Acceleration scaling and stochastic dynamics of a fluid particle in turbulence
2021
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. In this paper, I present the 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, in addition to the expected power law behavior with the dissipation rate. The exponential growth, which clearly departs from the previous propositions, reflects the additive nature of the kinetic energy, and gives the possibility to see the acceleration as a multiplicative cascade process integrating the effects of sweeps by the flow structures along the fluid-particle trajectory. I then propose scaling laws for the Reynolds number dependence of the conditional and unconditional acceleration variance using the Barenblatt's incomplete similarity framework. On the basis of these observations, we introduce a vectorial stochastic model for the dynamics of a tracer in turbulent flows. This model incorporates the non-Markovian log-normal model for the dissipation rate recently proposed by Chevillard, as well as an additional hypothesis regarding non-diagonal terms in the diffusion tensor. I will show that this model is in good agreement with direct numerical simulations and presents the essential characteristics of the "Lagrangian turbulence" highlighted in recent years, namely (i) non-Gaussian PDF of 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 power increments, reflecting the temporal irreversibility.
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