Cetacean propulsion by a periodic flapping motion of their fluke is considered and studied on a benchmark flexible straight wing. The aim of the study was to validate low-order models for this configuration. First, the two-dimensional rigid case is investigated, comparing the aerodynamic performance of the airfoil periodic motion vs the reduced frequency, with published data and unsteady Reynolds-averaged numerical simulation results. It appears that viscous drag modeling must be added to the discrete vortex method, in order to obtain sensible thrust results, for Garrick frequencies below 2. All high- and low-order models agree at the remarkable Garrick frequency of 1.82, although the experiment shows a lower efficiency of about 25%. The positions of the shed vortices match comparing the unsteady Reynolds-averaged numerical simulation and the discrete vortex method. Then, the three-dimensional leading-edge-suction-parameter modulated discrete vortex method is extended, by means of a lifting line theory. A modification of the method is proposed in order to consider wing dihedral, resulting from the spanwise flexibility. The configuration considers a reduced frequency of 1.82. Three types of spanwise wing flexibility are examined. For the inflexible and flexible cases, a reasonable agreement is observed between the different methods for each coefficient. The intermediate flexible wing provides a better thrust coefficient, while excessive flexibility proves to be detrimental. Vorticity fields are compared with previously published data for the three wings. For the highly flexible wing and the right choice of deformation parameters, the discrete vortex method produces reliable results.
References
1.
P.
Liu
and N.
Bose
, “Propulsive performance from oscillating propulsors with spanwise flexibility
,” Proc. R. Soc. London, Ser. A
453
, 1763
–1770
(1997
).2.
F. E.
Fish
, “Comparative kinematics and hydrodynamics of odontocete cetaceans: Morphological and ecological correlates with swimming performance
,” J. Exp. Biol.
201
, 2867
–2877
(1998
).3.
F. E.
Fish
, C. M.
Schreiber
, K. W.
Moored
, G.
Liu
, H.
Dong
, and H.
Bart-Smith
, “Hydrodynamic performance of aquatic flapping: Efficiency of underwater flight in the manta
,” Aerospace
3
, 20
(2016
).4.
F.
Ayancik
, F. E.
Fish
, and K. W.
Moored
, “Three-dimensional scaling laws of cetacean propulsion characterize the hydrodynamic interplay of flukes' shape and kinematics
,” J. R. Soc. Interface
17
, 20190655
(2020
).5.
X.
Wu
, X.
Zhang
, X.
Tian
, X.
Li
, and W.
Lu
, “A review on fluid dynamics of flapping foils
,” Ocean Eng.
195
, 106712
(2020
).6.
I. E.
Garrick
, “Propulsion of a flapping and oscillating airfoil
,” Technical Report No. NACA TN-D-85
, 1937
.7.
G. C.
Lewin
and H.
Haj-Hariri
, “Modelling thrust generation of a two-dimensional heaving airfoil in a viscous flow
,” J. Fluid Mech.
492
, 339
–362
(2003
).8.
K.
Kamrani Fard
, V.
Ngo
, and J. A.
Liburdy
, “A leading-edge vortex initiation criteria for large amplitude foil oscillations using a discrete vortex model
,” Phys. Fluids
33
, 115123
(2021
).9.
J.
Young
and J. C.
Lai
, “Oscillation frequency and amplitude effects on the wake of a plunging airfoil
,” AIAA J.
42
, 2042
–2052
(2004
).10.
J.
Young
, “Numerical simulation of the unsteady aerodynamics of flapping airfoils
,” Ph.D. thesis (University of New South Wales
, 2005
).11.
S.
Heathcote
, Z.
Wang
, and I.
Gursul
, “Effect of spanwise flexibility on flapping wing propulsion
,” J. Fluids Struct.
24
, 183
–199
(2008
).12.
R. E.
Gordnier
, S. K.
Chimakurthi
, C. E.
Cesnik
, and P. J.
Attar
, “High-fidelity aeroelastic computations of a flapping wing with spanwise flexibility
,” J. Fluids Struct.
40
, 86
–104
(2013
).13.
T.
Simonet
, K.
Roncin
, L.
Lapierre
, and L.
Daridon
, “Étude de l'efficience d'un système de propulsion maritime par foil oscillant souple
,” in 17èmes Journées de L'Hydrodynamique
(2020
).14.
J.
Deparday
and K.
Mulleners
, “Modeling the interplay between the shear layer and leading edge suction during dynamic stall
,” Phys. Fluids
31
, 107104
(2019
).15.
F. F.
Siala
, K. K.
Fard
, and J. A.
Liburdy
, “Experimental study of inertia-based passive flexibility of a heaving and pitching airfoil operating in the energy harvesting regime
,” Phys. Fluids
32
, 017101
(2020
).16.
E.
Bøckmann
, “Wave propulsion of ships
,” Ph.D. thesis (Norwegian University of Science and Technology
, 2015
).17.
T.
Faure
and C.
Leogrande
, “High angle-of-attack aerodynamics of a straight wing with finite span using a discrete vortex method
,” Phys. Fluids
32
, 104109
(2020
).18.
J. G.
Leishman
and T.
Beddoes
, “A semi-empirical model for dynamic stall
,” J. Am. Helicopter Soc.
34
, 3
–17
(1989
).19.
P. J.
Moriarty
and A. C.
Hansen
, “AeroDyn theory manual
,” Technical Report No. NREL/TP-500-36881
(National Renewable Energy Lab
., Golden, CO
, 2005
).20.
D.
Cleaver
, Z.
Wang
, and I.
Gursul
, “Vortex mode bifurcation and lift force of a plunging airfoil at low Reynolds numbers
,” in 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
(AIAA
, 2010
), p. 390
.21.
R. E.
Sheldahl
and P. C.
Klimas
, “Aerodynamic characteristics of seven symmetrical airfoil sections through 180-degree angle of attack for use in aerodynamic analysis of vertical axis wind turbines
,” Technical Report No. SAND-80-2114
(Sandia National Labs
., Albuquerque, NM
, 1981
).22.
C.
Duport
, J.-B.
Leroux
, K.
Roncin
, C.
Jochum
, and Y.
Parlier
, “Benchmarking of a 3D non-linear lifting line method against 3D RANSE simulations
,” La Houille Blanche
105
, 70
–73
(2019
).23.
C.
Duport
, J. B.
Leroux
, K.
Roncin
, C.
Jochum
, and Y.
Parlier
, “Comparison of 3D non-linear lifting line method calculations with 3D RANSE simulations and application to the prediction of the global loading on a cornering kite
,” in 15ème Journées de l'Hydrodynamique
(2016
).24.
W. F.
Phillips
and D.
Snyder
, “Modern adaptation of Prandtl's classic lifting-line theory
,” J. Aircr.
37
, 662
–670
(2000
).25.
M.
Galbraith
and M.
Visbal
, “Implicit large eddy simulation of low Reynolds number flow past the SD7003 airfoil
,” in 46th AIAA Aerospace Sciences Meeting and Exhibit
(AIAA
, 2008
), p. 225
.26.
P.
Catalano
and R.
Tognaccini
, “Influence of free-stream turbulence on simulations of laminar separation bubbles
,” in 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
(AIAA
, 2009
), p. 1471
.27.
P. R.
Spalart
and M. K.
Strelets
, “Mechanisms of transition and heat transfer in a separation bubble
,” J. Fluid Mech.
403
, 329
–349
(2000
).28.
A.
Crivellini
and V.
D'Alessandro
, “Spalart–Allmaras model apparent transition and RANS simulations of laminar separation bubbles on airfoils
,” Int. J. Heat Fluid Flow
47
, 70
–83
(2014
).29.
M.
Ol
, B.
McAuliffe
, E.
Hanff
, U.
Scholz
, and C.
Kähler
, “Comparison of laminar separation bubble measurements on a low Reynolds number airfoil in three facilities
,” in 35th AIAA Fluid Dynamics Conference and Exhibit
(AIAA
, 2005
), p. 5149
.30.
K.
Ramesh
, A.
Gopalarathnam
, K.
Granlund
, M. V.
Ol
, and J. R.
Edwards
, “Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding
,” J. Fluid Mech.
751
, 500
–538
(2014
).31.
T. M.
Faure
, L.
Dumas
, V.
Drouet
, and O.
Montagnier
, “A modified discrete-vortex method algorithm with shedding criterion for aerodynamic coefficients prediction at high angle of attack
,” Appl. Math. Modell.
69
, 32
–46
(2019
).32.
T. M.
Faure
, L.
Dumas
, and O.
Montagnier
, “Numerical study of two-airfoil arrangements by a discrete vortex method
,” Theor. Comput. Fluid Dyn.
34
, 79
–103
(2020
).33.
H.
Glauert
, The Elements of Aerofoil and Airscrew Theory
(Cambridge University Press
, 1926
).34.
J.
Katz
and A.
Plotkin
, Low-Speed Aerodynamics
(Cambridge University Press
, 2001
).35.
A.
Saini
, S.
Narsipur
, and A.
Gopalarathnam
, “Leading-edge flow sensing for detection of vortex shedding from airfoils in unsteady flows
,” Phys. Fluids
33
, 087105
(2021
).36.
N.
Chiereghin
, D.
Cleaver
, and I.
Gursul
, “Unsteady lift and moment of a periodically plunging airfoil
,” AIAA J.
57
, 208
–222
(2019
).37.
D.
Floryan
, T.
Van Buren
, and A. J.
Smits
, “Efficient cruising for swimming and flying animals is dictated by fluid drag
,” Proc. Natl. Acad. Sci. U.S.A.
115
, 8116
–8118
(2018
).38.
L.
Prandtl
, “Applications of modern hydrodynamics to aeronautics
,” Technical Report No. 116 NACA
(1923
).39.
K.
Ramesh
, T. P.
Monteiro
, F. J.
Silvestre
, A. B.
Guimaraes Neto
, T.
de Souza Siqueira Versiani
, and R. G.
Annes da Silva
, “Experimental and numerical investigation of post-flutter limit cycle oscillations on a cantilevered flat plate
,” in International Forum on Aeroelasticity and Structural Dynamics
, IFASD 2017, Como, Italy, 25–28 June 2017 (2017
).40.
K.
Ramesh
, A.
Gopalarathnam
, M. V.
Ol
, K.
Granlund
, and J. R.
Edwards
, “Augmentation of inviscid airfoil theory to predict and model 2D unsteady vortex dominated flows
,” in 41st AIAA Fluid Dynamics Conference and Exhibit
(AIAA
, Honolulu, Hawaii
, 2011
), AIAA Paper No. 2011-3578.41.
E. J.
Chae
and Y. L.
Young
, “Influence of spanwise flexibility on steady and dynamic responses of airfoils vs hydrofoils
,” Phys. Fluids
33
, 067124
(2021
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.