Investigation on Hydrodynamic Performance of Flapping Foil Interacting with Oncoming Von Kármán Wake of a D-Section Cylinder

Journal of Marine Science and Engineering

Article Investigation on Hydrodynamic Performance of Flapping Foil Interacting with Oncoming Von Kármán Wake of a D-Section Cylinder

Jian Li 1,* , Peng Wang 1,*, Xiaoyi An 1, Da Lyu 1 , Ruixuan He 1 and Baoshou Zhang 2

1 School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China; [email protected] (X.A.); [email protected] (D.L.); [email protected] (R.H.) 2 School of Aerospace Engineering, Tsinghua University, Beijing 100084, China; [email protected] * Correspondence: [email protected] (J.L.); [email protected] (P.W.)

Abstract: Flapping foils are studied to achieve an efficient propeller. The performance of the flapping foil is influenced by many factors such as oncoming vortices, heaving amplitude, and geometrical parameters. In this paper, investigations are performed on flapping foils to assess its performance in the wake of a D-section cylinder located half a diameter in front of the foil. The effects of heaving amplitude and foil thickness are examined. The results indicate that oncoming vortices facilitate the flapping motion. Although the thrust increases with the increasing heaving amplitude, the propelling efficiency decreases with it. Moreover, increasing thickness results in higher efficiency. The highest propelling efficiency is achieved when the heaving amplitude equals ten percent of the chord length with a symmetric foil type of NACA0050 foil. When the heaving amplitude is small, the influence of the thickness tends to be more remarkable. The propelling efficiency exceeds 100% and the heaving amplitude is 10% of the chord length when the commonly used equation is adopted. This result demonstrates that the flapping motion extracts some energy from the oncoming vortices. Citation: Li, J.; Wang, P.; An, X.; Lyu, Based on the numerical results, a new parameter, the energy transforming ratio (RET), is applied to D.; He, R.; Zhang, B. Investigation on explicate the energy transforming procedure. The R represents that the flapping foil is driven by Hydrodynamic Performance of ET Flapping Foil Interacting with the engine or both the engines and the oncoming vortices with the range of RET being (0, Infini) and Oncoming Von Kármán Wake of a (−1, 0), respectively. With what has been discussed in this paper, the oncoming wake of the D-section D-Section Cylinder. J. Mar. Sci. Eng. cylinder benefits the flapping motion which indicates that the macro underwater vehicle performs 2021, 9, 658. https://doi.org/ better following a bluff body. 10.3390/jmse9060658 Keywords: flapping foil; heaving amplitude; foil thickness; von Kármán wake; energy Received: 7 April 2021 transforming ratio Accepted: 11 June 2021 Published: 14 June 2021

Publisher’s Note: MDPI stays neutral 1. Introduction with regard to jurisdictional claims in By mimicking ocean creatures, researchers promote a myriad of new outstanding published maps and institutional affil- underwater vehicles. The flapping motion is believed to be an efficient self-propelling or iations. energy extracting method. The fluid dynamics of flapping foils have been concentrated on for several decades, especially on its propulsion performance, with a focus on trying to find a pitching strategy for maximizing the thrust [1] and to explain the relationship between the geometry and the spanwise components of the locally shed vortices [2] or Copyright: © 2021 by the authors. to find a semi-active flapping motion to improve the performance [3]. Factors that may Licensee MDPI, Basel, Switzerland. produce effects on the hydrodynamic performance have been studied since early periods. This article is an open access article It has been proven that a combined oscillating foil performs better than a pure heaving or a distributed under the terms and pure pitching foil in most conditions [4,5]. Moreover, the phase difference and pivot points conditions of the Creative Commons Attribution (CC BY) license (https:// do have a perfect value, which we applied here, for more efficient propulsion described creativecommons.org/licenses/by/ in the work of Lin and Wu [6]. The Reynolds number, foil thickness, and camber were 4.0/). found to play a certain part by working on changing the leading-edge vortex [7]. These

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studies indicate that the formation and evolution of the leading-edge vortex(LEV) and trailing edge vortex(TEV) play a significant role in improving the performance. Thus, the performance of the flapping foil is connected with the factors that may affect the LEV and TEV [8–10]. It is known that the environment of the flowing water is always complex, which includes various vortices and fluctuations, and the shedding of LEV and TEV are consequently affected. Thus, hydrodynamic performance could be different. The interaction of the coming wake and the fixed foil was studied, with a focus concerning the vortex structure and force characteristics [11]. A series of numerical works for dual-foil turbines [12–14] or propulsion wing [15–17] are carried out. However, the relation between the oncoming vortices and the improvement of their performance is not clear. A two-dimensional inviscid analysis was carried out, which indicates that the phase between foil motion and the arrival of oncoming vortices is a critical parameter, and the highest efficiency is observed when the phase is such that the foil moves in close proximity with respect to the oncoming vortices [18]. The encounter between oncoming inverted von Kármán vortices and the free-pitching foil was studied by Bao [19], which indicated that the interaction is positive for thrust efficiency. An excellent investigation was performed to reveal the passive propulsion in vortex wake (a von Kármán wake) via an experiment method [20] and it was shown that there was a suction area in a cylinder wake that could drive the dead fish moving forward. Recently, the drag of a NACA0012 is found negative at two and three diameters downstream of the cylinder due to large regions of flow recirculation and the formation of an LEV, respectively [21]. It can be inferred that the oncoming wake can be positive for the flapping foil. However, most of the works mentioned above are about the foil flapping in the inverted von Kármán wake or a fixed foil in the von Kármán wake. The present work focuses mainly on the questions: Whether the oncoming vortices are positive with respect to the thrust generation for a foil oscillating near the D-section cylinder (a gap of half diameter of the cylinder); how does the motion parameters and foil geometry affect the hydrodynamic performance? How does the confluent wake change? The present paper details a series of numerical simulations performed with various foil sections flapping in the wake of a D-section cylinder. The performance of the foil flapping in the open flow and the characteristic of the wake of the D-section cylinder are studied first. Then, the effects of the heaving amplitude and foil thickness are discussed. Finally, the energy transformation ratio is elaborated to explicate the energy transforming procedure, while, according to the results, the commonly used formula of propelling efficiency could not describe this interaction well when considering the oncoming wake.

2. Computational Approach 2.1. Problem Formulation In this study, the two-dimensional flow fields of a flapping foil in the wake of a fixed D-section cylinder arranged tandemly are numerically studied. The flapping foil oscillates in the wake of the D-section cylinder and the leading edge vortex interacts with the oncoming wake. The shedding vortices of the D-section cylinder are disturbed by the foil and its LEV. Based on previous research, the foil and the cylinder are the same scales, as shown in Figure1a, where U0 is the velocity of the incoming flow and it is adopted as the characteristic velocity. D is the diameter of the cylinder and L is the distance between the D-section cylinder and the flapping foil. One of the main influences of the oncoming vortices on the flapping foil is when the vortices encounter the foil, which is believed that the different heaving amplitudes are quite outstanding when flapping in the wake. Thus, the parameter L is not discussed. Further work will be conducted on the length scale. Here, the length scale L is set as L = D/2 and the foil chord length is C where C = D. Amax is the amplitude of the heaving motion. The foil chord length is selected as the reference length. The Reynolds number can be defined as Re = U0C/υ. Here, υ represents the kinematic viscosity of the fluid. In this article, the Reynolds number is fixed as J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 3 of 16

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number can be defined as Re = U0C/υ. Here, υ represents the kinematic viscosity of the fluid. In this article, the Reynolds number is fixed as Re = 2 × 104, which is identical to the 4 Renumber = 2 × 10adopted, which in isYoung identical et al. to simulation the number [7 adopted]. As for inthe Young coordinate et al. simulationsystem, the [7 origin]. As forpoint the is coordinate set behind system, the center the of origin the cylinder point is for set a behind length the of L center. The positive of the cylinder direction fors aof lengththe abscissa of L. The and positive ordinate directions axis are ofshown the abscissa in Figure and 1a ordinate. axis are shown in Figure1a.

(a) Sketch map for vortices interaction. (b) Flapping motion.

FigureFigure 1. 1.Diagrammatic Diagrammatic sketch sketch for for the the ﬂapping flapping foil. foil.

BasedBased on on the the reference reference length, length, the the non-dimensional non-dimensional frequency frequency can can be definedbe defined with with the followingthe following equation. equation. f C Sr = fC (1) Sr U0 (1) U The oscillating motion of the foil has been simplified0 by researchers from the behavior of marineThe oscillating organisms. motion One of of the the flapping foil has modes been simplified discussed byby DUresearchers [22], which from is confirmedthe behav- thatior of it ismarine a propeller organisms. motion One when of the Re =flapping 1100 as modes shown discussed in Figure1 byb, isDU applied [22], which here with is con- a higherfirmed Re that which it is a may propeller turn the motion motion when into Re a drag = 1100 one. as The shown formula in Figure can be 1b described, is applied as here the following:with a higher Re which may turn the motion into a drag one. The formula can be described Heaving : y(t) = A sin(2π f t) as the following: max (2) Pitching : θ(t) = θmax sin(2π f t + φ) Heaving: y t  Amax sin 2 ft where f is the frequency and T = 1/f. φ is the phase difference between the heaving motion(2) and pitching motion, which is fixedPitching as φ:=π t/2  accordingmax sin 2  ftto previous  studies. Meanwhile, the amplitude of pitching motion θmax equals to π/12. The pitch center is fixed at the where f is the frequency and T = 1/f.  is the phase difference between the heaving mo- position of Or, as shown Figure1b, which is C/3 from the leading edge on the chord. tion Withand pitching the known motion motion, which of flapping is fixed as foils shown = π/2 according in Equation to (2),previous the velocity studies. and Mean- the

angularwhile, the velocity amplitude can be of expressed pitching motion as follows: max equals to π/12. The pitch center is fixed at the position of Or, as shown Figure 1b, which is C/3 from the leading edge on the chord. v(t) = 2π f A cos(2π f t) With the known motion of flapping foilsmax shown in Equation (2), the velocity and (3)the (t) = f ( f t + ) angular velocity can be expressedω as2 followsπ θmax:cos 2π φ

where v(t) and ω(t) represent velocity and angular velocity, respectively. Symbols CT, v t  2 fAmax cos 2 ft CL and CM [23] are adapted to donate the instantaneous thrust coefficient, lift coefficient,(3) Fx and moment coefficient, respectively.t 2  The f max coefficients cos 2  ft are  as follows: CT = 2 , 0.5ρU0CS Fy M CL = 2 , CM = 2 2 and Fx, Fy, M, and S stands for the thrust, lift, torque, and where0.5 ρvtU0CS and 0.5tρU 0representC S velocity and angular velocity, respectively. Symbols foil span, respectively. , and C [23] are adapted to donate the instantaneous thrust coefficient, lift co- CT InC orderL to studyM the performance of flapping foils, the consuming power or the powerefficient, of energy and collectingmoment is coefficient, studied with respectively. the non-dimensional The coefficients coefficients. are Therefore, as follows: the time-meanF thrustx coefficientFy and the time-meanM power coefficient are described as follows: CCCTLM,,   and Fx, Fy, M, and S stands for the 0.5U2 CS 0.5  U 2 CS 0.5  U 2 C 2 S 0 01 R t0+nT 0 CTmean = CTdt thrust, lift, torque, and foil span,nT trespec0 tively. (4) 1 R t0+nT CPmean = (CL(t)v(t) + CMω(t)S)dt/U In order to study the performancenT t0 of flapping foils, the consuming0 power or the power of energy collecting is studied with the non-dimensional coefficients. Therefore, where n = 3 represents the cycles used to calculate the time-mean coefficient.

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The propulsive efﬁciency of the ﬂapping foil is described as follows.

CTmean ηT = (5) CPmean

2.2. Numerical Method The unsteady flow field around the cylinder and flapping foil, which is considered viscosity and incompressible, is simulated with an unsteady incompressible solver using the commercially available CFD code (ANSYS Fluent). Therefore, the continuity and momentum equations are described as follows:

∂ρ ∂ + (ρui) = 0 (6) ∂t ∂xi ! ∂(ρui) ∂ ∂pr ∂ ∂ui 0 0 + ρuiuj = − + µ − ρu iu j + Si (7) ∂t ∂xj ∂xi ∂xj ∂xj 0 where i, j = 1,2, ρ = constant, and ui uj are the transient velocity components, while u i and 0 u j are the fluctuating velocity components. The component pr is the transient pressure 0 0 0 0 and u iu j is the mean value of u iu j. The sum of all other force components that do not belong to the transient term is Si. Moreover, the subscripts i, j = 1,2 in Equations (6) and (7) identify the components along the X-axis and the Y-axis. The finite-volume method is applied to achieve the spatial discretization of the Navier– Stokes equations (Equations (6) and (7)) and temporal discretization is performed using the second-order accuracy backward-implicit scheme. The k-ω SST turbulence model includes the transverse dissipative derivative term, which has been commonly tested and believed to be suitable for the turbulence and shear flow, used with wall restriction, which J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 5 of 16 is employed here to solve the unsteady flow. The method is validated by Ashraf et al. [7] and the results agree very well with the published results shown in Figure2, where k = Srπ/C.

6 Figure 2. Variation of CTmean against k for a plunging NACA0012 airfoil h = 0.4C, Re = 3 ×6 10 , Figure 2. Variation of CTmean against k for a plunging NACA0012 airfoil h = 0.4C, Re = 3 × 10 , adapted from (Ashraf, et al. 2011), Copyright © 2010 Elsevier Ltd. adapted from (Ashraf, et al. 2011), Copyright © 2010 Elsevier Ltd. 3. Validation of the Numerical Method 3. Validation of the Numerical Method As described in the numerical study of Ashraf et al. [7], the turbulence model has been validatedAs described with quitein the an numerical accuracy study shown of in Ashraf Figure 2et. al. [7], the turbulence model has been validatedSix different with quite mesh an sizes accuracy (with shown different in Figure heights 2 of. the ﬁrst cell layer [24] are tested. TheSix inletdifferent velocity mesh is sizes 0.292 (with m/s, different the movement heights formulaof the first is thecell samelayer with[24] are Equation tested. (2), Theand inlet the velocity amplitude is 0.292 of them/s, pitching the movement motion isformulaπ/6. The is the results same are with listed Equation in Table (2),1, and where

the amplitude= CTmean ( ofi+ 1) the−C Tmean pitching(i) × motion is π/6. The results are listed in Table 1, where ∆ C (i) 100%. The result converges and the result of case 5 is close C i1Tmean C i  enoughTmean to case 6, Tmean which suggests 100% . that The the result method converges and size and of the case result 6 satisﬁes of case the 5 calculationis close of the presentCiTmean numerical  simulation. enough to case 6, which suggests that the method and size of case 6 satisfies the calculation of the present numerical simulation.

Table 1. Mesh validation for different sizes.

Number 1 2 3 4 5 6 Y+ 5 2 1 0.9 0.85 0.8 Size 150,000 230,000 270,000 360,000 380,000 410,000

CTmean 52.0207 52.5885 52.9898 53.0229 53.0165 53.0163 Δ 1.09% 0.76% 0.06% 0.01% 0.00004%

In order to confirm the accuracy of this mesh method with the k-ω SST turbulence model, it is tested again to compare with the experiment and linear and nonlinear theory results [25]. Figure 3 provides the thrust coefficient and power coefficient as a function of Sr for the angle of attack at 15°, the amplitude of heaving motion Amax/C = 0.75, and with  = π/2. The agreement between theory, experiment, and present simulation is fair overall.

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Table 1. Mesh validation for different sizes.

Number 1 2 3 4 5 6 Y+ 5 2 1 0.9 0.85 0.8 Size 150,000 230,000 270,000 360,000 380,000 410,000 CTmean 52.0207 52.5885 52.9898 53.0229 53.0165 53.0163 ∆ 1.09% 0.76% 0.06% 0.01% 0.00004%

In order to confirm the accuracy of this mesh method with the k-ω SST turbulence model, it is tested again to compare with the experiment and linear and nonlinear theory J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEWresults [25]. Figure3 provides the thrust coefficient and power coefficient as a function6 of of16 ◦ Sr for the angle of attack at 15 , the amplitude of heaving motion Amax/C = 0.75, and with φ = π/2. The agreement between theory, experiment, and present simulation is fair overall.

(a) Time-mean thrust coefficient as a function of Sr, for the an- (b) Power coefficient as a function of Sr, for the angle of attack gle of attack 15°, Amax/C = 0.75, and Re = 3.7 × 104 15°, Amax/C = 0.75, and Re = 3.7 × 104

Figure 3. ValidationFigure of 3. gridValidation strategy, of gridcompared strategy, with compared linear, nonlinear, with linear, and nonlinear, experiment and result experiments. results. The computational domain is considered as a square area in this study. The inlet The computational domain is considered as a square area in this study. The inlet ve- velocity boundary is at a distance of 20C from the pitching center of the foil, as u = (U,0) locity boundary is at a distance of 20C from the pitching center of the foil, as u = (U,0) and ∂p = andp ∂x 0. As for the no-stress outflow boundary condition, u = (0,0) and p = 0 are  0 . As for the no-stress outflow boundary condition, u = (0,0) and p = 0 are used and usedx and the downstream pressure outlet is at a distance of 40C from the pitching center theof thedownstream foil. The pressure slip wall outlet condition is at a is distance used for of the40Cupper from the and pitching lower flowcenter boundaries of the foil. Thelocated slip atwall a distance condition of 30is Cusedfrom for the the central upper line and of lower the cylinder. flow boundaries With the chosenlocated meshingat a dis- + tancestrategy of 30 caseC from 6, where the centraly = 0.8, line the of first the meshcylinder. layer With could the be chosen estimated. meshing The strategy flow field case is 6,divided where intoy+ = three0.8, the zones. first Themesh pitching layer could foil is be surrounded estimated. by The structured flow field grids is divided (the orange into threecolor) zones and the. The up-down pitching zonefoil is (gray surrounded color) andby structured the fixed zonegrids (blue (the orange and green color) color) and arethe both constructed using a structured mesh, which is shown in Figure4. Moreover, the up-down zone (gray color) and the fixed zone (blue and green color) are both constructed DEFINE_CG_MOTION function and User Defined Function(UDF) [26] are used to handle using a structured mesh, which is shown in Figure 4. Moreover, the DEFINE_CG_MO- the prescribed motion equations (Equation (2)) and regenerate the interface boundaries TION function and User Defined Function(UDF)[26] are used to handle the prescribed between the pitching zone and up-down zones. The formula presented by Kinsey and motion equations (Equation (2)) and regenerate the interface boundaries between the Dumas [27] is adopted to determine the time steps of the simulation and the formula is pitching zone and up-downn zones. oThe formula presented by Kinsey and Dumas [27] is = T C/kvk adoptedexpressed to asdetermine∆t min the2000 time, 100 steps, of where the simulationT is the oscillation and the periodformula and is expressed ||v|| is the as maximum instantaneous convective flux velocity in the computational domain. T Cv t min , , where T is the oscillation period and ||v|| is the maximum in- 2000 100 stantaneous convective flux velocity in the computational domain.

(a) Computational domain. (b) Mesh details of Section 1. Figure 4. Computational domain and mesh details.

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Figure 3. Validation of grid strategy, compared with linear, nonlinear, and experiment results.

The computational domain is considered as a square area in this study. The inlet velocity boundary is at a distance of 20C from the pitching center of the foil, as u = (U,0) and p  0 . As for the no-stress outflow boundary condition, u = (0,0) and p = 0 are used and x the downstream pressure outlet is at a distance of 40C from the pitching center of the foil. The slip wall condition is used for the upper and lower flow boundaries located at a distance of 30C from the central line of the cylinder. With the chosen meshing strategy case 6, where y+ = 0.8, the first mesh layer could be estimated. The flow field is divided into three zones. The pitching foil is surrounded by structured grids (the orange color) and the up-down zone (gray color) and the fixed zone (blue and green color) are both constructed using a structured mesh, which is shown in Figure 4. Moreover, the DEFINE_CG_MO- TION function and User Defined Function(UDF)[26] are used to handle the prescribed motion equations (Equation (2)) and regenerate the interface boundaries between the pitching zone and up-down zones. The formula presented by Kinsey and Dumas [27] is adopted to determine the time steps of the simulation and the formula is expressed as T Cv J. Mar. Sci. Eng. 2021, 9, 658 t min , , where T is the oscillation period and ||v|| is the maximum in-6 of 15 2000 100 stantaneous convective flux velocity in the computational domain.

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 8 of 16 (a) Computational domain. (b) Mesh details of Section 1.

FigureFigure 4. Computational 4. Computational domain domain and m andesh meshdetails. details.

4.4. ResultsResults andand DiscussionDiscussion InIn thisthis section,section, numerical results results are are presented presented and and discussed discussed concerning concerning the the perfor- per- formancemance of ofhydrofoils hydrofoils with with oscillatory oscillatory motion motion in inthe the wake wake of ofa D a- D-sectionsection cylinder. cylinder. The The re-

resultssults of of different different foil foil thickness thickness investigations investigations are are obtained as well asas thethe tendenciestendencies ofof consumingconsuming powerpower andand thrust.thrust. SimulationsSimulations areare carriedcarried outout withwith thethe ﬁxedfixed frequencyfrequency andand differentdifferent heaving heaving amplitudes. amplitudes. Hydrodynamic Hydrodynamic performance performance parameters parameters and and ﬂow flow structures struc- aretures studied. are studied.

4.1.4.1. TheThe OriginalOriginal FlappingFlapping FoilFoil MotionMotion andand FlowFlow FieldField ofofthe theD-Section D-Section CylinderCylinder TheThe flappingflapping foilfoil is described as as a a propulsion propulsion device device by by DU DU at atthe the condition condition that that Re Re= 1100. = 1100. However, However, we we are are concern concerneded with with the the situation situation declared declared by by M.A. Ashraf et et al., al., wherewhere thethe ReRe == 2 × 10104.4 The. The non non-dimensional-dimensional frequency frequency of ofthe the flapping flapping foil foil is fixed is fixed at S atr = Sr0.3. = 0.3.The Thetotal total flapping flapping cycle cycle and andthe vorticity the vorticity magnitude magnitude are shown are shown in Figure in Figure 5. 5.

Figure 5. Vorticity magnitude for a total flapping cycle of NACA0012 when Sr = 0.3 and Re = 2 × 104. Figure 5. Vorticity magnitude for a total flapping cycle of NACA0012 when Sr = 0.3 and Re = 2 × 104. The vortices shedding from the leading edge pair up and the vortices shedding from The vortices shedding from the leading edge pair up and the vortices shedding from the trailing edge form an inverted von Kármán wake (IVKW). According to Zhang’s the trailing edge form an inverted von Kármán wake (IVKW). According to Zhang’s con- conclusion, the thrust wake is most likely an IVKW, but an IVKW is not necessarily a thrust clusion, the thrust wake is most likely an IVKW, but an IVKW is not necessarily a thrust wake. This motion adapted by DU et al. is confirmed as a propelling motion with Re = 1100. However,wake. This when motion Re =adapted 2 × 104 byis appliedDU et al. here, is confirmed this motion as provides a propelling no thrust. motion The with instant Re = 4 thrust1100. coefficientHowever, when and time-mean Re = 2 × 10 thrust is applied coefficient here, in this two motion flapping provides periodic no processes thrust. areThe showninstant in thrust Figure coefficient6. and time-mean thrust coefficient in two flapping periodic pro- cessesAccording are shown to Figuresin Figure5 and 6. 6, the thrust generated by the foil lasts a short period time and drag dominates the horizontal force. The process of the flapping foil for the second half of the0.05 flapping cycle is symmetric with the first four positions. It indicates the same horizontal force peak in the second half퐶푇 of the flapping cycle. This phenomenon supports the view-0.15 that the growth and dissipation of the LEV and TEV play an important role in affecting-0.35 the performance of the flapping foil. The time-mean thrust is negative, which means that this kind of motion does푪푇푚푒푎푛 not provide thrust under the given Reynolds number. -0.55 -0.75

Thurst coefficient Thurst -0.95 -1.15 2.169 2.219 2.269Time/s2.319 2.369

Figure 6. Instant thrust coefficient and time-mean thrust coefficient in two flapping cycles.

According to Figures 5 and 6, the thrust generated by the foil lasts a short period time and drag dominates the horizontal force. The process of the flapping foil for the second half of the flapping cycle is symmetric with the first four positions. It indicates the same horizontal force peak in the second half of the flapping cycle. This phenomenon supports the view that the growth and dissipation of the LEV and TEV play an important role in

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4. Results and Discussion In this section, numerical results are presented and discussed concerning the performance of hydrofoils with oscillatory motion in the wake of a D-section cylinder. The results of different foil thickness investigations are obtained as well as the tendencies of consuming power and thrust. Simulations are carried out with the fixed frequency and different heaving amplitudes. Hydrodynamic performance parameters and flow structures are studied.

4.1. The Original Flapping Foil Motion and Flow Field of the D-Section Cylinder The flapping foil is described as a propulsion device by DU at the condition that Re = 1100. However, we are concerned with the situation declared by M.A. Ashraf et al., where the Re = 2 × 104. The non-dimensional frequency of the flapping foil is fixed at Sr = 0.3. The total flapping cycle and the vorticity magnitude are shown in Figure 5.

Figure 5. Vorticity magnitude for a total flapping cycle of NACA0012 when Sr = 0.3 and Re = 2 × 104.

The vortices shedding from the leading edge pair up and the vortices shedding from the trailing edge form an inverted von Kármán wake (IVKW). According to Zhang’s conclusion, the thrust wake is most likely an IVKW, but an IVKW is not necessarily a thrust wake. This motion adapted by DU et al. is confirmed as a propelling motion with Re = 4 J. Mar. Sci. Eng. 2021, 9, 658 1100. However, when Re = 2 × 10 is applied here, this motion provides no thrust. The 7 of 15 instant thrust coefficient and time-mean thrust coefficient in two flapping periodic processes are shown in Figure 6.

0.05 퐶푇 -0.15 -0.35 푪푇푚푒푎푛 -0.55

-0.75 Thurst coefficient Thurst J. Mar. Sci. Eng. 2021, 9, x FOR PEER -0.95REVIEW 9 of 16

-1.15 J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 9 of 16 2.169 2.219 2.269Time/s2.319 2.369

affecting the performance of the flapping foil. The time-mean thrust is negative, which Figure 6. Instant thrust coefficient and time-mean thrust coefficient in two flapping cycles. Figuremeans 6. Instant that this thrust kind coefficient of motion anddoes time-mean not provide thrust thrust coefficient under the in given two flappingReynolds cycles. number. affecting the performance of the flapping foil. The time-mean thrust is negative, which The oncoming wake is typically a von Kármán wake (VKW) as shown in Figure 7a AccordingmeansThe that oncoming to this Figure kinds wake5 ofand motion 6, is the typically thrustdoes not generated aprovide von Kby áthrust rmthe áfoiln under wakelasts thea short (VKW) given period Reynolds as showntime number in Figure. 7a and dragand dominates the vortices the shedding horizontal from force D. The-section process cylinder of the decay flapping while foil proceeding for the second downstream. and theThe vortices oncoming shedding wake is from typically D-section a von cylinderKármán wake decay (VKW) while as proceeding shown in Figure downstream. 7a half ofand the the flapping vortices cycle shedding is symmetric from withD-section the first cylinder four positions. decay while It indicates proceeding the same downstream. horizontal force peak in the second half of the flapping cycle. This phenomenon supports the view that the growth and dissipation of the LEV and TEV play an important role in

(a) Von Karman wake of the D-section cylinder. (b) Instant drag and lift coefficients. (a) Von KarmanFigureFigure wake 7. Wake 7. of Wake the of D theof- sectionthe D-section D-section cylinder cylinder cylinder. andand(b the ) Instant drag drag and anddrag lift lift andcoefficients. coefﬁcients. lift coefficients .

Figure The7. WakeThe instant instant of the drag D drag-section coefficient coefficient cylinder andand the lift drag coefficient coefficient and lift ofcoefficients. of the the D- D-sectionsection cylinder cylinder are shown are shown in in Figure 7b. Both the curves exhibit very good periodicity and the period of the lift variation Figure7Tb.he Both instant the drag curves coefficient exhibit and very lift good coefficient periodicity of the D and-section the periodcylinder of are the shown lift variation in is twice of that with respect to the drag variation. is twiceFigure of 7b that. Both with the curves respect exhibit to the very drag good variation. periodicity and the period of the lift variation The Fast Fourier Transformation (FFT) is applied for the vortex shedding frequency is twiceTheFast of th Fourierat with respect Transformation to the drag (FFT)variation. is applied for the vortex shedding frequency as shown in Figure 8. The main frequency of the forces represents the vortices shedding as shownThe inFast Figure Fourier8. TheTransformation main frequency (FFT) is of applied the forces for the represents vortex shedding the vortices frequency shedding frequency. The variation frequency of the lift coefficient equals exactly the frequency of frequency.as shown Thein Figure variation 8. The frequency main frequency of the of lift the coefficient forces represents equals the exactly vortices the shedding frequency of vortices shedding due to the vortices of the D-section cylinder shedding at the top or bot- frequency. The variation frequency of the lift coefficient equals exactly the frequency of vorticestom of shedding the cylinder due, which to the results vortices in a pressure of the D-section drop with cylinderrespect to sheddingit. The frequency at the of top or bottomvortices of theshedding cylinder, due whichto the vortices results of in the a pressure D-section drop cylinder with shedding respect toat it.the The top frequencyor bot- of the lift coefficient (marked as fL) is fL = 4.96 Hz, which indicates that the non-dimensional tom of the cylinder, which results in a pressure drop with respect to it. The frequency of the lift coefficient (marked as fL) is fL = 4.96 Hz, which indicates that the non-dimensional fDL thevortex lift coefficient shedding frequency(marked as of fL the) is DfL -=section 4.96 Hz, cylinder which isindicates Sr = fthatD =the 0.17. non -dimensional vortex shedding frequency of the D-section cylinder is Sr =U L = 0.17. fDU0 0 vortex shedding frequency of the D-section cylinder is Sr = L = 0.17. U0

(a) The frequency spectrum of the drag coefficient . (b) The frequency spectrum of the lift coefficient . (a) The frequency FigurespectrumFigure 8. Frequency of8. Frequencythe drag spectrum coefficient spectrum of of. the the dragdrag(b) The coefﬁcientcoefficient frequency and and liftsp lift ectrumcoefficient. coefﬁcient. of the lift coefficient.

Figure4.2. 8. Effect Frequency of Heaving spectrum Amplitude of the drag and coefficientThickness and lift coefficient.

4.2. EffectThe offlapping Heaving foil Amplitude is placed and in Thickness the wake generated by the D-section cylinder, with a gap of 0.5D. The vortices shedding from the flapping foil will interfere with the on-coming The flapping foil is placed in the wake generated by the D-section cylinder, with a VKW, which keeps growing. The interaction is mainly dependent on the heaving ampli- gap of 0.5D. The vortices shedding from the flapping foil will interfere with the on-coming tude. Here, the flapping motion is limited with a flapping frequency of Sr = 0.3. The hy- VKW, which keeps growing. The interaction is mainly dependent on the heaving ampli- drodynamic performance of the flapping foil is described by the time-mean thrust coeffi- tude. Here, the flapping motion is limited with a flapping frequency of Sr = 0.3. The hy- cient and the energy-consuming power coefficient. The two parameters changing with drodynamic performance of the flapping foil is described by the time-mean thrust coefficient and the energy-consuming power coefficient. The two parameters changing with

J. Mar. Sci. Eng. 2021, 9, 658 8 of 15

4.2. Effect of Heaving Amplitude and Thickness The ﬂapping foil is placed in the wake generated by the D-section cylinder, with a gap of 0.5D. The vortices shedding from the ﬂapping foil will interfere with the on- J. Mar. Sci. Eng. 2021, 9, x FOR PEERcoming REVIEW VKW, which keeps growing. The interaction is mainly dependent on the heaving10 of 16

amplitude. Here, the flapping motion is limited with a flapping frequency of Sr = 0.3. The hydrodynamic performance of the flapping foil is described by the time-mean thrust coefficient and the energy-consuming power coefficient. The two parameters changing respect to the heaving amplitude and foil thickness are shown in Figure 9 with 3D bar with respect to the heaving amplitude and foil thickness are shown in Figure9 with 3D graphs. bar graphs.

(a) Thrust coefficient with different amplitude. (b) Input power with different amplitude.

FigureFigure 9.9. Time-meanTime-mean thrustthrust coefﬁcientcoefficient andand energy-consumingenergy-consuming powerpower coefﬁcientcoefficient withwith variousvarious heavingheaving amplitudeamplitude andand × 4 thicknessthickness with with Sr Sr = = 0.3,0.3, ReRe == 22 × 104..

TheThe time-meantime-mean thrust increases significantly significantly with with the the heaving heaving amplitude amplitude for for cases cases of ofNACA0030, NACA0030, NACA0040, NACA0040, and and NACA0050 NACA0050 when when AmaxA >max 0.5>C, 0.5 whileC, while the NACA0020 the NACA0020 cases casesremain remain stable. stable. As for As cases for cases of NACA0012 of NACA0012 and andNACA0015, NACA0015, the thethrust thrust decreases decreases and and be- becomescomes negative, negative, which which means means that that the the flapping flapping foil foil produces produces drag and the dragdrag increasesincreases withwith thethe heavingheaving amplitude amplitude when whenA Amaxmax >> 0.5C.. TheThe thrustthrust coefficientcoefficient remainsremains steadysteady whenwhen thethe amplitudeamplitude isis lessless thanthan 0.50.5CC.. AsAs forfor thethe effectseffects ofof thethe foilfoil thicknessthickness onon thethe parameters,parameters, time-meantime-mean thrustthrust increases increases with with the the thickness thickness for for all all tested tested flapping flapping frequencies, frequencies, while while the inputthe input power power decreases. decreases. This indicatesThis indicates that the that thicker the thicker foil performs foil performs better than better the than thinner the foil,thinner as shown foil, as in shown Figure in9. Figure 9. WithWith the the definition definition of of the the propelling propelling efficiency efficiency in Equation in Equation (5), the(5), thrustthe thrust time’s time velocity’s ve- islocity higher is higher than the than energy-consuming the energy-consuming power, power which, which results results in the in propelling the propelling efficiency effi- exceedingciency exceeding 1, as shown 1, as inshown Figure in 10 Figure. This 10. ‘bizarre’ This ‘bizarre’ phenomenon phenomenon is caused is bycaused the suction by the areasuction behind area thebehind D-section the D cylinder.-section cylinder. The results The indicate results indicate that the movingthat the forwardmoving forward motion requiresmotion requires less energy less to energy maintain to maintain the moving the velocity moving when velocity the flappingwhen the propulsion flapping propul- device ission involved device inis theinvolved suction in areathe suction of a von area Ká rmof áan von wake Kármán with awake small with heaving a small amplitude. heaving Moreover,amplitude. it Moreover seems that, it the seems original that equationthe original used equation to calculate used the to calculate propelling the efficiency propelling is notefficiency proper is for not the proper flapping for motionthe flapping when motion tracking when a close tracking target. a close target. As shown in Figure 11, the vortex wake is a regular structure when the foil flaps in the wake with Amax < 0.5C. A pair of vortices are formed behind the foil during a flapping process. When the Amax > 0.5C, the vortex shedding from the foil will merge with the oncoming wake and will be disturbed by the foil. Consequently, a pair and one single vortex are formed, shown in Figure 11d. LEV is inhibited, which results in a low-pressure area in front of the foil and the thrust is enlarged when the amplitude increases.

Figure 10. Propelling efficiency changing with Amax with Sr = 0.3 and Re = 2 × 104.

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 10 of 16

respect to the heaving amplitude and foil thickness are shown in Figure 9 with 3D bar graphs.

(a) Thrust coefficient with different amplitude. (b) Input power with different amplitude. Figure 9. Time-mean thrust coefficient and energy-consuming power coefficient with various heaving amplitude and thickness with Sr = 0.3, Re = 2 × 104.

The time-mean thrust increases significantly with the heaving amplitude for cases of NACA0030, NACA0040, and NACA0050 when Amax > 0.5C, while the NACA0020 cases remain stable. As for cases of NACA0012 and NACA0015, the thrust decreases and becomes negative, which means that the flapping foil produces drag and the drag increases with the heaving amplitude when Amax > 0.5C. The thrust coefficient remains steady when the amplitude is less than 0.5C. As for the effects of the foil thickness on the parameters, time-mean thrust increases with the thickness for all tested flapping frequencies, while the input power decreases. This indicates that the thicker foil performs better than the thinner foil, as shown in Figure 9. With the definition of the propelling efficiency in Equation (5), the thrust time’s velocity is higher than the energy-consuming power, which results in the propelling efficiency exceeding 1, as shown in Figure 10. This ‘bizarre’ phenomenon is caused by the suction area behind the D-section cylinder. The results indicate that the moving forward motion requires less energy to maintain the moving velocity when the flapping propul- J. Mar. Sci. Eng. 2021, 9, 658 sion device is involved in the suction area of a von Kármán wake with a small heaving9 of 15 amplitude. Moreover, it seems that the original equation used to calculate the propelling efficiency is not proper for the flapping motion when tracking a close target.

4 4 FigureFigure 10.10. PropellingPropelling efﬁciencyefficiency changingchanging withwithA Amaxmax with S Srr = 0 0.3.3 and and R Ree = 2 2 ×× 1010. .

The vortices shedding performance changes with the heaving amplitude, which results in the fluctuation of hydrodynamic parameters. The instantaneous CT,CL, and CM in two flapping cycles for NACA0030 with different heaving amplitudes are demonstrated in Figure 12. The fluctuation amplitudes of the three parameters increase with flapping am-

plitude, the highest peak of the thrust appears when the foil moves around the symmetric line at t = 0.5T during one cycle, while Amax > 0.5C. As discussed above, when the foil flapping in the vortex street with heaving amplitude Amax < 0.5C, the vortex shedding is regular. The pressure distribution is symmetrical for the upwards and downwards processes, shown in Figure 12a, and so the peaks of the thrust coefficient are the same. While the heaving amplitude is larger than 0.5C, the confluent vortex wake could not remain as a pair. The pressure distribution is shown in Figure 12b, the highest pressure difference is observed when t = 0.5T, and the thrust coefficient reaches its peak as shown in Figure 12c. The pressure difference of the upper and lower surface reaches its peak when t is around 0.5T, which results in the peak of the lift coefficient. The center of the vortex becomes the center position of low pressure. Thus, the hydrodynamic performance of the flapping foil is closely related to the vortex shedding. The thickness of the flapping foil plays an important role in improving the propelling force. The thrust of the thicker foil is higher than the thinner foil, as shown in Figure9a. The vortex streets of different foils are shown in Figure 13. The results indicate that the thickness of the foil affects the shedding of LEV. The shedding of LEV is delayed when the foil thickness grows. The oncoming clockwise vortex is pushed down and inhibits the anticlockwise vortex for the thinner foil as shown in Figure 13. When the foil is thicker than NACA0020, the distance between the anticlockwise vortex and the clockwise vortex is magnified. Moreover, it can be seen that the leading edge separation is evident and the diffusion of the resultant vortex decreases with the increasing thickness. Pressure around the vortices is lower than the flow field, thus the pressure at the leading edge remains low with a longer period time for the thicker foil, which results in a bigger time-mean thrust. The differential pressure resulting from delayed vortices benefits the pitching motion.

4.3. Definition of the Energy Transforming Ratio The commonly used equation for the propelling efficiency is established by assuming that the flapping foil is driven by the engine to move forward in still water. However, the oncoming wake may enhance or depress the flapping motion. Thus, the propelling efficiency calculated by Equation (5) represents the ratio of the foil moving power and the engine power. In the present flapping system, there are three parts that should considered: (a) the engine power (P1), which donates energy consumption or harvesting by the flapping system; (b) the foil moving power(P2), which represents the propelling power of the flapping foil; (c) the flow energy(P3), which stands for the energy of the oncoming flow. The diagram of the energy system is shown in Figure 14. J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 11 of 16

As shown in Figure 11, the vortex wake is a regular structure when the foil flaps in the wake with Amax < 0.5C. A pair of vortices are formed behind the foil during a flapping J. Mar. Sci. Eng. 2021, 9, 658 process. When the Amax > 0.5C, the vortex shedding from the foil will merge with the on-10 of 15 coming wake and will be disturbed by the foil. Consequently, a pair and one single vortex are formed, shown in Figure 11 (d). LEV is inhibited, which results in a low-pressure area in front of the foil and the thrust is enlarged when the amplitude increases.

(a) Amax = 0.1C. (b) Amax = 0.3C.

4 Figure 11. Vorticity magnitude contours and vortex interactions at various Amax, with Sr = 0.3 and Re = 2 × 10 .

According to the results discussed in the former parts, there are three types of energy transformations, as shown in Figure 15. J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 12 of 16

Figure 11. Vorticity magnitude contours and vortex interactions at various Amax, with Sr = 0.3 and Re = 2 × 104.

The vortices shedding performance changes with the heaving amplitude, which results in the fluctuation of hydrodynamic parameters. The instantaneous CT, CL, and CM in two flapping cycles for NACA0030 with different heaving amplitudes are demonstrated J. Mar. Sci. Eng. 2021, 9, 658 in Figure 12. The fluctuation amplitudes of the three parameters increase with flapping11 of 15 amplitude, the highest peak of the thrust appears when the foil moves around the symmetric line at t = 0.5T during one cycle, while Amax > 0.5C.

FigureFigure 12.12. PressurePressure distributiondistribution during one ﬂappingflapping cycle for ( a) Amaxmax == 0.2 0.2CC andand (b (b) )AAmamaxx = 0=.7 0.7C; (Cc;) CT, (d) CL, and (e) CM in two flapping cycles stands for NACA0030 with different heaving ampli- (c)CT,(d)CL, and (e)CM in two ﬂapping cycles stands for NACA0030 with different heaving tudes under the condition that Sr = 0.3 and Re = 2 × 104. amplitudes under the condition that Sr = 0.3 and Re = 2 × 104.

TheAs discussed modes are above, given w ashen follows: the foil flapping in the vortex street with heaving ampli- tudeMode Amax < 1:0.5 theC, the flapping vortex foil shedding consumes is regular. energy fromThe pressure the oncoming distribution wakeand is symmetrical the engine, whichfor the works upwards as a and propulsion downwards device. processes, shown in Figure 12a, and so the peaks of the thrustMode coefficient 2: theflapping are the same. foil extracts While energythe heaving from theamplitude oncoming is larger wake andthan the 0.5 engineC, the con- acts asfluent a dynamo. vortex wake could not remain as a pair. The pressure distribution is shown in Fig- ure 1Mode2b, the 3: highest the engine pressure provides difference power is for observed the flapping when foil t = to 0 overcome.5T, and the resistance thrust coeffi- or to generatecient reaches thrust its and peak the as upstream shown in flow Figure is accelerated. 12c. The pressure difference of the upper and lowerSince surface we alwaysreaches focusits peak on when whether t is thearound foil could0.5T, which work asresults a propulsion in the peak device of the or lift as ancoefficient. energy harvester, The center the of the engine vortex power becomes (P1) the and center foil moving position power of low (P2)pressure. are applied Thus, the to calculatehydrodynamic the newly performance defined energy of the flapping transforming foil is ratio closely as follows. related to the vortex shedding. The thickness of the flapping foil plays an important role in improving the propelling force. The thrust of the thicker foil is higherP1 than− P2 the thinnerP1 − P2 foil, as shown in Figure 9a. RET = sign(P1) = (8) The vortex streets of different foils are shownP1 in Figure 13.|P1 | A negative value stands for the energy extraction from the oncoming wake by the foil and a positive value reflects that the engine’s power is driving the flapping foil. If 1 > RET > 0, it means that part of the engine’s energy is applied for the foil working as a propulsion device and part of it is applied to accelerate the oncoming flow as shown in mode 3. J. Mar. Sci. Eng. 2021, 9, 658 12 of 15 J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 13 of 16

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 14 of 16

Figure 13. Vorticity magnitude contours and vortex interaction at various thicknesses with Sr = 0.3, J. Mar. Sci. Eng. 2021, 9, x FOR PEER FigureREVIEW 13. Vorticity magnitude contours and vortex interaction at various thicknesses with Sr14 =of 0.3, 16 Amax = 0.2C, and Re = 2 × 104 at t/T = 1. 4 Amax = 0.2C, and Re = 2 × 10 at t/T = 1. The results indicate that the thickness of the foil affects the shedding of LEV. The shedding of LEV is delayed when the foil thickness grows. The oncoming clockwise vortex is pushed down and inhibits the anticlockwise vortex for the thinner foil as shown in Figure 13. When the foil is thicker than NACA0020, the distance between the anticlockwise vortex and the clockwise vortex is magnified. Moreover, it can be seen that the leading edge separation is evident and the diffusion of the resultant vortex decreases with the increasing thickness. Pressure around the vortices is lower than the flow field, thus the pressure at the leading edge remains low with a longer period time for the thicker foil, which results in a bigger time-mean thrust. The differential pressure resulting from delayed vortices benefits the pitching motion. Figure 14. Diagram4.3. of Definition the energy of the system. Energy Transforming Ratio The commonly used equation for the propelling efficiency is established by assuming AccordingFigurethat to the the 14. results flappingDiagram discussed foil of is the driven energy in the by system.theformer engine parts, to move there forward are three in still types water. of energy However , the transformationsFigureoncoming, as shown 14. D wakeiagram in Figure may of theenhance 15. energy or system.depress the flapping motion. Thus, the propelling efficiency calculated by Equation (5) represents the ratio of the foil moving power and the engineAccording power. In tothe the present results flapping discussed system, in ththeere former are three parts, parts there that should are three considered: types of energy transfor(a) the enginemations power, as shown(P1), which in Figure donates 15. energy consumption or harvesting by the flapping system; (b) the foil moving power(P2), which represents the propelling power of the flapping foil; (c) the flow energy(P3), which stands for the energy of the oncoming flow. The diagram of the energy system is shown in Figure 14.

(a) Mode 1 (b) Mode 2 (c) M ode 3

Figure 15. DiagramFigure of three 15. typesDiagram of energy of three transform types ofations. energy transformations. (a) Mode 1 (b) Mode 2 (c) Mode 3 The modes are given as follows: Mode 1: theFigure flapping 15. D iagramfoil consumes of three typesenergy of energyfrom the transform oncomingations. wake and the engine, which works as a propulsion device. Mode 2: the flappingThe modes foil areextract givens energy as follows: from the oncoming wake and the engine acts as a dynamo. Mode 1: the flapping foil consumes energy from the oncoming wake and the engine, Mode 3: thewhich engine works provides as a propulsion power for device.the flapping foil to overcome resistance or to generate thrust andMode the upstream 2: the flapping flow is accelerated.foil extracts energy from the oncoming wake and the engine Since we alwaysacts as afocus dynamo. on whether the foil could work as a propulsion device or as an energy harvester,M odethe engine3: the engine power provides (P1) and power foil moving for the powerflapping (P2) foil are to overcomeapplied to resistance or to calculate the newlygenerate defined thrust energy and the transforming upstream flow ratio is as accelerated. follows. Since we always focus on whether the foil could work as a propulsion device or as PPPP1 2 1 2 an energy harvester,RET  the sign engine( P 1) power (P1) and foil moving power (P2) are applied to P1 P1 (8) calculate the newly defined energy transforming ratio as follows. A negative value stands for the energy extraction fromPPPP 1the 2oncoming 1 2 wake by the RET  sign( P 1) foil and a positive value reflects that the engine’s power is drivingP1 the flappingP1 foil. (8) If 1 > RET > 0, it means that part of the engine’s energy is applied for the foil working as a propulsion deviceA negative and part value of it isstands applied for to the accelerate energy extractionthe oncoming from flow the asoncoming shown wake by the in mode 3. foil and a positive value reflects that the engine’s power is driving the flapping foil. If RET ≥ 1, the flappingIf 1 > RET foil > 0, provides it means nothat thrust part ofand the it engine’s is considered energy that is applied the flapping for the foil working foil is workingas as aa propulsionpump, as shown device in and mode part 3. of it is applied to accelerate the oncoming flow as shown If −1 < RETin ≤ mode 0, the 3 .flapping foil extracts energy from the oncoming wake and the engine and is workingIf R asET ≥a 1,propulsion the flapping device, foil providesas shown noin modethrust 1. and it is considered that the flapping If RET ≤ −1,foil this is system working could as a workpump, as as an shown energy in harvester, mode 3. as shown in mode 2. When applied Ifto −Equation1 < RET ≤ (8), 0, thethe flappingtendency foilof the extract energys energy transformation from the thatoncoming varies wake and the with thickness engineis shown and in is Figure working 16. as The a propulsion four condition device,s of asthe shown results in described mode 1. above are padded with differentIf RET ≤ colors.−1, this Thesystem bright could green work color as an reflects energy that harvester, the flapping as shown foil in in mode 2. the oncoming wakeW workshen applied as an energyto Equation harvester. (8), the Dark tendency green of means the energy that the transformation flapping that varies with thickness is shown in Figure 16. The four conditions of the results described above are padded with different colors. The bright green color reflects that the flapping foil in the oncoming wake works as an energy harvester. Dark green means that the flapping

J. Mar. Sci. Eng. 2021, 9, 658 13 of 15

If RET ≥ 1, the flapping foil provides no thrust and it is considered that the flapping foil is working as a pump, as shown in mode 3. If −1 < RET ≤ 0, the flapping foil extracts energy from the oncoming wake and the engine and is working as a propulsion device, as shown in mode 1. If RET ≤ −1, this system could work as an energy harvester, as shown in mode 2. When applied to Equation (8), the tendency of the energy transformation that varies J. Mar. Sci. Eng. 2021, 9, x FOR PEER withREVIEW thickness is shown in Figure 16. The four conditions of the results described15 above of 16

are padded with different colors. The bright green color reflects that the flapping foil in the oncoming wake works as an energy harvester. Dark green means that the flapping propulsion device extracts energy from the oncoming wake and the energy provided by propulsion device extracts energy from the oncoming wake and the energy provided by the engine. The orange stands for a common propeller driven by the engine. The red color the engine. The orange stands for a common propeller driven by the engine. The red color indicates that the flapping foil performs negatively under these conditions. indicates that the flapping foil performs negatively under these conditions.

FigureFigure 16.16. EnergyEnergy transformingtransforming ratioratio varyingvarying withwith thickness.thickness.

5.5. ConclusionsConclusions TheThe effectseffects of of heaving heaving amplitude amplitude and and foil foil thickness thickness on the on thrust the thrust and energy-consuming and energy-con- aresuming numerically are numerically studied. studied. The motion The motion can be energizedcan be energized in the wake.in the wake. The original The original drag- generatingdrag-generating flapping flapping motion motion provides provides thrust. thrust. TheThe heavingheaving amplitudeamplitude studystudy isis carriedcarried outout atat aa rangerange ofof 0.10.1CC~1.0~1.0CC onon 2D2D NACANACA symmetricsymmetric foilsfoils withwith aa fixedfixed frequency.frequency. TheThe resultsresults indicateindicate thatthat thethe increasingincreasing amplitudeamplitude resultsresults inin a higher energy energy-consuming-consuming power power and and horizontal horizontal force. force. Moreover, Moreover, when when oscil- os- cillation is enclosed in the wake (Amax < 0.5C), the horizontal force is quite steady and lation is enclosed in the wake (Amax < 0.5C), the horizontal force is quite steady and the theconfluent confluent wake wake remained remained a von a vonKármán Kárm type.án type. When When the oscillation the oscillation oversteps, oversteps, the hori- the horizontal force and energy-consuming power witnesses significant growth. The confluent zontal force and energy-consuming power witnesses significant growth. The confluent wake would change from VKW to IVKW and 1P+1S wake. wake would change from VKW to IVKW and 1P+1S wake. The thickness investigation was performed on the NACA symmetric foils with 12%, The thickness investigation was performed on the NACA symmetric foils with 12%, 15%, 20%, 30%, 40%, and 50% thick sections. The thrust increases with the thickness. 15%, 20%, 30%, 40%, and 50% thick sections. The thrust increases with the thickness. The The horizontal force tends to be a drag with a relatively thin foil. However, when the horizontal force tends to be a drag with a relatively thin foil. However, when the foil is foil is thicker than NACA0020, it turns out to be thrust. The energy-consuming power thicker than NACA0020, it turns out to be thrust. The energy-consuming power decreases decreases with growing thickness, which results in a higher propelling efficiency. The with growing thickness, which results in a higher propelling efficiency. The growing growing thickness delays the shedding of the LEV. thickness delays the shedding of the LEV. A newly defined parameter, energy transforming ratio (RET), is elaborated to solve A newly defined parameter, energy transforming ratio (RET), is elaborated to solve the problem of the improper use of common propelling efficiency, according to the results discussed.the problem This of newthe improper parameter use can of describe common the pro propellingpelling efficiency, efficiency according and energy to harvesting the results efficiencydiscussed. with This its new value parameter located can in different describe ranges. the propelling efficiency and energy harvesting efficiencyFinally, when with theits value micro located underwater in different vehicles ranges. propelled by the flapping foil chases a bluffFinally, body, a smallwhen heaving the micro amplitude underwater and avehicles thick section propelled foil areby preferredthe flapping to construct foil chase thes a flappingbluff body, propulsion a small heaving device. amplitude and a thick section foil are preferred to construct the flapping propulsion device.

Author Contributions: Conceptualization, Jian Li and Peng Wang; Methodology, Da Lyu and Jian Li; Software: Peng Wang; Validation, Xiaoyi An and Ruixuan He; Formal Analysis, Jian Li, Da Lyu and Baoshou Zhang; Investigation, Jian Li and Da Lyu; Resources, Peng Wang; Data curation, Jian Li; Writing—original draft preparation, Jian Li and Xiaoyi An; Writing—review and editing, Jian Li, Xiaoyi An, Ruixuan He, Da Lyu and Baoshou Zhang; Visualization, Jian Li and Baoshou Zhang; Supervision, Peng Wang; Project administration, Jian Li; Funding acquisition, Peng Wang. All authors have read and agreed to the published version of the manuscript. Funding: This work is supported by grants from the National Natural Science Foundation of China (Grant NO.51875466), (Grant NO.51805436), the Tsinghua University Shuimu Scholar Program (2020SM027), and the China Postdoctoral Science Foundation (BX20200187)

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Author Contributions: Conceptualization, J.L. and P.W.; Methodology, D.L. and J.L.; Software: P.W.; Validation, X.A. and R.H.; Formal Analysis, J.L., D.L. and B.Z.; Investigation, J.L. and D.L.; Resources, P.W.; Data curation, J.L.; Writing—original draft preparation, J.L. and X.A.; Writing— review and editing, J.L., X.A., R.H., D.L. and B.Z.; Visualization, J.L. and B.Z.; Supervision, P.W.; Project administration, J.L.; Funding acquisition, P.W. All authors have read and agreed to the published version of the manuscript. Funding: This work is supported by grants from the National Natural Science Foundation of China (Grant NO.51875466), (Grant NO.51805436), the Tsinghua University Shuimu Scholar Program (2020SM027), and the China Postdoctoral Science Foundation (BX20200187). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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