"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

The Fractal Geometry of Nature

Applications of second-moment turbulent closure hypotheses to geophysical fluid problems have developed rapidly since 1973, when genuine predictive skill in coping with the effects of stratification was demonstrated. The purpose here is to synthesize and organize material that has appeared in a number of articles and add new useful material so that a complete (and improved) description of a turbulence model from conception to application is condensed in a single article. It is hoped that this will be a useful reference to users of the model for application to either atmospheric or oceanic boundary layers.

http://fap.if.usp.br/~hbarbosa/uploads/Teaching/Modclim2010a/MellorYamada1982.pdf

Development of a turbulence closure model for geophysical fluid problems

A new k-ϵ eddy viscosity model for high reynolds number turbulent flows

The paper develops proposals for a model of turbulence in which the Reynolds stresses are determined from the solution of transport equations for these variables and for the turbulence energy dissipation rate E. Particular attention is given to the approximation of the pressure-strain correlations; the forms adopted appear to give reasonably satisfactory partitioning of the stresses both near walls and in free shear flows. Numerical solutions of the model equations are presented for a selection of strained homogeneous shear flows and for two-dimensional inhomogeneous shear flows including the jet, the wake, the mixing layer and plane channel flow. In addition, it is shown that the closure does predict a very strong influence of secondary strain terms for flow over curved surfaces.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112075001814

Progress in the development of a Reynolds-stress turbulence closure

A new subgrid scale model is proposed for Large Eddy Simulations in complex geometries. This model which is based on the square of the velocity gradient tensor accounts for the effects of both the strain and the rotation rate of the smallest resolved turbulent fluctuations. Moreover it recovers the proper y
3 near-wall scaling for the eddy viscosity without requiring dynamic procedure. It is also shown from a periodic turbulent pipe flow computation that the model can handle transition.

/pdf/subgrid-scale-stress-modelling-based-on-the-square-of-the-e05h19jine.pdf

Subgrid-scale stress modelling based on the square of the velocity gradient tensor

Space-time correlation measurements in the roughly isotropic turbulence behind a regular grid spanning a uniform airstream give the simplest Eulerian time correlation if we choose for the upstream probe signal a time delay which just ‘cancels’ the mean flow displacement. The correlation coefficient of turbulent velocities passed through matched narrow-band niters shows a strong dependence on nominal filter frequency (∼ wave-number at these small turbulence levels). With plausible scaling of the time separations, a scaling dependent on both wave-number and time, it is possible to effect a good collapse of the correlation functions corresponding to wave-numbers from 0·5 cm−1, the location of the peak in the three-dimensional spectrum, to 10 cm−1, about half the Kolmogorov wave-number. The spectrally local time-scaling factor is a ‘parallel’ combination of the times characterizing (i) gross strain distortion by larger eddies, (ii) wrinkling distortion by smaller eddies, (iii) convection by larger eddies and (iv) gross rotation by larger eddies.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112071001599

Simple Eulerian time correlation of full-and narrow-band velocity signals in grid-generated, ‘isotropic’ turbulence

It is found that when the average kinetic energies of normal velocity components in decaying, grid-generated turbulence are equilibrated by a symmetric contraction of the wind tunnel, this equality can persist downstream. A second result is further confirmation of the fact that the best power-law fit to the inverse turbulent energy during the early part of decay is near (x–x1)1·25, for both rod grids and disk grids. The Kolmogorov decay law is re-derived by a spectral method which is essentially equivalent to the original. Finally, a crude theoretical estimate of component energies in the straight duct after a weak contraction seems to support the experiments.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112066000338

The use of a contraction to improve the isotropy of grid-generated turbulence

A reasonably uniform mean temperature gradient has been superimposed upon a nearly homogeneous turbulent shear flow in a wind tunnel. The overheat is small enough to have negligible effect on the turbulence. Away from the wind-tunnel entrance, the transverse statistical homogeneity is good and the temperature fluctuations and their integral scales grow monotonically like the corresponding velocity fluctuations (Harris, Graham & Corrsin 1977). Measurements of several moments, one- and two-point correlation functions, spectra, integral scales, microscales, probability densities, and joint probability densities of the turbulent velocities, temperature fluctuations, and temperature-velocity products are reported. The heat-transport characteristics are much like those of momentum transport, with the turbulent Prandtl number nearly 1. The temperature fluctuation is better correlated with the streamwise than the transverse velocity component, and the cross-component D12 of the turbulent diffusivity tensor has sign opposite to and about twice the magnitude of the diagonal component D22. Some resemblance of directional properties (relative magnitudes of correlation functions, integral scales, microscales) of the temperature with those of the streamwise velocity is also observed. Comparisons of the present data with measurements in the inner part of a heated boundary layer and a fully turbulent pipe flow (x2/d = 0·25) show comparable magnitudes of temperature-velocity correlation coefficients, turbulent Prandtl numbers and ratios of turbulent diffusivities, and show similar shapes of two-point correlation functions.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112081002930

Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. Part 1

With a transverse array of channels of equal widths but differing resistances, we have generated an improved approximation to spatially homogeneous turbulent shear flow. The scales continue to grow with downstream distance, even in a region where the mean velocity gradient and one-point turbulence moments (component energies and shear stress) have attained essentially constant values. This implies asymptotic non-stationarity in the basic Eulerian frame convected with the mean flow, behaviour which seems to be inherent to homogeneous turbulent shear flow.Two-point velocity correlations with space separation and with space-time separation yield characteristic departures from isotropy, including clear ‘upstream–downstream’ unsymmetries which cannot be classified simply as axis tilting of ellipse-like iso-correlation contours.The high wave-number structure is roughly locally isotropic although the turbulence Reynolds number based on Taylor ‘microscale’ and r.m.s. turbulent velocity is only 130. Departures from isotropy in the turbulent velocity gradient moments are measurable.The approximation to homogeneity permits direct estimation of all components of the turbulent pressure/velocity-gradient tensor, which accounts for inter-component energy transfer and helps to regulate the turbulent shear stress. It is found that its principal axes are aligned with those of the Reynolds stress tensor. Finally, the Rotta (1951, 1962) linear hypothesis for intercomponent energy transfer rate is roughly confirmed.

Experiments on nearly homogeneous turbulent shear flow

Publisher Summary This chapter identifies the conditions necessary for the validity of a gradient transport model in a simple mean-free-path type of random transport process. A principal shortcoming of the telegraph equation as a turbulent diffusion model is that it possesses no Eulerian statistical properties, whereas Eulerian coordinates are those in which most experimental investigations and theoretical analysis of turbulent motion are conveniently carried out. By qualitative analogy with random walk transport, analyzed without rigor, the chapter presents a collection of conditions that may be necessary for the applicability of simple gradient transport models in turbulence. These conditions stipulate degrees of homogeneity and stationarity of the mean field being transported, and of the turbulence properties central to the transport mechanism. One or more of these conditions appear to be violated in each of the traditional turbulent flow boundary value problems, such as boundary layer and jet. There are many alternative approaches to turbulent transport, which are related to neither a gradient transport approximation, nor to a hypothetical, long-path, radiative transport analogy. Closure schemes for hierarchies of turbulence moment equations have been based on a variety of truncated expansion procedures, which often invoke no explicit physical or phenomenological images, but are more or less ad hoc, based sometimes on the intuitive belief that higher order correlation coefficients tend to be very much smaller than lower order ones. It is possible that one of these more formal closure methods will eventually succeed.

Stanley Corrsin

Papers

Simple Eulerian time correlation of full-and narrow-band velocity signals in grid-generated, ‘isotropic’ turbulence

The use of a contraction to improve the isotropy of grid-generated turbulence

Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. Part 1

Experiments on nearly homogeneous turbulent shear flow

Limitations of Gradient Transport Models in Random Walks and in Turbulence