The Lagrangian view of passive scalar turbulence has recently produced interesting results and interpretations. Innovations in theory, experiments, simulations and data analysis of Lagrangian turbulence are reviewed here in brief. Part of the review is closely related to the so-called Kraichnan model for the advection of the passive scalar in synthetic turbulence. Possible implications for a better understanding of the passive scalar mixing in Navier-Stokes turbulence are also discussed.
Deep Dive into Lagrangian views on turbulent mixing of passive scalars.
The Lagrangian view of passive scalar turbulence has recently produced interesting results and interpretations. Innovations in theory, experiments, simulations and data analysis of Lagrangian turbulence are reviewed here in brief. Part of the review is closely related to the so-called Kraichnan model for the advection of the passive scalar in synthetic turbulence. Possible implications for a better understanding of the passive scalar mixing in Navier-Stokes turbulence are also discussed.
Eulerian and Lagrangian points of view on the motion of fluids, and on the mixing of scalars, are equivalent (Prandtl & Tietjens 1934) but cannot be related to each other in analytically tractable ways except in a few special instances. The bridge between the two descriptions is formally obvious, at least for a conserved scalar. A batch of a massless tracer particle which is advected in an incompressible turbulent flow u, subject to the diffusion κ, follows the Langevin equation (Gardiner 2004) given by ẋ = u(x(t), t) + √ 2κ χ(t) , (1.1)
where χ(t) is vectorial white noise that is statistically independent in each of its three components. This equation is complementary to a Fokker-Planck equation for θ(x, t), the probability density function (PDF) of the tracer at time t at position x. The equation ∂θ ∂t
is exactly the advection-diffusion equation for a scalar field in the Eulerian frame.
The scalar diffusivity is denoted by κ. Its ratio to the kinematic viscosity ν of the fluid is the Schmidt number
or, for an advected temperature field, the Prandtl number, P r.
The scalar mixing process at a given Reynolds number is significantly different in the Kolmogorov-Obukhov-Corrsin regime given by Sc ≤ 1 (Kolmogorov 1941, Obukhov 1949, Corrsin 1951) from that in the Batchelor regime given by Sc ≫ 1 (Batchelor 1959). While the scalar is advected by the turbulent inertial range in the former, it is mainly advected in the latter by smooth velocity structures of the viscous range of turbulence.
In turbulent mixing of passive scalars, and in turbulence in general, Eulerian and Lagrangian views have their specialized places. For example, the former appears better suited for studying problems such as nonpremixed combustion in reacting flows (Linãn & Williams 1993, Peters 2000). Indeed, the comparative ease of making Eulerian measurements renders them more appealing in a number of instances. If, on the other hand, one has to calculate the dispersion of θ arising from a few localized sources, the Lagrangian view is more appropriate (Sawford 2001). Examples of such applications include fumigation (Sawford et al. 1998), the spread of buoyant plumes (Heinz & van Dop 1999), and the surface motion of buoys over the sea (Maurizi et al. 2004). Lagrangian studies are being recognized as increasingly important in geophysics, e.g. for the mixing of plankton and other biomatter in the upper ocean (Seuront & Schmitt 2004) and for droplet dynamics in cumulus clouds (Vaillancourt et al. 2002).
In spite of their wide-spread utility, Lagrangian data are harder to obtain experimentally and are more expensive to compute. For example, it is unclear if we can ever measure with satisfactory accuracy high-order moments of the particle acceleration along a trajectory in a turbulent flow of moderately high Reynolds number (Yakhot & Sreenivasan 2005); see also discussion of Fig. 1 in Sec. 3. The relevant point here is that measuring (as an example) the fourth moment of the Lagrangian acceleration is equivalent to measuring the twelfth moment of Eulerian velocity differences (which has not been obtained so far with impeccable accuracy). The basis for this connection is the relation for the Lagrangian acceleration expressed in terms of Eulerian velocity increments,
where η is the fluctuating small scale around the Kolmogorov length η K and the combination η × δ η u/ν = 1, relating the velocity increment δ η u on the length scale η to η. The difficulties are compounded by the tendency of most conventional Lagrangian tracers to cluster near solid walls, which produces a large mismatch of information between fluid particles and tracers. Large shear will have similar effects. Thus, one should ask if the additional complexities associated with Lagrangian calculations and measurements justify the investment and effort, leaving aside their relative novelty for the present: When is the Lagrangian perspective the method of choice? What makes it advantageous? What have we learnt in the big picture? New insights into these questions, especially on the scalar mixing problem, have become possible for the so-called Kraichnan model (Kraichnan 1968(Kraichnan , 1994)). For a review with an extensive list of references, see Falkovich et al. (2001). The Kraichnan model has been applied to diverse physical circumstances such as the mixing of passive scalars and their dynamics in compressible turbulence (Gawȩdzki & Vergassola 2000, Bec et al. 2004). In a further development, the claim has been made (with certain careful caveats) that the anomalous scaling of active scalars can be understood in terms of passive scalar fields (Ching et al. 2003).
All these developments deserve some comment. This is one of our purposes here. Secondly, we will comment on these theoretical achievements in the context of numerical and experimental work on the mixing of the passive scalar in Navier-Stokes turbulence. In the process, we draw attention to a few open questions that shoul
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