The spectrum of fractal dimensions of passively convected scalar gradients in chaotic fluid flows

Varosi, Frank; Antonsen, Thomas M.; Ott, Edward

doi:10.1063/1.858081

Cited by 86 publications

(55 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Multifractal scaling of gradients of undiffused tracers has been predicted by Varosi et al [15] but, apart from the nondiffusive limitation, there are some important differences between the approach of Varosi et al and that employed here. in their theoretical formula linking the generalized dimension spectrum to the PDF of Lyapunov exponents, Varosi et al employ a variable box-size cover designed to resolve each filament individually.…”

Section: Freely Decaying Evolutionmentioning

confidence: 86%

Tracer microstructure in the large-eddy dominated regime

Pierrehumbert

1994

Chaos, Solitons & Fractals

255

301

View full text Add to dashboard Cite

Abstract--We have re-examined the inertial range behavior of a passive scalar which is advected by a large-scale velocity field causing a cascade of tracer variance to small scales, where it is dissipated by diffusion. This has been done within the context of an idealized model based on mixing by a 2D area-preserving map alternating with a weak diffusion step; the model is a special case of the general advection-diffusion problem. Both freely decaying and forced equilibrium systems were considered. Our main interest in this concerns the validity of Batchelor's theory predicting a k -I tracer variance spectrum, but the tracer microstructure has been diagnosed in terms of concentration probability distribution functions, generalized dimensions of the dissipation field, structure functions, and cancellation exponents. 2D simulations carried out at 10242 resolution show that i;, the decaying case the evolution settles into a 'fractal eigenmode' in which the variance decays exponentially with time at a rate dependent on the Lyapunov exponent but independent of the diffusion coefficient. Although the concentration pattern is self-similar with time, the power spectrum is not algebraic. Concentration PDFs have exponential tails. The dissipation field is not multifractal, and formally has Dq = 2 for all q. The convergence of the squared-gradient PDFs under coarse-graining indicates some underlying fractal behavior, however, and we have introduced the notion of 'fractal degree of freedom' systems to describe such entities. Cancellation exponents and structure functions were also considered, and have a self-similarity which is compatible with a non-intermittent behavior of the dissipation field. These matters have also been addressed for the equilibrium case. The main difference is that the power spectrum of concentration variance in equilibrium exhibits a power-law inertial range, though it is steeper than k -1, but not as steep as k -2. Other features are similar to the decaying case. Very high resolution simulations of the undiffused problem indicate that a k -1 spectrum is approached asymptotically, but only at resolutions corresponding to 106 x 106. The passive scalar behavior is compared and contrasted with the behavior of scalar pseudo-vorticity (an 'active scalar') in the family of generalized 2D turbulence models introduced elsewhere in this issue.

show abstract

Section: Freely Decaying Evolutionmentioning

confidence: 86%

Tracer microstructure in the large-eddy dominated regime

Pierrehumbert

1994

Chaos, Solitons & Fractals

255

301

View full text Add to dashboard Cite

show abstract

“…Since |∇ω| 2 grows at widely varying exponential rates, only a few points would be visible if |∇ω| 2 was plotted directly using a linear scale. Therefore, we plot the following quantity instead [27],…”

Section: B Comparison Of Theory and Numerical Resultsmentioning

confidence: 99%

“…(15) has been shown to be true for the generalized baker's map [26] and numerically verified for cases where there are no KAM surfaces, for example, see Ref. [27]. The function G(h) is concave upward, G ′′ (h) ≥ 0.…”

Section: B Distribution Of Finite-time Lyapunov Exponentmentioning

confidence: 99%

Intermittency in two-dimensional turbulence with drag

et al. 2005

Self Cite

View full text Add to dashboard Cite

We consider the enstrophy cascade in forced two-dimensional turbulence with a linear drag force. In the presence of linear drag, the energy wavenumber spectrum drops with a power law faster than in the case without drag, and the vorticity field becomes intermittent, as shown by the anomalous scaling of the vorticity structure functions. Using previous theory, we compare numerical simulation results with predictions for the power law exponent of the energy wavenumber spectrum and the scaling exponents of the vorticity structure functions ζ2q. We also study, both by numerical experiment and theoretical analysis, the multifractal structure of the viscous enstrophy dissipation in terms of its Rényi dimension spectrum Dq and singularity spectrum f (α). We derive a relation between Dq and ζ2q, and discuss its relevance to a version of the refined similarity hypothesis. In addition, we obtain and compare theoretically and numerically derived results for the dependence on separation r of the probability distribution of δ r ω, the difference between the vorticity at two points separated by a distance r. Our numerical simulations are done on a 4096 × 4096 grid.

show abstract

“…For the closed flow we choose a sinusoidal shear flow with the direction of the shear periodically alternating along the x and y axis [29,30]. The velocity field is…”

Section: The Modelsmentioning

confidence: 99%

Chaotic mixing induced transitions in reaction–diffusion systems

Neufeld

Haynes

Tél

2002

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

We study the evolution of a localized perturbation in a chemical system with multiple homogeneous steady states, in the presence of stirring by a fluid flow. Two distinct regimes are found as the rate of stirring is varied relative to the rate of the chemical reaction. When the stirring is fast localized perturbations decay towards a spatially homogeneous state. When the stirring is slow (or fast reaction) localized perturbations propagate by advection in form of a filament with a roughly constant width and exponentially increasing length. The width of the filament depends on the stirring rate and reaction rate but is independent of the initial perturbation. We investigate this problem numerically in both closed and open flow systems and explain the results using a one-dimensional "mean-strain" model for the transverse profile of the filament that captures the interplay between the propagation of the reaction-diffusion front and the stretching due to chaotic advection.The problem of chemical or biological activity in fluid flows has recently become an area of active research [1][2][3][4][5][6][7][8]. Apart from theoretical interest this problem has a number of industrial [9] and environmental [10,11] applications. One of the simplest manifestation of non-linear behavior in reaction-diffusion systems is the possibility of travelling front solutions. In this paper we study the effect of chaotic mixing, by an unsteady laminar flow, in reaction-diffusion systems supporting front propagation.

show abstract

The spectrum of fractal dimensions of passively convected scalar gradients in chaotic fluid flows

Cited by 86 publications

References 20 publications

Tracer microstructure in the large-eddy dominated regime

Tracer microstructure in the large-eddy dominated regime

Intermittency in two-dimensional turbulence with drag

Chaotic mixing induced transitions in reaction–diffusion systems

Contact Info

Product

Resources

About