It has often been remarked that turbulence is a subject of great scientific and technological importance, and yet one of the least understood (e.g. McComb 1990). To an outsider this may seem strange, since the basic physical laws of fluid mechanics are well established, an excellent mathematical model is available in the Navier-Stokes equations, and the results of well over a century of increasingly sophisticated experiments are at our disposal. One major difficulty, of course, is that the governing equations are nonlinear and little is known about their solutions at high Reynolds number, even in simple geometries. Even mathematical questions as basic as existence and uniqueness are unsettled in three spatial dimensions (cf Temam 1988). A second problem, more important from the physical viewpoint, is that experiments and the available mathematical evidence all indicate that turbulence involves the interaction of many degrees of freedom over broad ranges of spatial and temporal scales. One of the problems of turbulence is to derive this complex picture from the simple laws of mass and momentum balance enshrined in the NavierStokes equations. It was to this that Ruelle & Takens (1971) contributed with their suggestion that turbulence might be a manifestation in physical

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The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows

It is the purpose of this tutorial paper to place bispectrum estimation in a digital signal processing framework in order to aid engineers in grasping the utility of the available bispectrum estimation techniques, to discuss application problems that can directly benefit from the use of the bispectrum, and to motivate research in this area Three general reasons are behind the use of bispectrum in signal processing and are addressed in the paper: to extract information due to deviations from normality, to estimate the phase of parametric signals, and to detect and characterize the properties of nonlinear mechanisms that generate time series

Bispectrum estimation: A digital signal processing framework

Publisher Summary This chapter presents an overview of the theory of homogeneous turbulence. Homogeneous isotropic turbulence is an idealized concept of turbulence, assumed to be governed by a statistical law that is invariant under arbitrary translation (homogeneity), rotation, or reflection (isotropy) of the coordinate system. This is an idealization of real turbulent motions, which are observed in nature or produced in a laboratory, and generally have much more complicated structures. This idealization was first introduced by Taylor to the theory of turbulence and used to reduce the formidable complexity of the statistical expression of turbulence, and thus make the subject feasible for theoretical treatment. Turbulence combines the difficulty of a strongly nonlinear dynamical system with that of a nonconservative system. The presence of energy dissipation due to viscosity deprives turbulence of the opportunity of being in a thermal equilibrium state. In this chapter, concepts of quasi-normality of large-scale motions are explained. A brief discussion on zero cumulant approximation and modified zero cumulant approximation is presented. Turbulence of other dimensions is also explained.

Theory of Homogeneous Turbulence

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

This paper argues that the active turbulence and coherent motions near the top of a vegetation canopy are patterned on a plane mixing layer, because of instabilities associated with the characteristic strong inflection in the mean velocity profile. Mixing-layer turbulence, formed around the inflectional mean velocity profile which develops between two coflowing streams of different velocities, differs in several ways from turbulence in a surface layer. Through these differences, the mixing-layer analogy provides an explanation for many of the observed distinctive features of canopy turbulence. These include: (a) ratios between components of the Reynolds stress tensor; (b) the ratio K H /K M of the eddy diffusivities for heat and momentum; (c) the relative roles of ejections and sweeps; (d) the behaviour of the turbulent energy balance, particularly the major role of turbulent transport; and (e) the behaviour of the turbulent length scales of the active coherent motions (the dominant eddies responsible for vertical transfer near the top of the canopy). It is predicted that these length scales are controlled by the shear length scale L s = U(h)/U′(h) (where h is canopy height, U(z) is mean velocity as a function of height z, and U′ = dU/dz). In particular, the streamwise spacing of the dominant canopy eddies Λ x = mL s , with m = 8.1. These predictions are tested against many sets of field and wind-tunnel data. We propose a picture of canopy turbulence in which eddies associated with inflectional instabilities are modulated by larger-scale, inactive turbulence, which is quasi-horizontal on the scale of the canopy.

Coherent eddies and turbulence in vegetation canopies: The mixing-layer analogy

The evolution of unsheared grid-generated turbulence in a stably stratified fluid was investigated in a closed-loop salt-stratified water channel. Simultaneous single-point measurements of the horizontal and vertical velocity and density fluctuations were obtained, including turbulent mass fluxes central in understanding the energetics of the fluctuating motion. When the buoyancy lengthscale was initially substantially larger than the largest turbulent scales, the initial behaviour of the velocity and density fields was similar to that in the non-stratified case. With further downstream development, the buoyancy lengthscale decreased while the turbulence scale grew. Deviations from neutral behaviour occurred when these two lengthscales became of the same order, after the initially larger inertial forces associated with the initial kinetic energy had become weaker and buoyancy forces became important.Buoyancy forces produced anisotropy in the largest scales first, preventing them from overturning, while smaller-scale isotropic turbulent motions remained embedded within the larger-scale wave motions. These small-scale motions exhibited classical turbulent behaviour and scaled universally with Kolmogorov length and velocity scales. Eventually even the smallest scales of the decaying turbulence were affected by buoyancy, all isotropic motions disappeared, and Kolmogorov scaling failed. The turbulent vertical mass flux decreased to zero for this condition, indicating that the original turbulent field had been completely converted to random internal wave motions.The transition from a fully turbulent state to one of internal waves occurred rapidly in a time less than the characteristic time of the turbulence based on the largest-scale eddies found in the flow at transition. The dissipation rate for complete transition to a wave field was found to be of the order of et = 24.5νN2, where ν is the kinematic viscosity and N the Brunt-Vaisala frequency. This is in fairly good agreement with the value 30νN2 predicted by Gibson (1980, 1981).The present experiments have determined quantitative limits on the range of active turbulent scales in homogeneous stratified turbulence, in terms of an upper limit near the buoyancy lengthscale and a lower limit determined by viscosity in the usual way. This description has been used here to help explain and assimilate the results from the earlier stratified grid-turbulence experiments of Lin & Veenhuizen (1975) and Dickey & Mellor (1980). While some of the features of the present observations may be qualitatively seen in the numerical simulations of the problem of Riley, Metcalfe & Weissman (1981), there are fundamental differences, probably due in part to large differences in initial lengthscale ratios and in the limited range of scales attainable in numerical simulations. The present experiments may serve as a useful test case for future modelling and interpretation of the behaviour of turbulence in stratified flows observed in the oceans and atmosphere.

Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid

The spatial decay and structural evolution of grid-generated turbulence under the effect of buoyancy was studied in a ten-layer, salt-stratified water channel. The various density gradients were chosen such that the initial overturning turbulent scale was slightly smaller than any of the respective buoyancy scales. The observed general evolution of the flow from homogeneous turbulence to a composite of fossil turbulence or quasi-two-dimensional turbulence and internal wavefield is in good agreement with the predictions of Gibson (1980) and the lengthscale model of Stillinger, Helland & Van Atta (1983). The effect of the initial size of the turbulent lengthscale compared with the buoyancy scale on the decay and evolution of the turbulence is investigated and the observed influence on the rate of decay of both longitudinal and vertical velocity fluctuations pointed out by Van Atta, Helland & Itsweire (1984) is shown to be related to the magnitude of the initial internal wavefield at the grid. An attempt is made to remove the wave-component kinetic energy from the vertical-velocity-fluctuation data of Stillinger, Helland & Van Atta (1983) in order to obtain the true decay of the turbulent fluctuations. The evolution of the resulting fluctuations is similar to that of the present large-grid data and several towed-grid experiments. The rate of destruction of the density fluctuations (active-scalar dissipation rate) is estimated from the evolution equation for the potential energy, and the deduced Cox numbers are compared with those obtained from oceanic microstructure measurements. The classical Kolmogorov and Batchelor scalings appear to collapse the velocity and density spectra better than the buoyancy scaling proposed by Gargett, Osborn & Nasmyth (1984). The rise of the velocity spectra at low wavenumbers found by Stillinger, Helland & Van Atta (1983) is shown to be related to internal waves.

The evolution of grid-generated turbulence in a stably stratified fluid

The behaviour of an evolving, stably stratified turbulent shear flow was investigated in a ten-layer, closed-loop, salt-stratified water channel. Simultaneous single-point measurements of the mean and fluctuating density and longitudinal and vertical velocities were made over a wide range of downstream positions. For strong stability, i.e. a mean gradient Richardson number Ri greater than a critical value of Ricr ≈ 0.25, there is no observed growth of turbulence and the buoyancy effects are similar to those in the unsheared experiments of Stillinger, Helland & Van Atta (1983) and Itsweire, Helland & Van Atta (1986). For values of Richardson number less than Ricr the turbulence grows at a rate depending on Ri and for large evolution times the ratio between the Ozmidov and turbulent lengthscale approaches a constant value which is also a function of Richardson number.Normalized velocity and density power spectra for the present experiments conform to normalized spectra from previous moderate- to high-Reynolds-number studies. With increasing or decreasing stability, the stratified shear spectra exhibit greater portions of the universal non-stratified spectrum curve. The shapes of the shear-stress and buoyancy-flux cospectra confirm that they act as sources and sinks for the velocity and density fluctuations.

Growth and decay of turbulence in a stably stratified shear flow

The influence of fluctuations in the rate of local turbulent energy dissipation on higher‐order structure functions for small separation distances r and on moments of turbulent velocity derivatives is examined using the hypotheses of Kolmogoroff and Obukhov for the probability density and variance of the dissipation fluctuations. The predicted variation of the skewness and flatness factors with Rλ represents the available experimental data fairly well over a relatively wide range of Rλ when the constant μ introduced in the variance hypothesis is suitably chosen. The predicted variation of S with K fits the data very well. The present analysis breaks down for moments of sufficiently high order probably due to basic shortcomings of the hypotheses.

Reynolds number dependence of skewness and flatness factors of turbulent velocity derivatives

Structure functions of turbulent velocity fluctuations up to fourth order have been measured at several heights in the atmospheric boundary layer over the open ocean, and the results are compared with theoretical predictions for separations in the inertial subrange. The behaviour of second- and third-order quantities shows substantial agreement with the predictions of Kolmogorov's original theory over a wide range of separations, but the results of a recent modification of the theory, attempting to account for intermittency in the local dissipation rate, are also consistent with the data over somewhat shorter separation intervals. The behaviour of the measured fourth-order structure function disagrees with that predicted from Kolmogorov's original work, but good agreement is found with the results of the modified theory.

C. W. Van Atta

Papers

Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid

The evolution of grid-generated turbulence in a stably stratified fluid

Growth and decay of turbulence in a stably stratified shear flow

Reynolds number dependence of skewness and flatness factors of turbulent velocity derivatives

Structure functions of turbulence in the atmospheric boundary layer over the ocean