A statistical conservation law in two and three dimensional turbulent flows

Particles in turbulence live complicated lives. It is nonetheless sometimes possible to find order in this complexity. It was proposed in [Falkovich et al., Phys. Rev. Lett. 110, 214502 (2013)] that pairs of Lagrangian tracers at small scales, in an incompressible isotropic turbulent flow, have a statistical conservation law. More specifically, in a d-dimensional flow the distance $R(t)$ between two neutrally buoyant particles, raised to the power $-d$ and averaged over velocity realizations, remains at all times equal to the initial, fixed, separation raised to the same power. In this work we present evidence from direct numerical simulations of two and three dimensional turbulence for this conservation. In both cases the conservation is lost when particles exit the linear flow regime. In 2D we show that, as an extension of the conservation law, a Evans-Cohen-Morriss/Gallavotti-Cohen type fluctuation relation exists. We also analyse data from a 3D laboratory experiment [Liberzon et al., Physica D 241, 208 (2012)], finding that although it probes small scales they are not in the smooth regime. Thus instead of \left, we look for a similar, power-law-in-separation conservation law. We show that the existence of an initially slowly varying function of this form can be predicted but that it does not turn into a conservation law. We suggest that the conservation of \left, demonstrated here, can be used as a check of isotropy, incompressibility and flow dimensionality in numerical and laboratory experiments that focus on small scales

Dynamical landscape of transitional pipe flow

The transition to turbulence in pipes is characterized by a coexistence of laminar and turbulent states. At the lower end of the transition, localized turbulent pulses, called puffs, can be excited. Puffs can decay when rare fluctuations drive them close to an edge state lying at the phase-space boundary with laminar flow. At higher Reynolds numbers, homogeneous turbulence can be sustained, and dominates over laminar flow. Here we complete this landscape of localized states, placing it within a unified bifurcation picture. We demonstrate our claims within the Barkley model, and motivate them generally. Specifically, we suggest the existence of an antipuff and a gap-edge -- states which mirror the puff and related edge state. Previously observed laminar gaps forming within homogeneous turbulence are then naturally identified as antipuffs nucleating and decaying through the gap edge

Two-dimensional turbulence with local interactions: statistics of the condensate

Two-dimensional turbulence self-organizes through a process of energy accumulation at large scales, forming a coherent flow termed a condensate. We study the condensate in a model with local dynamics, the large-scale quasi-geostrophic equation, observed here for the first time. We obtain analytical results for the mean flow and the two-point, second-order correlation functions, and validate them numerically. The condensate state requires parity+time-reversal symmetry breaking. We demonstrate distinct universal mechanisms for the even and odd correlators under this symmetry. We find that the model locality is imprinted in the small scale dynamics, which the condensate spatially confines.Comment: 5 pages, 3 figure

Statistics of inhomogeneous turbulence in large scale quasi-geostrophic dynamics

A remarkable feature of two-dimensional turbulence is the transfer of energy from small to large scales. This process can result in the self-organization of the flow into large, coherent structures due to energy condensation at the largest scales. We investigate the formation of this condensate in a quasi-geostropic flow in the limit of small Rossby deformation radius, namely the large scale quasi-geostrophic model. In this model potential energy is transferred up-scale while kinetic energy is transferred down-scale in a direct cascade. We focus on a jet mean flow and carry out a thorough investigation of the second order statistics for this flow, combining a quasi-linear analytical approach with direct numerical simulations. We show that the quasi-linear approach applies in regions where jets are strong and is able to capture all second order correlators in that region, including those related to the kinetic energy. This is a consequence of the blocking of the direct cascade by the mean flow in jet regions, suppressing fluctuation-fluctuation interactions. The suppression of the direct cascade is demonstrated using a local coarse-graining approach allowing to measure space dependent inter-scale kinetic energy fluxes, which we show are concentrated in between jets in our simulations. We comment on the possibility of a similar direct cascade arrest in other two-dimensional flows, arguing that it is a special feature of flows in which the fluid element interactions are local in spaceComment: 19 pages, 13 figure

Jets or vortices - what flows are generated by an inverse turbulent cascade?

An inverse cascade{energy transfer to progressively larger scales{is a salient feature of two-dimensional turbulence. If the cascade reaches the system scale, it creates a coherent ow expected to have the largest available scale and conform with the symmetries of the domain. In a doubly periodic rectangle, the mean ow with zero total momentum was therefore believed to be unidirec- tional, with two jets along the short side; while for an aspect ratio close to unity, a vortex dipole was expected. Using direct numerical simulations, we show that in fact neither the box symmetry is respected nor the largest scale is realized: the ow is never purely unidirectional since the inverse cascade produces coherent vortices, whose number and relative motion are determined by the aspect ratio. This spontaneous symmetry breaking is closely related to the hierarchy of averaging times. Long-time averaging restores translational invariance due to vortex wandering along one direction, and gives jets whose profile, however, can be deduced neither from the largest-available-scale argument, nor from the often employed maximum-entropy principle or quasilinear approximation

A mechanism for turbulence proliferation

The subcritical transition to turbulence, as occurs in pipe flow, is believed to generically be a phase transition in the directed percolation universality class. At its heart is a balance between the decay rate and proliferation rate of localized turbulent structures, called puffs in pipe flow. Here we propose the first-ever dynamical mechanism for puff proliferation -- the process by which a puff splits into two. In the first stage of our mechanism, a puff expands into a slug. In the second stage, a laminar gap is formed within the turbulent core. The notion of a split-edge state, mediating the transition from a single puff to a two puff state, is introduced and its form is predicted. The role of fluctuations in the two stages of the transition, and how splits could be suppressed with increasing Reynolds number, are discussed. Using numerical simulations, the mechanism is validated within the stochastic Barkley model. Concrete predictions to test the proposed mechanism in pipe and other wall bounded flows, and implications for the universality of the directed percolation picture, are discussed