Introduction to a quasi-linear potential theory

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.

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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

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.505 ◽

2018 ◽

Vol 851 ◽

pp. 344-373 ◽

Cited By ~ 7

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.

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The Obstacle Problem in a Non-Linear Potential Theory

Potential Theory ◽

10.1007/978-1-4613-0981-9_27 ◽

1988 ◽

pp. 209-213 ◽

Cited By ~ 2

Author(s):

P. Lehtola

Keyword(s):

Potential Theory ◽

Obstacle Problem ◽

Linear Potential ◽

Non Linear ◽

Linear Potential Theory

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

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.500 ◽

2017 ◽

Vol 826 ◽

pp. 781-796 ◽

Cited By ~ 10

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 $90^{\circ }$.

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