Structures In Turbulent Shear Flows Research Articles

On the logarithmic region in wall turbulence

AbstractConsiderable discussion over the past few years has been devoted to the question of whether the logarithmic region in wall turbulence is indeed universal. Here, we analyse recent experimental data in the Reynolds number range of nominally$2\times 1{0}^{4} \lt {\mathit{Re}}_{\tau } \lt 6\times 1{0}^{5} $for boundary layers, pipe flow and the atmospheric surface layer, and show that, within experimental uncertainty, the data support the existence of a universal logarithmic region. The results support the theory of Townsend (The Structure of Turbulent Shear Flow, Vol. 2, 1976) where, in the interior part of the inertial region, both the mean velocities and streamwise turbulence intensities follow logarithmic functions of distance from the wall.

Large-scale structures in turbulent and reverse-transitional sink flow boundary layers

Aspects of large-scale organized structures in sink flow turbulent and reverse-transitional boundary layers are studied experimentally using hot-wire anemometry. Each of the present sink flow boundary layers is in a state of ‘perfect equilibrium’ or ‘exact self-preservation’ in the sense of Townsend (The Structure of Turbulent Shear Flow, 1st and 2nd edns, 1956, 1976, Cambridge University Press) and Rotta (Progr. Aeronaut. Sci., vol. 2, 1962, pp. 1–220) and conforms to the notion of ‘pure wall-flow’ (Coles,J. Aerosp. Sci., vol. 24, 1957, pp. 495–506), at least for the turbulent cases. It is found that the characteristic inclination angle of the structure undergoes a systematic decrease with the increase in strength of the streamwise favourable pressure gradient. Detectable wall-normal extent of the structure is found to be typically half of the boundary layer thickness. Streamwise extent of the structure shows marked increase as the favourable pressure gradient is made progressively severe. Proposals for the typical eddy forms in sink flow turbulent and reverse-transitional flows are presented, and the possibility of structural self-organization (i.e. individual hairpin vortices forming streamwise coherent hairpin packets) in these flows is also discussed. It is further indicated that these structural ideas may be used to explain, from a structural viewpoint, the phenomenon ofsoftrelaminarization or reverse transition of turbulent boundary layers when subjected to strong streamwise favourable pressure gradients. Taylor's ‘frozen turbulence’ hypothesis is experimentally shown to be valid for flows in the present study even though large streamwise accelerations are involved, the flow being even reverse transitional in some cases. Possible conditions, which are required to be satisfied for the safe use of Taylor's hypothesis in pressure-gradient-driven flows, are also outlined. Measured convection velocities are found to be fairly close to the local mean velocities (typically 90% or more) suggesting that the structure gets convected downstream almost along with the mean flow.

Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers

In this paper we investigate the relationship between the large- and small-scale energy-containing motions in wall turbulence. Recent studies in a high-Reynolds-number turbulent boundary layer (Hutchins & Marusic, Phil. Trans. R. Soc. Lond. A, vol. 365, 2007a, pp. 647–664) have revealed a possible influence of the large-scale boundary-layer motions on the small-scale near-wall cycle, akin to a pure amplitude modulation. In the present study we build upon these observations, using the Hilbert transformation applied to the spectrally filtered small-scale component of fluctuating velocity signals, in order to quantify the interaction. In addition to the large-scale log-region structures superimposing a footprint (or mean shift) on the near-wall fluctuations (Townsend, The Structure of Turbulent Shear Flow, 2nd edn., 1976, Cambridge University Press; Metzger & Klewicki, Phys. Fluids, vol. 13, 2001, pp. 692–701.), we find strong supporting evidence that the small-scale structures are subject to a high degree of amplitude modulation seemingly originating from the much larger scales that inhabit the log region. An analysis of the Reynolds number dependence reveals that the amplitude modulation effect becomes progressively stronger as the Reynolds number increases. This is demonstrated through three orders of magnitude in Reynolds number, from laboratory experiments at Reτ ~ 103–104 to atmospheric surface layer measurements at Reτ ~ 106.

Experimental support for Townsend’s Reynolds number similarity hypothesis on rough walls

The Reynolds number similarity hypothesis of Townsend [The Structure of Turbulent Shear Flow (Cambridge University Press, Cambridge, UK, 1976)] states that the turbulence beyond a few roughness heights from the wall is independent of the surface condition. The underlying assumption is that the boundary layer thickness δ is large compared to the roughness height k. This hypothesis was tested experimentally on two types of three-dimensional rough surfaces. Boundary layer measurements were made on flat plates covered with sand grain and woven mesh roughness in a closed return water tunnel at a momentum thickness Reynolds number Reθ of ∼14000. The boundary layers on the rough walls were in the fully rough flow regime (ks+⩾100) with the ratio of the boundary layer thickness to the equivalent sand roughness height δ∕ks greater than 40. The results show that the mean velocity profiles for rough and smooth walls collapse well in velocity defect form in the overlap and outer regions of the boundary layer. The Reynolds stresses for the two rough surfaces agree well throughout most of the boundary layer and collapse with smooth wall results outside of 3ks. Higher moment turbulence statistics and quadrant analysis also indicate the differences in the rough wall boundary layers are confined to y<5ks. The present results provide support for Townsend’s Reynolds number similarity hypothesis for uniform three-dimensional roughness in flows where δ∕ks⩾40.

Coherent Structures Detected in Atmospheric Boundary-layer Turbulence using Wavelet transforms at Huaihe River Basin, China

The presence of coherent structures in turbulent shear flows suggests order in apparently random flows. These coherent structures play an important dynamical role in momentum and scalar transport. To develop dynamical models describing the evolution of such motion, it is necessary to detect and isolate the coherent structures from the background fluctuations. In this paper, we decomposed atmospheric turbulence time series into large-scale eddies, which include coherent structures and small eddies, which are stochastic by using Fourier digital filtering. The wavelet energy computed for the three components of the velocity fluctuations in the large-scale eddies appears to have local maximum values at certain time scales, which correspond to the scales or frequencies of coherent structures. We extract coherent signals from large-scale vortices at this scale by inverse wavelet transform formulae. This method provides an objective technique for examining the turbulence signal associated with coherent structures in the atmospheric boundary layer. The average duration of coherent structures in three directions based on Mexican hat wavelets are 33 s, 34 s and 25 s respectively. Symmetric andanti-symmetric wavelet basis functions give almost the same results. The main features of the structures during the day and night have little difference. The dimensionless durations for u, v and w have linear correlations with each other. These relationships are insensitive to the wavelet basis.

Universal behavior of entrainment due to coherent structures in turbulent shear flow.

A solution is suggested for a persistent mystery in the physics of turbulent flows: cumulus clouds rise to towering heights, practically without entraining the ambient medium, while apparently similar turbulent jets quickly lose their identity through entrainment and mixing. Dynamical system computations on a model vortical flow show that entrainment due to coherent structures depends sensitively on relative speeds of fluid parcels. Local heating, for example, can alter drastically the sizes of Kolmogorov-Arnol'd-Moser tori and chaotic mixing regions. The entrainment rate and, hence, the lifetime of a turbulent shear flow show a universal, nonmonotone dependence on the heating.

Review of Coherent Structures in Turbulent Free Shear Flows and Their Possible Influence on Computational Methods

A review of selected experiments on coherent structures in turbulent shear flows is performed. Different experimental approaches (conditional averages, filtering techniques, wavelets, linear stochastic estimation and proper orthogonal decomposition, etc.) are illustrated and their links with computations (LES, DNS, SDM, etc.) is emphasized. It is particularly shown that some kind of universal behavior of the background turbulence can be retrieved from these various experimental methods.

Identification of Coherent Structure in Turbulent Shear Flow With Wavelet Correlation Analysis

In order to identify coherent structure of turbulent shear flow, a new combination of familiar techniques of signal processing, called wavelet correlation analysis, is developed based on the wavelet transform. The wavelet correlation analysis provides the unique capability for decomposing the correlation of arbitrary signals over a two-dimensional time delay-period plane. By analyzing two superposition functions implicating several pure frequencies, the correlation of periodic oscillations at several frequencies can well be separated and observed clearly. Coherent structures in the intermediate region of a plane turbulent jet are investigated using the wavelet correlation method. It is shown that the wavelet correlation analysis can extract the most essential scales governing features of eddy motions. The coherent structure information and apparent flapping behaviors are clearly revealed over a two-dimensional time-period plane.

Wavelet Auto-Correlation Analysis Applied to Eddy Structure Identification of Free Turbulent Shear Flow.

To investigate the eddy structure of turbulent shear flow in both physical-space and Fourier-space, a new combination of conventional techniques of signal processing, called wavelet auto-correlation analysis, has been developed based on wavelet transform. The wavelet auto-correlation analysis provides the unique capability for describing the auto-correlation characteristics of arbitrary signals in terms of period and time delay. The wavelet auto-correlation technique was used to analyze the fluctuating velocities at various positions in a plane turbulent jet. It was found that the peaks of WR(α, τ) in the high period region correspond to the large peaks of R(τ), and the peaks of WR(α, τ) in the low period region are equivalent to the small peaks of R(τ). The branching structures in WR(α, τ) reveal that a periodic large eddy motion contains the periodic small eddy motions. In the shear layer at x/d=10, the periodic eddy motions with α=15, 40, 75, 100 and 150 ms are clearly observed.

Identification of coherent structures in turbulent shear flows with a fuzzy artmap neural network.

An implementation of a Fuzzy Artmap neural network is used to detect and to identify (recognise) structures (patterns) embedded in the velocity field of a turbulent wake behind a circular cylinder. The net is trained to recognise both clockwise and anticlockwise eddies present in the u and v velocity fields at 420 diameters downstream of the cylinder that generates the wake, using a pre-processed part of the recorded velocity data. The phase relationship that exists between the angles of the velocity vectors of an eddy pattern is used to reduce the number of classes contained in the data, before the start of the training procedure. The net was made stricter by increasing the vigilance parameter within the interval [0.90, 0.95] and a set of net-weights were obtained for each value. Full data files were scanned with the net classifying patterns according to their phase characteristics. The net classifies about 27% of the recorded signals as eddy motions, with the strictest vigilance parameter and without the need to impose external initial templates. Spanwise distances (homogeneous direction of the flow) within the centres of the eddies identified suggest that they form pairs of counter-rotating vortices (double rollers). The number of patterns selected with Fuzzy Artmap is lower than that reported for template matching because the net classifies eddies according to the recirculating pattern present at the core or central region, while template matching extends the region over which correlation between data and template is performed. In both cases, the topology of educed patterns is in agreement.

A model for large-scale structures in turbulent shear flows

A procedure based on energy stability arguments is presented as a method for extracting large-scale, coherent structures from fully turbulent shear flows. By means of two distinct averaging operators, the instantaneous flow field is decomposed into three components: a spatial mean, coherent field and random background fluctuations. The evolution equations for the coherent velocity, derived from the Navier–Stokes equations, are examined to determine the mode that maximizes the growth rate of volume-averaged coherent kinetic energy. Using a simple closure scheme to model the effects of the background turbulence, we find that the spatial form of the maximum energy growth modes compares well with the shape of the empirical eigenfunctions given by the proper orthogonal decomposition. The discrepancy between the eigenspectrum of the stability problem and the empirical eigenspectrum is explained by examining the role of the mean velocity field. A simple dynamic model which captures the energy exchange mechanisms between the different scales of motion is proposed. Analysis of this model shows that the modes which attain the maximum amplitude of coherent energy density in the model correspond to the empirical modes which possess the largest percentage of turbulent kinetic energy. The proposed method provides a means for extracting coherent structures which are similar to those produced by the proper orthogonal decomposition but which requires only modest statistical input.

Vorticity-based eduction of large-scale structures in turbulent shear flows

This paper is mainly concerned with a vorticity-based conditional sampling technique, which identifies large-scale vorticity-bearing flow events in turbulent shear flows using multiple X-wire probes. Basic ideas and procedures of the technique are described, and several examples of the results are presented. Advantages and limitations of the technique are also discussed from an experimental point of view.

The Proper Orthogonal Decomposition, wavelets and modal approaches to the dynamics of coherent structures

We present brief precis of three related investigations. Fuller accounts can be found elsewhere. The investigations bear on the identification and prediction of coherent structures in turbulent shear flows. A second unifying thread is the Proper Orthogonal Decomposition (POD), or Karhunen-Loeve expansion, which appears in all three investigations described. The first investigation demonstrates a close connection between the coherent structures obtained using linear stochastic estimation, and those obtained from the POD. Linear stochastic estimation is often used for the identification of coherent structures. The second investigation explores the use (in homogeneous directions) of wavelets instead of Fourier modes, in the construction of dynamical models; the particular problem considered here is the Kuramoto-Sivashinsky equation. The POD eigenfunctions, of course, reduce to Fourier modes in homogeneous situations, and either can be shown to converge optimally fast; we address the question of how rapidly (by comparison) a wavelet representation converges, and how the wavelet-wavelet interactions can be handled to construct a simple model. The third investigation deals with the prediction of POD eigenfunctions in a turbulent shear flow. We show that energy-method stability theory, combined with an anisotropic eddy viscosity, and erosion of the mean velocity profile by the growing eigenfunctions, produces eigenfunctions very close to those of the POD, and the same eigenvalue spectrum at low wavenumbers.

Polarized vorticity dynamics on a vortex column

It is shown that vortex core dynamics results from the interaction of two slowly deforming, but overlapping, helical vortex structures. These are the left- and right-handed components of the vortex, and are obtained by a generalized Helmholtz decomposition (the complex helical wave decomposition) of the vorticity field. This decomposition is based on a Fourier expansion in eigenfunctions of the curl operator, which has only real eigenvalues λ. The sum of eigenmodes with λ≳0 (λ<0) constitutes the right (left) polarized component, and the vector lines of the field (e.g., vortex lines) are locally right (left) handed helixes. It is found that for a localized vortex the polarized structures are also localized, a crucial result for physical space applications. The polarized vortex structures deform slowly (compared to unpolarized structures) and behave almost like solitary waves when isolated. It is shown that this is because the nonlinearity in the Navier–Stokes equations is largely suppressed between eigenmodes of the same polarity. Moreover, the helicity of polarized structures is very high. The interaction between overlapping polarized structures however gives each structure a different propagation velocity and also results in some additional deformation. The latter is shown to occur mainly in two places: at the front of the structure where a low enstrophy bubble forms (which is a permanent feature in each of the polarized packets), and at the back where a tail develops. Otherwise, the deformation occurs on a much slower time scale compared to that for unpolarized vortices. Thus the rapid changes in the total vorticity field result from the superposition of two slowly deforming structures moving in opposite directions, as is the case for the one-dimensional (1-D) wave equation. The decomposition can also be applied to turbulent flows. In fact, it offers new insight into the structure of turbulent shear flows. The organization of small-scale vortical threads forming in the neighborhood of a coherent structure and their polarization serve as a prime example.

Instabilities and large structures in reattaching boundary layers

The underlying premise in this work is that large structures in turbulent shear flows are not universal and similar large structures in different flows owe their origins to basically similar instabilities. The turbulent boundary-layer flow over a backward-facing step and also the reattaching transitional boundary layer behind a large spanwise rod have been studied experimentally in parallel primarily to understand the origin of the curious large structures observed in the far downstream region (after 39 step heights) of the former flowfield. Simultaneous flow visualization and hot-wire anemometry and also plain-flow visualization have been carried out. In the backward-facing step, the three-dimensional mixing layer developing after detachment persists even after reattachment. The flow does not recover fully back to a regular turbulent boundary layer even far downstream of reattachment. The turbulence-producing instability of a turbulent boundary layer continues to compete with the wall-bounded mixing layer instability, giving rise to the curious schismatic arrays of large structures.

Evolution and dynamics of shear-layer structures in near-wall turbulence

Near-wall flow structures in turbulent shear flows are analysed, with particular emphasis on the study of their space–time evolution and connection to turbulence production. The results are obtained from investigation of a database generated from direct numerical simulation of turbulent channel flow at a Reynolds number of 180 based on half-channel width and friction velocity. New light is shed on problems associated with conditional sampling techniques, together with methods to improve these techniques, for use both in physical and numerical experiments. The results clearly indicate that earlier conceptual models of the processes associated with near-wall turbulence production, based on flow visualization and probe measurements need to be modified. For instance, the development of asymmetry in the spanwise direction seems to be an important element in the evolution of near-wall structures in general, and for shear layers in particular. The inhibition of spanwise motion of the near-wall streaky pattern may be the primary reason for the ability of small longitudinal riblets to reduce turbulent skin friction below the value for a flat surface.

The outer region of a turbulent boundary layer

Matched asymptotic expansions are used as a framework from which to derive differential equations that describe the mean velocity and turbulence fields in the outer region of a zero-pressure-gradient turbulent boundary layer. Attention is focused upon solutions to these equations in the very outer region or superlayer, where the boundary layer merges with the outer flow. It is found that both the velocity and turbulence fields approach their free-stream values exponentially fast as Townsend [The Structure of Turbulent Shear Flow (Cambridge U.P., Cambridge, 1976)] had foreseen, but not necessarily in the detailed manner he conjectured. These details are used to help construct approximate solutions for the mean velocity and turbulence fields in the outer region that display the correct asymptotic form both in the logarithmic region and the superlayer. The resulting solution for mean velocity is closely in accord with Coles’ law of the wake and accurately reproduces data over the complete Reynolds number range for which the boundary layer is turbulent; likewise the profiles for the turbulence intensities. It is further shown that the turbulence intensities conform to a law analogous to the law of the wake.

Toward full-field measurements of instantaneous visualizations of coherent structures in turbulent shear flows

One goal of contemporary fluid mechanics research is to obtain an adequate physical model of turbulent flows that can be used as a basis for modeling realistic base for boundary layer flow that would consist of the entire instantaneous vector velocity field in all of space for a specific period of time must be established. This information would then be used to test modeling efforts. This paper illustrates, for the specific task of particle tracking to visualize fluid motions, the steps the authors have taken to solve the imaging problems and to provide quantitative estimates. It demonstrates how far the authors have progressed in the two years since they last reported on their efforts (Brodkey, R. S. Chem. Eng. Educ., 20, 202, 1986). Although final results are not yet available, future research needs are indicated.

Influence of density variations on the structure of low-speed turbulent flows: a report on Euromech 237

The European Mechanics Colloquium number 237 was held at the Institut de Mécanique Statistique de la Turbulence (Université d'Aix-Marseille II) from 18 to 21 July 1988. This was the first meeting to consider the influence of density variations on turbulent flows for both non-reacting and reacting flows in a variety of configurations. Several new experiments, computational models and theoretical analyses were presented for non-reacting flows. All these approaches showed the marked effects of density variations on coherent structures in turbulent shear flows. It was found that the approximate models used for turbulent Reynolds stresses in homogeneous flows do not need to be changed for the mean flow field but predictions for variances and correlations are still rather uncertain: in fact experimental results providing such information in variable-density flows are rare. The special problem of measuring turbulence in flows with density variations was discussed. In the discussions it was agreed that there are in fact strong similarities in the effects of density gradients on the dynamics of non-reacting flows and reacting flows despite the differences in the distributions of density gradients. Participants at the meeting from industry emphasized the importance of these flows to many different kinds of industrial and environmental problems.

A thermodynamical theory of turbulence. I. Basic developments

This paper is concerned with the construction of a thermodynamical theory for turbulence based on a continuum model consistent with a wide range of experimental results and observations. A complete theory with appropriate constitutive equations is developed for viscous turbulent flow but the special case of (rate-independent) inviscid turbulent flow is also discussed. The theoretical results obtained readily account for such mechanical aspects of turbulent flow as anisotropy, as well as the energetic effects of turbulent fluctuations, in addition to the more standard thermomechanical effects. More specifically, three different scales of motion and modelling, namely molecular, microscopic and macroscopic, are considered in the construction of the basic theory. Whereas the ordinarythermal effects(such as temperature) on the macroscopic scale represent the manifestation of vibratory motions at the molecular level, similar variables are used to represent the energeticturbulent effectson the macroscopic level that arise from turbulent fluctuations at the microscopic level. The various ingredients of the thermodynamical aspects (both due to thermal and turbulent effects) of the continuum model are incorporated into the theory by means of a recent procedure to thermodynamics by Green & Naghdi (Proc. R. Soc. Lond.A 357, 253 (1977)). The mechanical aspects of the model for a turbulent fluid requires admission of additional balance laws for eddy concentration and for a kinematical variable which represents the effect of alignment of these eddies (at the microscopic level) along a particular direction on the macroscopic scale, in accordance with observations by Townsend (The structure of turbulent shear flow, Cambridge University Press (1976)) and others.