No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Clustering based on geometry and interactions of turbulence bursting rate processes in a trough region
Abstract
This paper aims at understanding turbulence occurring due to fluid flow in the trough region between two artificial adjacent asymmetric waveforms in an open-channel using velocity data collected by 3-D Acoustic Doppler Velocimeter (ADV), concentrating on interactions among the turbulence bursting phenomena across different spatial locations. The Statistical Learning stage of the analysis begins with the identification and extraction of statistically informative and physically interpretable features in the geometry of the Bursting Rate Processes (BRP). Statistical measures characterising the differences among the concerned processes have been developed and used for splitting the trough region into different regions based on the geometry, structure and randomness in the BRP, using the principles of statistical clustering involving parametric, non-parametric techniques and ideas of information theoretic entropy. Experimental observations support the existence of certain BRP which may be considered to be dominant over the others in an almost global sense. The issue of identifying a single bursting phenomenon that changes its orientation strongly relative to others or its closest neighbour across all spatial locations or stretches of vertical heights for every horizontal location and its importance in the entire physical scenario in the light of the spatial clustering problem has been addressed and settled too. Copyright © 2007 John Wiley & Sons, Ltd.
... In turbulent boundary layers, coherent structures with large flux events have been proposed to explain the "bursting" phenomena responsible for two types of eddy motions name as "ejections" and "sweeps" (Cantwell, 1981;Robinson, 1991). These events are traditionally detected by conditional sampling through quadrant analysis in the (x, z)-plane (Willmarth & Lu, 1972) and their statistics have been investigated for a variety of flows and wall-roughness conditions, for example, experiments in open-channel (Hurther & Lemmin, 2000;Hurther, Lemmin, & Terray, 2007;Mazumder, 2007;Mazumder, Pal, Ghoshal, & Ojha, 2009;Nakagawa & Nezu, 1977;Nelson, Shreve, McLean, & Drake, 1995;Ojha & Mazumder, 2008;Venditti & Bauer, 2005); in wind-tunnel (Raupach, 1981); under an-ice boundary layer (Fer, McPhee, & Sirevaag, 2004); in atmospheric boundary layers (Hurther & Lemmin, 2003;Katul, Kuhn, Schieldge, & Hsieh, 1997;Katul, Poggi, Cava, & Finnigan, 2006;Sterk, Jacobs, & van Boxel, 1998), and in scour around vertical circular cylinders (Debnath, Manik, & Mazumder, 2012;Kirkil, Constantinscu, & Ettema, 2008;Sarkar, Chakraborty, & Mazumder, 2015;Sarkar, Chakraborty, & Mazumder, 2016). Besides the dominant role of the sweeps close to a rough wall, an "equilibrium region" is often observed in fully developed turbulent flows (Krogstad, Antonia, & Browne, 1992). ...
In this paper, we intend to quantify the contribution of turbulent events to the total Reynolds shear stresses u ′ v ′ , u ′ w ′ , and v ′ w ′ from the four different quadrants of three different planes (xy, xz, yz); and to make a comparative study among the planes in the scour geometry developed by short circular cylinder of fixed length with a fixed diameter placed over the sand bed transverse to the flow. We also intend to predict the magnitude of covariance terms u ′ v ′ , u ′ w ′ , and v ′ w ′ and their contributions in the four quadrants by making use of the conditional probability distribution of the Reynolds shear stresses −u ′ v ′ , −u ′ w ′ , and −v ′ w ′ , which can be derived by applying the cumulant-discard method to the Gram-Charlier probability distribution of the two variables. This consideration motivates the work on the flow over the obstacle marks generated on sand bed using different short cylinders. The contributions of burst-sweep cycles to the Reynolds shear stresses from the planes over and within the scour around the obstacle are computed using the quadrant analysis to identify the leading shear stress plane, which are responsible to form the scour geometry. It is discovered that the yz and xy-planes are much more important in the scouring regions, whereas xz-plane is important for the smooth surface. Using cumulant-discard method (taking into account the cumulants of less than fourth order), it is shown that the qualitative behaviours of turbulent events agree well with experimental data. Thus, it is confirmed that even the third-order probability distribution of the Reynolds stresses can describe the experimental results very well. KEYWORDS conditional probability, covariance terms, Gaussian distribution, open channel flow, scour-bed, turbulence 1 INTRODUCTION Turbulent flow has always been a challenge for scientists, that is common in nature and has an important role in several geophysical processes related to a variety of phenomena such as river morphology, landscape modelling, atmospheric dynamics, and ocean currents. As the turbulent flows are Nomenclature: a r = D c ∕L, cylinder aspect ratio; d 50 , mean sediment size; D c , diameter of cylinder; F s , sediment Froude number; H w , water depth; h ′ , thickness of sand bed; h = H w − h ′ , water depth over the sand bed; L, length of cylinder; Q, flow discharge; Re, flow Reynolds number; u, v, w, flow velocities along stream-wise, transverse and vertical to the flow; u m , maximum flow velocity; ̄ u, ̄ v, ̄ w, time-averaged flow velocities; u ′ , v ′ , w ′ , fluctuations in u, v and w; w s , width of the scour hole; x, y, z, Cartesian coordinates; í µí¼, kinematic viscosity of the fluid; í µí¼, fluid density; í µí¼ g , geometric standard deviation of the grain size distribution; Fr, froude number; í µí¼ xy , í µí¼ xz , í µí¼ yz , shear stresses; F ku , stream-wise flux of turbulent kinetic energy; F kw , vertical flux of turbulent kinetic energy; S u , coefficients of skewness in the direction of u; S w , coefficients of skewness in the direction of w; σ u = √ u ′2 , turbulence intensity (r.m.s value) in x direction; σ v = √ v ′2 , turbulence intensity (r.m.s value) in y direction; σ w = √ w ′2 , turbulence intensity (r.m.s value) in z direction. irregular, seemingly random (chaotic) and complex, till today no analytical solutions exist for turbulent flows. We believe that even after 516 years (Leonardo da Vinci around 1500, see Gad-El-Hak, 2000), turbulence studies are still in their infancy. We are still discovering how turbulence behaves, in many respects. We do have a crude, practical, working Environmetrics. 2017;28:e2442. wileyonlinelibrary.com/journal/env
... The turbulent flow and related bursting phenomena over an isolated asymmetric waveform structure were studied statistically by Mazumder and Mazumder [14] using the analysis of multivariate normal distribution. Mazumder [15,16] studied the turbulence of fluid flow in the trough region between a pair of adjacent asymmetric waveform structures using a statistical clustering technique based on geometry and interactions of turbulence bursting rate. Recently, Paul [17] made a comparative study of the turbulence and fractional contributions of bursting events to the Reynolds shear stress over the trough regions formed by a pair of scalene and a pair of isosceles triangular-shaped waveform structures. ...
... We have thus posed the problem as an unsupervised learning or statistical clustering problem (Hastie et al., 2001). Similar principles have been used in turbulence applications (Mazumder, 2007). ...
Flume experiments were carried out to study the turbulence and its impact on suspension and segregation of grain-sizes under unidirectional flow conditions over the sand-gravel mixture bed. The components of fluid velocity with fluctuations were measured vertically using 3-D Micro-acoustic Doppler velocimeter (ADV). The theoretical models for velocity and sediment suspension have been developed based on the concept of mixing length that includes the damping effect of turbulence due to sediment suspension in the flow over the sand-gravel mixture bed. Statistical analysis of segregation of grain-sizes along downstream of the bed has been performed using the principle of unsupervised learning or clustering problem. Exploratory data analysis suggests that there is a progressive downstream fining of sediment sizes with selective depositions of gravels, sand-gravels and sand materials along the stream, which may be segmented into three regions such as, the upstream, the transitional and the downstream respectively. This contribution is relevant to understand the direction of ancient rivers, the bed material character in the river form, sorting process and its role in controlling the sediment flux through landscape.
This paper aims at identifying statistically different circulation patterns characterising fluid flow in the trough region between two adjacent asymmetric waveforms, using the velocity data collected by 3D acoustic Doppler velocimeter. Statistical clustering has been performed using ideas originating from information theory and scale space theory in computer vision for splitting the trough region into different spatially connected segments (identifying the circulation bubble in the process) on the basis of circulation patterns. The paper attempts to visualise the fluid fluctuations in the trough region, with emphasis on the circulation region, by simulating the directional fluctuations of fluid particles from the kernel density estimates learned from the experimental data. The image representation of the estimate of the spatial turbulent kinetic energy (TKE) function reveals interesting features corresponding to the regions of high TKE, suggesting the possibilities for further research in this area along the lines of feature extraction and image analysis.
Results are presented examining the interaction between two sandy bed forms under low-sediment transport conditions in a small laboratory flume. The initial artificially made bed forms were out of equilibrium with the flow field. Temporal evolution of bed forms was monitored using time-lapse photography in order to characterize bed form adjustment to the imposed flow. Velocity measurements were collected using an acoustic Doppler velocimeter to characterize both mean flow and turbulence associated with different bed form geometries. Sandy bed forms all had identical initial geometries; however, the initial distance between bed form crests was varied between experiments. Overall deformation of the bed varied as a function of initial bed form spacing; however, bed forms evolved unpredictably as periods of relatively slow change were punctuated by periods of rapidly changing geometry. Subtle changes in bed form trough geometry were found to have a strong influence on turbulence and therefore sediment transport. Comparison with field studies suggests that the mechanisms described herein are active in natural systems.
The prediction of flow resistance and sediment transport is reviewed, in the context of the general behaviour of alluvial channels.
In the use of smoothing methods in data analysis, an important question is which observed features are "really there," as opposed to being spurious sampling artifacts. An approach is described based on scale-space ideas originally developed in the computer vision literature. Assessment of Significant ZERo crossings of derivatives results in the SiZer map, a graphical device for display of significance of features with respect to both location and scale. Here "scale" means "level of resolution"; that is, "bandwidth.".
We consider the problem of nonparametric estimation of a d-dimensional probability density and its `principal directions' in the independent component analysis model. A new method of estimation based on diagonalization of nonparametric estimates of certain matrix functionals of the density is suggested. We show that the proposed estimators of principal directions are -consistent and that the corresponding density estimators converge at the optimal rate.
Kullback-Leibler divergence and the Neyman-Pearson lemma are two fundamental concepts in statistics. Both are about likelihood ratios: Kullback-Leibler divergence is the expected log-likelihood ratio, and the Neyman-Pearson lemma is about error rates of likelihood ratio tests. Exploring this connection gives another statistical interpretation of the Kullback-Leibler divergence in terms of the loss of power of the likelihood ratio test when the wrong distribution is used for one of the hypotheses. In this interpretation, the standard non-negativity property of the Kullback-Leibler divergence is essentially a restatement of the optimal property of likelihood ratios established by the Neyman-Pearson lemma. The asymmetry of Kullback-Leibler divergence is overviewed in information geometry.
Measurements of mean and turbulence characteristics in uniform open-channel flows of constant mean depth over artificial fixed one-dimensional periodic bed features were obtained using two-component laser-Doppler velocimetry. Two types of bed forms, based on triangular elements of the same height (almost-equal-to 20% of the flow depth) and wavelength (almost-equal-to 12.5 times the height) but of different shape, were studied: (1) 45-degrees triangle ridges separated by a flat region; and (2) triangular elements approximating dunes with a mildly sloping upstream face instead of a flat region. The role of the total shear velocity, based on the mean energy slope and the mean flow depth, as a possible velocity scale is studied. Velocity defect and the Reynolds shear stress profiles were generally well approximated by the flat-bed profiles for regions of flow far from the bed. Sufficiently far from the recirculation and reattachment zones, turbulence characteristics collapsed fairly well when scaled with the total shear velocity over most of the depth, but do not follow the flat-bed behavior in the lower part of the flow. Possible implications of the clear-water measurements for the description of suspended sediment concentration profiles and total suspended load computations are explored.
A hydrodynamic model is developed for the calculation of shear stress and pressure along a sinusoidal boundary that is fixed or propagating very slowly in a wide rectangular open or closed channel. The model uses curvilinear coordinates that follow the wavey boundary, and it is similar to that for two-dimensional wavy boundary flows of infinite extent proposed by Benjamin (1959), except that the present model takes account of turbulent stresses on the basis of the turbulent eddy viscosity concept employed by Hussain and Reynolds (1970). The compact analytical expressions for the perturbed shear stress and pressure on the wavy boundary are derived for the cases of hydraulically smooth turbulent and fully rough turbulent flows. The developed model is shown to be in good agreement with the experiments conducted by Zilker et al. (1977). The developed hydrodynamic model is applied to the problem of the formation of ripples in erodible channels by Kobayashi and Madsen (1985).
The turbulent-flow field above dunes is predicted with equations describing the transport of the kinetic energy of turbulence (k), its rate of dissipation (∈), and algebraic relations derived from a second-moment turbulence closure (Algebraic Stress Model). The resulting set of expressions jointly with the identified boundary conditions are solved with a computer code based on a finite-difference solution that uses the PISO algorithm to handle the coupling between the continuity and momentum equations to obtain mean velocity and turbulent-stress profiles in the flow field, turbulent kinetic energy, and pressure and shear stress distributions over the dune surface. Estimates of flow resistance are obtained by integrating the pressure and shear stress distributions acting on the dune surface. Predictions compared well with detailed data of experiments on turbulent open-channel flow over dunes and with experimental data of total flow resistance.
Extensive visual and quantitative studies of turbulent boundary layers are described. Visual studies reveal the presence of surprisingly well-organized spatially and temporally dependent motions within the so-called ‘laminar sublayer’. These motions lead to the formation of low-speed streaks in the region very near the wall. The streaks interact with the outer portions of the flow through a process of gradual ‘lift-up’, then sudden oscillation, bursting, and ejection. It is felt that these processes play a dominant role in the production of new turbulence and the transport of turbulence within the boundary layer on smooth walls.
Quantitative data are presented providing an association of the observed structure features with the accepted ‘regions’ of the boundary layer in non-dimensional co-ordinates; these data include zero, negative and positive pressure gradients on smooth walls. Instantaneous spanwise velocity profiles for the inner layers are given, and dimensionless correlations for mean streak-spacing and break-up frequency are presented.
Tentative mechanisms for formation and break-up of the low-speed streaks are proposed, and other evidence regarding the implications and importance of the streak structure in turbulent boundary layers is reviewed.
Controlled experiments have shown that the grain-size distribution of suspended sediments is related to bed material, flow velocity and height of suspension above the sand bed in an open channel flow. A theoretical model has been developed for computation of suspended grain-size distribution on the basis of continuity equations of sediment and water, using the computed bed-layer concentration as a reference. The proposed model includes the effect of suspension concentration into the mean velocity, turbulent and viscous shear stresses owing to the dynamic coupling between the flow and sediments in suspension. The effect of hindered settling due to the increased concentration in suspension is also taken into account. The model is considered to be a more general one than the existing models, and the results of the present model compare well with the experimental data. Copyright © 2004 John Wiley & Sons, Ltd.
The objective of this paper is to analyze statistically the turbulence characteristics of flow over an artificial waveform geometry in an open channel, using velocity data collected by 3-D acoustic doppler velocimeter (ADV). The joint distribution of velocity fluctuations have been studied and modeled using multivariate normal distribution, and its effectiveness is analyzed in the light of explaining turbulent bursting events, like sweeping and ejection. The relationship between the randomness in the directional variations of the 3-D velocity fluctuations and the intensity of circulation has been studied. We have exploited this relationship and used it in conjunction with the principles of statistical clustering for splitting the region of flow, into different spatially connected segments based on circulation patterns. Directional data analysis has been adopted to visualize the flow pattern over the entire test section, in particular, the circulation bubbles in the region of separated flow. Copyright © 2006 John Wiley & Sons, Ltd.
Half-title pageSeries pageTitle pageCopyright pageDedicationPrefaceAcknowledgementsContentsList of figuresHalf-title pageIndex
There has been major progress in recent years in data-based bandwidth selection for kernel density estimation. Some “second generation” methods, including plug-in and smoothed bootstrap techniques, have been developed that are far superior to well-known “first generation” methods, such as rules of thumb, least squares cross-validation, and biased cross-validation. We recommend a “solve-the-equation” plug-in bandwidth selector as being most reliable in terms of overall performance. This article is intended to provide easy accessibility to the main ideas for nonexperts.
EM Algorithm and its extensions
- M Geofferey
- Krishnan
Geofferey M, Krishnan T. 1997. EM Algorithm and its extensions. John Wiley: New York.
Stochastic Geometry: Likelihood and Computation. Chapman and Hall: London. Chaudhuri PC, Marron JS. 1999. SiZer for exploration of structures in curves
- Kendall O W Barndorff-Nielsen
- Lieshout
- Mnm
Barndorff-Nielsen O, Kendall W, Lieshout MNM. 1999. Stochastic Geometry: Likelihood and Computation. Chapman and Hall: London. Chaudhuri PC, Marron JS. 1999. SiZer for exploration of structures in curves. Journal of the American Statistical Association 94(447): 807–823.
Turbulence characteristics over artificial waveforms and its implication on sediment transport
- Mazumder Bs
- K Dk
- Ojha
- Sp
Mazumder BS, Pal DK, Ghoshal K, Ojha SP. 2004. Turbulence characteristics over artificial waveforms and its implication on sediment transport, In: ICON-HERP-2004, International Conference on Hydraulic Engineering: Research and Practice, IIT, Roorkee, 204–214.
Interactions between bedforms: topography, turbulence and transport DOI:10 A brief Survey of bandwidth selection for density estimation
- D Jerolmack
- Mohrig D Jones Mc
- Marron Js
- Sheather
- Js
Jerolmack D, Mohrig D. 2005. Interactions between bedforms: topography, turbulence and transport. Journal of Geophysical Research 110(F02014), DOI:10.1029/2004JF000126. Jones MC, Marron JS, Sheather JS. 1996. A brief Survey of bandwidth selection for density estimation. Journal of the American Statistical Association 91: 401–407.
Interpreting Kullback-Leibler Divergence with the Neyman-Pearson Lemma ISM (preprint www
- Eguchis Copasj