A Lagrangian fluctuationdissipation relation for scalar turbulence. Part II. Wallbounded flows
Abstract
We derive here Lagrangian fluctuationdissipation relations for advected scalars in wallbounded flows. The relations equate the dissipation rate for either passive or active scalars to the variance of scalar inputs from the initial values, boundary values, and internal sources, as those are sampled backward in time by stochastic Lagrangian trajectories. New probabilistic concepts are required to represent scalar boundary conditions at the walls: the boundary localtime density at points on the wall where scalar fluxes are imposed and the boundary first hittingtime at points where scalar values are imposed. These concepts are illustrated both by analytical results for the problem of pure heat conduction and by numerical results from a database of channelflow flow turbulence, which also demonstrate the scalar mixing properties of nearwall turbulence. As an application of the fluctuationdissipation relation, we examine for wallbounded flows the relation between anomalous scalar dissipation and Lagrangian spontaneous stochasticity, i.e. the persistent nondeterminism of Lagrangian particle trajectories in the limit of vanishing viscosity and diffusivity. In the first paper of this series, we showed that spontaneous stochasticity is the only possible mechanism for anomalous dissipation of passive or active scalars, away from walls. Here it is shown that this remains true when there are no scalar fluxes through walls. Simple examples show, on the other hand, that a distinct mechanism of nonvanishing scalar dissipation can be thin scalar boundary layers near the walls. Nevertheless, we prove for general wallbounded flows that spontaneous stochasticity is another possible mechanism of anomalous scalar dissipation.
 Publication:

Journal of Fluid Mechanics
 Pub Date:
 October 2017
 DOI:
 10.1017/jfm.2017.571
 arXiv:
 arXiv:1703.08133
 Bibcode:
 2017JFM...829..236D
 Keywords:

 Physics  Fluid Dynamics
 EPrint:
 Journal of Fluid Mechanics 829, 236279 (2017)