Acceleration scaling and stochastic dynamics of a fluid particle in turbulence

TL;DR

This paper investigates the statistical dependence of fluid-particle acceleration on dissipation rate and kinetic energy in turbulence, revealing exponential and power-law behaviors, and introduces a stochastic model aligning well with simulations.

Contribution

It uncovers the exponential dependence of acceleration on kinetic energy and proposes a multiplicative cascade model incorporating intermittency and flow structure effects.

Findings

Acceleration depends exponentially on kinetic energy with Reynolds number independence.

Conditional acceleration variance scales with Reynolds number using incomplete similarity.

The stochastic model matches direct numerical simulation results well.

Abstract

It is well known that the fluid-particle acceleration is intimately related to the dissipation rate of turbulence, in line with the Kolmogorov assumptions. On the other hand, various experimental and numerical works have reported as well its dependence on the kinetic energy, which is generally attributed to intermittency and non-independence of the small-scale dynamics from large-scale ones. The analyses given in this paper focus on statistics of the fluid-particle acceleration conditioned on both the local dissipation rate and the kinetic energy. It is shown that this quantity presents an exponential dependence on the kinetic energy with a growth rate independent of the Reynolds number, in addition to the expected power law behavior with the dissipation rate. The exponential growth, which clearly departs from the previous propositions, reflects additional kinematic effects of the flow structures on the acceleration. Regarding intermittency, to account for the persistence of the effect of the large-scales on the dissipation rate, it is further proposed scaling laws for the Reynolds number dependence of the conditional and unconditional acceleration variance using Barenblatt's incomplete similarity framework. It is then shown that both these intermittency and kinematic effects can be combined in a multiplicative cascade process for the acceleration depending on the kinetic energy and the dissipation rate. On the basis of these observations, we introduce a vectorial stochastic model for the dynamics of a tracer in turbulent flows. This model incorporates a fractional log-normal process for the dissipation rate recently proposed, as well as an additional hypothesis regarding non-diagonal terms in the diffusion tensor which naturally leads to the decomposition between tangential and centripetal acceleration. This model is shown to be in good agreement with direct numerical simulations and presents the essential characteristics of the Lagrangian turbulence highlighted in recent years, namely (i) non-Gaussian acceleration, (ii) scale separation between the norm of the acceleration and its components, (iii) anomalous scaling law for the Lagrangian velocity spectra, and (iv) negative skewness of the increments of the mechanical power, reflecting the temporal irreversibility of the dynamics.