We present numerical solutions to the extended Doering-Constantin variational principle for upper bounds on the energy dissipation rate in plane Couette flow, bridging the entire range from low to asymptotically high Reynolds numbers. Our variational bound exhibits structure, namely a pronounced minimum at intermediate Reynolds numbers, and recovers the Busse bound in the asymptotic regime. The most notable feature is a bifurcation of the minimizing wave numbers, giving rise to simple scaling of the optimized variational parameters, and of the upper bound, with the Reynolds number.

The dynamics of curved vortex filaments is studied analytically and numerically in the framework of a three-dimensional complex Ginzburg-Landau equation (CGLE). It is proved that a straight vortex line is unstable with respect to spontaneous stretching and bending in a certain range of parameters of the CGLE, resulting in formation of persistent entangled vortex configurations. The analysis shows that the standard approach relating the velocity of the filament with the local curvature is insufficient to describe the instability and stretching of vortex lines.

The onset of the wave resistance, $R$, via generation of surface gravity waves by a long body moving with a velocity $V$, is considered. We demonstrate that for a special shape of the disturbance moving uniformly with velocity $V$, the wave resistance $R\left(V\right)$ has a well-defined threshold (perfect bifurcation) at the critical value $V_c$, i.e., $R\sim \sqrt{{V-V}_c}$ for $V>\mathrm{}{V}_c$ and equals zero for $V<\mathrm{}{V}_c$. For arbitrary shape of a moving disturbance the bifurcation becomes imperfect. It results in a small nonzero $R$ usually observed at $0<V<\mathrm{}{V}_c$. Using this idea, we have succeeded in describing quantitatively experimental data on the wave resistance of ship models obtained by D. W. Taylor in 1908.

We employ results from differential-geometry based nonlinear control theory and bifurcation theory to develop a new technique for the synchronization and control of chaotic systems. This enables us to obtain a simple condition for predicting the occurrence of the intermittent bursts of desynchronization in unidirectionally coupled chaotic systems.

We study the stability and breakup of invariant tori in Hamiltonian flows using a combination of Kolmogorov-Arnold-Moser (KAM) theory and renormalization-group techniques. We implement the scheme numerically for a family of Hamiltonians quadratic in the actions to analyze the strong coupling regime. We show that the KAM iteration converges up to the critical coupling at which the torus breaks up. Adding a renormalization consisting of a rescaling of phase space and a shift of resonances allows us to determine the critical coupling with higher accuracy. We determine a nontrivial fixed point and its universality properties.

The phase of a chaotic trajectory in autonomous flows is often ignored because of the wide use of the extremely popular Poincaré surface-of-section technique in the study of chaotic systems. We present evidence that, in general, a chaotic flow is practically composed of a small number of intrinsic modes of proper rotations from which the phase can be computed via the Hilbert transform. The fluctuations of the phase about that of a uniform rotation can be described by fractional Brownian random processes. Implications to nonlinear digital communications are pointed out.

Through an investigation of the dynamics of solitons in three-level atoms, we demonstrate the possibility of optical pulse control and shaping in coherently driven media. It is also shown that solitons generated in three-level atoms, in contrast to two-level atoms, can propagate at the speed of light.

We perform experiments and numerical simulations on the mean streamwise velocity field of an unstable cylinder wake. We show that the mean velocity exhibits two weak secondary minima as the consequence of nonlinear interactions, resulting in a strong mean flow correction of the unstable mode. This correction, dominating the basic flow, governs the decrease in the length of the recirculation region $\Delta L_r$ in the supercritical regime. This explains the early classical observations of M. Nishioka and H. Sato [J. Fluid Mech. 89, 49 (1978)].

The vortex structure of immiscible liquid-liquid Taylor-Couette-Poiseuille flow was studied using photographic techniques. Several parameters were considered, including the feed composition and the inner cylinder rotation rate. For certain feed compositions and sufficiently large rotation rates a translating banded structure, which consisted of alternating aqueous and organic-rich vortices, persisted indefinitely. At lower rotation rates, either a spatially homogeneous emulsion evolved or sustained oscillations between the banded and homogeneous structures developed.

We investigate fluid flow through disordered porous media by direct simulation of the Navier-Stokes equations in a two-dimensional percolation structure. We find, in contrast to the log-normal distribution for the local currents found in the analog random resistor network, that over roughly 5 orders of magnitude the distribution $n\left(E\right)$ of local kinetic energy $E$ follows a power law, with $n\left(E\right)\propto E^{-\alpha }$, where $\alpha =0.90\mathrm{}\pm 0.03$ for the entire cluster, while $\alpha =0.64\mathrm{}\pm 0.05$ for fluid flow in the backbone only. Thus the “stagnant” zones play a significant role in transport through porous media, in contrast to the dangling ends for the analogous electrical problem.

The short time revivals of initially localized wave packets are well known in simple, closed, 1-degree-of-freedom (1D) systems. In 2D or higher, if the system is integrable or has exclusively periodic dynamics, a generalization is possible. If the dynamics are chaotic, revivals have not been previously seen and are, a priori, not expected. Nevertheless, we have found that some stretched wave packets in a chaotic system experience, very early, surprisingly large recurrences. We extend a semiclassical theory founded on summing over heteroclinic orbits to determine a set of necessary conditions. The most important one is an Einstein-Brillouin-Keller-like quantization of classical flux crossing the turnstile formed by the stable and unstable manifolds of the initial wave packet's underlying central orbit.

Evidence of noise-sustained patterns in nonlinear optical systems is given. They are found in passive optical cavities, filled by Kerr type nonlinear media, when the angle of incidence of the pump beam is not zero, in a regime of convective instability. These patterns arise as a macroscopic manifestation of dynamically amplified noise, with amplification factors of up to ${10}^5$. We characterize the difference between noise-sustained and deterministic patterns in terms of statistical properties of the field spectral intensity.

We report the first measurements of dynamic transport properties of an acoustic pulsed wave propagating in a strongly disordered 2D medium by means of time-dependent coherent backscattering. The coherent backscattering peaks are recorded using an array of transducers. Both transport speed and diffusion coefficient have been determined accurately by comparison to a real-space generalization of coherent backscattering in optics.

Until now, it has been observed that a collective handling of uniform spheres could only lead to a random close packing of 0.64 maximum volume fraction. In this Letter, we show that denser crystalline arrangements can be obtained when beads are poured at low flow rates into a horizontally shaken container. A parallel is suggested between this process and that of colloidal sedimentation which also yields crystalline structure.

Kolmogorov's refined similarity hypothesis (RSH) is extended to study the inertial-range scaling of a passive scalar advected by a rapidly changing incompressible velocity field in $d$ dimensions. For ${\zeta }_2>d$, the non-negativity of the scalar dissipation rate constrains the $2n$th order scaling exponents, ${\zeta }_{2n}$, to be linear in $n$ asymptotically. With the RSH formulated in terms of a stochastic variable $\theta$, the molecular-diffusion terms are evaluated in general $d$ dimensions. For $d\ge 2$, the exponents are found to be ${\zeta }_{2n}=\frac{1}{2}\sqrt{[d-{\zeta }_2-g(n){\zeta }_2{]}^2+4\mathrm{ng}(n\left){\zeta }_2\right(d-{\zeta }_2)}-\frac{1}{2}[d-{\zeta }_2-g(n\left){\zeta }_2\right]$, where $g\left(n\right)=(2n-1)〈{\theta }^{2n-2}〉〈{\theta }^2〉/〈{\theta }^{2n}〉$.

In a low temperature He gas Rayleigh-Bénard experiment, Rayleigh numbers from ${10}^3$ to more than ${10}^{14}$ are explored. Local velocity is estimated through the time lag between two closeby temperature probes. This allows characterizing of the high Rayleigh regime $(\mathrm{Ra}>{10}^{11})$ as a fully turbulent one, possibly corresponding to the asymptotic regime predicted by R. Kraichnan [Phys. Fluids 5, 1374 (1962)].

We have observed the fluidlike rotation of a pair of identical optical vortices (OV's) as they propagate through free space. Similar to vortex filaments in a fluid, the initial rotation rate is found to be inversely proportional to the squared distance of separation. Owing to unusually small vortex cores, we obtained rotation rates that were 2 orders of magnitude larger than expected for “conventional” large core OV's.

We give experimental and theoretical evidence that two-wave mixing light amplification in electro-optic and photoconductive materials is dramatically influenced by an anisotropic photoexcitation cross section of mobile charge carriers. An enhancement of the two-wave mixing gain is predicted and an increase of 90% is observed in a dichroic Ni doped ${\mathrm{KNbO}}_3$ crystal in a geometry with large angles between the two wave propagation directions. The enhancement is possible because in anisotropic materials the photogeneration can lead to higher modulation depths as compared to the depth of light intensity modulation.

We report measurements by diffusing-wave spectroscopy of particle motions in a gas-fluidized bed. The homogenous state of the bed known as the uniformly fluidized state is actually a weak solid in which particles are at rest. The only truly fluid state is an inhomogenous mixture of gas bubbles and of liquid regions in which microscopic particle dynamics are shown to be collisional. The motion of macroscopic bubbles is the source of particle motions in the bed. Measurements of mean free paths, collision rates, and velocity fluctuations (the “granular temperature”) show that dissipation occurs nonuniformly in the medium.

Direct numerical simulation results are presented which indicate that the balance between coherence and randomness in small-scale motion is such that a conditional-averaging procedure may be an appropriate mathematical operation to treat the problem of fluid turbulence as it is used in the two-field theory of turbulence [W. D. McComb and A. G. Watt, Phys. Rev. Lett. 65, 3281 (1990)] and in large eddy simulation models. Important results on phase and scale coherence suggest an interpretation based on random sweeping and vortex straining/stretching.