# A Non-Linear BEM–FEM Coupled Scheme for the Performance of Flexible Flapping-Foil Thrusters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description

- the space-fixed frame $(x,z)$, with respect to which the foil moves in the negative direction of $x-axis$ with constant cruising speed $U$
- the body-fixed (non-inertial) $({x}^{\prime},{z}^{\prime})$ positioned at the foil’s center of rotation with ${x}^{\prime}-axis$ in the direction of the un-deformed chord line
- the body-fixed (non-inertial) $({x}^{\u2033},{z}^{\u2033})$ position at the leading edge (LE). This frame is exclusively used for the structural response problem.

## 3. Mathematical Formulation

#### 3.1. Structural Dynamics of the Foil

#### 3.2. Modelling the Fluid Flow around the Foil

#### 3.3. Hydrodynamic Pressure and Force

## 4. Numerical Methods

#### 4.1. Finite Element Method (FEM)

#### 4.1.1. Discretization Scheme for FEM

#### 4.1.2. Time Integration

#### 4.2. Boundary Element Method (BEM)

#### 4.2.1. Boundary Integral Equation (BIE) & Discretization

#### 4.2.2. Solution Schemes

**BEM–ABM**: The DtN operator, derived from the BIE, acts as a constraint to the dynamical system evolution equation that is constructed using the pressure-type Kutta condition. We consider ${\mu}_{W1}$ as the dynamic variable of the problem, and thus, the formulation allows for the treatment of an initial value problem (IVP). In order to express the pressure-type Kutta condition as a function of $\upsilon ={\mu}_{W1},$ we use the DtN map in Equation (34), in conjunction with the discretized form of Equation (14), to obtain a (spatially and temporarily) nonlocal differential equation, with explicit and implicit nonlinearities with respect to $\upsilon ={\mu}_{W1}.$ The latter is finally put in the following form,

**BEM–NR**: The BIE along with the discretized form of the pressure-type Kutta condition, detailed in Appendix A, is used to construct the complete system of equations, with the boundary fields ${\Phi}_{B}$ and ${\mu}_{W1}$ as unknowns. A set of ${N}_{B}+1$ equations can be solved for the unknown values of ${\Phi}_{Bi}$ and ${\mu}_{w1}$ at each time step, which can be written in a compact form

#### 4.3. Non-Linear BEM-FEM

## 5. Results

#### 5.1. Convergence Characteristics of the Numerical Scheme

#### 5.2. The Case of a Flexible Flapping Foil

#### 5.3. The Case of a Flexible Heaving Foil

#### 5.4. Effects of Flexural Rigidity on Froude Efficiency

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BEM | Boundary element method |

FEM | Finite element method |

TE | Trailing edge |

LE | Leading edge |

GIM | General iterative method |

TSM | Time stepping method |

IBVP | Initial boundary value problem |

IVP | Initial value problem |

BIE | Boundary integral equation |

$c$ | Chord length |

$g$ | Nominal gravitational acceleration |

${x}_{R}$ | Location of pivot axis from leading edge |

$h$ | Heave response |

$\theta $ | Pitch angle |

${\theta}_{o}$ | Pitch amplitude |

${h}_{o}$ | Heave amplitude |

$\omega $ | Angular frequency of motion |

${\rho}_{f}$ | Water density |

${\rho}_{s}$ | Material density |

$m$ | Foil mass |

$a,b$ | Proportional damping coefficients |

$L(t)$ | Instantaneous lift force |

$\eta $ | Froude efficiency |

$\Phi $ | Potential field |

## Appendix A

## Appendix B

## References

- Yogang Singh, S.K.; Bhattacharyya, V.G. Idichandy, “CFD approach to modelling, hydrodynamic analysis and motion characteristics of a laboratory underwater glider with experimental results”. J. Ocean Eng. Sci.
**2017**, 2, 90–119. [Google Scholar] [CrossRef] - Wu, X.; Zhang, X.; Tian, X.; Li, X.; Lu, W. A review on fluid dynamics of flapping foils. Ocean Eng.
**2019**. [Google Scholar] [CrossRef] - Sfakiotakis, M.; Lane, D.; Bruce, J.; Davies, C. Review of Fish Swimming Modes for Aquatic Locomotion. IEEE J. Ocean. Eng.
**1999**, 24, 237–252. [Google Scholar] [CrossRef][Green Version] - Politis, G.K.; Tsarsitalides, V.T. Flapping wing propulsor design: An approach based on systematic 3D-BEM. Ocean Eng.
**2014**, 84, 98–123. [Google Scholar] [CrossRef] - Silva, L.D.; Yamaguchi, H. Numerical study on active wave devouring propulsion. J. Mar. Technol. (Jpn.)
**2012**, 17, 261–275. [Google Scholar] [CrossRef][Green Version] - Belibassakis, K.; Politis, G. Hydrodyanamic performance of flapping wings for augmenting ship propulsion. J. Ocean Eng.
**2013**, 72, 227–240. [Google Scholar] [CrossRef] - Belibassakis, K.; Filippas, E. Ship propulsion in waves by actively controlled flapping foils. Appl. Ocean Res.
**2015**, 52, 1–11. [Google Scholar] [CrossRef] - Bøckmann, E.; Steen, S. Model test and simulation of a ship with wavefoils. Appl. Ocean Res.
**2016**, 57, 8–18. [Google Scholar] [CrossRef] - Bowker, J.A. Coupled Dynamics of a Flapping Foil Powered Vessel. Ph.D. Thesis, University of Southampton, Southampton, UK, 2018. [Google Scholar]
- BIO-PROPSHIP: Augmenting Ship Propulsion in Rough Sea by Biomimetic-Wing System. Research Project Funded by National Strategic Reference Framework NSRF Greece 2007–2013. Available online: http://arion.naval.ntua.gr/~biopropship/index_en.html (accessed on 2 December 2019).
- Xiao, Q.; Zhu, Q. A review on flow energy harvesters based on flapping foils. J. Fluids Struct.
**2014**, 46, 174–191. [Google Scholar] [CrossRef] - Jeanmonod, G.; Olivier, M. Effects of chordwise flexibility on 2D flapping foils used as an energy extraction device. J. Fluids Struct.
**2017**, 70, 327–345. [Google Scholar] [CrossRef][Green Version] - Koutsogiannakis, P.E.; Filippas, E.S.; Belibassakis, K.A. A study of Multi-Component Oscillating-Foil Hydrokinetic Turbines with a GPU-Accelerated Boundary Element Method. J. Mar. Sci. Eng.
**2019**, 7, 424. [Google Scholar] [CrossRef][Green Version] - Filippas, E.; Gerostathis, T.; Belibassakis, K. Semi-activated oscillating hydrofoil as a nearshore biomimetic energy device system in waves and currents. J. Ocean Eng.
**2018**, 154, 396–415. [Google Scholar] [CrossRef] - Triantafyllou, M.S.; Triantafyllou, G.S.; Yue, D. Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech.
**2000**, 32, 33–53. [Google Scholar] [CrossRef] - Rozhdestvensky, K.; Ryzhov, V. Aerodynamics of flapping-wing propulsors. Prog. Aerosp. Sci.
**2003**, 39, 585–633. [Google Scholar] [CrossRef] - Shyy, W.; Aono, H.; Chimakurthi, S.K.; Trizilia, P.; Kang, C.K.; Liu, H.; Cesnik, C.E.S. Recent progress in flapping wing aerodynamics and aeroelasticity. Prog. Aerosp. Sci.
**2010**, 46, 284–327. [Google Scholar] [CrossRef] - Tay, W.B.; Lim, K.B. Numerical analysis of active chordwise flexibility on the performance of non-symmetrical flapping airfoils. J. Fluids Struct.
**2010**, 26, 74–91. [Google Scholar] [CrossRef] - Olivier, M.; Dumas, G. A parametric investigation of the propulsion of 2D chordwise-flexible flapping wings at low Reynolds number using numerical simulations. J. Fluids Struct.
**2016**, 63, 201–237. [Google Scholar] [CrossRef][Green Version] - Garg, N.; Kenway, G.K.; Martins, J.R.; Young, Y.L. High-fidelity multipoint hydrostructural optimization of a 3-D hydrofoil. J. Fluids Struct.
**2017**, 71, 15–39. [Google Scholar] [CrossRef] - Weihs, D. Hydrodynamic propulsion by large amplitude oscillation of an airfoil with chordwise flexibility. J. Fluid Mech.
**1978**, 8, 486–497. [Google Scholar] - Katz, N.J.; Weihs, D. Large amplitude unsteady motion of a flexible slender propulsor. J. Fluid Mech.
**1979**, 90, 713–723. [Google Scholar] [CrossRef] - Alben, S. Optimal flexibility of a flapping appendage in an inviscid fluid. J. Fluid Mech.
**2008**, 614, 355–380. [Google Scholar] [CrossRef][Green Version] - Alben, S.; Witt, C.; Baker, T.V.; Anderson, E.; Lauder, G.V. Dynamics of freely swimming flexible bodies. Phys. Fluids
**2012**, 24, 051901. [Google Scholar] [CrossRef][Green Version] - Zhu, Q. Numerical Simulation of a flapping foil with chordwise and spanwise flexibility. AIAA J.
**2007**, 45, 2448–2457. [Google Scholar] [CrossRef] - Zhu, Q.; Shoele, K. Propulsion performance of a skeleton-strengthened fin. J. Exp. Biol.
**2008**, 11, 2087–2100. [Google Scholar] [CrossRef] [PubMed][Green Version] - Priovolos, K.; Flippas, E.S.; Belibassakis, K.A. A vortex-based method for improved flexible flapping-foil thruster performance. Eng. Anal. Bound. Elem.
**2018**, 95, 69–84. [Google Scholar] [CrossRef] - Richards, A.J.; Oshkai, P. Effects of the stiffness, inertia and oscillation kinematics on the thrust generation and efficiency of an oscillating-foil propulsion system. J. Fluids Struct.
**2015**, 57, 357–374. [Google Scholar] [CrossRef] - Egan, B.C.; Brownell, C.J.; Murray, M.M. Experimental assessment of performance characteristics for pitching flexible propulsors. J. Fluids Struct.
**2015**, 67, 22–33. [Google Scholar] [CrossRef] - Fernandez-Prats, R. Effect of chordwise flexibility on pitching foil propulsion in a uniform current. Ocean Eng.
**2017**, 145, 24–33. [Google Scholar] [CrossRef] - Cleaver, D.J.; Gursul, I.; Calderon, D.E.; Wang, Z. Thrust enhancement due to flexible trailing-edge of plunging foils. J. Fluids Struct.
**2014**, 51, 402–412. [Google Scholar] [CrossRef][Green Version] - Prempraneerach, P.; Hover, F.; Triantafyllou, M. The effect of chordwise flexibility on the thrust and efficiency of a flapping foil. In Proceedings of the 13th International Symposium on Unmanned Untetherd Submersible Technology: Special Session on Bioengineering Research Related to Autonomous Underwater Vehicles, Durham, NH, USA, 11–13 August 2013. [Google Scholar]
- Barannyk, O.; Buckham, B.J.; Oshkai, P. On the performance of an oscillating plate underwater propulsion system with variable chordwise flexibility at different depths of submergence. J. Fluids Struct.
**2012**, 28, 152–166. [Google Scholar] [CrossRef] - Paraz, F.; Eloy, C.; Schouveiler, L. Experimental study of the response of a flexible plate to a harmonic forcing in a flow. Comptes Rendus Mec.
**2014**, 342, 532–538. [Google Scholar] [CrossRef] - Paraz, F.; Schouveiler, L.; Eloy, C. Thrust generation by a heaving flexible foil: Resonance, nonlinearities and optimality. Phys. Fluids
**2016**, 28, 011903. [Google Scholar] [CrossRef] - Huera-Huarte, F.J.; Gharib, M. On the effects of tip deflection in flapping propulsion. J. Fluids Struct.
**2017**, 71, 217–233. [Google Scholar] [CrossRef] - Papadakis, G.; Voutsinas, S.G. A strongly coupled Eulerian Lagrangian method verified in 2D external compressible flows. J. Comput. Fluids
**2019**, 195, 104325. [Google Scholar] [CrossRef] - Papadakis, G.; Filippas, E.; Ntouras, D.; Belibassakis, K.A. Effects of viscosity and nonlinearity on 3D flapping-foil thruster for marine applications. In Proceedings of the Oceans 2019 MTS/IEEE, Marseille, France, 17–20 June 2019. [Google Scholar]
- Reddy, J.N. Theory and Analysis of Elastic Plates and Shells; CRC Press Taylor & Francis Group: Boca Raton, FL, USA, 2007. [Google Scholar]
- Hughes, T.J.R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis; Prentice-Hall INC: Englewood Cliffs, NJ, USA, 1987. [Google Scholar]
- Politis, G.K. Simulation of unsteady motion of a propeller in a fluid including free wake modelling. Eng. Anal. Bound. Elem.
**2004**, 28, 633–653. [Google Scholar] [CrossRef] - Politis, G. Unsteady wake rollup modeling using a molifier based filtering technique. Dev. Appl. Ocean. Eng. DAOE
**2016**, 5. [Google Scholar] [CrossRef] - Mantia, M.L.; Dadnichki, P. Unsteady panel method for flapping foil. Eng. Anal. Bound. Elem.
**2009**, 33, 572–580. [Google Scholar] [CrossRef] - Filippas, E. Hydrodynamic Analysis of Ship and Marine Biomimetic Systems in Waves Using GPGPU Programming. Ph.D. Thesis, NTUA, Athens, Greece, 2019. [Google Scholar]
- Chowdhury, I.; Dasgupta, S. Computation of Rayleigh Damping Coefficients for Large Systems. Electron. J. Geotech. Eng.
**2003**, 43, 6855–6868. [Google Scholar] - Moran, J. An Introduction to Theoretical and Computational Aerodynamics; Wiley & Sons: New York, NY, USA, 1984. [Google Scholar]
- Kress, R. Linear Integral Equations; Springer: Berlin, Germany, 1989. [Google Scholar]
- Katz, J.; Plotkin, A. Low Speed Aerodynamics; McGraw-Hill: New York, NY, USA, 1991. [Google Scholar]
- Filippas, E.S.; Belibassakis, K.A. Hydrodynamic analysis of flapping-foil thrusters operating beneath the free surface and in waves. Eng. Anal. Bound. Elem.
**2014**, 41, 47–59. [Google Scholar] [CrossRef] - Mabie, H.H.; Rogers, C.B. Transverse vibrations of double-tapered cantilever beams with end support and with end mass. J. Acoust. Soc. Am.
**1974**, 55, 986–991. [Google Scholar] [CrossRef][Green Version] - Beltempo, A.; Balduzzi, G.; Alfano, G.; Auricchio, F. Analytical derivation of a general 2D non-prismatic beam model based on the Hellinger-Reissner principle. Eng. Struct.
**2015**, 101, 88–98. [Google Scholar] [CrossRef] - Warburton, G.B. The Dynamic Behaviour of Structures; Pergamon Press: Oxford, UK, 1976; p. 131. [Google Scholar]
- Mathieu Olivier, G.D. Effect of mass and chordwise flexibility on 2D self-propelled flapping wings. J. Fluids Struct.
**2016**, 64, 46–66. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Basic notation: (

**a**) flexible flapping-foil kinematics with respect to a space-fixed Cartesian coordinate system (x, z) and (

**b**) thickness profile in a body-fixed cartesian coordinate system (x’’, z’’).

**Figure 2.**Unsteady hydrodynamics notation for the chord-wise flexible flapping foil, introducing the inertial and body-fixed coordinate systems for the hydrodynamics problem formulation.

**Figure 3.**Comparison of the time-evolution of trailing vortex sheet for four periods of time. The simplified wake model (upper) and the free wake model (lower) are shown with the blue vectors denoting the potential jump. In the lower figure the simplified wake model (dashed line) is included for comparison purposes.

**Figure 4.**Static response for non-prismatic beams and comparison with data from [51]. The transverse displacement is presented as a function of beam length in the upper subplots for two values of taper ratio. The corresponding normalized beam profiles are shown in the lower subplots, where the maximum thickness is 0.25 m. (

**a**) Deflection and (

**b**) shape of variable thickness beam. (

**c**) Deflection and (

**d**) shape of beam with linear thickness distribution.

**Figure 5.**Dynamic response of the tip of a cantilever beam under tip point load for one period of motion and comparison against the analytic solution. Data from Warburton [52]. (

**a**) Deflection, (

**b**) tip loading of the beam.

**Figure 6.**Relative error (%) of (

**a**) the thrust efficiency coefficient ${C}_{T}$ (left) and (

**b**) the propulsive efficiency ${\eta}_{F}$ (right) with respect to the value of the finest discretization simulated with BEM–ABM. The error is denoted by the color-bar, and the contours display the iso-λ values. For the following kinematic parameters ${h}_{o}/c=0.75,{\alpha}_{eff}={22}^{\circ},\mathrm{Str}=0{.2,\mathrm{x}}_{R}={x}_{LE},\psi =-{90}^{\circ}$.

**Figure 7.**Relative error (%) of (

**a**) the thrust efficiency coefficient ${C}_{T}$ (left) and (

**b**) the propulsive efficiency ${\eta}_{F}$ (right) with respect to the value of the finest discretization simulated with BEM–NR. The error is denoted by the color-bar, and the contours display the iso-λ values. For the following kinematic parameters ${h}_{o}/c=0.75,{\alpha}_{eff}={22}^{\circ},\mathrm{Str}=0{.3,\mathrm{x}}_{R}={x}_{LE},\psi =-{90}^{\circ}$.

**Figure 8.**Relative error (%) of (

**a**) the thrust efficiency coefficient ${C}_{T}$ (left) and (

**b**) the propulsive efficiency ${\eta}_{F}$ (right) with respect to the value of the finest discretization simulated with BEM–NR. The error is denoted by the color-bar, and the contours display the iso-λ values. For the following kinematic parameters ${h}_{o}/c=0.75,{\alpha}_{eff}={22}^{\circ},\mathrm{Str}=0{.3,\mathrm{x}}_{R}={x}_{LE},\psi =-{90}^{\circ}$.

**Figure 9.**Time history of foil pitch and heave against thrust and lift forces for the chordwise rigid foil with simple harmonic motion and comparison with the experimental data from [32]. (

**a**) Pitch angle, (

**b**) Heave, (

**c**) Horizontal force, (

**d**) Vertical force of the foil.

**Figure 10.**Time history of foil pitch and heave against thrust and lift forces for chordwise flexible foil with simple harmonic motion profile and comparison with the experimental data from [32]. (

**a**) Pitch angle, (

**b**) Heave, (

**c**) Horizontal force, (

**d**) Vertical force of the foil.

**Figure 11.**Comparison with experimental data (denoted by circles) from [34] for the case of a flexible flat plate with $D=0.018\mathrm{Nm},{h}_{o}=0.004\mathrm{m}$ and $\mathrm{Re}=6000$: (

**a**) TE/LE amplitude response ratio and (

**b**) maximum pressure difference recorded at the TE during simulations.

**Figure 12.**Pressure coefficient instantaneous distribution comparison between (

**a**) BEM–ABM and (

**b**) BEM–NR, for the purely-heaving flexible plate with $\omega /{\omega}_{o}=3.$ The deformed foil deflection is added for illustration purposes.

**Figure 13.**(

**a**) Phase lag as a function of the frequency and comparison with experimental data (denoted by black triangles) from [34]. (

**b**) Thrust nondimensionalized by the characteristic elastic force, as a function of the frequency.

**Figure 14.**Envelopes of the foil’s elastic deflection during the last period of motion for two frequency values, one corresponding to the first resonance (upper figure) and the other (lower figure) for an intermediate frequency.

**Figure 15.**Propulsive performance of a chord-wise flexible NACA 0012 flapping foil in terms of the (

**a**) thrust coefficient and (

**b**) Froude efficiency as functions of Young’s modulus for the following parameters $Str=0.3,U=0.4\mathrm{m}/\mathrm{s},{h}_{o}/c=0.75,\psi ={90}^{\circ},{x}_{R}=1/3c,c=0.1\mathrm{m}.$

**Figure 16.**Typical configurations of the un-deformed (black) and deformable camber line (blue curves) on the trajectory of the leading edge (LE; dashed line) for a purely heaving foil.

**Figure 17.**Propulsive performance of a chord-wise flexible foil with $E=3.45{10}^{5}\mathrm{Pa}$ NACA 0012 flapping foil (solid lines) compared to the equivalent rigid foil (dashed lines). (

**a**) Thrust coefficient and (

**b**) Froude efficiency for $U=0.3\mathrm{m}/\mathrm{s},{\theta}_{o}={10}^{\circ},\psi ={90}^{\circ},{x}_{R}=1/3c,$ as functions of Strouhal number.

Mabie et al. [50] | Relative Error with FEM (% ×10^{−3}) | |||
---|---|---|---|---|

frequency | α = 5.0 | α = 10.0 | α = 5.0 | α = 10.0 |

Ω_{1} | 30.9820 | 72.0487 | 0.1451 | 0.0231 |

Ω_{2} | 91.9273 | 186.802 | 0.0410 | 0.2007 |

Ω_{3} | 199.1682 | 371.238 | 0.0197 | 0.1413 |

Ω_{4} | 356.2088 | 635.049 | 0.0198 | 0.1609 |

Ω_{5} | 564.1394 | 981.657 | 0.0835 | 0.5628 |

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**MDPI and ACS Style**

Anevlavi, D.E.; Filippas, E.S.; Karperaki, A.E.; Belibassakis, K.A. A Non-Linear BEM–FEM Coupled Scheme for the Performance of Flexible Flapping-Foil Thrusters. *J. Mar. Sci. Eng.* **2020**, *8*, 56.
https://doi.org/10.3390/jmse8010056

**AMA Style**

Anevlavi DE, Filippas ES, Karperaki AE, Belibassakis KA. A Non-Linear BEM–FEM Coupled Scheme for the Performance of Flexible Flapping-Foil Thrusters. *Journal of Marine Science and Engineering*. 2020; 8(1):56.
https://doi.org/10.3390/jmse8010056

**Chicago/Turabian Style**

Anevlavi, Dimitra E., Evangelos S. Filippas, Angeliki E. Karperaki, and Kostas A. Belibassakis. 2020. "A Non-Linear BEM–FEM Coupled Scheme for the Performance of Flexible Flapping-Foil Thrusters" *Journal of Marine Science and Engineering* 8, no. 1: 56.
https://doi.org/10.3390/jmse8010056