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Force generation in avian and aquatic species is of considerable interest for possible engineering applications. The aim of this work is to highlight the theoretical and physical foundations of a new formulation of the unsteady Kutta condition, which postulates a finite pressure difference at the trailing edge of the foil. The condition, necessary to obtain a unique solution and derived from the unsteady Bernoulli equation, implies that the energy supplied for the wing motion generates trailing-edge vortices and their overall effect, which depends on the motion initial parameters, is a jet of fluid that propels the wing. The postulated pressure difference (the value of which should be experimentally obtained) models the trailing-edge velocity difference that generates the thrust-producing jet. Although the average thrust values computed by the proposed method are comparable to those calculated by assuming null pressure difference at the trailing edge, the latter (commonly used) approach is less physically meaningful than the present one, as there is a singularity at the foil trailing edge. Additionally, in biological applications, that is, for autonomous flapping, the differences ought to be more significant, as the corresponding energy requirements should be substantially altered, compared to the studied oscillatory motions.

Generating force by active flapping of fins and wings has become a focus of research in multiple disciplines from zoology through fluid mechanics and mathematics. Strong interest in the field started back in the 1950s with the landmark studies of Weis-Fogh culminating in the paper “Unusual Mechanisms for the Generation of Lift in Flying Animals” [

However, experiments on their own are not sufficient to understand the underlying physical mechanisms as they also rely to some extent on assumptions. Over the years mathematical and computational models were developed and dedicated experiments performed, resulting in a steady flow of high quality scientific works, of which we mention here only a small but relevant sample.

Triantafyllou et al. [

Basu and Hancock [

In a broad review of various panel methods Hess [

In summary, there is no experimental evidence that conclusively supports the notion of the pressure difference at the trailing edge being equal to zero for unsteady motions but, despite the lack of such evidence, it has been commonly assumed in the literature (e.g., see again [

The first step in the formulation of a comprehensive unsteady Kutta condition appears consequently to be the relaxation of the postulated null pressure difference at the trailing edge. This allows considering the variation in direction and magnitude of the velocities in the vicinity of the trailing edge, that is, the formation and shedding of trailing-edge vortices, resulting in the generation of the experimentally observed thrust-producing jet. Such a modified condition, postulating a finite pressure difference at the trailing edge of the oscillating wing, was already implemented by us in two-dimensional and three-dimensional panel methods, presented in La Mantia and Dabnichki [

The current work extends the numerical study reported in our previous publication on the influence of the wake model on thrust generation [

Theoretically it has been firmly established that the potential flow approach is well suited for the prediction of gross hydrodynamic forces acting on moving bodies (see, e.g., [

Following Katz and Plotkin [

Exemplary heaving and pitching oscillatory motion of the chosen NACA 0012 hydrofoil in the

The discretized form of (

As previously stated, the wake has to be modelled. It was indeed chosen to study how the generated forces are affected by the use of two different (typical) modelling approaches (see below the corresponding results). More specifically, the positions of the shed wake panels were obtained either (i) by estimating the influence on each wake panel of the source and doublet distributions of the foil and wake (“Estimated Wake” approach, following [

Additionally, since the foil is moving, the positions of the wake collocation points in the body-fixed coordinate system have to be calculated at each time step starting from their positions in the inertia frame of reference. The body-fixed position of the wake panels closest to the trailing edge, that is, those added at each time step, was set parallel to the chord.

Besides, in our studies the wake panel length

To summarize, the numerical results discussed below are obtained, in exemplary two-dimensional cases, as specified, by using options (i) or (ii) for the wake shape and (iii) for the wake panel length. In the earlier study [

Equation (

The trailing-edge condition, necessary to obtain a unique solution, is derived from the unsteady Bernoulli equation, that is, the conservation of momentum equation for incompressible fluid and irrotational flow. It implies that the second derivative of every involved function, such as the fluid velocity and pressure, exists and is continuous (^{3} is the fluid (water) density, which is assumed to be constant and uniform. At the trailing edge the unsteady Bernoulli equation can be written as

It is important to underline that as the trailing edge is the domain singularity point, the velocity and pressure do not have there a unique value; that is, a pressure difference can be present there. Besides, (

As already anticipated, it is here postulated that the pressure difference at the trailing edge could be finite rather than zero, for unsteady motion. Following La Mantia and Dabnichki [

More generally, such an assumption implies that the energy supplied for the wing motion generates time-dependent trailing-edge vortices. Their overall effect, which depends on the motion initial parameters, is a jet of fluid that propels the wing. As the kinetic energy is transferred from the jet back to the wing, the vortices would disappear and it is consequently not necessary to assume the fluid to be viscous.

The postulated pressure difference at the trailing edge is then fundamental for such a model as it can justify the velocity difference that generates the thrust-producing jet. It has to be noted that the trailing-edge vortices could also rotate in the opposite direction; that is, depending on the initial conditions the generated force can propel the wing or oppose its forward motion.

The mentioned pressure gradient (perpendicular to the wake panel at the trailing edge) can be viewed mathematically as a

The thrust production can consequently be modelled by taking into account the singularity at the foil trailing edge (which exists as the domain is a

Following our previous work [

The wake panel length

In this section we initially show how a different choice of ^{−3}) at the trailing edge of the oscillating foil, at the first time step. A Newton-Raphson iterative procedure was developed to fulfil the constraint. Following (^{−3}, the iterative procedure stops. If ^{−3}, the iterative procedure continues similarly till the imposed condition is fulfilled. The corresponding convergence was reached within a few cycles, for values of

In Figure

Pressure coefficients, at the first time step, as a function of the collocation point position in the body-fixed coordinate system, in the proximity of the trailing edge of the foil. The legend indicates the wake panel lengths.

Figure

Thrust coefficient

As the length of the wake panels increases, the thrust coefficient magnitude becomes larger; that is, if a wake panel is larger, its influence on the unsteady motion results in being larger. It can also be noted that the relation between the thrust coefficient and wake panel length is approximately linear, as it is expected from the equations governing the problem, and that a 50% increase of

For the sake of completeness, in Figure

Thrust coefficient

The size of the linear wake panels has therefore a noticeable influence on the computational results. In some previous studies (see especially La Mantia and Dabnichki [

The latter outcome is also evident in Figure

Thrust coefficient

Similar motion parameters were used to analyse the influence of the wake shape on the numerical results for a sample case (St = 0.3,

If the wake shape is estimated by using the potential formulas, the evaluated thrust coefficient is −0.45 (“Estimated Wake,” option (i) above). If it is imposed that the wake follows the foil path (i.e., the wake panels remain static in the inertia coordinate system),

Figure

Instantaneous thrust

It should be emphasised once more that in the computer program the wake shape, that is, the positions of the wake panels in the global or local frame of reference, has to be set or estimated on the basis of the chosen assumptions, for example, the potential flow theory. However, as it can be seen in Figure

As mentioned above, we believe that the numerical implementation of the flapping-wing problem should be defined on a physical basis. This means that all the assumptions needed in the code development were related to experimental evidence or to theoretical (physically based) models. Those that could not be experimentally verified or theoretically justified were chosen for the sake of simplicity, for example, to have faster computations.

There is then a need for considering the differences between the actual flow conditions and potential assumptions. Such a consideration would provide an appropriate option to properly choose the mathematical model able to better represent the physical problem, once relevant experimental data on the pressure and velocity distribution over flapping wings are available.

It should be noted that the employed numerical procedure affects specifically the local pressure coefficients while the average thrust values seem to be less dependent on the chosen parameters; see also Ol et al. [

The improved understanding of force generation and related energy requirements in flapping flight needs accurate solutions for the corresponding physical problems. Potential methods yield results very close to the experimental ones but often utilise simplified descriptions of the physical phenomena. An appropriate Kutta condition offers the opportunity to more accurately describe existing physical effects by incorporating relevant experimental results. In this work we discussed a novel approach to bridge this gap in the case of unsteady flows around an oscillating hydrofoil. The proposed Kutta condition takes into account an existing domain singularity at the foil trailing edge (generally neglected in the literature) to model closely the physics of the problem, that is, the trailing-edge pressure difference and the resulting thrust-producing jet. The computational results obtained by methods that do not account for such physical features agree well with experimental data but do not explain the thrust-generation mechanisms that are ultimately due to the trailing-edge pressure difference. Further research is needed in order to define theoretically how the pressure in the trailing-edge region should be defined and how this integral measure (in effect a Heaviside step function) could be represented as a function of the foil edge geometry, fluid properties, and flapping kinematics. Such a research should also allow solving the dynamics problem of autonomous flapping-wing movement, as a result of thrust generation, and should be applied to three-dimensional flows, too, on the basis, for example, of the related approach presented in La Mantia and Dabnichki [

The authors declare that there is no conflict of interests regarding the publication of this paper.