k Spectrum of Passive Scalars in Lagrangian Chaotic Fluid Flows

Slow decay of concentration variance due to no-slip walls in chaotic mixing

Chaotic mixing in a closed vessel is studied experimentally and numerically in different two-dimensional (2D) flow configurations. For a purely hyperbolic phase space, it is well known that concentration fluctuations converge to an eigenmode of the advection-diffusion operator and decay exponentially with time. We illustrate how the unstable manifold of hyperbolic periodic points dominates the resulting persistent pattern. We show for different physical viscous flows that, in the case of a fully chaotic Poincaré section, parabolic periodic points at the walls lead to slower (algebraic) decay. A persistent pattern, the backbone of which is the unstable manifold of parabolic points, can be observed. However, slow stretching at the wall forbids the rapid propagation of stretched filaments throughout the whole domain, and hence delays the formation of an eigenmode until it is no longer experimentally observable. Inspired by the baker's map, we introduce a 1D model with a parabolic point that gives a good account of the slow decay observed in experiments. We derive a universal decay law for such systems parametrized by the rate at which a particle approaches the no-slip wall.

Exponential decay of chaotically advected passive scalars in the zero diffusivity limit

The time asymptotic decay of the variance of a passive scalar in a chaotic flow is studied. Two mechanisms for this decay, which involve processes at short and long length scales, respectively, are considered. The validity of the short length scale mechanism, which is based on Lagrangian stretching theory, is discussed. We also investigate the regimes of applicability and observable signatures of the two mechanisms. Supporting evidence is provided by high resolution numerical experiments.

Spectral properties and transport mechanisms of partially chaotic bounded flows in the presence of diffusion

The spectral properties of the Poincaré operator associated with the advection-diffusion equation for partially chaotic periodic flows defined in bounded domains are analyzed in this Letter. For vanishingly small diffusivities (i.e., for the Peclet number tending to infinity) the dominant eigenvalue Lambda exhibits the scaling Lambda approximately Pe-alpha, where the exponent alpha in (0,1) depends on the global property of the flow (shape, geometry, and symmetry of quasiperiodic islands). The value of the exponent alpha is an indicator of qualitatively different transport mechanisms and depends on the localization properties of the corresponding eigenfunctions.

Decay of scalar turbulence revisited

We demonstrate that at long times the rate of passive scalar decay in a turbulent, or simply chaotic, flow is dominated by regions where mixing is less efficient. We examine two situations. The first is of a spatially homogeneous stationary turbulent flow with both viscous and inertial scales present. It is shown that at large times scalar fluctuations decay algebraically in time at all spatial scales. The second example explains chaotic stationary flow in a disk/pipe. The boundary region of the flow controls the long-time decay, which is algebraic at some transient times, but becomes exponential, with the decay rate dependent on the scalar diffusion coefficient, at longer times.

Scalings of scalar structure functions in a velocity field with coherent vortical structures

In planar turbulence modeled as an isotropic and homogeneous collection of two-dimensional noninteracting compact vortices, the structure functions S(p)(r) of a statistically stationary passive scalar field have the following scaling behavior in the limit where the Péclet number Pe-->infinity: S(p)(r) approximately const+ln(r/L Pe(-1/3)) for L Pe(-1/3)<<r<<L, S(p)(r) approximately (r/L Pe(-1/3))(6(1-D)) for L Pe(-1/2)<<r<<L Pe(-1/3), where L is a large scale and D is the fractal codimension of the spiral scalar structures generated by the vortices (1/2 < or =D < 2/3). Note that L Pe(-1/2) is the scalar Taylor microscale that stems naturally from our analytical treatment of the advection-diffusion equation. The essential ingredients of our theory are the locality of interscale transfer and Lundgren's time average assumption. A phenomenological theory explicitly based only on these two ingredients reproduces our results and a generalization of this phenomenology to spatially smooth chaotic flows yields (k ln k)(-1) generalized power spectra for the advected scalar fields.

Multifractal structure of chaotically advected chemical fields

The structure of the concentration field of a decaying substance produced by chemical sources and advected by a smooth incompressible two-dimensional flow is investigated. We focus our attention on the nonuniformities of the Holder exponent of the resulting filamental chemical field. They appear most evidently in the case of open flows where irregularities of the field exhibit strong spatial intermittency as they are restricted to a fractal manifold. Nonuniformities of the Holder exponent of the chemical field in closed flows appears as a consequence of the nonuniform stretching of the fluid elements. We study how this affects the scaling exponents of the structure functions, displaying anomalous scaling, and relate the scaling exponents to the distribution of local Lyapunov exponents of the advection dynamics. Theoretical predictions are compared with numerical experiments.

Power spectrum of passive scalars in two dimensional chaotic flows

In this paper the power spectrum of passive scalars transported in two dimensional chaotic fluid flows is studied theoretically. Using a wave-packet method introduced by Antonsen et al., several model flows are investigated, and the fact that the power spectrum has the k(-1)-scaling predicted by Batchelor is confirmed. It is also observed that increased intermittency of the stretching tends to make the roll-off of the power spectrum at the high k end of the k(-1) scaling range more gradual. These results are discussed in light of recent experiments where a k(-1) scaling range was not observed. (c) 2000 American Institute of Physics.

Simple model of intermittent passive scalar turbulence

The passive scalar convection by chaotic two-dimensional incompressible flow is studied. Analytically solvable equations are suggested to describe the evolution of the probability density functions of tracer gradients and power spectra. The parameters of the model are expressed explicitly via the correlation functions of the velocity field. The multifractal spectrum f(alpha) of the scalar dissipation field is calculated; strict multifractality holds only for small values of alpha. Stationary and exponentially decaying power spectra of the scalar are obtained.