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Fluid Dynamics, Turbulence

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Fluid flows are open systems far from equilibrium. Fluid motion is sustained by energy injected at a certain scale, the so‐calledintegral scale and is dissipated by viscosity mainly in small-scale structures. If the integral scale and the dissipative scale are widely separated andthe motions on the integral scale are sufficiently strong, the fluid develops a range of spatio‐temporal structures. In three‐dimensionalflows these structures steadily decay into smaller structures and are generated by the instability of larger structures. This leads to a cascadingprocess which transports energy across scales. Turbulence appears if the fluid motion is driven far away from equilibrium. It develops through sequencesof instabilities and processes of selforganization. From this respect, turbulence is a highly ordered phenomenon, whose spatio‐temporalcomplexity, however, has still to be explored.

Introduction

Turbulence is one...

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Abbreviations

Basic equation of fluid dynamics:

Fluid motion is mathematically treated on the basis of a continuum theory. The fundamental evolution equations are the Euler equation for ideal fluids and the Navier–Stokes equation for Newtonian fluids.

Vortex motions:

Numerical calculations of turbulent flow fields show that the flows are dominated by coherent structures in form of vortex sheets or tube-like vortices. The question, why vorticity tends to be condensed in localized objects, is one of the central issues of fluid dynamics. Regarding two‐dimensional flows there are attempts to approximate fluid motion by a collection of point vortices. This allows one to investigate properties of flows on the basis of a finite dimensional (Hamiltonian) dynamical system.

Turbulence modeling and large eddy simulations:

The evolution equation for the average velocity field of turbulent flows contains the Reynolds stresses, whose origin are the turbulent pulsations. Turbulence modeling consists of relating the Reynolds stresses to averaged quantities of the fluid motion. This allows one to perform numerical computations of large-scale flows without resolving the turbulent fine structure.

Phenomenological theories of the fine structure of turbulence:

Phenomenological theories play an important role in physics, and are quite often formulated before a microscopic understanding of the physical problem has been achieved. Phenomenological theories have been developed for the fine structure of turbulence. Of great importance is the theory of Kolmogorov , which he formulated in the year 1941 and refined in 1962. The so‐called K41 and K62 theories focus on the self‐similar behavior of statistical properties of velocity increments, i. e. the velocity difference between two points with a spatial distance r. Recently, phenomenological theories have been developed that consider the joint probabilities of velocity increments at different scales. It is expected that multiple scale analysis of turbulence will provide new insights into the spatio‐temporal complexity of turbulence.

Turbulent cascades:

Fluid motions are dissipative systems. Stationary flows can only be maintained by a constant energy input in the form of shear flows or body forces. Usually, the length and time scales related to the energy input are widely separated from the ones on which energy is dissipated. A consequence is the establishment of an energy transport across scales. It is believed that this energy transport is local in scale leading to the so‐called energy cascades. These cascades are related to the emergence of scaling behavior. There is a direct energy cascade in three dimensions from large to small scales and an inverse cascade of energy from small scales to large scales in two‐dimensional flows.

Analytical theories of turbulence:

Analytical theories of turbulence try to assess the experimental results on turbulent flows directly from a statistical treatment of the basic fluid dynamical equations. Analytical theories rely on renormalized perturbation expansions and use methods from quantum field theory and renormalization group methods. However, no generally accepted theory has emerged so far.

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Acknowledgments

We thank all members of our working groups for fruitful collaboration. We thank T. Christen and M. Wilczekfor crictically reading the manuscript.

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Authors and Affiliations

  1. Institute for Theoretical Physics, University of Münster, Münster, Germany

    Rudolf Friedrich

  2. Institute of Physics, Carl-von‐Ossietzky University Oldenburg, Oldenburg, Germany

    Joachim Peinke

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  1. Rudolf Friedrich
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Friedrich, R., Peinke, J. (2009). Fluid Dynamics, Turbulence. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_215

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