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Register a new userAccurate predictions of chaotic motion of a free fall disk
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It is important to know the accurate trajectory of a free fall object in fluid (such as a spacecraft), whose motion might be chaotic in many cases. However, it is impossible to accurately predict its chaotic trajectory in a long enough duration by traditional numerical algorithms in double precision. In this paper, we give the accurate predictions of the same problem by a new strategy, namely the Clean Numerical Simulation (CNS). Without loss of generality, a free fall disk in water is considered, whose motion is governed by the Andersen-Pesavento-Wang model. We illustrate that convergent and reliable trajectories of a chaotic free fall disk in a long enough interval of time can be obtained by means of the CNS, but different traditional algorithms in double precision give disparate trajectories. Besides, unlike the traditional algorithms in double precision, the CNS can predict the accurate posture of the free fall disk near the vicinity of the bifurcation point of some physical parameters in a long duration. Therefore, the CNS can provide reliable prediction of chaotic systems in a long enough interval of time.
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Dynamical and statistical properties of tracer advection are studied in a family of flows produced by three point-vortices of different signs. A collapse of all three vortices to a single point is then possible. Tracer dynamics is analyzed by numerical construction of Poincar{e} sections, and is found to be strongly chaotic: advection pattern in the region around the center of vorticity is dominated by a well developed stochastic sea, which grows as the vortex system approaches the collapse; at the same time, the islands of regular motion around vortices, known as vortex cores, shrink. An estimation of the cores radii from the minimum distance of vortex approach to each other is obtained. Tracer transport was found to be anomalous: for all of the three numerically investigated cases, the variance of the tracer distribution grows faster than a linear function of time, corresponding to a super-diffusive regime. The transport exponent varies with time decades, implying the presence of multi-fractal transport features. Yet, its value is never too far from 3/2, indicating some kind of universality. Statistics of Poincar{e} recurrences is non-Poissonian: distributions have long power-law tails. The anomalous properties of tracer statistics are the result of the complex structure of the advection phase space, in particular, of strong stickiness on the boundaries between the regions of chaotic and regular motion. The role of the different phase space structures involved in this phenomenon is analyzed. Based on this analysis, a kinetic description is constructed, which takes into account different time and space scalings by using a fractional equation.
I consider the problem of weakly nonlinear stability of three-dimensional parity-invariant magnetohydrodynamic systems to perturbations, involving large scales. I assume that the MHD state, the stability of which I investigate, does not involve large spatio-temporal scales, and it is stable to perturbations involving the same small spatial scales, as the perturbed MHD state. Mean-field equations, which I derive for the perturbation using asymptotic techniques for multiscale systems, are a generalization of the standard equations of magnetohydrodynamics (the Navier-Stokes equation with the Lorentz force and the magnetic induction equation). In them, the operator of combined eddy diffusivity emerges, which is in general anisotropic and not necessarily negatively defined, and new quadratic terms, analogous to the ones describing advection. A method for efficient computation of coefficients of the eddy diffusivity tensor and eddy advection terms in the mean-field equations is proposed.
We consider the motion of a particle subjected to the constant gravitational field and scattered inelasticaly by hard boundaries which possess the shape of parabola, wedge, and hyperbola. The billiard itself performs oscillations. The linear dependence of the restitution coefficient on the particle velocity is assumed. We demonstrate that this dynamical system can be either regular or chaotic, which depends on the billiard shape and the oscillation frequency. The trajectory calculations are compared with the experimental data; a good agreement has been achieved. Moreover, the properties of the system has been studied by means of the Lyapunov exponents and the Kaplan-Yorke dimension. Chaotic and nonuniform patterns visible in the experimental data are interpreted as a result of large embedding dimension.
The Random Phase and Amplitude Formalism (RPA) has significantly extended the scope of weak turbulence studies. Because RPA does not assume any proximity to the Gaussianity in the wavenumber space, it can predict, for example, how the fluctuation of the complex amplitude of each wave mode grows through nonlinear interactions with other modes, and how it approaches the Gaussianity. Thus, RPA has a great potential capability, but its validity has been assessed neither numerically nor experimentally. We compare the theoretical predictions given by RPA with the results of direct numerical simulation (DNS) for a three-wave Hamiltonian system, thereby assess the validity of RPA. The predictions of RPA agree quite well with the results of DNS in all the aspects of statistical characteristics of mode amplitudes studied here.
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