# Flow Interactions Between Low Aspect Ratio Hydrofoils in In-line and Staggered Arrangements

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup and Methods

## 3. Results

#### 3.1. Isolated Hydrofoil Performance and Wake Measurements

#### 3.2. Wake Measurements of Two Interacting Hydrofoils

#### 3.3. Linear Unsteady Hydrofoil Theory

#### 3.4. Follower Performance Comparison with Theory

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Wynne-Edwards, V.C. Animal Dispersion: In Relation to Social Behaviour; Oliver and Boyd: London, UK, 1962. [Google Scholar]
- Tinbergen, J. Social Behaviour in Animals: With Special Reference to vertebrates; Springer Science & Business Media: Berlin, Germany, 2012. [Google Scholar]
- Weihs, D. Hydromechanics of fish schooling. Nature
**1973**, 241, 290–291. [Google Scholar] [CrossRef] - Weimerskirch, H.; Martin, J.; Clerquin, Y.; Alexandre, P.; Jiraskova, S. Energy saving in flight formation. Nature
**2001**, 413, 697. [Google Scholar] [CrossRef] - Drucker, E.G.; Lauder, G.V. Locomotor function of the dorsal fin in teleost fishes: Experimental analysis of wake forces in sunfish. J. Exp. Biol.
**2001**, 204, 2943–2958. [Google Scholar] - Standen, E.; Lauder, G.V. Hydrodynamic function of dorsal and anal fins in brook trout (Salvelinus fontinalis). J. Exp. Biol.
**2007**, 210, 325–339. [Google Scholar] [CrossRef] [Green Version] - Partridge, B.L.; Pitcher, T.; Cullen, J.M.; Wilson, J. The three-dimensional structure of fish schools. Behav. Ecol. Sociobiol.
**1980**, 6, 277–288. [Google Scholar] [CrossRef] - Portugal, S.J.; Hubel, T.Y.; Fritz, J.; Heese, S.; Trobe, D.; Voelkl, B.; Hailes, S.; Wilson, A.M.; Usherwood, J.R. Upwash exploitation and downwash avoidance by flap phasing in ibis formation flight. Nature
**2014**, 505, 399–402. [Google Scholar] [CrossRef] [Green Version] - Ashraf, I.; Bradshaw, H.; Ha, T.T.; Halloy, J.; Godoy-Diana, R.; Thiria, B. Simple phalanx pattern leads to energy saving in cohesive fish schooling. Proc. Natl. Acad. Sci. USA
**2017**, 114, 9599–9604. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Akhtar, I.; Mittal, R.; Lauder, G.V.; Drucker, E. Hydrodynamics of a biologically inspired tandem flapping foil configuration. Theor. Comput. Fluid Dyn.
**2007**, 21, 155–170. [Google Scholar] [CrossRef] - Rival, D.; Hass, G.; Tropea, C. Recovery of energy from leading-and trailing-edge vortices in tandem-airfoil configurations. J. Aircr.
**2011**, 48, 203–211. [Google Scholar] [CrossRef] [Green Version] - Boschitsch, B.M.; Dewey, P.A.; Smits, A.J. Propulsive performance of unsteady tandem hydrofoils in an in-line configuration. Phys. Fluids
**2014**, 26, 051901. [Google Scholar] [CrossRef] - Broering, T.M.; Lian, Y.; Henshaw, W. Numerical investigation of energy extraction in a tandem flapping wing configuration. AIAA J.
**2012**, 50, 2295–2307. [Google Scholar] [CrossRef] [Green Version] - Broering, T.M.; Lian, Y.S. The effect of phase angle and wing spacing on tandem flapping wings. Acta Mech. Sin.
**2012**, 28, 1557–1571. [Google Scholar] [CrossRef] - Liu, G.; Ren, Y.; Dong, H.; Akanyeti, O.; Liao, J.C.; Lauder, G.V. Computational analysis of vortex dynamics and performance enhancement due to body–fin and fin–fin interactions in fish-like locomotion. J. Fluid Mech.
**2017**, 829, 65–88. [Google Scholar] [CrossRef] [Green Version] - Gong, W.Q.; Jia, B.B.; Xi, G. Experimental study on mean thrust of two plunging wings in Tandem. AIAA J.
**2015**. [Google Scholar] [CrossRef] - Gong, W.Q.; Jia, B.B.; Xi, G. Experimental study on instantaneous thrust and lift of two plunging wings in tandem. Exp. Fluids
**2016**, 57, 8. [Google Scholar] [CrossRef] - Muscutt, L.; Weymouth, G.; Ganapathisubramani, B. Performance augmentation mechanism of in-line tandem flapping foils. J. Fluid Mech.
**2017**, 827, 484–505. [Google Scholar] [CrossRef] [Green Version] - Kurt, M.; Moored, K.W. Flow interactions of two-and three-dimensional networked bio-inspired control elements in an in-line arrangement. Bioinspiration Biomimetics
**2018**, 13, 045002. [Google Scholar] [CrossRef] [Green Version] - Dewey, P.A.; Quinn, D.B.; Boschitsch, B.M.; Smits, A.J. Propulsive performance of unsteady tandem hydrofoils in a side-by-side configuration. Phys. Fluids
**2014**, 26, 041903. [Google Scholar] [CrossRef] - Dong, G.J.; Lu, X.Y. Characteristics of flow over traveling wavy foils in a side-by-side arrangement. Phys. Fluids
**2007**, 19, 057107. [Google Scholar] [CrossRef] - Kurt, M.; Moored, K.W. Unsteady Performance of Finite-Span Pitching Propulsors in Side-by-Side Arrangements. In Proceedings of the 2018 Fluid Dynamics Conference, Atlanta, GA, USA, 25–29 June 2018; p. 3732. [Google Scholar]
- Godoy-Diana, R.; Vacher, J.; Raspa, V.; Thiria, B. On the Fluid Dynamical Effects of Synchronization in Side-by-Side Swimmers. Biomimetics
**2019**, 4, 77. [Google Scholar] [CrossRef] [Green Version] - Shoele, K.; Zhu, Q. Performance of synchronized fins in biomimetic propulsion. Bioinspiration Biomimetics
**2015**, 10, 026008. [Google Scholar] [CrossRef] [PubMed] - Daghooghi, M.; Borazjani, I. The hydrodynamic advantages of synchronized swimming in a rectangular pattern. Bioinspiration Biomimetics
**2015**, 10, 056018. [Google Scholar] [CrossRef] [PubMed] - Maertens, A.P.; Gao, A.; Triantafyllou, M.S. Optimal undulatory swimming for a single fish-like body and for a pair of interacting swimmers. J. Fluid Mech.
**2017**, 813, 301–345. [Google Scholar] [CrossRef] [Green Version] - Novati, G.; Verma, S.; Alexeev, D.; Rossinelli, D.; van Rees, W.M.; Koumoutsakos, P. Synchronisation through learning for two self-propelled swimmers. Bioinspiration Biomimetics
**2017**, 12, 036001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Filella, A.; Nadal, F.; Sire, C.; Kanso, E.; Eloy, C. Model of collective fish behavior with hydrodynamic interactions. Phys. Rev. Lett.
**2018**, 120, 198101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lighthill, M. Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. B Biol. Sci.
**1971**, 179, 125–138. [Google Scholar] - Gopalkrishnan, R.; Triantafyllou, M.; Triantafyllou, G.; Barrett, D. Active vorticity control in a shear flow using a flapping foil. J. Fluid Mech.
**1994**, 274, 1–21. [Google Scholar] [CrossRef] [Green Version] - Warkentin, J.; DeLaurier, J. Experimental aerodynamic study of tandem flapping membrane wings. J. Aircr.
**2007**, 44, 1653–1661. [Google Scholar] [CrossRef] [Green Version] - Verma, S.; Novati, G.; Koumoutsakos, P. Efficient collective swimming by harnessing vortices through deep reinforcement learning. Proc. Natl. Acad. Sci. USA
**2018**, 115, 5849–5854. [Google Scholar] [CrossRef] [Green Version] - Liao, J.C.; Beal, D.N.; Lauder, G.V.; Triantafyllou, M.S. The Kármán gait: Novel body kinematics of rainbow trout swimming in a vortex street. J. Exp. Biol.
**2003**, 206, 1059–1073. [Google Scholar] [CrossRef] [Green Version] - Lauder, G.V.; Tytell, E.D. Hydrodynamics of undulatory propulsion. Fish Physiol.
**2005**, 23, 425–468. [Google Scholar] - Tytell, E.D.; Lauder, G.V. The hydrodynamics of eel swimming: I. Wake structure. J. Exp. Biol.
**2004**, 207, 1825–1841. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Buchholz, J.H.; Smits, A.J. The wake structure and thrust performance of a rigid low-aspect-ratio pitching panel. J. Fluid Mech.
**2008**, 603, 331–365. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dong, H.; Mittal, R.; Najjar, F. Wake topology and hydrodynamic performance of low-aspect-ratio flapping foils. J. Fluid Mech.
**2006**, 566, 309–343. [Google Scholar] [CrossRef] [Green Version] - Von Ellenrieder, K.D.; Parker, K.; Soria, J. Flow structures behind a heaving and pitching finite-span wing. J. Fluid Mech.
**2003**, 490, 129–138. [Google Scholar] [CrossRef] - Senturk, U.; Smits, A.J. Numerical simulations of the flow around a square pitching panel. J. Fluids Struct.
**2018**, 76, 454–468. [Google Scholar] [CrossRef] - Jeong, J.; Hussain, F. On the identification of a vortex. J. Fluid Mech.
**1995**, 285, 69–94. [Google Scholar] [CrossRef] - Wagner, H. Über die Entstehung des dynamischen Auftriebes von Tragflügeln. J. Appl. Math. Mech. Für Angew. Math. Und Mech.
**1925**, 5, 17–35. [Google Scholar] [CrossRef] [Green Version] - von Kármán, T.; Sears, W.R. Airfoil theory for non-uniform motion. J. Aeronaut. Sci.
**1938**, 5, 379–390. [Google Scholar] [CrossRef] - Theodorsen, T. General Theory of Aerodynamic Instability and the Mechanism of Flutter; Technical report; Ames Research Center, NASA: Mountain View, CA, USA, 1935.
- Garrick, I.E. Propulsion of a Flapping and Oscillating Airfoil; Technical Report; Langley Memorial Aeronautical Laboratory: Langley Field, VA, USA, 1937. [Google Scholar]
- Bisplinghoff, R.L.; Ashley, H.; Halfman, R.L. Aeroelasticity; Courier Corporation, Dover Publications: Minoela, NY, USA, 2013. [Google Scholar]
- Brennen, C.E. A Review of Added Mass and Fluid Inertial Forces; Technical Report; Naval Civil Engineering Laboratory: Sierra Madre, CA, USA, 1982. [Google Scholar]
- Ayancik, F.; Zhong, Q.; Quinn, D.B.; Brandes, A.; Bart-Smith, H.; Moored, K.W. Scaling laws for the propulsive performance of three-dimensional pitching propulsors. J. Fluid Mech.
**2019**, 871, 1117–1138. [Google Scholar] [CrossRef] [Green Version] - Helmbold, H.B. Der unverwundene ellipsenflugel als tragende flanche. Jahrbuch
**1942**, I111–I113. [Google Scholar] - Anderson, J.D. Aircraft Performance and Design; WCB/McGraw-Hill: Boston, MA, USA, 1999. [Google Scholar]
- Taira, K.; Colonius, T. Three-dimensional flows around low-aspect-ratio flat-plate wings at low Reynolds numbers. J. Fluid Mech.
**2009**, 623, 187–207. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Schematics of the canonical arrangements; in-line (

**a**), side-by-side (

**b**), and tip-to-tip (

**c**).

**Figure 2.**Schematics of two different wake topologies behind a finite-span pitching hydrofoil where (

**a**) vortex rings bifurcate away from each other in the cross-stream direction as they advect downstream, commonly seen in eel-like swimming, and (

**b**) interconnected vortex rings advect downstream, commonly seen in fish-like swimming.

**Figure 3.**(

**a**) A schematic of the tomo PIV experimental facility. (

**b**) A schematic of the force measurement facility. (

**c**) A detailed schematic of the actuation mechanism. (

**d**) A schematic of the interacting hydrofoils’ spatial arrangement showing the streamwise, ${X}^{*}$, and cross-stream spacing, ${Y}^{*}$, respectively.

**Figure 4.**Wake structures shedding from the trailing edge of an isolated wing shown with ${\lambda}_{2}=-0.07$ isocontours at the dimensionless time of ${t}^{*}=0.25$, as an isometric (

**a**), side (

**b**), and top (

**c**) view of the flowfield, and vorticity contours (${\omega}_{z}$) in the mid-span plane (

**d**).

**Figure 5.**Time averaged velocity field components, $(u,v)$, normalized with the free-stream velocity, U, for the isolated wing in the mid-span plane (

**a**,

**b**), and different YZ planes (

**c**,

**d**).

**Figure 6.**Isometric (top row;

**a**,

**b**) and side view (bottom row;

**c**,

**d**) of three-dimensional vortex structures for the in-line (left column;

**a**,

**c**) and the staggered case (right column;

**b**,

**d**) shown at the time instant, ${t}^{*}=1$. The vortex structures are defined by the isosurface ${\lambda}_{2}=-0.07$ and are colored with corresponding values of spanwise vorticity.

**Figure 7.**Time-averaged velocity for the in-line arrangement (left column;

**a**,

**c**,

**e**) and the staggered arrangement (right column;

**b**,

**d**,

**f**). The top row (

**a**,

**b**) presents the $|{v}^{*}|=0.5$ isocontour colored with their corresponding time-averaged velocity, ${u}^{*}$. The middle row (

**c**,

**d**) presents the time-averaged streamwise velocity at the mid-span plane, while the bottom row (

**e**,

**f**) presents the time-averaged cross-stream velocity at the mid-span. Arcs and lines with arrows show the expansion of the streamwise accelerated flow branches, and the distance between the leading edge of the follower and the time-averaged downwash region, respectively.

**Figure 8.**Locations of the leading edge of the follower foil (x markers) relative to the time-averaged velocity jets from the flowfield of the isolated leader.

**Figure 9.**Lift of the follower hydrofoil as a function of the synchrony and cross-stream spacing from the experiments (

**a**) and the three-dimensional quasi-steady theory (

**b**). The follower is at a fixed streamwise spacing of ${X}^{*}=0.75$ for all the cases presented here.

**Figure 10.**Normalized thrust coefficients of the follower hydrofoil acquired from the experiments (

**a**) and the theoretical solution (

**b**) as a function of cross stream spacing and synchrony. Three high thrust regions are denoted on the experimental data with white circles. The temporal variation of thrust coefficient within one oscillation cycle for the marked peak thrust cases compared to isolated foil case are given in (

**c**). For each case, instantaneous thrust forces were phase averaged over 100 oscillation cycles.

${X}^{*}$ | $0.75$ | |

${Y}^{*}$ | 0–0.6 | $0.2$ increments |

$\varphi $ | 0–2$\pi $ | $\pi /12$ increments |

k | $3.1$ | |

$St$ | $0.8$ | |

${\theta}_{0}$ | 7.5° | ($A/c=0.26$) |

f | 1.5 |

**Table 2.**Time-averaged net thrust, power, lift and drag coefficients, as well as propulsive efficiency of an isolated hydrofoil at $St=0.8$ and $k=3.1$. $\pm (\xb7)$ represents the standard deviation calculated from 10 experimental trials.

Coefficients | |
---|---|

${C}_{T,\mathrm{iso}}$ | $0.47\pm 0.13$ |

${C}_{P,\mathrm{iso}}$ | $7.65\pm 0.11$ |

${C}_{L,\mathrm{iso}}$ | $0.26\pm 0.11$ |

${C}_{D,\mathrm{iso}}$ | $0.43\pm 0.04$ |

${\eta}_{,\mathrm{iso}}$ | $0.06\pm 0.01$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kurt, M.; Eslam Panah, A.; Moored, K.W.
Flow Interactions Between Low Aspect Ratio Hydrofoils in In-line and Staggered Arrangements. *Biomimetics* **2020**, *5*, 13.
https://doi.org/10.3390/biomimetics5020013

**AMA Style**

Kurt M, Eslam Panah A, Moored KW.
Flow Interactions Between Low Aspect Ratio Hydrofoils in In-line and Staggered Arrangements. *Biomimetics*. 2020; 5(2):13.
https://doi.org/10.3390/biomimetics5020013

**Chicago/Turabian Style**

Kurt, Melike, Azar Eslam Panah, and Keith W. Moored.
2020. "Flow Interactions Between Low Aspect Ratio Hydrofoils in In-line and Staggered Arrangements" *Biomimetics* 5, no. 2: 13.
https://doi.org/10.3390/biomimetics5020013