Application of the finite-size Lyapunov exponent to particle tracking velocimetry in fluid mechanics experiments

Abstract

A finite-size (or scale) Lyapunov exponent (FSLE), lambdaa(x), is presented in a statistical mechanical framework and employed to characterize mixing in a variety of laboratory and computational fluid mechanics experiments. The FSLE is the exponential rate at which two particles separate from a distance x to ax. Laboratory particle tracking experiments are used to study penetrative convection and flow in porous media while computational experiments are used to study Lévy processes and deterministic diffusion. The apparent scaling relation lambaa(x) approximately Cax(-beta(a)) of the FSLE holds over intermediate initial separations where the laboratory experiment data is most accurate and asymptotically for the computational experiments. The dependence of the exponent beta on a decreases with increasing a. In the matched index porous system, Ca is also a function of mean fluid velocity. The exponent beta is alpha when the Lévy process is alpha-stable and in this case beta is independent of a.

On upscaling operator-stable Lévy motions in fractal porous media

“…Since an a-stable Lévy motion is a fractal with divider dimension a, we may without loss of generality model the drift velocity or acceleration in a fractal medium as a Lévy process. The degree of stability can be obtained experimentally by using particle tracking velocimetry (PTV) in conjunction with the finite-size Lyapunov exponent (FSLE) [6].…”

Analysis of dispersion in porous media via matched-index particle tracking velocimetry experiments

“…We note that the particles could not be tracked sufficiently long to obtain high quality a-times for larger separations where an asymptotic regime would be reached. Issues such as the effect of changing the threshold value and a more in depth analysis of the FSLE can be found in [15].…”

“…This quantity was first introduced by Aurell et al [2] and here a slightly more general version is utilized. The actual statistical mechanical definition is complex and technical and we refer the reader to [15] for details. Here a heuristic definition is provided.…”

Penetrative convection in stratified fluids: velocity and temperature measurements

“…In the simplest treatment of mixing, one prescribes a dispersion coefficient which scales with characteristic size and velocity (Moroni et al, 2003;Kleinfelter et al, 2005;Cushman et al, 2005). The dispersion coefficient is typically much greater than diffusivity arising from microscopic processes alone (Miesch et al, 2000).…”

“…In the simplest treatment of mixing, one prescribes a dispersion coefficient which scales with characteristic size and velocity (Moroni et al, 2003;Kleinfelter et al, 2005;Cushman et al, 2005).…”