Session H8: Nonlinear Dynamics: Chaos and Bifurcations

Stochastic versus chaotic behaviour in the noisy generalized Kuramoto-Sivashinsky equation

Random fluctuations are well-known to have significant impact on the formation of complex spatiotemporal patterns in a wide spectrum of biological, engineering and physical environments, including fluid systems such Rayleigh-B\'enard convection, contact line dynamics, or waves in free-surface thin film flows. Many of these systems can be modeled by stochastic partial differential equations in large or unbounded domains, a simple prototype of which is the generalised Kuramoto-Sivashinsky (gKS) equation. Its deterministic version has been used in a wide variety of fluid flow contexts, such as two-phase flows with surfactants, free falling films and films in the presence chemical reactions, heating effects and curved substrates, amongst others. Here we study the dynamical states of the noisy gKS equation by making use of time series techniques based on chaos theory, in particular permutation entropy and nonlinear forecasting. We focus on analyzing temporal signals of global measure in the spatiotemporal pattern as the dispersion parameter of the gKS equation and the strength of the noise are varied, observing a rich variety of different emerging regimes, from high-dimensional chaos to purely stochastic behaviour.   

Lagrangian chaos in three- dimensional steady buoyancy-driven flows

Natural convection plays a key role in fluid dynamics owing to its ubiquitous presence in nature and industry. Buoyancy-driven flows are prototypical systems in the study of thermal instabilities and pattern formation. The differentially heated cavity problem has been widely studied for the investigation of buoyancy-induced oscillatory flow. However, far less attention has been devoted to the three-dimensional Lagrangian transport properties in such flows. This study seeks to address this by investigating Lagrangian transport in the steady flow inside a cubic cavity differentially-heated from the side. The theoretical and numerical analysis expands on previously reported similarities between the current flow and lid-driven flows. The Lagrangian dynamics are controlled by the P\'{e}clet number (Pe) and the Prandtl number (Pr). Pe controls the behaviour qualitatively in that growing Pe progressively perturbs the integable state (Pe$=$0), thus paving the way to chaotic dynamics. Pr plays an entirely quantitative role in that Pr\textless 1 and Pr\textgreater 1 amplifies and diminishes, respectively, the perturbative effect of non-zero Pe.   

Slip induced mixing in a model slug flow

Mixing of reactants in microfluidic slugs has a significant influence on the performance of processes. We discuss how mixing can be enhanced in slug flows by introducing periodic hydrophobicity on the confining walls. We consider a rectangular slug moving in a straight microchannel constructed by a shift-reflect transform of a unit cell with finite slip on one wall. This leads to alternating regions of slip and no-slip on each wall. The velocity field within the 2D slug is approximated as that in a driven cavity and computed by a Chebyshev spectral collocation. We go beyond a blinking flow model by capturing the velocity field under the discontinuous boundary conditions of inter-cell transit using domain decomposition. Thus,advection is described by a sequence of maps. It is seen that the hydrophobic sections reduce the size of the closer vortex and locally attract the separatrix. This permits "crossing" of streamlines in adjacent unit cells ,opening up the possibility of chaotic mixing. "Good crossing", as quantified by an Eulerian indicator called "transversality", seems to occur in a larger area when slug length is comparable to unit cell length. We quantify mixing and the internal structures that result using different Lagrangian techniques to reach a holistic consensus.   

Nonlinear Dynamic Stability of the Viscoelastic Plate Considering Higher Order Modes

-The dynamic stability of viscoelastic plates is investigated in this paper by using chaotic and fractal theory. The nonlinear integro-differential dynamic equation is changed into an autonomic 4-dimensional dynamical system. The numerical time integrations of equations are obtained by using the fourth order Runge-Kutta method. And the Lyapunov exponent spectrum, the fractal dimension of strange attractors and the time evolution of deflection are obtained. The influence of viscoelastic parameter on dynamic buckling of viscoelastic plates is discussed. The effect of higher order modes on dynamic stability of viscoelastic plate is obtained, the necessity of considering higher order modes is discussed.   

Topology of azimuthally travelling waves in thermocapillary liquid bridges

The topology of the laminar three-dimensional flow in a cylindrical liquid bridge driven by thermocapillary forces is investigated. Attention is focussed on travelling hydrothermal waves which are analysed in a co-rotating frame of reference in which the flow becomes steady. Chaotic and regular regions in form of KAM tori are found as well as closed streamlines. The flow features are discussed in terms of shape, location and period of closed orbits, KAM structures, their relation to the basic-state toroidal vortex flow and the dependence on the Marangoni number.   

Transitional behavior of convective patterns in porous media: Insights from basin stability analysis

The present study investigates the transitional behavior of convective modes in Horton-Rogers-Lapwood convection (HRLC). We first provide new pore-scale numerical and experimental evidences on the variation of the stability level of single-cell and double-cell convection modes in a 2D HRLC problem. In order to interpret this transitional behavior, we employ the concept of basin stability and develop a basin stability diagram of the first four convection modes in HRLC. This is in contrast to the standard bifurcation analysis of HRLC using linear stability analysis and continuation techniques, which only provides local information about the (range of) existence, and any possible co-existence of different convection modes. The present basin stability analysis of HRLC not only provides the local information about the (co-)existence of different patterns, but also, it determines their relative stability as well as how the basin of stability of these modes contract or expands as the Rayleigh number varies. The results of the present study show how establishing the dependence of basin stability on the Rayleigh number is essential to analyze the transition between different convection patterns observed experimentally and numerically.   

Forced Snaking

We study spatial localization in the real subcritical Ginzburg-Landau equation $u_t=m_0 u+ m_1\cos\left(\frac{2\pi}{\ell}x\right) u+u_{xx}+d|u|^2u-|u|^4u$ with spatially periodic forcing. When $d>0$ and $m_1 =0$ this equation exhibits bistability between the trivial state $u=0$ and a homogeneous nontrivial state $u=u_0$ with stationary localized structures which accumulate at the Maxwell point $m_0=-3d^2/16$. When spatial forcing is included its wavelength is imprinted on $u_0$ creating conditions favorable to front pinning and hence spatial localization. We use numerical continuation to show that under appropriate conditions such forcing generates a sequence of localized states organized within a snakes-and-ladders structure centered on the Maxwell point, and refer to this phenomenon as \textit{forced snaking}. We determine the stability properties of these states and show that longer lengthscale forcing leads to stationary trains consisting of a finite number of strongly localized, weakly interacting pulses exhibiting \textit{foliated snaking}.   

Noise induced transitions in rugged energy landscapes

External or internal random fluctuations are ubiquitous in many physical and technological systems and can play a key role in their dynamics often inducing a wide variety of complex spatiotemporal phenomena, including noise-induced spatial patterns and noise-induced phase transitions. Many of these phenomena can be modelled by noisy multiscale systems characterized by the presence of a wide range of different time- and lengthscales interacting nontrivially with each other. Here we analyse the effects of additive noise on systems that are described in terms of a rugged energy landscape, modelled as a slowly-varying multiscale potential perturbed by periodic multiscale fluctuations. Some examples of this problem include the dynamics of sessile droplets on heterogeneous substrates, crystallization and the evolution of protein folding. We demonstrate that the interplay between noise and the small scale fluctuations in the potential can give rise to a dramatically different bifurcation structure and dynamical behaviour compared to that of the original, unperturbed model. For instance, we observe several nontrivial and largely unexpected dynamic-state transitions controlled by the noise intensity. We characterize these transitions in terms of critical exponents.   

Onset of chaos in helical vortex breakdown at low Reynolds number

Swirling jet flows are generally characterized by two non-dimensional parameters: the swirl and the Reynolds number. Bubble, spiral or double spiral vortex breakdown as well as columnar vortex are part of the observed dynamics when these two control parameters are varied. This rich dynamic produces strong mixing that is traditionally investigated in the framework of Lagrangian chaos, with typical applications to combustion chambers. In contrast to chaotic advection, Eulerian chaos has not been reported for such open flows. Here, Eulerian chaos is studied through direct numerical flow simulations of an unconfined Grabowsky and Berger vortex using the incompressible Navier-Stokes solver NEK5000. At a fixed swirl number, a sequence of periodic, quasiperiodic, chaotic, quasiperiodic and periodic states is observed as the Reynolds number increases from 200 to 300. Therefore, Fourier spectrum, Poincar\'{e} section map, sensitivity to initial condition and largest Lyapunov exponent are computed to identify the chaotic window which results from the nonlinear interaction between a self-sustained single helical mode, triggered by an upstream bubble breakdown, and other helical modes. Finally, a route to chaos in the incompressible Navier-Stokes equations is sketched.   

Large-scale columnar vortices in rotating turbulence

In the rotating turbulence, flow structures are affected by the angular velocity of the system's rotation. When the angular velocity is small, three-dimensional statistically-isotropic flow, which has the Kolmogorov spectrum all over the inertial subrange, is formed. When the angular velocity increases, the flow becomes two-dimensional anisotropic, and the energy spectrum has a power law $k^{-2}$ in the small wavenumbers in addition to the Kolmogorov spectrum in the large wavenumbers. When the angular velocity decreases, the flow returns to the isotropic one. It is numerically found that the transition between the isotropic and anisotropic flows is hysteretic; the critical angular velocity at which the flow transitions from the anisotropic one to the isotropic one, and that of the reverse transition are different. It is also observed that the large-scale columnar structures in the anisotropic flow depends on the external force which maintains a statistically-steady state. In some cases, small-scale anticyclonic structures are aligned in a columnar structure apart from the cyclonic Taylor column. The formation mechanism of the large-scale columnar structures will be discussed.