Hydromechanics of lunate-tail swimming propulsion

1974 ◽  
Vol 64 (2) ◽  
pp. 375-392 ◽  
Author(s):  
M. G. Chopra
Keyword(s):  
Aspect Ratio ◽  
Leading Edge ◽  
Vortex Sheet ◽  
The Body ◽  
Uniform Motion ◽  
Physical Parameters ◽  
Thrust Coefficient ◽  
Propulsive Efficiency ◽  
Reduced Frequency ◽  
General Motion

This paper investigates the non-uniform motion of a thin plate of finite aspect ratio, with a rounded leading edge and sharp trailing edge, executing heaving and pitching oscillations at zero mean lift. Such vertical motions characterize the horizontal lunate tails with which cetacean mammals propel themselves, and the same motions, turned through 90° to become horizontal motions of sideslip and yaw, characterize the vertical lunate tails of certain fast-swimming fishes. An oscillating vortex sheet consisting of streamwise and spanwise components is shed to trail behind the body and it is this additional feature of the streamwise component resulting from the finiteness of the plate that makes this study a generalization of the two-dimensional treatment of lunate-tail propulsion by Lighthill (1970). The forward thrust, the power required, the energy imparted to the wake and the hydromechanical propulsive efficiency are determined for this general motion as functions of the physical parameters defining the problem: namely the aspect ratio, the reduced frequency, the feathering parameter and the position of the pitching axis. The dependence of the thrust coefficient and propulsive efficiency on these physical parameters, for the complete range of variation consistent with the assumptions of the problem, has been depicted graphically.

Note on optimum propulsion of heaving and pitching airfoils from linear potential theory

2017 ◽  
Vol 826 ◽  
pp. 781-796 ◽  
Author(s):  
R. Fernandez-Feria
Keyword(s):  
Phase Shift ◽  
Potential Theory ◽  
Linear Theory ◽  
Leading Edge ◽  
Linear Potential ◽  
Thrust Coefficient ◽  
Propulsive Efficiency ◽  
Pitching Motion ◽  
Reduced Frequency ◽  
Linear Potential Theory

The conditions that maximize the propulsive efficiency of a heaving and pitching airfoil are analysed using a novel formulation for the thrust force within the linear potential theory. Stemming from the vortical impulse theory, which correctly predicts the decay of the thrust efficiency as the inverse of reduced frequency$k$for large$k$(Fernandez-Feria,Phys. Rev. Fluids, vol. 1, 2016, 084502), the formulation is corrected here at low frequencies by adding a constant representing the viscous drag. It is shown first that the thrust coefficient and propulsive efficiency thus computed agree quite well with several sets of available experimental data, even for not so small flapping amplitudes. For a pure pitching motion, it is found that the maximum propulsion efficiency is reached for the airfoil pitching close to the three-quarter chord point from the leading edge with a relatively large reduced frequency, corresponding to a relatively low thrust coefficient. According to the theory, this efficiency peak may approach unity. For smaller$k$, other less pronounced local maxima of the propulsive efficiency are attained for pitching points ahead of the leading edge, with larger thrust coefficients. The linear theory also predicts that no thrust is generated at all for a pitching axis located between the three-quarter chord point and the trailing edge. These findings contrast with the results obtained from the classical linear thrust by Garrick, with the addition of the same quasi-static thrust, which are also computed in the paper. For a combined heaving and pitching motion, the behaviour of the propulsive efficiency in relation to the pitching axis is qualitatively similar to that found for a pure pitching motion, for given fixed values of the feathering parameter (ratio between pitching and heaving amplitudes) and of the phase shift between the pitching and heaving motions. The peak propulsive efficiency predicted by the linear theory is for an airfoil with a pitching axis close to, but ahead of, the three-quarter chord point, with a relatively large reduced frequency, a feathering parameter of approximately$0.9$and a phase shift slightly smaller than $90^{\circ }$.

Supersonic Flow Past Wing-Body Combinations

1953 ◽  
Vol 4 (3) ◽  
pp. 287-314 ◽  
Author(s):  
W. Chester
Keyword(s):  
Supersonic Flow ◽  
Aspect Ratio ◽  
Leading Edge ◽  
The Body ◽  
Infinite Cylinder ◽  
Main Stream ◽  
Exact Equation ◽  
Body Of Revolution ◽  
Flow Past ◽  
General Conditions

SummaryThe supersonic flow past a combination of a thin wing and a slender body of revolution is discussed by means of the linearised equation of motion. The exact equation is first established so that the linearised solution can be fed back and the order of the error terms calculated. The theory holds under quite general conditions which should be realised in practice.The wing-body combination considered consists of a wing symmetrically situated on a pointed body of revolution and satisfying the following fairly general conditions. The wing leading edge is supersonic at the root, and the body is approximately cylindrical downstream of the leading edge. The body radius is of an order larger than the wing thickness, but is small compared with the chord or span of the wing.It is found that if the wing and body are at the same incidence, and the aspect ratio of the wing is greater than 2 (M2-1)-½, where M is the main stream Mach number, the lift is equivalent to that of the complete wing when isolated. If the wing only is at incidence then the lift is equivalent to that of the part of the wing lying outside the body.The presence of the body has a more significant effect on the drag. If, for example, the body is an infinite cylinder of radius a, and the wing is rectangular with aspect ratio greater than 2(M2-1)-½, then the drag of the wing is decreased by a factor (1-2a/b), where 2b is the span of the wing.When these conditions do not hold the results are not quite so simple but are by no means complicated.

Experimental Study of Very Low Aspect Ratio Wings in Slender Bodies

2012 ◽  
Author(s):  
M. A. Arevalo-Campillos ◽  
S. Tuling ◽  
L. Parras ◽  
C. del Pino ◽  
L. Dala
Keyword(s):  
Experimental Study ◽  
Aspect Ratio ◽  
Numerical Simulations ◽  
Water Channel ◽  
Vortex Sheet ◽  
The Body ◽  
Low Aspect Ratio ◽  
Moderate Reynolds Number ◽  
Slender Bodies ◽  
Low Aspect Ratio Wings

The dynamics of very low aspect ratio wings (or strakes) vortices in slender bodies are complex due to the interaction of the shed vortex sheet and the body vortex. For missiles at supersonic speeds these interactions are not easily predicted using engineering level tools. To shed some new light onto this problem, an experimental study in a water channel for moderate Reynolds number (Re = 1000) was performed for a 19D body and strake configuration with strakes having a span to body diameter ratio of 1.25. Comparisons to numerical simulations in supersonic flow are also performed. Flow visualisation has been carried out to characterize the vortex dynamics at different angles of attack; these being 11°, 16°, 22° and 27°. The comparison between a slender body without strakes and the body-strake configuration has given some key indicators in relation to the vortex position of the core. Furthermore, unsteady wing-body interference has been observed at angles of attack above 20° for both experimental and numerical simulations. Consequently, the average position of the vortex core is located at larger distances from the missile in comparison to the body without strakes. The numerical simulations show good correlation with the experimental tests even though the dynamic convective interactions between the body vortex and strake vortex sheet are not predicted.

Optimal flexibility of a flapping appendage in an inviscid fluid

2008 ◽  
Vol 614 ◽  
pp. 355-380 ◽  
Author(s):  
SILAS ALBEN
Keyword(s):  
Power Law ◽  
Leading Edge ◽  
Vortex Sheet ◽  
Input Power ◽  
The Body ◽  
Flexible Body ◽  
Inviscid Fluid ◽  
Driving Frequency ◽  
Pitch Frequency ◽  
New Formulation

We present a new formulation of the motion of a flexible body with a vortex-sheet wake and use it to study propulsive forces generated by a flexible body pitched periodically at the leading edge in the small-amplitude regime. We find that the thrust power generated by the body has a series of resonant peaks with respect to rigidity, the highest of which corresponds to a body flexed upwards at the trailing edge in an approximately one-quarter-wavelength mode of deflection. The optimal efficiency approaches 1 as rigidity becomes small and decreases to 30–50% (depending on pitch frequency) as rigidity becomes large. The optimal rigidity for thrust power increases from approximately 60 for large pitching frequency to ∞ for pitching frequency 0.27. Subsequent peaks in response have power-law scalings with respect to rigidity and correspond to higher-wavenumber modes of the body. We derive the power-law scalings by analysing the fin as a damped resonant system. In the limit of small driving frequency, solutions are self-similar at the leading edge. In the limit of large driving frequency, we find that the distribution of resonant rigidities ~k−5, corresponding to fin shapes with wavenumber k. The input power and output power are proportional to rigidity (for small-to-moderate rigidity) and to pitching frequency (for moderate-to-large frequency). We compare these results with the range of rigidity and flapping frequency for the hawkmoth forewing and the bluegill sunfish pectoral fin.

Numerical Study on Thrust Generation Performance of Plunging Airfoils

2013 ◽  
Vol 312 ◽  
pp. 235-238
Author(s):  
Ji Gao ◽  
Rui Shan Yuan ◽  
Ming Hui Zhang ◽  
Yong Hui Xie
Keyword(s):  
Viscous Flow ◽  
Numerical Study ◽  
Angle Of Attack ◽  
Leading Edge ◽  
Thrust Coefficient ◽  
Propulsive Efficiency ◽  
Propulsion Performance ◽  
Incompressible Viscous Flow ◽  
Thrust Generation ◽  
The Difference

In this paper, the effects of angle of attack, camber and camber location on propulsion performance of flapping airfoils undergoing plunging motion were numerically studied at Re=20000 and h=0.175. The unsteady incompressible viscous flow around four different airfoil sections was simulated applying the dynamic mesh. The results show that the time averaged thrust coefficient CTmean and propulsive efficiency η of the symmetric airfoil decrease with the increasing angle of attack, and the variation of CTmean is more obvious than that of CPmean. Both CTmean and η for NACA airfoils studied in this paper decrease with the increasing camber and the difference between the propulsion performances of different airfoils is not obvious, and the thrust generation and power of various NACA airfoils gradually increase during the downstroke and decrease during the upstroke. Under the same conditions, the airfoil with a further distance between the maximum camber location and the chord of the leading edge leads to higher propulsive efficiency.

scholarly journals Influence of Thickness Variation on the Flapping Performance of Symmetric NACA Airfoils in Plunging Motion

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Liangyu Zhao ◽  
Shuxing Yang
Keyword(s):  
Lift Coefficient ◽  
Parametric Method ◽  
Leading Edge ◽  
Thickness Variation ◽  
Thrust Coefficient ◽  
Propulsive Efficiency ◽  
Maximum Thickness ◽  
Class Function ◽  
Edge Vortex ◽  
The Impact

In order to investigate the impact of airfoil thickness on flapping performance, the unsteady flow fields of a family of airfoils from an NACA0002 airfoil to an NACA0020 airfoil in a pure plunging motion and a series of altered NACA0012 airfoils in a pure plunging motion were simulated using computational fluid dynamics techniques. The “class function/shape function transformation“ parametric method was employed to decide the coordinates of these altered NACA0012 airfoils. Under specified plunging kinematics, it is observed that the increase of an airfoil thickness can reduce the leading edge vortex (LEV) in strength and delay the LEV shedding. The increase of the maximum thickness can enhance the time-averaged thrust coefficient and the propulsive efficiency without lift reduction. As the maximum thickness location moves towards the leading edge, the airfoil obtains a larger time-averaged thrust coefficient and a higher propulsive efficiency without changing the lift coefficient.

Hydromechanics of lunate-tail swimming propulsion. Part 2

1977 ◽  
Vol 79 (1) ◽  
pp. 49-69 ◽  
Author(s):  
M. G. Chopra ◽  
T. Kambe
Keyword(s):  
Swimming Performance ◽  
Leading Edge ◽  
Angular Frequency ◽  
Optimum Shape ◽  
Physical Parameters ◽  
Sweep Angle ◽  
Propulsive Efficiency ◽  
Generalized Force ◽  
Total Thrust ◽  
The Mean

This paper investigates the propulsive performance of the lunate tails of aquatic animals achieving high propulsive efficiency (the hydromechanical efficiency being defined as the ratio of the work done by the mean forward thrust to the mean rate at which work is done by the tail movements on the surrounding fluid). Small amplitude heaving and pitching motions of a finite flat-plate wing of general planform with a rounded leading edge and a sharp trailing edge are considered. This is a generalization of Chopra's (1974) work on model rectangular tails. This motion characterizes vertical oscillations of the horizontal tail flukes of some cetacean mammals. The same oscillations, turned through a right angle to become horizontal motions of side-slip and yaw, characterize the caudal fins of certain fast-swimming fishes; viz. wahoo, tunny, wavyback skipjack, etc., from the Percomorphi and whale shark, porbeagle, etc., from the Selachii. Davies’ (1963, 1976) method of finding the loading distribution on the wing and generalized force coefficients, through approximate solution of an integral equation relating the loading and the upwash (lifting-surface theory), is used to find the total thrust and the rate of working of the tail, which in turn specify the hydromechanical swimming performance of the animals. The physical parameters concerned are the tail aspect ratio ((span)2/planform area), the reduced frequency (angular frequency x typical length/forward speed), the feathering parameter (the ratio of the tail slope to the slope of the path of the pitching axis), the position of the pitching axis, and the curved shapes of the leading and trailing edges. The variation of the thrust and the propulsive efficiency with these parameters has been discussed to indicate the optimum shape of the tail. It is found that, compared with a rectangular tail, a curved leading edge as in lunate tails gives a reduced thrust contribution from the leading-edge suction for the same total thrust; however, a sweep angle of the leading edge exceeding about 30° leads to a marked reduction of efficiency. Another implication of the present analysis is that no negative work is involved in the actual oscillation of the tail.The present results are used to obtain an estimate of the drag coefficient for the motion of the animals, based on observed data and the computed thrust. The results show some evidence of differences between the CD's for cetacean mammals and scombroid fish respectively. Some discussion of this difference is also given.

scholarly journals Dynamic Stall Characteristics of Pitching Swept Finite-Aspect-Ratio Wings

Fluids ◽  
2021 ◽  
Vol 6 (12) ◽  
pp. 457
Author(s):  
Al Habib Ullah ◽  
Kristopher L. Tomek ◽  
Charles Fabijanic ◽  
Jordi Estevadeordal
Keyword(s):  
Aspect Ratio ◽  
Separation Bubble ◽  
Leading Edge ◽  
Dynamic Stall ◽  
Trailing Edge ◽  
Sweep Angle ◽  
Pitching Motion ◽  
Naca0012 Airfoil ◽  
Phase Average ◽  
Reduced Frequency

An experimental investigation regarding the dynamic stall of various swept wing models with pitching motion was performed to analyze the effect of sweep on the dynamic stall. The experiments were performed on a wing with a NACA0012 airfoil section with an aspect ratio of AR = 4. The experimental study was conducted for chord-based Reynolds number Rec =2×105 and freestream Mach number Ma=0.1. First, a ‘particle image velocimetry’ (PIV) experiment was performed on the wing with three sweep angles, Λ=0o, 15o, and 30o, to obtain the flow structure at several wing spans. The results obtained at a reduced frequency showed that a laminar separation bubble forms at the leading edge of the wing during upward motion. As the upward pitching motion continues, a separation burst occurs and shifts towards the wing trailing edge. As the wing starts to pitch downward, the growing dynamic stall vortex (DSV) vortex sheds from the wing’s trailing edge. With the increasing sweep angle of the wing, the stall angle is delayed during the dynamic motion of the wing, and the presence of DSV shifts toward the wingtip. During the second stage, a ‘turbo pressure-sensitive paint’ (PSP) technique was deployed to obtain the phase average of the surface pressure patterns of the DSV at a reduced frequency, k=0.1. The phase average of pressure shows a distinct pressure map for two sweep angles, Λ=0o, 30o, and demonstrates a similar trend to that presented in the published computational studies and the experimental data obtained from the current PIV campaign.

scholarly journals The wake structure and thrust performance of a rigid low-aspect-ratio pitching panel

2008 ◽  
Vol 603 ◽  
pp. 331-365 ◽  
Author(s):  
JAMES H. J. BUCHHOLZ ◽  
ALEXANDER J. SMITS
Keyword(s):  
Aspect Ratio ◽  
Strouhal Number ◽  
Three Dimensional ◽  
Leading Edge ◽  
Horseshoe Vortex ◽  
Dimensional Structure ◽  
Peak Efficiency ◽  
Thrust Coefficient ◽  
Wake Structure ◽  
Thrust Performance

Thrust performance and wake structure were investigated for a rigid rectangular panel pitching about its leading edge in a free stream. For ReC = O(104), thrust coefficient was found to depend primarily on Strouhal number St and the aspect ratio of the panel AR. Propulsive efficiency was sensitive to aspect ratio only for AR less than 0.83; however, the magnitude of the peak efficiency of a given panel with variation in Strouhal number varied inversely with the amplitude to span ratio A/S, while the Strouhal number of optimum efficiency increased with increasing A/S. Peak efficiencies between 9% and 21% were measured. Wake structures corresponding to a subset of the thrust measurements were investigated using dye visualization and digital particle image velocimetry. In general, the wakes divided into two oblique jets; however, when operating at or near peak efficiency, the near wake in many cases represented a Kármán vortex street with the signs of the vortices reversed. The three-dimensional structure of the wakes was investigated in detail for AR = 0.54, A/S = 0.31 and ReC = 640. Three distinct wake structures were observed with variation in Strouhal number. For approximately 0.20 < St < 0.25, the main constituent of the wake was a horseshoe vortex shed by the tips and trailing edge of the panel. Streamwise variation in the circulation of the streamwise horseshoe legs was consistent with a spanwise shear layer bridging them. For St > 0.25, a reorganization of some of the spanwise vorticity yielded a bifurcating wake formed by trains of vortex rings connected to the tips of the horseshoes. For St > 0.5, an additional structure formed from a perturbation of the streamwise leg which caused a spanwise expansion. The wake model paradigm established here is robust with variation in Reynolds number and is consistent with structures observed for a wide variety of unsteady flows. Movies are available with the online version of the paper.

scholarly journals Prandtl–Batchelor flow past a flat plate with a forward-facing flap

1984 ◽  
Vol 143 ◽  
pp. 351-365 ◽  
Author(s):  
P. G. Saffman ◽  
S. Tanveer
Keyword(s):  
Flat Plate ◽  
Leading Edge ◽  
Vortex Sheet ◽  
Inviscid Flow ◽  
The Body ◽  
Two Dimensional ◽  
Rear Edge ◽  
Flow Past ◽  
Range Of Values ◽  
Streamline Method

Two-dimensional steady inviscid flow past an inclined flat plate with a forward-facing flap attached to the rear edge is considered for the case when a vortex sheet separates from the leading edge of the flat plate and reattaches at the leading edge of the flap, with uniform vorticity distributed between the vortex sheet and the body. Solutions are found for a particular geometry and a range of values of the vorticity. The method used to calculate the flow is an extension of a free-streamline method widely used in cases where the velocity is a constant on the separating streamline.

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