Fifth International Symposium on Bifurcations and Instabilities in Fluid Dynamics (BIFD2013) (original) (raw)

Numerical BifurcationMethods and their Application to Fluid Dynamics: Analysis beyond Simulation

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 arementioned 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.

Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics

Computational methods in applied sciences, 2019

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution of a single scalar field like a density or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn-and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dipcoating. Through the different examples we illustrate how to employ the numerical tools provided by the packages AUTO07P and PDE2PATH to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues.

Global stability analysis of complex fluids

2013

The bifurcations and control of the flow in a planar X-junction are studied via linear stability analysis and direct numerical simulations. This study reveals the instability mechanisms in a symmetric channel junction and shows how these can be stabilized or destabilized by boundary modification. We observe two bifurcations as the Reynolds number increases. They both scale with the inlet speed of the two side channels and are almost independent of the inlet speed of the main channel. Equivalently, both bifurcations appear when the recirculation zones reach a critical length. A two-dimensional stationary global mode becomes unstable first, changing the flow from a steady symmetric state to a steady asymmetric state via a pitchfork bifurcation. The core of this instability, whether defined by the structural sensitivity or by the disturbance energy production, is at the edges of the recirculation bubbles, which are located symmetrically along the walls of the downstream channel. The en...

The numerical analysis of bifurcation problems with application to fluid mechanics

Acta Numerica 2000, 2000

In this review we discuss bifurcation theory in a Banach space setting using the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation points. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and finite difference methods. These results are used to provide a convergence theory for the mixed finite element method applied to the steady incompressible Navier–Stokes equations. Numerical methods for the calculation of several common bifurcations are described and the performance of these methods is illustrated by application to several problems in fluid mechanics. A detailed description of the Taylor–Couette problem is given, and extensive numerical and experimental results are provided for comparison and discussion.

Pattern dynamics near inverse homoclinic bifurcation in fluids

Physical Review E

We report for the first time the pattern dynamics in the vicinity of an inverse homoclinic bifurcation in an extended dissipative system. We observe, in direct numerical simulations of three dimensional Rayleigh-Bénard convection, a spontaneous breaking of a competition of two mutually perpendicular sets of oscillating cross rolls to one of two possible sets of oscillating cross rolls as the Rayleigh number is raised above a critical value. The time period of the cross-roll patterns diverges, and shows scaling behavior near the bifurcation point. This is an example of a transition from nonlocal to local pattern dynamics near an inverse homoclinic bifurcation. We also present a simple four-mode model that captures the pattern dynamics quite well. PACS numbers: 47.20.Ky, 47.55.pb, 47.20.Bp Extended dissipative systems driven away from thermodynamic equilibrium often form patterns, if the driving force exceeds a critical value . Competing instabilities may lead to interesting pattern dynamics, which helps in understanding the underlying instability mechanism. Several patterns are observed in continuum mechanical systems, such as Rayleigh-Bénard systems [2], Bénard-Marangoni systems [3], magnetohydrodynamics [4], ferrofluids [5], binary fluids [6], granular materials [7] under shaking, biological systems [8], etc. Symmetries and dissipation play a very significant role in pattern selection in such systems . The selection of a pattern is a consequence of at least one broken symmetry of the system. Unbroken symmetries often introduce multiple patterns, which may lead to a transition from local to global pattern dynamics. The gluing [10] of two limit cycles on two sides of a saddle point in the phase space of a given system is an example of a local to nonlocal bifurcation. It occurs when two limit cycles simultaneously become homoclinic orbits of the same saddle point. This phenomenon has been recently observed in a variety of systems including liquid crystals , fluid dynamical systems [12], biological systems [13], optical systems , and electrical circuits , and is a topic of current research. The pattern dynamics in the vicinity of a homoclinic bifurcation has, however, not been investigated in a fluid dynamical system.

Nonlinear instability in an ideal fluid

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 1997

Linearized instability implies nonlinear instability under certain rather general conditions. This abstract theorem is applied to the Euler equations governing the motion of an inviscid fluid. In particular this theorem applies to all 20 space periodic flows without stagnation points as well as 20 space-periodic shear flows.