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We present an iterative method allowing to synthetize a semi-numerical solution for the equations of motion of the resonant Saturn's satellites Titan-Hyperion (limited now to the planar problem). The current theory of Hyperion by Taylor, Sinclair & Message (1987) gives the greatest terms of the long-period part of the solution (depending on two angles: the libration angle τ, and the angular distance of the pericenters ζ). Using it as a first approximation, this solution is substituted numerically in the exact Lagrange equations of motion for Titan and Hyperion, computed for many values of the three angles: τ, ζ and ϕ (the mean synodic longitude). Then, a multivariable Fourier transform allows to reconstruct the equations in these three angles, that is in same form as the initial one with, in addition, the short-period terms. Then, a solution may be obtained and used as a better approximation in an iterative process. Besides a complete determination of the short-period perturbations of Hyperion obtained here completely for the first time, some long-period perturbations of Titan by Hyperion are also found which would be non negligible at the 10 km level.
We explain the high values of the acceleration (≈ 2° cy−2) found in the longitude of Mimas by Kozai and Dourneau when they fit to observations their current theory of the Mimas' motion. In fact, we have found that very long-period terms are missing in these theories; their expansion in powers of t well agrees with the observed acceleration. Effects of tidal dissipation are far smaller and could be determined only after accounting of these long-period terms.
We have recently built a coherent theory of the motion of the satellites Mimas, Enceladus, Tethys, Dione, Rhea, Titan and Iapetus. The final form of the “Théorie Analytique des Satellites de Saturne” (TASS1.6) is presented in Vienne & Duriez (1995). The internal precision of TASS is a few kilometers over three years and some tens kilometers over one century. The root-mean-square residuals of the adjustment of TASS over one century of Earth based observations reach 0″.12 for the best data sets, until 0″.015 for the few mutual phenomenas of 1981.
The state of knowledge of the motions of all Saturn's satellites is presented (excluding however the rings and their relating shepherding satellites). In particular, it appears that the theory of motion of the major satellites is now more precise than the available Earth-based observations, allowing to expect new progress with the next observations from mutual events and then with those from the Cassini mission.
Many previous studies have shown that the turbulent mixing layer under periodic forcing tends to adopt a lock-on state, where the major portion of the fluctuations in the flow are synchronized at the forcing frequency. The goal of this experimental study is to apply closed-loop control in order to provoke the lock-on state, using information from the flow itself. We aim to determine the range of frequencies for which the closed-loop control can establish the lock-on, and what mechanisms are contributing to the selection of a feedback frequency. In order to expand the solution space for optimal closed-loop control laws, we use the genetic programming control (GPC) framework. The best closed-loop control laws obtained by GPC are analysed along with the associated physical mechanisms in the mixing layer flow. The resulting closed-loop control significantly outperforms open-loop forcing in terms of robustness to changes in the free-stream velocities. In addition, the selection of feedback frequencies is not locked to the most amplified local mode, but rather a range of frequencies around it.
We compare the two current representations (TASS and that of Harper and Taylor) of the motion of Mimas, Enceladus, Tethys, Dione, Rhea, Titan, Hyperion and Japetus. Although both theories produce almost the same (o-c) residuals with the available observations, we show that they may give positions significantly different at dates corresponding to a gap in the distribution of these observations. In the interval 2000–2020 (corresponding to the CASSINI mission) these differences can reach 5000 km for Mimas, 8000km for Hyperion, 6 000 km for Japetus. So, to test a theory of motions it is necessary to use other criteria among which the most important ones are its dynamical consistency and its internal accuracy.
In order to improve the determination of the mixed terms in classical theories, we show how these terms may be derived from a general theory developed with the same variables (of a keplerian nature). We find that the general theory of the first order in the masses already allows us to develop the mixed terms which appear at the second order in the classical theory. We also show that a part of the constant perturbation of the semi-major axis introduced in the classical theory is present in the general theory as very long-period terms; by developing these terms in powers of time, they would be equivalent to the appearance of very small secular terms (in t, t2, …) in the perturbation of the semi-major axes from the second order in the masses. The short period terms of the classical theory are found the same in the general theory, but without the numerical substitution of the values of the variables.
We present the first closed-loop separation control experiment using a novel, model-free strategy based on genetic programming, which we call ‘machine learning control’. The goal is to reduce the recirculation zone of backward-facing step flow at
manipulated by a slotted jet and optically sensed by online particle image velocimetry. The feedback control law is optimized with respect to a cost functional based on the recirculation area and a penalization of the actuation. This optimization is performed employing genetic programming. After 12 generations comprised of 500 individuals, the algorithm converges to a feedback law which reduces the recirculation zone by 80 %. This machine learning control is benchmarked against the best periodic forcing which excites Kelvin–Helmholtz vortices. The machine learning control yields a new actuation mechanism resonating with the low-frequency flapping mode instability. This feedback control performs similarly to periodic forcing at the design condition but outperforms periodic forcing when the Reynolds number is varied by a factor two. The current study indicates that machine learning control can effectively explore and optimize new feedback actuation mechanisms in numerous experimental applications.
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