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 $90^{\circ }$.

Unsteady thrust, lift and moment of a two-dimensional flapping thin airfoil in the presence of leading-edge vortices: a first approximation from linear potential theory

2018 ◽  
Vol 851 ◽  
pp. 344-373 ◽  
Author(s):  
R. Fernandez-Feria ◽  
J. Alaminos-Quesada
Keyword(s):  
Potential Theory ◽  
Leading Edge ◽  
Input Power ◽  
Linear Potential ◽  
Two Dimensional ◽  
Thin Airfoil ◽  
Reduced Frequency ◽  
Large Phase ◽  
Linear Potential Theory ◽  
Oscillating Foil

The effect of a leading-edge vortex (LEV) on the lift, thrust and moment of a two-dimensional heaving and pitching thin airfoil is analysed within the unsteady linear potential theory. First, general expressions that take into account the effect of any set of unsteady point vortices interacting with the oscillating foil and unsteady wake are derived. Then, a simplified analysis, based on the Brown–Michael model, of the initial stages of the growing LEV from the sharp leading edge during each half-stroke is used to obtain simple expressions for its main contribution to the unsteady lift, thrust and moment. It is found that the LEV contributes to the aerodynamic forces and moment provided that a pitching motion exists, while its effect is negligible, in the present approximation, for a pure heaving motion, and for some combined pitching and heaving motions with large phase shifts which are also characterized in the present work. In particular, the effect of the LEV is found to decrease with the distance of the pivot point from the trailing edge. Further, the time-averaged lift and moment are not modified by the growing LEVs in the present approximation, and only the time-averaged thrust force is corrected, decreasing slightly in most cases in relation to the linear potential results by an amount proportional to$a_{0}^{2}k^{3}$for large$k$, where$k$is the reduced frequency and$a_{0}$is the pitching amplitude. The time-averaged input power is also modified by the LEV in the present approximation, so that the propulsion efficiency changes by both the thrust and the power, these corrections being relevant only for pivot locations behind the midchord point. Finally, the potential results modified by the LEV are compared with available experimental data.

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.

scholarly journals Computational Analysis of Propulsion Performance of Modified Pitching Motion Airfoils in Laminar Flow

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Yonghui Xie ◽  
Kun Lu ◽  
Di Zhang ◽  
Gongnan Xie
Keyword(s):  
Laminar Flow ◽  
Cfd Modeling ◽  
Shape Effect ◽  
Optimal Range ◽  
Propulsive Efficiency ◽  
Pitching Motion ◽  
Propulsion Performance ◽  
Reduced Frequency ◽  
Thrust Generation ◽  
The Cost

The thrust generation performance of airfoils with modified pitching motion was investigated by computational fluid dynamics (CFD) modeling two-dimensional laminar flow at Reynolds number of 104. The effect of shift distance of the pitch axis outside the chord line(R), reduced frequency(k), pitching amplitude(θ), pitching profile, and airfoil shape (airfoil thickness and camber) on the thrust generated and efficiency were studied. The results reveal that the increase inRandkleads to an enhancement in thrust generation and a decrease in propulsive efficiency. Besides, there exists an optimal range ofθfor the maximum thrust and the increasingθinduces a rapid decrease in propulsive efficiency. Six adjustable parameters(K)were employed to realize various nonsinusoidal pitching profiles. An increase inKresults in more thrust generated at the cost of decreased propulsive efficiency. The investigation of the airfoil shape effect reveals that there exists an optimal range of airfoil thickness for the best propulsion performance and that the vortex structure is strongly influenced by the airfoil thickness, while varying the camber or camber location of airfoil sections offers no benefit in thrust generation over symmetric airfoil sections.

Linear potential theory of steady internal supersonic flow with quasi-cylindrical geometry. Part 1. Flow in ducts

1994 ◽  
Vol 281 ◽  
pp. 159-191 ◽  
Author(s):  
Andreas Dillmann
Keyword(s):  
Supersonic Flow ◽  
Potential Theory ◽  
Broad Class ◽  
Three Dimensional ◽  
Double Series ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linear Potential ◽  
Linear Potential Theory ◽  
Initial Boundary Value

Based on linear potential theory, the general three-dimensional problem of steady supersonic flow inside quasi-cylindrical ducts is formulated as an initial-boundary-value problem for the wave equation, whose general solution arises as an infinite double series of the Fourier–Bessel type. For a broad class of solutions including the general axisymmetric case, it is shown that the presence of a discontinuity in wall slope leads to a periodic singularity pattern associated with non-uniform convergence of the corresponding series solutions, which thus are unsuitable for direct numerical computation. This practical difficulty is overcome by extending a classical analytical method, viz. Kummer's series transformation. A variety of elementary flow fields is presented, whose complex cellular structure can be qualitatively explained by asymptotic laws governing the propagation of small perturbations on characteristic surfaces.

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.

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