[Paper Review] Easy turbulence

Krzysztof Gawędzki|arXiv (Cornell University)|Jul 16, 1999

Fluid Dynamics and Turbulent Flows8 citations

TL;DR

This paper presents an introductory course on fully developed turbulence, covering the Navier-Stokes equations, Kolmogorov's theory of 3D turbulence, Kraichnan-Batchelor theory for 2D turbulence, Richardson dispersion, and cascades in passive advection. It establishes foundational concepts in turbulence through statistical descriptions, energy cascades, and intermittency effects, providing a coherent framework for understanding turbulent dynamics across dimensions.

ABSTRACT

This is an introductory course on fully developed turbulence. It discusses: in Lecture 1: the Navier Stokes equations, existence of solutions, statistical description, energy balance and cascade picture; in Lecture 2: the Kolmogorov theory of three-dimensional turbulence versus intermittency, the Kraichnan-Batchelor theory of two-dimensional turbulence; in Lecture 3: the Richardson dispersion law and the breakdown of the Lagrangian flow; in Lecture 4: direct and inverse cascades and intermittency in the Kraichnan model of passive advection.

Motivation & Objective

• To provide a foundational understanding of fully developed turbulence for researchers new to the field.
• To explain the statistical description and energy cascade mechanisms in turbulent flows.
• To contrast three-dimensional turbulence (Kolmogorov theory) with two-dimensional turbulence (Kraichnan-Batchelor theory).
• To analyze the Richardson dispersion law and its implications for Lagrangian flow breakdown.
• To explore direct and inverse cascades and intermittency in the Kraichnan model of passive scalar advection.

Proposed method

• Formal derivation and analysis of the Navier-Stokes equations as the governing framework for incompressible fluid flow.
• Application of statistical mechanics to describe turbulent flows through ensemble averages and structure functions.
• Use of the energy balance and cascade picture to explain how energy transfers across scales.
• Adaptation of Kolmogorov's 1941 theory to describe 3D turbulence with a -5/3 energy spectrum.
• Extension to 2D turbulence via the Kraichnan-Batchelor theory, emphasizing inverse energy cascade and enstrophy transfer.
• Modeling of passive scalar advection using the Kraichnan model to study intermittency and cascade dynamics.

Experimental results

Research questions

• RQ1 How do the Navier-Stokes equations govern the statistical behavior of fully developed turbulence?
• RQ2 What is the nature of energy cascade in three-dimensional turbulent flows according to Kolmogorov's theory?
• RQ3 How does two-dimensional turbulence differ from three-dimensional turbulence in terms of energy and enstrophy cascades?
• RQ4 What causes the breakdown of Lagrangian flow in turbulent dispersion, and how does Richardson's law describe this?
• RQ5 How do direct and inverse cascades emerge in the Kraichnan model of passive scalar advection, and what role does intermittency play?

Key findings

• The Navier-Stokes equations provide the fundamental dynamical framework for modeling incompressible turbulent flows.
• In three-dimensional turbulence, energy cascades forward from large to small scales, following Kolmogorov's -5/3 power law in the inertial range.
• In two-dimensional turbulence, energy cascades inversely to larger scales while enstrophy cascades forward to smaller scales, as described by the Kraichnan-Batchelor theory.
• Richardson's dispersion law describes the quadratic growth of particle separation in turbulent flows, indicating a breakdown of Lagrangian predictability.
• In the Kraichnan model of passive advection, direct and inverse cascades emerge depending on the dimensionality and correlation structure of the velocity field.
• Intermittency effects are evident in the scaling of structure functions, particularly in the deviation from self-similar scaling in both 2D and 3D turbulence.