Studies of Phase Turbulence in the One Dimensional Complex Ginzburg-Landau Equation

The phase-turbulent (PT) regime for the one dimensional complex Ginzburg-Landau equation (CGLE) is carefully studied, in the limit of large systems and long integration times, using an efficient new integration scheme. Particular attention is paid to solutions with a non-zero phase gradient. For fixed control parameters, solutions with conserved average phase gradient $\nu$ exist only for $|\nu|$ less than some upper limit. The transition from phase to defect-turbulence happens when this limit becomes zero. A Lyapunov analysis shows that the system becomes less and less chaotic for increasing values of the phase gradient. For high values of the phase gradient a family of non-chaotic solutions of the CGLE is found. These solutions consist of spatially periodic or aperiodic waves travelling with constant velocity. They typically have incommensurate velocities for phase and amplitude propagation, showing thereby a novel type of quasiperiodic behavior. The main features of these travelling wave solutions can be explained through a modified Kuramoto-Sivashinsky equation that rules the phase dynamics of the CGLE in the PT phase. The latter explains also the behavior of the maximal Lyapunov exponents of chaotic solutions.Comment: 16 pages, LaTeX (Version 2.09), 10 Postscript-figures included, submitted to Phys. Rev.

Coarsening dynamics in one dimension: The phase diffusion equation and its numerical implementation

Many nonlinear partial differential equations (PDEs) display a coarsening dynamics, i.e., an emerging pattern whose typical length scale $L$ increases with time. The so-called coarsening exponent $n$ characterizes the time dependence of the scale of the pattern, $L(t)\approx t^n$, and coarsening dynamics can be described by a diffusion equation for the phase of the pattern. By means of a multiscale analysis we are able to find the analytical expression of such diffusion equations. Here, we propose a recipe to implement numerically the determination of $D(\lambda)$, the phase diffusion coefficient, as a function of the wavelength $\lambda$ of the base steady state $u_0(x)$. $D$ carries all information about coarsening dynamics and, through the relation $|D(L)| \simeq L^2 /t$, it allows us to determine the coarsening exponent. The main conceptual message is that the coarsening exponent is determined without solving a time-dependent equation, but only by inspecting the periodic steady-state solutions. This provides a much faster strategy than a forward time-dependent calculation. We discuss our method for several different PDEs, both conserved and not conserved

A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates

Numerical computations of stationary states of fast-rotating Bose-Einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices. In this paper we propose a low-order finite element method with mesh adaptivity by metric control, as an alternative approach to the commonly used high order (finite difference or spectral) approximation methods. The mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states: an imaginary time propagation method and a Sobolev gradient descent method. We first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that simultaneously refinement using the real and imaginary part of the solution is successful. Mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases. Then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state. Considerable computational time saving is obtained compared to uniform mesh computations. The new method is applied to compute difficult cases relevant for physical experiments (large nonlinear interaction constant and high rotation rates).Comment: to appear in J. Computational Physic

Numerical studies on quantized vortex dynamics in superfludity and superconductivity

Ph.DDOCTOR OF PHILOSOPH

Nonlinear dynamics of the viscoelastic Kolmogorov flow

The weakly nonlinear regime of a viscoelastic Navier--Stokes fluid is investigated. For the purely hydrodynamic case, it is known that large-scale perturbations tend to the minima of a Ginzburg-Landau free-energy functional with a double-well (fourth-order) potential. The dynamics of the relaxation process is ruled by a one-dimensional Cahn--Hilliard equation that dictates the hyperbolic tangent profiles of kink-antikink structures and their mutual interactions. For the viscoelastic case, we found that the dynamics still admits a formulation in terms of a Ginzburg--Landau free-energy functional. For sufficiently small elasticities, the phenomenology is very similar to the purely hydrodynamic case: the free-energy functional is still a fourth-order potential and slightly perturbed kink-antikink structures hold. For sufficiently large elasticities, a critical point sets in: the fourth-order term changes sign and the next-order nonlinearity must be taken into account. Despite the double-well structure of the potential, the one-dimensional nature of the problem makes the dynamics sensitive to the details of the potential. We analysed the interactions among these generalized kink-antikink structures, demonstrating their role in a new, elastic instability. Finally, consequences for the problem of polymer drag reduction are presented.Comment: 26 pages, 17 figures, submitted to The Journal of Fluid Mechanic

On non-normality and classification of amplification mechanisms in stability and resolvent analysis

We seek to quantify non-normality of the most amplified resolvent modes and predict their features based on the characteristics of the base or mean velocity profile. A 2-by-2 model linear Navier-Stokes (LNS) operator illustrates how non-normality from mean shear distributes perturbation energy in different velocity components of the forcing and response modes. The inverse of their inner product, which is unity for a purely normal mechanism, is proposed as a measure to quantify non-normality. In flows where there is downstream spatial dependence of the base/mean, mean flow advection separates the spatial support of forcing and response modes which impacts the inner product. Success of mean stability analysis depends on the normality of amplification. If the amplification is normal, the resolvent operator written in its dyadic representation reveals that the adjoint and forward stability modes are proportional to the forcing and response resolvent modes. If the amplification is non-normal, then resolvent analysis is required to understand the origin of observed flow structures. Eigenspectra and pseudospectra are used to characterize these phenomena. Two test cases are studied: low Reynolds number cylinder flow and turbulent channel flow. The first deals mainly with normal mechanisms and quantification of non-normality using the inverse inner product of the leading forcing and response modes agrees well with the product of the resolvent norm and distance between the imaginary axis and least stable eigenvalue. In turbulent channel flow, structures result from both normal and non-normal mechanisms. Mean shear is exploited most efficiently by stationary disturbances while bounds on the pseudospectra illustrate how non-normality is responsible for the most amplified disturbances at spatial wavenumbers and temporal frequencies corresponding to well-known turbulent structures

Mechanism-enabled population balances and the effects of anisotropies in the complex Ginzburg-Landau equation

2019 Fall.Includes bibliographical references.This paper considers two problems. The first is a chemical modeling problem which makes use of ordinary differential equations to discover a minimum mechanism capable of matching experimental data in various metal nanoparticle nucleation and growth systems. This research has led to the concept of mechanism-enabled population balance modeling (ME-PBM). This is defined as the use of experimentally established nucleation mechanisms of particle formation to create more rigorous population balance models. ME-PBM achieves the goal of connecting reliable experimental mechanisms with the understanding and control of particle-size distributions. The ME-PBM approach uncovered a new and important 3-step mechanism that provides the best fits to experimentally measured particle-size distributions (PSDs). The three steps of this mechanism are slow, continuous nucleation and two surface growth steps. The importance of the two growth steps is that large particles are allowed to grow more slowly than small particles. This finding of large grow more slowly than small is a paradigm-shift away from the notion of needing nucleation to stop, such as in LaMer burst nucleation, in order to achieve narrow PSDs. The second is a study of the effects of anisotropy on the dynamics of spatially extended systems through the use of the anisotropic Ginzburg-Landau equation (ACGLE) and its associated phase diffusion equations. The anisotropy leads to different types of solutions not seen in the isotropic equation, due to the ability of waves to simultaneously be stable and unstable, including transient spiral defects together with phase chaotic ripples. We create a phase diagram for initial conditions representing both the longwave k = 0 case, and for wavevectors near the circle |k| = μ using the average L² energy

Transition to turbulence through steep global-modes cascade in an open rotating cavity

International audienceThe transition to turbulence in a rotating boundary layer is analysed via direct numerical simulation (DNS) in an annular cavity made of two parallel corotating discs of finite radial extent, with a forced inflow at the hub and free outflow at the rim. In a former numerical investigation (Viaud, Serre & Chomaz J. Fluid Mech., vol. 598, 2008, pp. 451-464) realized in a sectorial cavity of azimuthal extent 2p68, we have established the existence of a primary bifurcation to nonlinear global mode with angular phase velocity and radial envelope coherent with the so-called elephant mode theory. The former study has demonstrated the subcritical nature of this primary bifurcation with a base flow that keeps being linearly stable for all Reynolds numbers studied. The present work investigates the stability of this elephant mode by extending the cavity both in the radial and azimuthal direction. When the Reynolds number based on the forced throughflow is increased above a threshold value for the existence of the nonlinear global mode, a large-amplitude impulsive perturbation gives rise to a self-sustained saturated wave with characteristics identical to the 68-fold global elephant mode obtained in the smaller cavity. This saturated wave is itself globally unstable and a second front appears in the lee of the primary where small-scale instability develops. These secondary instabilities are identical for the 2p/68 and the 2p/4 long sectorial cavities, indicating that transition involves a Floquet mode of zero azimuthal wavenumber. This secondary instability leads to a very disorganized state, defining the transition to turbulence. The observed transition to turbulence linked to the secondary instability of a global mode confirms, for the first time on a real flow, the possibility of a direct transition to turbulence through an elephant mode cascade, a scenario that was up to now only observed on the Ginzburg-Landau model. © 2011 Cambridge University Press