Recent research is making progress in framing more precisely the basic dynamical
and statistical questions about turbulence and in answering them. It is helping both to
define the likely limits to current methods for modelling industrial and environmental
turbulent flows, and to suggest new approaches to overcome these limitations. Our
selective review is based on the themes and new results that emerged from more than
300 presentations during the Programme held in 1999 at the Isaac Newton Institute,
Cambridge, UK, and on research reported elsewhere. A general conclusion is that,
although turbulence is not a universal state of nature, there are certain statistical
measures and kinematic features of the small-scale flow field that occur in most
turbulent flows, while the large-scale eddy motions have qualitative similarities within
particular types of turbulence defined by the mean flow, initial or boundary conditions,
and in some cases, the range of Reynolds numbers involved. The forced transition
to turbulence of laminar flows caused by strong external disturbances was shown to
be highly dependent on their amplitude, location, and the type of flow. Global and
elliptical instabilities explain much of the three-dimensional and sudden nature of the
transition phenomena. A review of experimental results shows how the structure of
turbulence, especially in shear flows, continues to change as the Reynolds number
of the turbulence increases well above about 104 in ways that current numerical
simulations cannot reproduce. Studies of the dynamics of small eddy structures and
their mutual interactions indicate that there is a set of characteristic mechanisms in
which vortices develop (vortex stretching, roll-up of instability sheets, formation of
vortex tubes) and another set in which they break up (through instabilities and self-
destructive interactions). Numerical simulations and theoretical arguments suggest
that these often occur sequentially in randomly occurring cycles. The factors that
determine the overall spectrum of turbulence were reviewed. For a narrow distribution
of eddy scales, the form of the spectrum can be defined by characteristic forms of
individual eddies. However, if the distribution covers a wide range of scales (as in
elongated eddies in the ‘wall’ layer of turbulent boundary layers), they collectively
determine the spectra (as assumed in classical theory). Mathematical analyses of
the Navier–Stokes and Euler equations applied to eddy structures lead to certain
limits being defined regarding the tendencies of the vorticity field to become infinitely
large locally. Approximate solutions for eigen modes and Fourier components reveal
striking features of the temporal, near-wall structure such as bursting, and of the very
elongated, spatial spectra of sheared inhomogeneous turbulence; but other kinds of
eddy concepts are needed in less structured parts of the turbulence. Renormalized
perturbation methods can now calculate consistently, and in good agreement with
experiment, the evolution of second- and third-order spectra of homogeneous and
isotropic turbulence. The fact that these calculations do not explicitly include high-order moments and extreme events, suggests that they may play a minor role in the
basic dynamics. New methods of approximate numerical simulations of the larger
scales of turbulence or ‘very large eddy simulation’ (VLES) based on using statistical
models for the smaller scales (as is common in meteorological modelling) enable some
turbulent flows with a non-local and non-equilibrium structure, such as impinging or
convective flows, to be calculated more efficiently than by using large eddy simulation
(LES), and more accurately than by using ‘engineering’ models for statistics at a single
point. Generally it is shown that where the turbulence in a fluid volume is changing
rapidly and is very inhomogeneous there are flows where even the most complex
‘engineering’ Reynolds stress transport models are only satisfactory with some special
adaptation; this may entail the use of transport equations for the third moments or
non-universal modelling methods designed explicitly for particular types of flow. LES
methods may also need flow-specific corrections for accurate modelling of different
types of very high Reynolds number turbulent flow including those near rigid surfaces.This paper is dedicated to the memory of George Batchelor who was the inspiration
of so much research in turbulence and who died on 30th March 2000. These results
were presented at the last fluid mechanics seminar in DAMTP Cambridge that he
attended in November 1999.
Enstrophy Cascade and Smagorinsky Model of 2D Turbulent Flows