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The rate of change ε ( k , t ) = ∂ E ( k , t ) ∕ ∂ t of the energy spectrum E ( k , t ) of a weak turbulence in a system with a decay dispersion relation is investigated with particular interests in the initial stage of evolution such as the first several tens of periods. The theoretical predictions given by the kinetic equation of Hasselmann and that of Janssen which have been derived by the weak turbulence theory are compared with the results of direct numerical simulation (DNS) with a great accuracy. It is shown that ε ( k , t ) predicted by Janssen’s equation correctly reproduces the rapid variation of ε ( k , t ) with the linear time-scale which is observed in DNS, while that predicted by Hasselmann’s equation does not. It is also shown that the transient behavior of the asymptotic approach of Janssen’s ε ( k , t ) to Hasselmann’s one strongly depends on the wavenumber. For wavenumbers smaller than the spectral peak, the approach is exponential, while for wavenumbers larger than the spectral peak it contains a damped oscillation whose amplitude decays in time like 1 ∕ t . A reasonable explanation for the origin of this damped oscillation is given in terms of the frequency mismatch of the three-wave “sum” interactions. [ABSTRACT FROM AUTHOR]

A variety of statistically steady energy spectra in elastic wave turbulence have been reported in numerical simulations, experiments, and theoretical studies. Focusing on the energy levels of the system, we perform direct numerical simulations according to the Föppl-von Kármán equation, and successfully reproduce the variability of the energy spectra by changing the magnitude of external force systematically. When the total energies in wave fields are small, the energy spectra are close to a statistically steady solution of the kinetic equation in the weak turbulence theory. On the other hand, in large-energy wave fields, another self-similar spectrum is found. The coexistence of the weakly nonlinear spectrum in large wave numbers and the strongly nonlinear spectrum in small wave numbers is also found in moderate energy wave fields. [ABSTRACT FROM AUTHOR]

Forward flights of a bilaterally symmetrically flapping butterfly modeled as a four-link rigid-body system consisting of a thorax, an abdomen, and left and right wings are numerically simulated. The joint motions of the butterflies are adopted from experimental observations. Three kinds of the simulations, distinguished by ways to determine the position and attitude of the thorax, are carried out: a tethered simulation, a prescribed simulation, and free-flight simulations. The upward and streamwise forces as well as the wake structures in the tethered simulation, where the thorax of the butterfly is fixed, reasonably agree with those in the corresponding tethered experiment. In the prescribed simulation, where the thoracic trajectories as well as the joint angles are given by those observed in a free-flight experiment, it is confirmed that the butterfly can produce enough forces to achieve the flapping flights. Moreover, coherent vortical structures in the wake and those on the wings are identified. The generation of the aerodynamic forces due to the vortical structures are also clarified. In the free-flight simulation, where only the joint angles are given as periodic functions of time, it is found that the free flight is longitudinally unstable because the butterfly cannot maintain the attitude in a proper range. Focusing on the abdominal mass, which largely varies owing to feeding and metabolizing, we have shown that the abdominal motion plays an important role in periodic flights. The necessity of control of the thoracic attitude for periodic flights and maneuverability is also discussed. [ABSTRACT FROM AUTHOR]

Abstract: The objective of this paper is to clarify the principle of stabilization in flapping-of-wing flight of a butterfly, which is a rhythmic and cyclic motion. For this purpose, a dynamics model of a butterfly is derived by Lagrange’s method, where the butterfly is considered as a rigid multi-body system. For the aerodynamic forces, a panel method is applied. Validity of the mathematical models is shown by an agreement of the numerical result with the measured data. Then, periodic orbits of flapping-of-wing flights are searched in order to fly the butterfly models. Almost periodic orbits are obtained, but the model in the searched flapping-of-wing flight is unstable. This research, then, studies how the wake-induced flow and the flexibly torsional wing’s effect on the flight stability. Numerical simulations demonstrate that both the wake-induced flow and the flexible torsion reduces the flight instability. Because the obtained periodic flapping-of-wing flight is unstable, a feedback control system is designed, and a stable flight is realized. [Copyright &y& Elsevier]