Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

An introduction to chaotic dynamical systems (2nd edition), by Robert L. Devaney. Pp 336. £34·95. 1989. ISBN 0-201-13046-7 (Addison-Wesley)

A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes

We simulate forced quasi-static magnetohydrodynamic turbulence and investigate the anisotropy, energy spectrum, and energy flux of the flow, specially for large interaction parameters (N). We show that the angular dependence of the energy spectrum is well quantified using Legendre polynomials. For large N, the energy spectrum is exponential. Our direct computation of energy flux reveals an inverse cascade of energy at low wavenumbers, similar to that in two-dimensional turbulence. We observe the flow be two-dimensional (2D) for moderate N (N ∼ 20), and two-dimensional three-component (2D-3C) type for N ⩾ 27. In our forced simulation, the transition from 2D to 2D-3C occurs at higher value of N than that in Favier et al. [“On the two-dimensionalization of quasistatic magnetohydrodynamic turbulence,” Phys. Fluids 22, 075104 (2010)] who employ decaying simulations.

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Strong anisotropy in quasi-static magnetohydrodynamic turbulence for high interaction parameters

Magnetohydrodynamic (MHD) turbulence at low magnetic Reynolds number is experimentally investigated by studying a liquid metal flow in a cubic domain. We focus on the mechanisms that determine whether the flow is quasi-two-dimensional, three-dimensional or in any intermediate state. To this end, forcing is applied by injecting a DC current through one wall of the cube only, to drive vortices spinning along the magnetic field. Depending on the intensity of the externally applied magnetic field, these vortices extend part or all of the way through the cube. Driving the flow in this way allows us to precisely control not only the forcing intensity but also its dimensionality. A comparison with the theoretical analysis of this configuration singles out the influences of the walls and of the forcing on the flow dimensionality. Flow dimensionality is characterised in several ways. First, we show that when inertia drives three-dimensionality, the velocity near the wall where current is injected scales as . Second, we show that when the distance over which momentum diffuses under the action of the Lorentz force (Sommeria & Moreau, J. Fluid Mech., vol. 118, 1982, pp. 507–518) reaches the channel width , the velocity near the opposite wall follows a similar law with a correction factor that measures three-dimensionality. When , by contrast, the opposite wall has less influence on the flow and . The central role played by the ratio is confirmed by experimentally verifying the scaling put forward by Sommeria & Moreau ( is the interaction parameter) and, finally, the nature of the three-dimensionality involved is further clarified by distinguishing weak and strong three-dimensionalities previously introduced by Klein & Potherat (Phys. Rev. Lett., vol. 104 (3), 2010, 034502). It is found that both types vanish only asymptotically in the limit . This provides evidence that because of the no-slip walls, (i) the transition between quasi-two-dimensional and three-dimensional turbulence does not result from a global instability of the flow, unlike in domains with non-dissipative boundaries (Boeck et al. Phys. Rev. Lett., vol. 101, 2008, 244501), and (ii) it does not occur simultaneously at all scales.

Why, how and when MHD turbulence at low becomes three-dimensional

MHD turbulence at low Magnetic Reynolds number is experimentally investigated by studying a liquid metal flow in a cubic domain. We focus on the mechanisms that determine whether the flow is quasi-2D, 3D or in any intermediate state. To this end, forcing is applied by injecting a DC current $I$ through one wall of the cube only, to drive vortices spinning along the magnetic field. Depending on the intensity of the externally applied magnetic field, these vortices extend part or all of the way through the cube. Driving the flow in this way allows us to precisely control not only the forcing intensity but also its dimensionality. A comparison with the theoretical analysis of this configuration singles out the influences of the walls and of the forcing on the flow dimensionality, which is characterised in several ways. First, when inertia drives three-dimensionality, the velocity near the wall where current is injected scales as $U_b\sim I^{2/3}$. Second, when the distance $l_z$ over which momentum diffuses under the action of the Lorentz force reaches the channel width $h$, the velocity near the opposite wall $U_t$ follows a similar law with a correction factor $(1-h/l_z)$. When $l_z<h$, by contrast, the opposite wall has less influence on the flow and $U_t\sim I^{1/2}$. The central role played by the ratio $l_z/h$ is confirmed by experimentally verifying Sommeria & Moreau (1982)'s scaling $l_z\sim N^{1/2}$ ($N$ is the interaction parameter) and finally, the nature of the three-dimensionality is further clarified by distinguishing weak and strong three-dimensionalities. It is found that both vanish only asymptotically in the limit $N\rightarrow\infty$. This provides evidence that because of the no-slip walls, 1) the transition between quasi-2D and 3D turbulence does not result from a global instability of the flow, and 2) it doesn't occur simultaneously at all scales.

Why, how and when MHD turbulence at low $Rm$ becomes three-dimensional

This paper presents two- and three-dimensional direct numerical simulations of the flow around a circular cylinder placed symmetrically in a plane channel. Results are presented in the Reynolds number range (based on the cylinder diameter and centerline velocity) of 10 to 390 for a blockage ratio (ratio of the cylinder diameter to the channel height) of 0.2. The aim of this work was to investigate in detail the confinement effect due to the channel’s stationary walls on the force coefficients and the associated Strouhal numbers, as well as on the generated flow regimes. Present results suggest a transition from a 2-D to a 3-D shedding flow regime between Re = 180 and Re = 210. This transition was found to be dominated by mode A and mode B three dimensional instabilities, similar to those observed in the case of an unconfined circular cylinder. This is the first time that the existence of the two modes, and of naturally occurring vortex dislocations, has been confirmed via full 3-D simulations for the case...

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Three dimensional flow around a circular cylinder confined in a plane channel

This paper presents three-dimensional direct numerical simulations of liquid metal flow around a circular cylinder placed symmetrically in a rectangular duct, under a wide range of magnetic field intensities. Results are presented for values of the Hartmann number (based on the duct width) in the range of 0 ⩽ Ha ⩽ 1120, and the Reynolds number (based on the cylinder diameter and centerline velocity) in the range 0 ⩽ Rec ⩽ 5000. The generated flow regimes and the associated critical values of parameters are investigated in detail through full three-dimensional simulations. The effect of the magnetic field on the wake structure is discussed in relation to the possible mechanisms for the generation or suppression of vortices, and to previous attempts to model magnetohydrodynamic flows using simplified two-dimensional models. Present results reveal a non-monotonic dependance of the critical Reynolds number for the onset of vortex shedding, with respect to the Hartmann number. For certain combinations of Ha an...

Three-dimensional numerical simulations of magnetohydrodynamic flow around a confined circular cylinder under low, moderate, and strong magnetic fields

On the transition to turbulence of a viscoplastic fluid past a confined cylinder: A numerical study

The focus of the work is the “phase lock-on”, a well-known phenomenon observed in oscillating dynamical systems where the forcing and output frequencies are synchronized. This phenomenon appears to be universal with the frequencies related by a rational ratio. We considered the case of an incompressible flow past a confined cylinder where the incident bulk flow is subjected to periodic velocity perturbations. While we considered both Newtonian and non-Newtonian (Bingham) fluids, the main interest was on the Bingham case. Numerical simulations were conducted at a fixed Reynolds number of Re = 100 based on the bulk fluid velocity and cylinder diameter. The relative amplitude of the periodic velocity perturbation was considered in the range 0.10–0.30 and the forcing frequency relative to the natural unforced shedding frequency was in the range 1.4–2.3. These choices yielded fixed lock-on periodic states that correspond to the number 2 on the horizontal axis of an Arnold tongue graph. Yield stress effects are shown for Bingham numbers in the range 0–0.5.

Nicolas Kanaris

Papers

Three dimensional flow around a circular cylinder confined in a plane channel

Three-dimensional numerical simulations of magnetohydrodynamic flow around a confined circular cylinder under low, moderate, and strong magnetic fields

On the transition to turbulence of a viscoplastic fluid past a confined cylinder: A numerical study

Phase lock-on in periodically excited Bingham flow past a confined cylinder