Phase Transition to Turbulence in Spatially Extended Shear Flows

A new conception of subcritical transition to turbulence in unbounded smooth shear flows is discussed. According to this scenario, the transition to turbulence is caused by the interplay between the four basic phenomena: (a) linear “drift” of spatial Fourier harmonics (SFH) of disturbances in wave-number space (k-space); (b) transient growth of SFH; (c) viscous dissipation; (d) nonlinear process that closes a feedback loop of transition by angular redistribution of SFH in k-space; The key features of the concept are: transition to turbulence only by the finite amplitude vortex disturbances; anisotropy of the process in k-space; onset on chaos due to the dynamic (not stochastic) process. The evolution of 2D small-scale vortex disturbances in the parallel flows with uniform shear of velocity is analyzed in the framework of the weak turbulence approach. This numerical test analysis is carried out to prove the most problematic statement of the conception—existence of positive feedback caused by the nonlinear process (d). Numerical calculations also show the existence of a threshold: if amplitude of the initial disturbance exceeds the threshold value, the self maintenance of disturbances becomes realistic. The latter, in turn, is the characteristic feature of the flow transition to the turbulent state and its self maintenance.

Download Full-text

Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.317 ◽

2013 ◽

Vol 731 ◽

pp. 1-45 ◽

Cited By ~ 37

Author(s):

A. Riols ◽

F. Rincon ◽

C. Cossu ◽

G. Lesur ◽

P.-Y. Longaretti ◽

...

Keyword(s):

Magnetic Field ◽

Shear Flow ◽

Shear Flows ◽

Three Dimensional ◽

Transition To Turbulence ◽

Global Bifurcations ◽

Magnetic Field Generation ◽

Dynamo Action ◽

Systems Research ◽

Hydrodynamic Shear

AbstractMagnetorotational dynamo action in Keplerian shear flow is a three-dimensional nonlinear magnetohydrodynamic process, the study of which is relevant to the understanding of accretion processes and magnetic field generation in astrophysics. Transition to this form of dynamo action is subcritical and shares many characteristics with transition to turbulence in non-rotating hydrodynamic shear flows. This suggests that these different fluid systems become active through similar generic bifurcation mechanisms, which in both cases have eluded detailed understanding so far. In this paper, we build on recent work on the two problems to investigate numerically the bifurcation mechanisms at work in the incompressible Keplerian magnetorotational dynamo problem in the shearing box framework. Using numerical techniques imported from dynamical systems research, we show that the onset of chaotic dynamo action at magnetic Prandtl numbers larger than unity is primarily associated with global homoclinic and heteroclinic bifurcations of nonlinear magnetorotational dynamo cycles born out of saddle-node bifurcations. These global bifurcations are found to be supplemented by local bifurcations of cycles marking the beginning of period-doubling cascades. The results suggest that nonlinear magnetorotational dynamo cycles provide the pathway to injection of both kinetic and magnetic energy for the problem of transition to turbulence and dynamo action in incompressible magnetohydrodynamic Keplerian shear flow in the absence of an externally imposed magnetic field. Studying the nonlinear physics and bifurcations of these cycles in different regimes and configurations may subsequently help to understand better the physical conditions of excitation of magnetohydrodynamic turbulence and instability-driven dynamos in a variety of astrophysical systems and laboratory experiments. The detailed characterization of global bifurcations provided for this three-dimensional subcritical fluid dynamics problem may also prove useful for the problem of transition to turbulence in hydrodynamic shear flows.

Download Full-text

Definition and Application of a Five-Parameter Characterization of One-Dimensional Cellular Automata Rule Space

Artificial Life ◽

10.1162/106454601753238645 ◽

2001 ◽

Vol 7 (3) ◽

pp. 277-301 ◽

Cited By ~ 33

Author(s):

Gina M. B. Oliveira ◽

Pedro P. B. de Oliveira ◽

Nizam Omar

Keyword(s):

Phase Transition ◽

Dynamical Systems ◽

Cellular Automata ◽

Dynamic Behavior ◽

Discrete Dynamical Systems ◽

Transition Diagram ◽

One Dimensional ◽

Rule Space ◽

Spatially Extended

Cellular automata (CA) are important as prototypical, spatially extended, discrete dynamical systems. Because the problem of forecasting dynamic behavior of CA is undecidable, various parameter-based approximations have been developed to address the problem. Out of the analysis of the most important parameters available to this end we proposed some guidelines that should be followed when defining a parameter of that kind. Based upon the guidelines, new parameters were proposed and a set of five parameters was selected; two of them were drawn from the literature and three are new ones, defined here. This article presents all of them and makes their qualities evident. Then, two results are described, related to the use of the parameter set in the Elementary Rule Space: a phase transition diagram, and some general heuristics for forecasting the dynamics of one-dimensional CA. Finally, as an example of the application of the selected parameters in high cardinality spaces, results are presented from experiments involving the evolution of radius-3 CA in the Density Classification Task, and radius-2 CA in the Synchronization Task.

Download Full-text

Transient growth and minimal defects: Two possible initial paths of transition to turbulence in plane shear flows

Physics of Fluids ◽

10.1063/1.1775194 ◽

2004 ◽

Vol 16 (10) ◽

pp. 3515-3529 ◽

Cited By ~ 28

Author(s):

Damien Biau ◽

Alessandro Bottaro

Keyword(s):

Shear Flows ◽

Transition To Turbulence ◽

Transient Growth ◽

Plane Shear

Download Full-text

The influence of invariant solutions on the transient behaviour of an air bubble in a Hele-Shaw channel

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2019.0434 ◽

2019 ◽

Vol 475 (2232) ◽

pp. 20190434

Author(s):

Jack S. Keeler ◽

Alice B. Thompson ◽

Grégoire Lemoult ◽

Anne Juel ◽

Andrew L. Hazel

Keyword(s):

Flow Rate ◽

Solution Structure ◽

Shear Flows ◽

Stable Solution ◽

Transition To Turbulence ◽

Edge States ◽

Unstable Periodic Orbits ◽

Nonlinear Stability Analysis ◽

Transient Behaviour ◽

Air Bubble

We hypothesize that dynamical systems concepts used to study the transition to turbulence in shear flows are applicable to other transition phenomena in fluid mechanics. In this paper, we consider a finite air bubble that propagates within a Hele-Shaw channel containing a depth-perturbation. Recent experiments revealed that the bubble shape becomes more complex, quantified by an increasing number of transient bubble tips, with increasing flow rate. Eventually, the bubble changes topology, breaking into multiple distinct entities with non-trivial dynamics. We demonstrate that qualitatively similar behaviour to the experiments is exhibited by a previously established, depth-averaged mathematical model and arises from the model’s intricate solution structure. For the bubble volumes studied, a stable asymmetric bubble exists for all flow rates of interest, while a second stable solution branch develops above a critical flow rate and transitions between symmetric and asymmetric shapes. The region of bistability is bounded by two Hopf bifurcations on the second branch. By developing a method for a numerical weakly nonlinear stability analysis we show that unstable periodic orbits (UPOs) emanate from the first Hopf bifurcation. Moreover, as has been found in shear flows, the UPOs are edge states that influence the transient behaviour of the system.

Download Full-text