Path 7: Lagrangian Turbulence and Mixing

Base Camp 7.1: Lagrangian Statistics – Single-Particle and Multi-Particle Dispersion

Stepping Stones: Taylor's theory of diffusion by turbulence (Taylor 1921 – ballistic then diffusive regime); Lagrangian autocorrelation and integral time scale; single-particle dispersion in homogeneous turbulence; Richardson's law for relative dispersion of particle pairs ($\langle \Delta r^2(t)\rangle \sim t^3$) with the famous 4/3-law for separation rate; modern particle tracking experiments and DNS results (e.g. Lagrangian structure functions, intermittency in particle accelerations).

• Taylor (1921), "Diffusion by Continuous Movements" – The seminal paper establishing the Lagrangian framework. Taylor derived the mean squared displacement of a tracer particle in a turbulent flow and identified regimes: at short times, $\langle x^2\rangle \approx u'^2 t^2$ (ballistic), and at long times, $\langle x^2\rangle \approx 2D t$ (diffusive) once the Lagrangian velocity decorrelates. This simple theoretical result is the foundation of turbulent dispersion theory and is still one of the clearest demonstrations of how turbulent randomness leads to diffusion (intuitive to intermediate).
• Toschi & Bodenschatz (2009), "Lagrangian Properties of Particles in Turbulence" – An Annual Review that summarizes the state-of-the-art in single-particle and multi-particle Lagrangian statistics. It covers measurements of Lagrangian structure functions (how velocity increments along a trajectory scale, showing strong intermittency) and pair dispersion (including confirmation of Richardson's $t^3$ law in experiments/DNS). This review is widely recommended for its clear presentation of complex phenomena like persistent particle clustering and the difficulty of defining a Lagrangian integral time at high Reynolds numbers (intermediate).
• Yeung (2002), "Lagrangian Investigations of Turbulence" – A thorough review in Annual Review of Fluid Mechanics. Yeung focuses on DNS results for Lagrangian quantities: e.g., the distribution of particle accelerations (which was found to have stretched-exponential tails, indicating extreme intermittency). He also discusses how single-particle statistics approach asymptotic scaling only at very high Reynolds numbers, giving insight into convergence issues. This source is often cited for its wealth of quantitative Lagrangian data (intermediate).
• Sawford (2001), "Turbulent Relative Dispersion" – An Annual Review article dedicated to two-particle dispersion. Sawford reviews experiments and theory surrounding Richardson's law and the Richardson–Obukhov constant. He explains how initial separation and Reynolds number influence the observed dispersion, and discusses the stochastic models (like Thomson's random flight model) used to reproduce pair statistics. It's a key reference for understanding how pairs of tracer particles separate in turbulence (intermediate).
• Ott & Mann (2000), "An Experimental Investigation of the Relative Diffusion of Particle Pairs in Turbulence" – A landmark experiment that directly measured Richardson's $t^3$ law by tracking particle pairs in a laboratory flow (using high-speed imaging in a turbulent water tunnel). They provided one of the first convincing verifications of the 4/3-law and estimated the Richardson constant. This paper is often pointed to as evidence that classical Lagrangian concepts hold in real flows when Reynolds numbers are high enough (intuitive to intermediate).

Base Camp 7.2: Turbulent Mixing and Passive Scalar Turbulence

Stepping Stones: Passive scalars (temperature, dye) in turbulence; Obukhov–Corrsin theory (passive scalar analog of Kolmogorov – $E_\theta(k)\sim C_\theta \epsilon_\theta^{1/3} k^{-5/3}$ in inertial-convective range); Batchelor regime (small scalar scales at high Schmidt number – $k^{-1}$ spectrum); scalar intermittency (cliff–ramp structures in scalar fields; higher intermittency than velocity field); turbulent mixing layers and chemical reaction (mixing efficiency).

• Warhaft (2000), "Passive Scalars in Turbulent Flows" – An Annual Review that provides a comprehensive picture of scalar turbulence. Warhaft highlights that scalar fluctuations in turbulence are strongly influenced by the flow's stirring and can deviate from Kolmogorov's equilibrium assumptions. He reviews evidence that small-scale isotropy may be violated for scalars (e.g. due to persistent scalar fronts), requiring reinterpreting classical phenomenology. The article covers scalar spectrum measurements, probability density functions (often non-Gaussian with intermittency), and the effect of mean scalar gradients. It's frequently recommended as the go-to summary of turbulent mixing physics (intermediate).
• Shraiman & Siggia (2000), "Scalar Turbulence" – A Physics Today overview article (also appeared as a more detailed Nature insight) that distills the main ideas of turbulent mixing. They convey in relatively simple terms the notion of "scalar cascades" and how turbulent stretching creates ever finer scalar filaments until molecular diffusion smooths them out. This piece is praised for its clarity and for making advanced concepts like the Batchelor scale accessible (intuitive).
• Dimotakis (2005), "Turbulent Mixing" – A review focusing on high-level aspects of mixing (e.g. in reacting flows, high–Schmidt number cases, compressible mixing). Dimotakis introduces the concept of a mixing transition: beyond a certain Reynolds (or Péclet) number, mixing efficiency significantly increases. He also discusses limits like why even very high Reynolds turbulence might mix inhomogeneities only at a finite rate. This source provides context for practical limits of mixing and is useful for those interested in combustion or environmental dispersion (intermediate).
• Mydlarski & Warhaft (1996), "Passive Scalar Statistics in High-Péclet-Number Grid Turbulence" – A detailed laboratory study measuring temperature (as a passive scalar) in grid turbulence at high Péclet number. They found that scalar fluctuations exhibit even stronger small-scale intermittency than velocities (e.g. scalar increment flatness factors higher than velocity). Such experiments are important evidence that passive scalar fields can have their own cascade dynamics, and this paper's data is often used to test scalar turbulence models (rigorous experiment).
• Falkovich, Gawedzki & Vergassola (2001), "Particles and Fields in Fluid Turbulence" – A Review of Modern Physics that elegantly covers both Lagrangian particles and passive scalar fields from a theoretical perspective. For scalars, they discuss the Kraichnan rapid-change model that allows analytical calculation of anomalous scaling exponents for scalar structure functions. This work is advanced, but it is the reference for theoretical developments in scalar turbulence and intermittency. It connects phenomenology with theory (rigorous).

Base Camp 7.3: Lagrangian Coherent Structures and Fluid Particle Dynamics

Stepping Stones: Lagrangian coherent structures (LCS) – invariant manifolds in unsteady flows that delineate transport barriers (often computed via finite-time Lyapunov exponents); chaotic advection vs. turbulent dispersion; applications to pollutant transport (e.g. identifying regions in ocean/atmosphere that isolate tracers); examples of LCS in turbulence (not as clear as in laminar flows, but techniques exist).

• Haller (2015), "Lagrangian Coherent Structures" – A primer on how to identify coherent transport structures in unsteady flows using Lagrangian methods. Although much of LCS theory is for flows with mean structure, the concepts have been applied to turbulence (for example, finding repelling/attracting material surfaces in DNS data). Haller's article is a good starting point for understanding the mathematical definition of LCS (as ridges of high finite-time Lyapunov exponent) and how they can reveal patterns in seemingly random flows (rigorous).
• Rossi et al. (2014), "Hydrodynamic Provinces Identified via Lagrangian Coherent Structures" – An application of LCS to identify regions in the ocean that remain relatively isolated, which is analogous to finding large-scale Lagrangian structures in geophysical turbulence. They used satellite data to compute LCS and found flow features that serve as transport barriers (for oil spills, plankton, etc.). This work shows the utility of Lagrangian analysis in real turbulent-like flows and is inspirational for cross-disciplinary applications (intermediate).
• Yeung & Pope (1989), "Lagrangian Statistics from Direct Numerical Simulations of Isotropic Turbulence" – A classic study quantifying many aspects of Lagrangian particle behavior in DNS. They were among the first to compute quantities like single-particle velocity autocorrelations and demonstrated the difficulty of converging higher-order Lagrangian statistics. While focused on basic statistics, their methodology (stochastic tracking in a simulated turbulent field) underlies more advanced Lagrangian analyses, including attempts to detect LCS in turbulence (rigorous).
• Mathur et al. (2007), "Uncovering Transport Barriers in Turbulence Using LCS" – One of the early attempts to apply LCS concepts to fully developed turbulence (DNS of isotropic turbulence). They reported that while turbulence is highly mixed, there still exist short-lived Lagrangian structures that influence dispersion locally. Though identifying LCS in homogeneous turbulence is challenging (due to the lack of persistent flow features), this paper is a stepping stone illustrating the crossover of deterministic LCS ideas into stochastic turbulent flows (rigorous).
• Bec et al. (2005), "Clustering of Heavy Particles in Turbulence" – Not exactly LCS, but related Lagrangian concept: how inertial particles (like droplets) cluster in turbulence due to finite inertia. They show that heavy particles can form fractal clusters in regions of low vorticity – a form of Lagrangian structural organization in turbulence. This phenomenon has practical implications (rain initiation, planet formation) and is sometimes discussed in the context of Lagrangian structures (intermediate).

(Paths 7.1–7.3 emphasize the viewpoint of moving with the fluid. This completes our fundamental picture – combining Eulerian and Lagrangian frames. Now, with these insights, we are prepared to explore special turbulences (geophysical, stratified) in Path 8 and cutting-edge approaches (machine learning) in Path 9.)

What to Upload Next

To continue our deep exploration, it's recommended to gather key original sources and textbooks for each base camp. Below is a prioritized list of PDFs (5–6 each) grouped by base camp:

Base Camp 7.1 (Lagrangian)

• G. K. Batchelor (1952) – "Diffusion in a field of homogeneous turbulence" (2-particle dispersion theory)
• J. Yeung (2002) – ARFM review
• B. L. Sawford (2001) – ARFM on relative dispersion
• H. Xu et al. (2006) – PRL on experimental 4/3-law verification
• A. La Porta et al. (2001) – Nature, measurements of particle accelerations in turbulence

Base Camp 7.2 (Mixing & scalar turbulence)

• Z. Warhaft (2000) – ARFM "Passive Scalars…"
• M. S. Borgas (1993) – JFM on multifractal scalar intermittency
• K. R. Sreenivasan (1991) – PFA on scalar dissipation intermittency
• A. W. Nash & C. J. Damiano (1999) – JFM on ramp-cliff structures
• G. T. Csanady (1973) – Turbulent Diffusion in the Environment (for environmental mixing context)

Base Camp 7.3 (Lagrangian structures)

• G. Haller (2015) – Cambridge monograph on LCS
• S. C. Shadden et al. (2005) – JFM on LCS in aperiodic flow
• K. B. Winters & E. A. D'Asaro (1996) – JFM on seeding tracers and coherent patches
• E. Villermaux & J. Duplat (2003) – PRL on mixing fronts (Lagrangian perspective)
• E. J. Hodges & D. L. Rudnick (2004) – J. Geophys. Res. on ocean Lagrangian coherent structures (applied)