We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as 'tipping points', is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods are mentioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems
A numerical study of the nonlinear torsional solutions of convection in rotating, internally heated, self-gravitating fluid spheres is presented. Their dependence on the Rayleigh number has been found for two pairs of Ekman (E) and small Prandtl (Pr) numbers in the region of parameters where, according to Zhang et al. [J. Fluid Mech. 813, R2 (2017)], the linear stability of the conduction state predicts that they can be preferred at the onset of convection. The bifurcation to periodic torsional solutions is supercritical for sufficiently small Pr. They are not rotating waves, unlike the nonaxisymmetric case. Therefore they have been computed by using continuation methods for periodic orbits. Their stability with respect to axisymmetric perturbations and physical characteristics have been analyzed. It was found that the time-and space-averaged equatorially antisymmetric part of the kinetic energy of the stable orbits splits into equal poloidal and toroidal parts, while the symmetric part is much smaller. Direct numerical simulations for E = 10 −4 at higher Rayleigh numbers (Ra) show that this trend is also valid for the nonperiodic flows and that the mean values of the energies remain almost constant with Ra. However, the modulated oscillations bifurcated from the quasiperiodic torsional solutions reach a high amplitude, compared with that of the periodic, increasing slowly and decaying very fast. This repeated behavior is interpreted as trajectories near heteroclinic chains connecting unstable periodic solutions. The torsional flows give rise to a meridional propagation of the kinetic energy near the outer surface and an axial oscillation of the hot nucleus of the metallic fluid sphere.
This study elucidates the origin of the multiplicity of stable oscillatory flows detected by time integration in tall rectangular cavities heated from the side. By using continuation techniques for periodic orbits, it is shown that initially unstable branches, arising at Hopf bifurcations of the basic steady flow, become stable after crossing Neimark-Sacker points. There are no saddlenode or pitchfork bifurcations of periodic orbits, which could have been alternative mechanisms of stabilization. According to the symmetries of the system, the orbits are either fixed cycles, which retain at any time the center-symmetry of the steady flow, or symmetric cycles involving a time shift in the global invariance of the orbit. The bifurcation points along the branches of periodic flows are determined. By using time integrations, with unstable periodic solutions as initial conditions, it is found out which of the bifurcations at the limits of the intervals of stable periodic orbits are sub or supercritical.PACS numbers: 47.15.-x, 47.20.-k
Abstract. A tutorial on continuation and bifurcation methods for the analysis of truncated dissipative partial differential equations is presented. It focuses on the computation of equilibria, periodic orbits, their loci of codimension-one bifurcations, and invariant tori. To make it more self-contained, it includes some definitions of basic concepts of dynamical systems, and some preliminaries on the general underlying techniques used to solve non-linear systems of equations by inexact Newton methods, and eigenvalue problems by means of subspace or Arnoldi iterations.
The thermal convection of rotating fluids in spherical geometry is a classical problem with application to many geophysical and astrophysical problems. The study of the transition to periodic solutions from the steady conduction state of a rotating and self-gravitating fluid sphere, heated uniformly from the inside, is discussed here. The continuation of double Hopf points is used to determine the region of the parameter space in which the first bifurcation is to solutions independent of the longitude (axisymmetric solutions). It is limited by three segments of curves separated by two triple Hopf points. This type of so-called torsional solutions was recently found, and it is shown here that they are the preferred solutions at the onset of convection for a wide range of fluids of Prandtl numbers, Pr, extending from Pr = 0 to Pr \approx 0.9, which includes, for instance, liquid metals and gases. Although the corresponding interval of Ekman numbers, E, narrows when Pr \rightar 0, it is shown that there is always a small gap of parameters, relevant to geophysics and astrophysics, where the torsional solutions are preferred. The limits of the double Hopf curves when Pr \rightar 0 follow linear laws of the form E = c(m1, m2)Pr, c(m1, m2) being constant depending on the two azimuthal wavenumbers, m1 and m2, of the eigenfunctions that define the double Hopf problem.
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