Small-scale turbulence has been an area of especially active research in the recent past, and several useful research directions have been pursued. Here, we selectively review this work. The emphasis is on scaling phenomenology and kinematics of small-scale structure. After providing a brief introduction to the classical notions of universality due to Kolmogorov and others, we survey the existing work on intermittency, refined similarity hypotheses, anomalous scaling exponents, derivative statistics, intermittency models, and the structure and kinematics of small-scale structure—the latter aspect coming largely from the direct numerical simulation of homogeneous turbulence in a periodic box.

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The phenomenology of small-scale turbulence

The structure of the intense-vorticity regions is studied in numerically simulated homogeneous, isotropic, equilibrium turbulent flow fields at four different Reynolds numbers, in the range Re, = 35-170. In accordance with previous investigators this vorticity is found to be organized in coherent, cylindrical or ribbon-like, vortices (‘worms’). A statistical study suggests that they are simply especially intense features of the background, O(o’), vorticity. Their radii scale with the Kolmogorov microscale and their lengths with the integral scale of the flow. An interesting observation is that the Reynolds number y/v, based on the circulation of the intense vortices, increases monotonically with ReA, raising the question of the stability of the structures in the limit of Re, --z co. Conversely, the average rate of stretching of these vortices increases only slowly with their peak vorticity, suggesting that self-stretching is not important in their evolution. One- and two-dimensional statistics of vorticity and strain are presented; they are non-Gaussian and the behaviour of their tails depends strongly on the Reynolds number. There is no evidence of convergence to a limiting distribution in this range of Re,, even though the energy spectra and the energy dissipation rate show good asymptotic properties in the higher-Reynolds-number cases. Evidence is presented to show that worms are natural features of the flow and that they do not depend on the particular forcing scheme.

/pdf/the-structure-of-intense-vorticity-in-isotropic-turbulence-2ruhxp7i29.pdf

The structure of intense vorticity in isotropic turbulence

A direct numerical simulation at resolution 2403 is used to obtain a statistically stationary three-dimensional homogeneous and isotropic turbulent field at a Reynolds number around 1000 (Rλ ≈ 150). The energy spectrum displays an inertial subrange. The velocity derivative distribution, known to be strongly non-Gaussian, is found to be close to, but not, exponential. The nth-order moments of this distribution, as well as the velocity structure functions, do not scale with n as predicted by intermittency models. Visualization of the flow confirms the previous finding that the strongest vorticity is organized in very elongated thin tubes. The width of these tubes is of the order of a few dissipation scales, while their length can reach the integral scale of the flow.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112091001957

The spatial structure and statistical properties of homogeneous turbulence

The intermittency of the rate of turbulent energy dissipation e is investigated experimentally, with special emphasis on its scale-similar facets. This is done using a general formulation in terms of multifractals, and by interpreting measurements in that light. The concept of multiplicative processes in turbulence is (heuristically) shown to lead to multifractal distributions, whose formalism is described in some detail. To prepare proper ground for the interpretation of experimental results, a variety of cascade models is reviewed and their physical contents are analysed qualitatively. Point-probe measurements of e are made in several laboratory flows and in the atmospheric surface layer, using Taylor's frozen-flow hypothesis. The multifractal spectrum f(α) of e is measured using different averaging techniques, and the results are shown to be in essential agreement among themselves and with our earlier ones. Also, long data sets obtained in two laboratory flows are used to obtain the latent part of the f(α) curve, confirming Mandelbrot's idea that it can in principle be obtained from linear cuts through a three-dimensional distribution. The tails of distributions of box-averaged dissipation are found to be of the square-root exponential type, and the implications of this finding for the f(α) distribution are discussed. A comparison of the results to a variety of cascade models shows that binomial models give the simplest possible mechanism that reproduces most of the observations. Generalizations to multinomial models are discussed.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112091001830

The multifractal nature of turbulent energy dissipation

Velocity probability density functions of high Reynolds number turbulence

DNS of turbulent heat transfer in channel flow with low to medium-high Prandtl number fluid

The decaying isotropic turbulence starting with a typical initial energy spectrum was numerically investigated by the spectral method for Reynolds numbers 50–500, using up to 128 3 grid points. The microscale Reynolds number reached at a fully developed state was \({\lesssim}100\). Only the periodic boundary condition was set on the surfaces of the cube which contained the turbulence. As a result, the Kolmogorov spectrum law and other interesting characteristics of the turbulence were found, including the perspective of intermittent strong-vorticity regions randomly distributed in a fully developed turbulence.

A Decaying Isotropic Turbulence Pursued by the Spectral Method

The q -th order intermittency exponents and generalized dimensions of isotropic turbulence are calculated for | q |≤30 from the data of the direct numerical simulation, which was recently executed on the 128 3 grid of a supercomputer to investigate a decaying isotropic turbulence. These values are compared with the theoretical and experimental values so far known. The corresponding f -α spectrum is also presented.

Intermittency of Dissipation in a Directly Simulated Fully-Developed Turbulence

In order to reliably compute the longitudinal structure functions in decaying and forced turbulence, local isotropy is examined with the aid of the isotropic expression of the incompressible condit...

Longitudinal structure functions in decaying and forced turbulence

The fine structure of the isotropic turbulence, which has recently been realized in a supercomputer by the authors, is studied at its fully-developed state. The geometrical relationship between vorticity-concentrated regions and high-strain regions and a variety of mutual orientations of the velocity and vorticity vectors in these regions are illustrated. A common exponential nature and a delicate difference of the space distributions of the velocity (longitudinal and lateral) gradients, along with the magnitude of vorticity and the square-root of local dissipation, are exposed.

Kiyoshi Yamamoto

Papers

DNS of turbulent heat transfer in channel flow with low to medium-high Prandtl number fluid

A Decaying Isotropic Turbulence Pursued by the Spectral Method

Intermittency of Dissipation in a Directly Simulated Fully-Developed Turbulence

Longitudinal structure functions in decaying and forced turbulence

Fine Structure of a Directly Simulated Isotropic Turbulence