe coefficient.fficient.

3. Results and Discussions Table 2. Main characteristics for computations. In this paper, we first analyze the propulsion performance of the full-active flapping foil in the Parameters Range parameter space of reduced frequency and pitching amplitude at a uniform freestream. Then, we study Chord length, c 0.17 m the effects of the oscillating amplitude, oscillating frequency coefficient, and simulation time of the Hydrofoil shape 2D NACA0012 time-varying freestream on the propulsion performance of the full-active flapping foil. Finally, based on Hydrofoil thickness, d 0.02 m the evolution of the instantaneous thrust coefficient, wake vortex, velocity and pressure near the foil, Constant flow velocity, 𝑈 0.25 m/s we reveal the mechanism of the time-varying freestream on the hydrodynamic performance of the Heaving amplitude, 𝑦 1c full-active flapping foil. The range of the oscillating amplitude and frequency of the time-varying Pitching amplitude, 𝜃 10–80° freestream is selected based on the literature [30]. The other physical parameters of the flapping foil Reduced frequency, 𝑓∗ 0.1–0.68 refer to the physical parameters of wave glider [10]. The main parameters are shown in Table2. Oscillating amplitude, 𝛾 0–0.15 Oscillating frequency coefficient, 𝜏 0–1 Table 2. Main characteristics for computations. Simulation Period, 𝑡 10–20T Parameters Range 3.1. The Propulsion PerformanceChord of Flapping length, cFoil at Uniform Freestream 0.17 m Hydrofoil shape 2D NACA0012 The amplitude and frequency of the heaving and pitching motions are the main factors affecting Hydrofoil thickness, d 0.02 m the hydrodynamic performanceConstant flow of the velocity, full-activeU flapping foil at uniform0.25 m/s freestream [9,29,37]. We ∞ first illustrate the contourHeaving of the amplitude, mean propulsiony0 power coefficient1c of the flapping foil in the parameter space of reducedPitching frequency amplitude, and θpitching0 amplitude, as shown10–80◦ in Figure 4. We note that the reduced frequency andReduced pitching frequency, amplitudef ∗ have a significant 0.1–0.68effect on the mean propulsion Oscillating amplitude,γ 0–0.15 power coefficient. Further, we observe that the effect of the pitching amplitude on the propulsion Oscillating frequency coefficient, τ ∗ 0–1 performance is related to Simulationthe reduced Period, frequency.t When 𝑓 > 0.13610–20T, the mean propulsion power coefficient of the flapping foil increases first and then decreases with the variation of the pitching amplitude. The optimal pitching amplitude appears at the middle of the pitching amplitude range, 3.1. The Propulsion Performance of Flapping Foil at Uniform Freestream where the mean propulsion power coefficient of the flapping foil reaches the maximum value. When 𝑓∗ =0.34The amplitude and 𝑓∗ =0.68 and frequency, the maximum of the heavingvalues of and the pitching mean propulsion motions are power the main coefficients factorsa areffecting 0.51 theand hydrodynamic 2.32, respectively, performance and the corresponding of the full-active optimal flapping pitching foil at uniformamplitudes freestream are both [9 ,2930,°37. According]. We first illustrateto Figure the4, we contour further of notice the mean that propulsionwith the redu powerction coein thefficient reduced of the frequency, flapping foilthe inoptimal the parameter pitching space of reduced frequency and pitching amplitude, as shown in Figure4. We note that the reduced frequency and pitching amplitude have a significant effect on the mean propulsion power coefficient. Further, we observe that the effect of the pitching amplitude on the propulsion performance is related to the reduced frequency. When f ∗ > 0.136, the mean propulsion power coefficient of the flapping foil increases first and then decreases with the variation of the pitching amplitude. The optimal pitching amplitude appears at the middle of the pitching amplitude range, where the mean propulsion power coefficient of the flapping foil reaches the maximum value. When f ∗ = 0.34 and f ∗ = 0.68, Appl. Sci. 2020, 10, x 7 of 18

amplitude of the flapping foil is decreased gradually. When 𝑓∗ =0.23, the optimal pitching amplitude is 20° , and when 𝑓∗ =0.11, the optimal pitching amplitude is reduced to 10° . Furthermore, we realize that when 𝑓∗ ≤0.11, the mean propulsion power coefficient of the flapping foil decreases monotonically with the variation of the pitching amplitude. When 𝑓∗ = 0.068, the ° ° mean propulsion power coefficients corresponding to 𝜃 =10 and 𝜃 =80 are 0.027 and 0.32, respectively. The contour of the propulsion efficiency of the full-active flapping foil on the parametric space Appl. Sci. 2020, 10, 6226 7 of 18 of reduced frequency and pitching amplitude is illustrated in Figure 5. The dark blue region of the Figure 5 indicates that the mean propulsion power coefficient and propulsion efficiency in this range theare maximum less than zero. values According of the mean to Figure propulsion 5, we power reveal coe thatffi cientswith the are variation 0.51 and of 2.32, the respectively, pitching amplitude, and the correspondingthe propulsion optimalefficiency pitching decreases amplitudes monotonically. are both When30◦ . 𝑓 According∗ =0.68, the to propulsion Figure4, we efficiencies further notice of the ° ° ∗ thatflapping with thefoil reductioncorresponding in the to reduced 𝜃 =10 frequency, and 𝜃 =60 the optimal are 48.1% pitching and 14.1%, amplitude respectively. of the flapping When 𝑓 foil= ° ° is0.23 decreased, the propulsion gradually. efficiencies When f ∗ = of0.23 the, theflapping optimal foil pitching corresponding amplitude to is𝜃20=10◦ , and and when 𝜃 f=50∗ = 0.11 are, the4.7% optimal and 0.5%, pitching respectively. amplitude According is reduced to Figure to 10 5,◦ . we Furthermore, also note that we with realize the variation that when of thef ∗ reduced0.11, ° ≤ thefrequency, mean propulsion the propulsion power coe efficienfficientcy of increases the flapping monotonically. foil decreases When monotonically 𝜃 =10 with, the the propulsion variation ∗ ∗ ofefficiencies the pitching of amplitude.the flapping When foilf ∗corresponding= 0.068, the mean to 𝑓 propulsion=0.27 and power 𝑓 coe=0.45fficients are corresponding6.4% and 19.3%, to θrespectively.0 = 10◦ and θ0 = 80◦ are 0.027 and 0.32, respectively.

Figure 4. The mean output power coefficient contour on the parametric space of reduced frequency andFigure pitching 4. The amplitude. mean output power coefficient contour on the parametric space of reduced frequency and pitching amplitude. The contour of the propulsion efficiency of the full-active flapping foil on the parametric space of reduced frequency and pitching amplitude is illustrated in Figure5. The dark blue region of the Figure5 indicates that the mean propulsion power coe fficient and propulsion efficiency in this range are less than zero. According to Figure5, we reveal that with the variation of the pitching amplitude, the propulsion efficiency decreases monotonically. When f ∗ = 0.68, the propulsion efficiencies of the flapping foil corresponding to θ0 = 10◦ and θ0 = 60◦ are 48.1% and 14.1%, respectively. When f ∗ = 0.23, the propulsion efficiencies of the flapping foil corresponding to θ0 = 10◦ and θ0 = 50◦ are 4.7% and 0.5%, respectively. According to Figure5, we also note that with the variation of the reduced frequency, the propulsion efficiency increases monotonically. When θ0 = 10◦ , the propulsion efficiencies of the flapping foil corresponding to f ∗ = 0.27 and f ∗ = 0.45 are 6.4% and 19.3%, respectively. Based on the above research, we conclude that when the reduced frequency is small, the smaller pitching amplitude can improve the propulsion performance of the full-active flapping foil. When the reduced frequency is large, the proper pitching amplitude can reach the best propulsion performance. Hover et al. [7] studied the influence mechanism of the leading-edge vortex on the hydrodynamic performance of the flapping foil. They reveal that the proper pitching amplitude is beneficial to the generated of the leading-edge vortex and reverse Karman vortex, which can improve the propulsion performance of the flapping foil. Meanwhile, we also note that when the pitching amplitude is large enough, the mean propulsion power coefficient is less than zero. Thus, the flapping foil cannot generate thrust in this condition. Appl.Appl. Sci.Sci. 20202020,, 1010,, 6226x 88 ofof 1818

Figure 5. The propulsion efficiency contour on the parametric space of reduced frequency and pitchingFigure 5. amplitude. The propulsion efficiency contour on the parametric space of reduced frequency and pitching amplitude. 3.2. The Influence of the Time Varying Freestream Based on the above research, we conclude that when the reduced frequency is small, the smaller pitchingIn the amplitude previous section,can improve we discuss the propulsion the propulsion performance performance of the of full-active the full-active flapping flapping foil. foilWhen at uniformthe reduced freestream frequency and analyzeis large, the the reduced proper frequency pitching and amplitude pitching can amplitude reach onthe the best foil. propulsion Based on theperformance. above study, Hover the oscillatinget al. [7] studied amplitude the andinfluenc oscillatinge mechanism frequency of the coe ffileading-edgecient of the vortex time-varying on the freestreamhydrodynamic on the performance hydrodynamic of the performance flapping foil. of the They flapping reveal foil that is the investigated proper pitching in this section.amplitude First, is webeneficial analyze to the the e ffgeneratedect of the of time-varying the leading-edge flow velocity vortex and on thereverse flapping Karman foil invortex, one period, which can which improve is the periodthe propulsion of the heaving performance and pitching of the motionflapping of foil. the flappingMeanwhile, foil. we Then, also we note study that the when influence the pitching of the time-varyingamplitude is freestreamlarge enough, on the mean propulsion ofpowe ther foil coefficient at a different is less period. than zero. Finally, Thus, we the recalculate flapping thefoil meancannot propulsion generate thrust power in coe thisfficient condition. of the flapping foil over 10 periods and analyze the influence of the time-varying flow on the propulsion performance of the flapping foil. 3.2. The Influence of the Time Varying Freestream 3.2.1. The Effect of the Time-Varying Flow in One Period In the previous section, we discuss the propulsion performance of the full-active flapping foil at uniformAccording freestream to Figure and 4analyze, we choose the reduced di fferent frequency parameters and of pitching reduced amplitude frequency on ( f ∗ the= 0.34 foil./ 0.68Based) and on pitchingthe above amplitude study, the (θ 0oscillating= 10◦ /30◦ /amplitude60◦ ) to study and the osci influencellating frequency of oscillating coefficient amplitude of theγ and time-varying oscillating frequencyfreestream coe on ffithecient hydrodynamicτ of the time-varying performance flow of the on flappi the propulsionng foil is investigated performance in ofthis the section. full-active First, flappingwe analyze foil. the The effect evolution of the oftime-varying the mean propulsion flow veloci powerty on coetheffi flappingcient of foil the flappingin one period, foil with which diff erentis the oscillatingperiod of the amplitude heaving andand oscillatingpitching motion frequency of the coe flappingfficient isfoil. shown Then, in we Figure study6. the We influence notice that of the the oscillatingtime-varying frequency freestream coeffi cienton the of theme time-varyingan propulsion flow of hasthe a significantfoil at a different effect on theperiod. mean Finally, propulsion we powerrecalculate coeffi thecient. mean In particular,propulsion the power effect coefficient of the oscillating of the flapping frequency foil coe overfficient 10 periods on the flappingand analyze foil isthe related influence to the of the pitching time-varying amplitude flow and on reducedthe propulsion frequency performance of the flapping of the flapping foil. When foil. the motion parameters are large, the time-varying flow has a more significant effect on the flapping performance of the3.2.1. flapping The Effect foil. of When the Time-Varyingf ∗ = 0.68, the maximumFlow in One change Period rate of the mean propulsion power coefficient caused by the oscillating frequency coefficient τ for θ = 10◦ and θ = 60◦ is 9.8% and 13%, respectively. 0 0 𝑓∗ =0.340.68 According= to Figure 4, we choose different parameters of reduced frequency ( / ) When f ∗ 0.34, the maximum change° ° rate° of the mean propulsion power coefficient caused by the and pitching amplitude (𝜃 =10/30 /60 ) to study the influence of oscillating amplitude 𝛾 and oscillating frequency coefficient τ for θ0 = 10◦ and θ0 = 30◦ is 4.3% and 5.3%, respectively. Besides, oscillating frequency coefficient 𝜏 of the time-varying flow on the propulsion performance of the we also found that when the vibration coefficient, the impact of the oscillating frequency coefficient on full-active flapping foil. The evolution of the mean propulsion power coefficient of the flapping foil the propulsion performance of active flapping is significantly reduced. We further notice that when the with different oscillating amplitude and oscillating frequency coefficient is shown in Figure 6. We Appl. Sci. 2020, 10, x 9 of 18 notice that the oscillating frequency coefficient of the time-varying flow has a significant effect on the mean propulsion power coefficient. In particular, the effect of the oscillating frequency coefficient on the flapping foil is related to the pitching amplitude and reduced frequency of the flapping foil. When the motion parameters are large, the time-varying flow has a more significant effect on the flapping performance of the flapping foil. When 𝑓∗ =0.68, the maximum change rate of the mean propulsion ° ° power coefficient caused by the oscillating frequency coefficient 𝜏 for 𝜃 =10 and 𝜃 =60 is 9.8% and 13%, respectively. When 𝑓∗ =0.34, the maximum change rate of the mean propulsion ° ° power coefficient caused by the oscillating frequency coefficient 𝜏 for 𝜃 =10 and 𝜃 =30 is 4.3% and 5.3%, respectively. Besides, we also found that when the vibration coefficient, the impact of the oscillating frequency coefficient on the propulsion performance of active flapping is significantly reduced. We further notice that when the oscillating frequency coefficient 𝜏>0.5, the impact of the time-varying flow on the mean propulsion performance of the flapping foil is significantly reduced, as shown in Figure 6. Ma [30] and Chen [35] have obtained similar conclusions when studying the effect of time-varying freestream on the energy harvesting performance of the full-active flapping foil. Referring to Figure 6, we also notice that the oscillating amplitude of the time-varying flow also affects the hydrodynamic performance of the full-active flapping foil. With the increase in the oscillating amplitude, the variation amplitude of the mean propulsion power coefficient of the flappingAppl. Sci. 2020 foil, 10caused, 6226 by the oscillating frequency coefficient increases proportionally, but the variation9 of 18 ∗ ° tendency does not change. When 𝑓 =0.68 and 𝜃 =10, the maximum change rate of the mean propulsion power coefficient caused by the oscillating frequency coefficient 𝜏 for 𝛾=0.05 and 𝛾= oscillating frequency coefficient τ > 0.5, the impact of the time-varying flow on the mean propulsion 0.15 are 1.5% and 5.3%, respectively. Since the oscillating amplitude does not affect the tendency of performance of the flapping foil is significantly reduced, as shown in Figure6. Ma [ 30] and Chen [35] the time-varying flow on the propulsion performance of the full-active flapping foil, in order to have obtained similar conclusions when studying the effect of time-varying freestream on the energy simplify the simulation calculation, the oscillating amplitudes in the subsequent calculations are all harvesting performance of the full-active flapping foil. fixed values, 𝛾=0.1.

Figure 6. The evolution of the mean propulsion power coefficient with different oscillating amplitude Figure 6. The evolution of the mean propulsion power coefficient with different oscillating amplitude and oscillating frequency coefficient for one period. and oscillating frequency coefficient for one period. Referring to Figure6, we also notice that the oscillating amplitude of the time-varying flow 3.2.2.also a Theffects Effect the hydrodynamicof the Time-Varying performance Flow in ofDifferent the full-active Periods flapping foil. With the increase in the oscillatingTo observe amplitude, the effect the variationof the time-varying amplitude flow of the on mean the propulsion propulsion performa power coencefficient of the of flapping the flapping foil infoil different caused byperiods, the oscillating the variation frequency of the coe meanfficient propulsion increases power proportionally, coefficient but in thedifferent variation periods tendency and does not change. When f ∗ = 0.68 and θ0 = 10◦ , the maximum change rate of the mean propulsion power coefficient caused by the oscillating frequency coefficient τ for γ = 0.05 and γ = 0.15 are 1.5% and 5.3%, respectively. Since the oscillating amplitude does not affect the tendency of the time-varying flow on the propulsion performance of the full-active flapping foil, in order to simplify the simulation calculation, the oscillating amplitudes in the subsequent calculations are all fixed values, γ = 0.1.

3.2.2. The Effect of the Time-Varying Flow in Different Periods To observe the effect of the time-varying flow on the propulsion performance of the flapping foil in different periods, the variation of the mean propulsion power coefficient in different periods and oscillating frequency coefficient is illustrated in Figure7. When f ∗ = 0.34, θ0 = 10◦ and γ = 0.1, the numerical simulation obtains a stable periodic solution after 10 cycles. Therefore, we choose the data within 10–20 cycles for comparison. According to Figure7, the oscillating frequency coe fficient of the time-varying flow has an apparent regularity for the impact of the mean propulsion power coefficient of the flapping foil. When τ = 0.5 and τ = 1.0, with the variation of the oscillating frequency coefficient and simulation period, the mean propulsion power coefficient is almost unchanged. According to Equation (3), when the reduced frequency is an integer multiple of the oscillating frequency coefficient of the time-varying flow, the average flow velocity of the time-varying flow in one cycle is equal to the constant term U . Therefore, the time-varying freestream has no significant effect on the hydrodynamic ∞ performance of the full-active flapping foil. When the relationship between the oscillating frequency Appl. Sci. 2020, 10, x 10 of 18

∗ ° oscillating frequency coefficient is illustrated in Figure 7. When 𝑓 =0.34, 𝜃 =10 and 𝛾=0.1, the numerical simulation obtains a stable periodic solution after 10 cycles. Therefore, we choose the data within 10–20 cycles for comparison. According to Figure 7, the oscillating frequency coefficient of the time-varying flow has an apparent regularity for the impact of the mean propulsion power coefficient of the flapping foil. When 𝜏=0.5 and 𝜏=1.0, with the variation of the oscillating frequency coefficient and simulation period, the mean propulsion power coefficient is almost unchanged. According to Equation (3), when the reduced frequency is an integer multiple of the oscillating frequency coefficient of the time-varying flow, the average flow velocity of the time-varying flow in one cycle is equal to the constant term 𝑈∞. Therefore, the time-varying freestream has no significant effect on the hydrodynamic performance of the full-active flapping foil. When the relationship Appl. Sci. 2020, 10, 6226 10 of 18 between the oscillating frequency coefficient of the time-varying freestream and the reduced frequency is not an integer multiple, with the variation of the simulation period, the oscillating coefrequencyfficient of coefficient the time-varying directly freestreamaffects the andmean the propul reducedsion frequency power coefficient is not an integerof the full-active multiple, withflapping the variationfoil. When of 𝜏<0.5 the simulation, the mean period, propulsion the oscillatingpower coefficient frequency of the coe flappingfficient foil directly fluctuates affects significantly, the mean propulsionand when power𝜏>0.5 coe, theffi cientmean of propulsion the full-active power flapping coefficient foil. Whenis almostτ < unchanged0.5, the mean with propulsion the variation power of coetheffi period,cient of as the shown flapping in Figure foil fluctuates 7. The influence significantly, of the oscillating and when τfrequency > 0.5, the coefficient mean propulsion on the flapping power coefoilffi cancient also is almost be found unchanged in Figure with 6. the The variation reason of of the the period, above as phenomenon shown in Figure is 7that. The the influence oscillating of thefrequency oscillating coefficient frequency of coe theffi cienttime-varying on the flapping flow af foilfects can the also average be found flow in Figurevelocity6. Thein reasonone cycle. of theEspecially above phenomenon when 𝜏<0.5 is, thatthe thechange oscillating of the frequency average flow coeffi velocitycient of thein different time-varying periods flow is a ffmoreects theprominent. average The flow average velocity flow in one velocity cycle. leads Especially to differences when τin < the0.5 hydrodynamic, the change of performance the average of flow the velocityflapping in foil diff inerent different periods periods. is more prominent. The average flow velocity leads to differences in the hydrodynamic performance of the flapping foil in different periods.

Figure 7. The variation of the mean propulsion power coefficient in different periods and oscillating ◦ frequencyFigure 7. The coeffi variationcients for off ∗the= mean0.34, θ propulsion0 = 10 . power coefficient in different periods and oscillating ∗ ° frequency coefficients for 𝑓 =0.34, 𝜃 =10. The effect of the time-varying flow on the propulsion performance of the flapping foil for f ∗ = 0.34 and θ = 10◦ is described above. Then, we analyze the influence of the time-varying flow on the The0 effect of the time-varying flow on the propulsion performance of the flapping foil for 𝑓∗ = flapping foil with the° variation of the period under different motion parameters (reduced frequency 0.34 and 𝜃 =10 is described above. Then, we analyze the influence of the time-varying flow on and pitching amplitude). The evolution of the mean propulsion power coefficient for different reduced the flapping foil with the variation of the period under different motion parameters (reduced frequency and pitching amplitude at τ = 0.2 and τ = 0.8 is shown in Figure8. We note that with the frequency and pitching amplitude). The evolution of the mean propulsion power coefficient for increase in the pitching amplitude of the flapping foil, the variation amplitude of the mean propulsion power coefficient of the flapping foil caused by the oscillating frequency coefficient increases gradually. When f ∗ = 0.68 and τ = 0.2, the maximum change rates of the mean propulsion power coefficient caused by the time-varying freestream for θ0 = 10◦ ,20◦ ,30◦ and 60◦ are 2.3%, 3.7%, 6.8% and 23.6%, respectively. When f ∗ = 0.68 and τ = 0.8, the maximum change rates of the mean propulsion power coefficient caused by the time-varying freestream for θ0 = 10◦ , 20◦ , 30◦ and 60◦ are 2%, 2.1%, 4.1% and 21.4%, respectively. Furthermore, we reveal that with the variation of the reduced frequency, the maximum change rates of the mean propulsion power coefficient caused by the time-varying freestream increase gradually. When θ0 = 30◦ and τ = 0.2, the maximum change rates of the mean propulsion power coefficient caused by the time-varying freestream for f ∗ = 0.34, 0.45 and 0.68 are 3.4%, 6.2% and 6.8%, respectively. When θ0 = 30◦ and τ = 0.8, the maximum change rates of the mean Appl. Sci. 2020, 10, x 11 of 18 different reduced frequency and pitching amplitude at 𝜏=0.2 and 𝜏=0.8 is shown in Figure 8. We note that with the increase in the pitching amplitude of the flapping foil, the variation amplitude of the mean propulsion power coefficient of the flapping foil caused by the oscillating frequency coefficient increases gradually. When 𝑓∗ =0.68 and 𝜏=0.2, the maximum change rates of the mean ° ° ° ° propulsion power coefficient caused by the time-varying freestream for 𝜃 =10,20 ,30 and 60 are 2.3%, 3.7%, 6.8% and 23.6%, respectively. When 𝑓∗ =0.68 and 𝜏=0.8, the maximum change ° rates of the mean propulsion power coefficient caused by the time-varying freestream for 𝜃 =10, 20°, 30° and 60° are 2%, 2.1%, 4.1% and 21.4%, respectively. Furthermore, we reveal that with the variation of the reduced frequency, the maximum change rates of the mean propulsion power ° Appl.coefficient Sci. 2020 caused, 10, 6226 by the time-varying freestream increase gradually. When 𝜃 =30 and 𝜏=0.211 of 18, the maximum change rates of the mean propulsion power coefficient caused by the time-varying ∗ ° freestream for 𝑓 =0.34, 0.45 and 0.68 are 3.4%, 6.2% and 6.8%, respectively. When 𝜃 =30 and propulsion𝜏=0.8, the powermaximum coeffi changecient caused rates of by the the mean time-varying propulsion freestream power coefficient for f ∗ = 0.34 caused, 0.45 by and the 0.68 time- are 2.4%,varying 3.9% freestream and 4.1%, for respectively. 𝑓∗ =0.34, 0.45 and 0.68 are 2.4%, 3.9% and 4.1%, respectively.

(a)

(b)

Figure 8. The evolution of the mean propulsion power coefficient for different reduced frequency and pitching amplitude, (a) τ = 0.2, (b) τ = 0.8.

3.2.3. The Effect of the Time-Varying Flow in Multiple Periods In the previous section, the influence of the time-varying flow on the propulsion performance of the full-active flapping foil in different periods is investigated. We note that the simulation time can affect the hydrodynamic performance of the flapping foil under the action of the time-varying freestream. The oscillating amplitude and frequency determine the average velocity of the time-varying flow in different periods, which affects the propulsion performance of the foil. Therefore, we infer that the average velocity of the time-varying flow under different oscillating amplitude and oscillating Appl. Sci. 2020, 10, x 12 of 18

Figure 8. The evolution of the mean propulsion power coefficient for different reduced frequency and pitching amplitude, (a) 𝜏=0.2, (b) 𝜏=0.8.

3.2.3. The Effect of the Time-Varying Flow in Multiple Periods In the previous section, the influence of the time-varying flow on the propulsion performance of the full-active flapping foil in different periods is investigated. We note that the simulation time can affect the hydrodynamic performance of the flapping foil under the action of the time-varying freestream. The oscillating amplitude and frequency determine the average velocity of the time- Appl.varying Sci. 2020 flow, 10 in, 6226 different periods, which affects the propulsion performance of the foil. Therefore,12 of we 18 infer that the average velocity of the time-varying flow under different oscillating amplitude and frequencyoscillating coefrequencyfficient incoefficient multiple in periods multiple will periods tend to will the tend constant to the velocity. constant Thus, velocity. with Thus, the extension with the ofextension simulation of simulation time, the etime,ffect ofthe the effect time-varying of the time-varying flow on the flow flapping on the foilflapping will graduallyfoil will gradually weaken. Kinseyweaken. [29 Kinsey] and Ma[29] [and30] alsoMa [30] found also a similarfound a phenomenon similar phenomenon when studying when studying the effect the of effect time-varying of time- flowvarying on theflow energy on the harvesting energy harvesting performance performance of the full-active of the full-active flapping flapping foil. The foil. average The average value ofvalue the propulsionof the propulsion power coepowerfficient coefficien in 10 cyclest in 10 with cycles the variationwith the ofvariation the oscillating of the frequencyoscillating coefrequencyfficient iscoefficient shown in is Figure shown9. in Referring Figure 9. to Referring Figure8 ,to we Figure know 8, that we aknow larger that pitching a larger amplitude pitching amplitude and reduced and frequencyreduced frequency can increase can increase the maximum the maximum change ratechange of therate mean of the propulsion mean propulsion power power coefficient coefficient of the of the flapping foil affected by the time-varying flow. Referring to Figure 9, when 𝑓∗ =0.68 and flapping foil affected by the time-varying flow. Referring to Figure9, when f ∗ = 0.68 and θ0 = 60◦ , 𝜃 =60° the maximum, the maximum change rate change of the flappingrate of the foil flapping is only 4.8%. foil is Therefore, only 4.8%. we Therefore, conclude we that conclude the time-varying that the freestreamtime-varying has freestream little effect has on little the propulsioneffect on the performance propulsion performance of the full-active of the flapping full-active foil flapping at different foil pitchingat different amplitude pitching and amplitude reduced frequency.and reduced When frequency. we calculate When the we hydrodynamic calculate the performancehydrodynamic of theperformance flapping foilof the by usingflapping numerical foil by simulation,using numerica we canl simulation, simplify thewe time-varyingcan simplify freestreamthe time-varying to the freestream to the steady flow. steady flow.

Figure 9. The evolution of the mean propulsion power coefficient with the variation of the oscillating frequencyFigure 9. The coe ffievolutioncient. of the mean propulsion power coefficient with the variation of the oscillating frequency coefficient. 3.3. The Mechanism of the Time Varying Freestream 3.3. The Mechanism of the Time Varying Freestream In the previous section, we analyze the influence of the time-varying flow on the flapping foil. Then,In we the study previous the eff section,ects of the we time-varying analyze the flowinfluence on the of evolution the time-varying of instantaneous flow on thrust the flapping coefficients, foil. vortexThen, we topology, study velocitythe effects field, of andthe thetime-varying pressure distribution flow on the near evolution the foil, of and instantaneous further explain thrust the ecoefficients,ffect mechanism vortex of topology, the time-varying velocity flow.field, Referring and the pressure to Figure distribution8, we notice near that whenthe foil,f ∗ and= 0.68 furtherand θexplain0 = 60 ◦the, the effect oscillating mechanism frequency of the coe time-varyingfficient of the fl time-varyingow. Referring flow to Figure has a significant8, we notice eff thatect onwhen the mean propulsion performance of the flapping. Therefore, we analyze the evolution of the instantaneous thrust coefficient under these motion parameters. The evolution of the instantaneous thrust coefficient and freestream velocity over three periods is shown in Figure 10. We notice that the instantaneous thrust coefficient has two peaks and troughs within one period. The peaks and troughs are the maximum instantaneous resistance and thrust of the flapping foil. Comparing the evolution of instantaneous thrust coefficient and time-varying flow velocity at different oscillating frequencies of the time-varying flow, we find that there is a correlation between the instantaneous velocity and the variation of the instantaneous thrust coefficient of the flapping foil. When t = 15.15T, the instantaneous thrust coefficients for τ = 0, 0.2 and 0.8 are 1.9, 2.15 and 2.25, respectively. The corresponding instantaneous Appl. Sci. 2020, 10, x 13 of 18

∗ ° 𝑓 =0.68 and 𝜃 =60 , the oscillating frequency coefficient of the time-varying flow has a significant effect on the mean propulsion performance of the flapping. Therefore, we analyze the evolution of the instantaneous thrust coefficient under these motion parameters. The evolution of the instantaneous thrust coefficient and freestream velocity over three periods is shown in Figure 10. We notice that the instantaneous thrust coefficient has two peaks and troughs within one period. The peaks and troughs are the maximum instantaneous resistance and thrust of the flapping foil. Comparing the evolution of instantaneous thrust coefficient and time-varying flow velocity at different oscillating frequencies of the time-varying flow, we find that there is a correlation between the instantaneous velocity and the variation of the instantaneous thrust coefficient of the flapping Appl. Sci. 2020, 10, 6226 13 of 18 foil. When 𝑡 = 15.15𝑇, the instantaneous thrust coefficients for 𝜏=0, 0.2 and 0.8 are 1.9, 2.15 and 2.25, respectively. The corresponding instantaneous velocity are 2.5 m/s, 2.65 m/s and 2.87 m/s, respectively.velocity are 2.5 When m/s, 𝑡 2.65 = m17.65𝑇/s and, the 2.87 instantaneous m/s, respectively. thrust Whencoefficientst = 17.65 for Tτ, the= 0 instantaneous, 0.2 and 0.8 are thrust 1.9, 2.3coe ffiandcients 2.4, forrespectively.τ = 0, 0.2 andThe 0.8corresponding are 1.9, 2.3 andinstantaneous 2.4, respectively. velocities The are corresponding 2.5 m/s, 2.65 instantaneous m/s and 2.87 m/s,velocities respectively. are 2.5 m /Ins, 2.65the mvicinity/s and 2.87of the m/s, peak, respectively. the instantaneous In the vicinity resistance of the peak, of the instantaneousflapping foil graduallyresistance increases of the flapping with the foil increase gradually in the increases instantaneous with the velocity. increase When in the𝜏=0.2 instantaneous, the instantaneous velocity. Whenthrust τcoefficients= 0.2, the for instantaneous 𝑡 = 15.35𝑇, thrust 16.35𝑇 coe andfficients 17.35𝑇 for tare= 15.35−5.41,T ,−16.355.85 Tandand −5.75,17.35 Trespectively.are 5.41, The5.85 − − correspondingand 5.75, respectively. instantaneous The corresponding velocities are instantaneous2.79 m/s, 2.41 velocitiesm/s and 2.39 are 2.79m/s, mrespectively./s, 2.41 m/s andWe 2.39can also m/s, − noterespectively. that the Weinstantaneous can also note thrust that of the the instantaneous flapping foil thrust gradually of the increases flapping with foil graduallythe increase increases in the withinstantaneous the increase velocity in the in instantaneous the vicinity of velocity the trough. in the vicinity of the trough.

Figure 10. The evolution of the instantaneous thrust coefficient and freestream velocity over three Figure 10. The evolution of the instantaneous thrust coefficient and freestream velocity over three ◦ periods for f ∗∗= 0.68, θ0 = 60 .° periods for 𝑓 =0.68, 𝜃 =60. In order to further explain the mechanism of the influence of time-varying flow on the flapping In order to further explain the mechanism of the influence of time-varying flow on the flapping foil, the three snapshots of the vorticity contours and the corresponding velocity contours for different foil, the three snapshots of the vorticity contours and the corresponding velocity contours for oscillating frequency coefficient (τ = 0, 0.2 and 0.8) are shown in Figure 11. According to Figure 10, different oscillating frequency coefficient (𝜏=0, 0.2 and 0.8) are shown in Figure 11. According to we choose the moment when the instantaneous thrust coefficient reaches the peak. We notice that the Figure 10, we choose the moment when the instantaneous thrust coefficient reaches the peak. We vortexes near the foil for the three oscillating frequency coefficient is basically the same, which shows notice that the vortexes near the foil for the three oscillating frequency coefficient is basically the that the time-varying flow does not significantly change the vortex topology of the flapping foil. same, which shows that the time-varying flow does not significantly change the vortex topology of However, we find differences between the velocity distribution near the foil for different oscillating the flapping foil. However, we find differences between the velocity distribution near the foil for frequencies. Referring to Figure 10, we note that with the variation of the oscillating frequency coefficient, the instantaneous velocities increases. Comparing Figure 11a–c, we find that as the instantaneous velocity increases, the velocity close to the lower surface of the foil gradually increases. According to the Bernoulli equation, the pressure gradient drops as the velocity increases. Therefore, comparing τ = 0.8 with τ = 0 and τ = 0.2, a more significant negative pressure region on the lower surface of the foil is generated, which leads to an increase in the resistance and reduces the propulsion performance of the flapping foil. Referring to Figure 11, the pressure distribution of the flapping foil at τ = 0, 0.2 and 0.8 is shown in Figure 12. We note that the pressure distributions on the upper surface of the foil for different oscillating frequencies are almost the same. However, there is a clear difference in the pressure distributions on the lower surface of the foil for different oscillating frequencies. Appl. Sci. 2020, 10, x 14 of 18 different oscillating frequencies. Referring to Figure 10, we note that with the variation of the oscillating frequency coefficient, the instantaneous velocities increases. Comparing Figure 11a–c, we find that as the instantaneous velocity increases, the velocity close to the lower surface of the foil gradually increases. According to the Bernoulli equation, the pressure gradient drops as the velocity increases. Therefore, comparing 𝜏=0.8 with 𝜏=0 and 𝜏=0.2, a more significant negative pressure region on the lower surface of the foil is generated, which leads to an increase in the resistance and reduces the propulsion performance of the flapping foil. Referring to Figure 11, the Appl. Sci. 2020, 10, 6226 14 of 18 pressure distribution of the flapping foil at 𝜏=0, 0.2 and 0.8 is shown in Figure 12. We note that the pressure distributions on the upper surface of the foil for different oscillating frequencies are almost theThe same. instantaneous However, velocity there is of a theclear lower difference surface in of the foilpressure for τ =distributions0.8 is more significanton the lower than surfaceτ = 0 andof theτ = foil0.2 for, as different shown inoscillating Figure 11 frequencies.. The strength The ofinstantaneous the corresponding velocity negative of the lower pressure surface region of the on foil the forlower 𝜏=0.8 surface is more of the significant foil for τ = than0.8 is𝜏=0 greater and than 𝜏=0.2τ = 0, asand shownτ = 0.2 in, whichFigure increases 11. The strength the resistance of the of correspondingthe flapping foil, negative as shown pressure in Figure region 10. on the lower surface of the foil for 𝜏=0.8 is greater than 𝜏=0 and 𝜏=0.2, which increases the resistance of the flapping foil, as shown in Figure 10.

𝜏=0 (a)

(b) 𝜏=0.2

(c) 𝜏=0.8 ∗ ° Figure 11. The snapshots of the vorticity contours and velocity contours for 𝑓 =0.68, 𝜃 =60,𝑡=◦ Appl.Figure Sci. 2020 11., 10The, x snapshots of the vorticity contours and velocity contours for f ∗ = 0.68, θ0 = 60 ,15 of 18 15.15𝑇t = 15.15, (a)T 𝜏=0,(a) τ, =(b)0, 𝜏=0.2 (b) τ =, (0.2,c) 𝜏=0.8 (c) τ =. 0.8.

Figure 12. The pressure distribution of the flapping foil at different oscillating frequency coefficient for Figure 12. The pressure distribution of the flapping foil at different oscillating frequency coefficient ◦ f ∗ = 0.68,∗ θ0 = 60 , t = 15.15° T for 𝑓 =0.68, 𝜃 =60, 𝑡 = 15.15𝑇

According to Equation (3), we note that the oscillating frequency coefficient of the time-varying flow determines the fluctuation of the instantaneous velocity. When 𝜏=0.2, the instantaneous velocity for 𝑡 = 15.35𝑇, 16.35𝑇 and 17.35𝑇 are 2.79 m/s, 2.41 m/s and 2.39 m/s, respectively, as shown in Figure 10. The three snapshots of the vorticity contours and the corresponding velocity contours for 𝜏=0.2 at different simulation times are shown in Figure 13. The corresponding pressure distribution of the flapping foil is shown in Figure 14. Due to the little fluctuation range of the time-varying flow, the vortex topology of the flapping foil at a different time is slightly different. However, the velocity distribution near the foil can also be observed. The velocity of the leading edge of the flapping foil at 𝑡 = 16.35𝑇 is slightly greater than 𝑡 = 15.35𝑇 and 17.35𝑇, as shown in Figure 13. According to the Bernoulli equation, the intensity of the negative pressure region generated by the leading edge of the lower surface of the flapping foil at 𝑡 = 16.35𝑇 is greater than 𝑡 = 15.35𝑇 and 17.35𝑇, as shown in Figure 14. The strong negative pressure region at the leading edge of the flapping foil increases the thrust. Referring to Figure 10, we note that the maximum value of the instantaneous thrust coefficient at 𝑡 = 16.35𝑇. We can conclude that the time-varying flow affects the pressure distribution of the flapping foil. However, its impact on the propulsion performance of the flapping foil is also related to the relative position between the negative pressure area and the foil.

(a) 𝑡 = 15.35𝑇

(b) 𝑡 = 16.35𝑇 Appl. Sci. 2020, 10, x 15 of 18

Appl. Sci.Figure2020 ,12.10, The 6226 pressure distribution of the flapping foil at different oscillating frequency coefficient15 of 18 ∗ ° for 𝑓 =0.68, 𝜃 =60, 𝑡 = 15.15𝑇 According to Equation (3), we note that the oscillating frequency coefficient of the time-varying According to Equation (3), we note that the oscillating frequency coefficient of the time-varying flow determines the fluctuation of the instantaneous velocity. When τ = 0.2, the instantaneous velocity flow determines the fluctuation of the instantaneous velocity. When 𝜏=0.2, the instantaneous for t = 15.35T, 16.35T and 17.35T are 2.79 m/s, 2.41 m/s and 2.39 m/s, respectively, as shown in Figure 10. velocity for 𝑡 = 15.35𝑇, 16.35𝑇 and 17.35𝑇 are 2.79 m/s, 2.41 m/s and 2.39 m/s, respectively, as The three snapshots of the vorticity contours and the corresponding velocity contours for τ = 0.2 shown in Figure 10. The three snapshots of the vorticity contours and the corresponding velocity at different simulation times are shown in Figure 13. The corresponding pressure distribution of contours for 𝜏=0.2 at different simulation times are shown in Figure 13. The corresponding the flapping foil is shown in Figure 14. Due to the little fluctuation range of the time-varying flow, pressure distribution of the flapping foil is shown in Figure 14. Due to the little fluctuation range of the vortex topology of the flapping foil at a different time is slightly different. However, the velocity the time-varying flow, the vortex topology of the flapping foil at a different time is slightly different. distribution near the foil can also be observed. The velocity of the leading edge of the flapping foil However, the velocity distribution near the foil can also be observed. The velocity of the leading edge at t = 16.35T is slightly greater than t = 15.35T and 17.35T, as shown in Figure 13. According to the of the flapping foil at 𝑡 = 16.35𝑇 is slightly greater than 𝑡 = 15.35𝑇 and 17.35𝑇, as shown in Figure Bernoulli equation, the intensity of the negative pressure region generated by the leading edge of the 13. According to the Bernoulli equation, the intensity of the negative pressure region generated by lower surface of the flapping foil at t = 16.35T is greater than t = 15.35T and 17.35T, as shown in the leading edge of the lower surface of the flapping foil at 𝑡 = 16.35𝑇 is greater than 𝑡 = 15.35𝑇 Figure 14. The strong negative pressure region at the leading edge of the flapping foil increases the and 17.35𝑇, as shown in Figure 14. The strong negative pressure region at the leading edge of the thrust. Referring to Figure 10, we note that the maximum value of the instantaneous thrust coefficient flapping foil increases the thrust. Referring to Figure 10, we note that the maximum value of the at t = 16.35T. We can conclude that the time-varying flow affects the pressure distribution of the instantaneous thrust coefficient at 𝑡 = 16.35𝑇. We can conclude that the time-varying flow affects the flapping foil. However, its impact on the propulsion performance of the flapping foil is also related to pressure distribution of the flapping foil. However, its impact on the propulsion performance of the the relative position between the negative pressure area and the foil. flapping foil is also related to the relative position between the negative pressure area and the foil.

(a) 𝑡 = 15.35𝑇

Appl. Sci. 2020, 10, x (b) 𝑡 = 16.35𝑇 16 of 18

(c) 𝑡 = 17.35𝑇 ∗ ° 𝑓 =0.68 𝜃 =60 𝜏=◦ FigureFigure 13. 13. TheThe snapshots snapshots of the of thevorticity vorticity contours contours and andpressure pressure contours contours for for f ∗ = ,0.68 , θ0 =, 60 , 0.2τ ,= (a0.2,) 𝑡 (a) = t 15.35𝑇= 15.35, (b)T ,(𝑡b) =t =16.35𝑇16.35, (cT) ,(𝑡c) t == 17.35𝑇17.35. T.

∗ ° Figure 14. The pressure distribution of the flapping foil at different moments for 𝑓 =0.68, 𝜃 =60, 𝜏=0.8.

4. Conclusions In this paper, the propulsion performance and hydrodynamic characteristics of the full-active flapping foil in the time-varying freestream are systematically investigated via two-dimensional URANS and the finite volume method. We define the time-varying freestream as a superposition of steady flow and sinusoidal pulsating flow. Through numerical simulation results, we summarize several conclusions as follows: We first analyze the propulsion performance of the flapping foil at uniform freestream, and note that when the reduced frequency is small, the smaller pitching amplitude can obtain the higher propulsion performance of the full-active flapping foil. When the reduced frequency is large, the mean propulsion power coefficient increases first and then decreases with the variation of the pitching amplitude. We also reveal that with the increase in the pitching amplitude, the propulsion efficiency of the foil decreases monotonically. Then, we examine the influence of the time-varying freestream on the propulsion performance of the flapping foil for various oscillating amplitude and oscillating frequency coefficient, and find that the influence of the time-varying flow is time dependent. For one period, we notice that the oscillating amplitude does not affect the tendency of the time-varying flow on the propulsion performance of the flapping foil. With the increase in the oscillating amplitude, the variation amplitude of the mean propulsion power coefficient caused by the time-varying flow increases proportionally. We reveal that the effect of the time-varying flow on the flapping foil depends on the range of the oscillating frequency coefficient. When 𝜏<0.5, the mean propulsion power coefficient of the flapping foil fluctuates significantly, and when 𝜏>0.5, the mean propulsion power coefficient is almost unchanged with the variation of the period. In particular, when the oscillation frequency of the time-varying flow is an integer multiple of the reduced frequency, the time-varying flow has almost no effect on the flapping foil. Otherwise, we also note that the effect of the time-varying flow is related to the motion parameters of the flapping foil. The larger the motion parameters, the more significant the impact of propulsion performance of Appl. Sci. 2020, 10, x 16 of 18

(c) 𝑡 = 17.35𝑇 ∗ ° Figure 13. The snapshots of the vorticity contours and pressure contours for 𝑓 =0.68, 𝜃 =60,𝜏= Appl. Sci.0.22020, (a,)10 𝑡, 6226 = 15.35𝑇, (b) 𝑡 = 16.35𝑇, (c) 𝑡 = 17.35𝑇. 16 of 18

Figure 14. The pressure distribution of the flapping foil at different moments for f ∗= 0.68, θ = 60◦ , ° Figure 14. The pressure distribution of the flapping foil at different moments for 𝑓∗ =0.68, 0𝜃 =60, τ 𝜏=0.8= 0.8. . 4. Conclusions 4. Conclusions In this paper, the propulsion performance and hydrodynamic characteristics of the full-active In this paper, the propulsion performance and hydrodynamic characteristics of the full-active flapping foil in the time-varying freestream are systematically investigated via two-dimensional flapping foil in the time-varying freestream are systematically investigated via two-dimensional URANS and the finite volume method. We define the time-varying freestream as a superposition of URANS and the finite volume method. We define the time-varying freestream as a superposition of steady flow and sinusoidal pulsating flow. Through numerical simulation results, we summarize steady flow and sinusoidal pulsating flow. Through numerical simulation results, we summarize several conclusions as follows: We first analyze the propulsion performance of the flapping foil at several conclusions as follows: We first analyze the propulsion performance of the flapping foil at uniform freestream, and note that when the reduced frequency is small, the smaller pitching amplitude uniform freestream, and note that when the reduced frequency is small, the smaller pitching can obtain the higher propulsion performance of the full-active flapping foil. When the reduced amplitude can obtain the higher propulsion performance of the full-active flapping foil. When the frequency is large, the mean propulsion power coefficient increases first and then decreases with the reduced frequency is large, the mean propulsion power coefficient increases first and then decreases variation of the pitching amplitude. We also reveal that with the increase in the pitching amplitude, with the variation of the pitching amplitude. We also reveal that with the increase in the pitching the propulsion efficiency of the foil decreases monotonically. Then, we examine the influence of the amplitude, the propulsion efficiency of the foil decreases monotonically. Then, we examine the time-varying freestream on the propulsion performance of the flapping foil for various oscillating influence of the time-varying freestream on the propulsion performance of the flapping foil for amplitude and oscillating frequency coefficient, and find that the influence of the time-varying flow is various oscillating amplitude and oscillating frequency coefficient, and find that the influence of the time dependent. For one period, we notice that the oscillating amplitude does not affect the tendency time-varying flow is time dependent. For one period, we notice that the oscillating amplitude does of the time-varying flow on the propulsion performance of the flapping foil. With the increase in the not affect the tendency of the time-varying flow on the propulsion performance of the flapping foil. oscillating amplitude, the variation amplitude of the mean propulsion power coefficient caused by the With the increase in the oscillating amplitude, the variation amplitude of the mean propulsion power time-varying flow increases proportionally. We reveal that the effect of the time-varying flow on the coefficient caused by the time-varying flow increases proportionally. We reveal that the effect of the flapping foil depends on the range of the oscillating frequency coefficient. When τ < 0.5, the mean time-varying flow on the flapping foil depends on the range of the oscillating frequency coefficient. propulsion power coefficient of the flapping foil fluctuates significantly, and when τ > 0.5, the mean When 𝜏<0.5, the mean propulsion power coefficient of the flapping foil fluctuates significantly, and propulsion power coefficient is almost unchanged with the variation of the period. In particular, when 𝜏>0.5, the mean propulsion power coefficient is almost unchanged with the variation of the when the oscillation frequency of the time-varying flow is an integer multiple of the reduced frequency, period. In particular, when the oscillation frequency of the time-varying flow is an integer multiple the time-varying flow has almost no effect on the flapping foil. Otherwise, we also note that the of the reduced frequency, the time-varying flow has almost no effect on the flapping foil. Otherwise, effect of the time-varying flow is related to the motion parameters of the flapping foil. The larger we also note that the effect of the time-varying flow is related to the motion parameters of the flapping the motion parameters, the more significant the impact of propulsion performance of the flapping foil. The larger the motion parameters, the more significant the impact of propulsion performance of foil. For multiple periods, we note that the time-varying freestream has little effect on the propulsion performance of the full-active flapping foil at different pitching amplitude and reduced frequency. In conclusion, we reveal that the time-varying incoming flow has little effect on the flapping propulsion performance for multiple periods. Therefore, we can simplify the time-varying flow to a steady flow field to a certain extent for numerical simulation. Meanwhile, we also realize that only the small oscillating amplitude of the time-varying flow is considered in this paper. The impact of large oscillating amplitude has not been discussed. Otherwise, the hydrological conditions in the marine environment are more complicated; their impact on the propulsion performance of the full-active flapping foil cannot be predicted accurately. Appl. Sci. 2020, 10, 6226 17 of 18

Author Contributions: Conceptualization, Z.Q.; formal analysis, Z.Q. and L.J.; funding acquisition, L.J. and J.Z; investigation, Z.Q., Y.Q. and J.S.; methodology, Z.Q.; validation, Z.Q. and L.J.; writing—original draft, Z.Q.; writing—review and editing, J.Z. All authors have read and agreed to the published version of the manuscript. Funding: This study was funded by the Natural Science Foundation of Tianjin (grant number 18JCYBJC24000), the Key Laboratory of Submarine Geosciences, MNR (grant number KLSG1907) and the National Key Research and Development Project (grant number 2017YFC0305900). Acknowledgments: The authors would like to thank the anonymous reviewers for their constructive comments and suggestions. Conflicts of Interest: The authors declare no conflict of interest.

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