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Computing the Karhunen–Loève dimension of an extensively chaotic flow field given a finite amount of data
Abstract
The use of Karhunen–Loève decomposition (KLD) to explore the complex fluid flows that are common in engineering applications is increasing and has yielded new physical insights. However, for most engineering systems the dimension of the dynamics is expected to be very large yet the flow field data is available only for a finite time. In this context, it is important to establish the amount of data required to compute the asymptotic value of the Karhunen–Loève dimension given a finite amount of data. Using direct numerical simulations of Rayleigh–Bénard convection in a finite cylindrical geometry we compute the asymptotic value of the Karhunen–Loève dimension. The amount of time required for the Karhunen–Loève dimension to reach a steady value is very slow in comparison with the time scale of the convection rolls. We show that the asymptotic value of the Karhunen–Loève dimension can be determined using much less data if one uses the azimuthal symmetry of the governing equations prior to performing a KLD. The Karhunen–Loève dimension is found to be extensive as the system size is increased and for a dimension measurement that captures 90% of the variance in the data the Karhunen–Loève dimension is approximately 20 times larger than the Lyapunov dimension.
... A similar procedure can be used for snapshot POD, as done by Duggleby and Paul (2010), which drastically increases the convergence of the modes. However, the underlying ansatz still assumes that the velocity field can be separated in to a space and a time function. ...
... However, it is still an attractive approach as it is a less computationally expensive method than classical POD. The effect of mode mixing can be reduced by decomposing the azimuthal direction using a Fourier series expansion, as proposed by (Duggleby and Paul, 2010). The POD equations are derived and explained more in detail in Chapter 2, but to summarize, the POD equation in pipe coordinates can written as r ′ S(m; r, r ′ ) Φ n (m; r ′ )dr ′ = λ n (m) Φ n (m; r), ...
... This is called mode mixing, and it can be avoided entirely by using cPOD instead of sPOD. Alternatively,Duggleby et al. (2007) acknowledged the known symmetries by azimuthally decomposing the flow into a Fourier series before solving the eigenvalue problem, a method that allows for faster convergence than regular sPOD,(Duggleby and Paul, 2010). However, as the modes presented byDuggleby et al. (2007) has a constant azimuthal mode number throughout the domain, the modes still exhibit non-optimal behavior, where the radial profile still changes its behavior with azimuthal phase. ...
The presence of organized motions in turbulent wall-bounded flows has been explored in progressively greater detail for more than half a century. It is only within recent years that the largest of these structures – the so-called very large-scale motions – has been recognized for its important energetic and shear stress content. As we gain more understanding of these motions and their contribution to the production of turbulence and their linkage to the mean flow and turbulence intensities, their further exploration become even more intriguing.
The work presented in this thesis investigates the two largest of the known coherent structures – the large-scale motions (LSM) and the very large-scale motions (VLSM) – and their relationship and contribution to the known statistical view of turbulence. To this end, I report an analysis of experimentally acquired data sets in turbulent pipe flow. The turbulent flow is broken down into a set of energetic modes using proper orthogonal decomposition (POD), where each mode can be argued to represent a coherent structure, or at least one phase of its evolution. The results support the existing understanding that these structures are energetically important with a large shear stress contribution. The work also provides a clear link between the large-scale and very large-scale motions, suggesting that the latter is composed of a streamwise pseudo-alignment of the shorter large-scale motions. This result is the principal conclusion of the thesis.
The POD analysis is also expanded to include flow structures induced by pipe curvature. These structures are shown to completely overwhelm the underlying turbulent structures and are shown to exhibit an unsteady behavior, governed by a single cell vortex structure of alternating rotational direction, referred to as “swirl switching”. This motion is suggested to be a governed by an alternating suppression of one of the cells of the Dean motion; a steady, dual cell solution with each cell located on either side of the bend symmetry plane.
... The application of POD is similar to that of Fourier analysis, except that it normally requires far less modes to represent the system within a desired level. This procedure can be used in place of Fourier-type of analysis of the original partial differential equations governing the fluid flow [4] [6]. POD expansion can be used to represent the Fourier coefficients [1]. ...
... The dyna shots idea poin vecto (4) To obtain the POD modes, the following eigenvalue problem needs to be solved. ...
... The application of POD is similar to that of Fourier analysis, except that it normally requires far less modes to represent the system within a desired level. This procedure can be used in place of Fouriertype of analysis of the original partial differential equations governing the fluid flow [4] [6]. POD expansion can be used to represent the Fourier coefficients [1]. ...
Large Eddy Simulations (LES) contains 3D, instantaneous, spatially filtered velocity fields as well as reactive/passive scalar concentration field describing the coherent flow structures. The Dynamics of coherent flow structures has a major impact on the mixing of the fuel and the oxidiser. The analysis of these structures in a turbulent jet is essential in understanding the fundamentals of fluid dynamics. Therefore there is a need for methods that can identify and analyse these structures. In this paper, we use machine-learning methods such as Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) to analyse the coherent flow structures. We used 2D LES of subsonic jets as our data, with Reynolds number corresponding to Re: 6000 (low pressure), 10,000 (medium pressure), and 13,000 (high pressure). Results for POD modes and DMD modes are discussed and compared.
... While POD has also been applied to DNS of turbulent pipes (e.g. Duggleby & Paul 2010), there have been no recent three-dimensional applications to turbulent boundary layer flows, although structures in transition were studied with POD by Rempfer & Fasel (1994). Recent incompressible zero-pressure-gradient flat-plate turbulent boundary layer simulations of Wu & Moin (2009) periodically introduced blocks of isotropic turbulence into the laminar flow at the inlet and allowed the boundary layer to progress through transition. ...
... This procedure has been employed in several applications (e.g., Freund & Colonius 2009;Duggleby & Paul 2010), and has been shown to improve the statistical convergence of POD because it incorporates information from all possible shifts of the homogeneous coordinate. Then, a POD expansion is used to represent the Fourier coefficientsû ...
Proper orthogonal decomposition (POD) is applied to the direct numerical simulation (DNS) of a turbulent boundary layer performed by Wu & Moin (2010), and the resulting POD modes of various scales are examined. The modes include structures resembling those observed in instantaneous flow fields, such as large-scale motions of streamwise velocity with ramp-like wall-normal growth. Other modes correspond closely to near-wall streaks. In addition, POD modes that are constant across the spanwise domain width are observed to grow from the wall with the mean boundary layer thickness. The results support the existence of boundary layer coherent motions described by the hairpin packet model (Adrian 2007).
... They found that snapshot POD may suffer from mode mixing, resulting in some non-optimal modes composed of parts of optimal modes. The effect of mode mixing can be reduced by decomposing the azimuthal direction using a Fourier series expansion, as proposed by Duggleby & Paul (2010). The POD equation in pipe coordinates can be written as ...
A dual-plane snapshot proper orthogonal decomposition (POD) analysis of turbulent pipe flow at a Reynolds number of 104 000 is presented. The high-speed particle image velocimetry data were simultaneously acquired in two planes, a cross-stream plane (2D–3C) and a streamwise plane (2D–2C) on the pipe centreline. The cross-stream plane analysis revealed large structures with a spatio-temporal extent of 1–2R, where R is the pipe radius. The temporal evolution of these large-scale structures is examined using the time-shifted correlation of the cross-stream snapshot POD coefficients, identifying the low-energy intermediate modes responsible for the transition between the large-scale modes. By conditionally averaging based on the occurrence/intensity of a given cross-stream snapshot POD mode, a complex structure consisting of wall-attached and -detached large-scale structures is shown to be associated with the most energetic modes. There is a pseudo-alignment of these large structures, which together create structures with a spatio-temporal extent of approximately 6R, which appears to explain the formation of the very-large-scale motions previously observed in pipe flow.
... Alternatively, Duggleby et al. 7 acknowledged the known symmetries by azimuthally decomposing the flow into a Fourier series before solving the eigenvalue problem, a method that allows for faster convergence than regular sPOD. 8 However, as the modes presented by Ref. 7 have a constant azimuthal mode number throughout the domain, the modes still exhibit non-optimal behavior, where the radial profile still changes its behavior with azimuthal phase. ...
Snapshot and classical proper orthogonal decomposition (POD) are used to examine the large-scale, energetic motions in fully developed turbulent pipe flow at ReD
= 47,000 and 93,000. The snapshot POD modes come in pairs, representing the same azimuthal mode number but with a simple phase shift. The first 10 snapshot POD modes, associated with the very large scale motions (VLSMs), contribute 43% of the average Reynolds shear stress, and for first 80 modes u′ and v′ are anti-correlated so that they all contribute to positive shear stress events. The attached motions are contained in the lower order modes, and detached motions do not appear until snapshot POD mode numbers ≥15. We find that snapshot POD can introduce mode mixing, which is avoided in classical POD. Classical POD also gives frequency information, confirming that the low order modes capture well the behavior of the very large scale motions.
... The helical form would be gradually approached with many data samples (snapshots) owing to the homogeneity in x and θ (as the eigenvalue spectra of simpler systems suggest (e.g. Duggleby & Paul 2010)). For the set of data in Hellström et al. (2011), the two most energetic modes contain long regions of u that are streamwise-aligned (helix angle of zero). ...
The physical structures of velocity are examined from a recent direct numerical simulation of fully developed incompressible turbulent pipe flow [X. Wu, J. R. Baltzer and R. J. Adrian, J. Fluid Mech. 698, 235–281 (2012; Zbl 1250.76116)] at a Reynolds number of Re D =24 580 (based on bulk velocity) and a Kármán number of R + =685. In that work, the periodic domain length of 30 pipe radii R was found to be sufficient to examine long motions of negative streamwise velocity fluctuation that are commonly observed in wall-bounded turbulent flows and correspond to the large fractions of energy present at very long streamwise wavelengths (≥3R). In this paper we study how long motions are composed of smaller motions. We characterize the spatial arrangements of very large-scale motions (VLSMs) extending through the logarithmic layer and above, and we find that they possess dominant helix angles (azimuthal inclinations relative to streamwise) that are revealed by two- and three-dimensional two-point spatial correlations of velocity. The correlations also reveal that the shorter, large-scale motions (LSMs) that concatenate to comprise the VLSMs are themselves more streamwise aligned. We show that the largest VLSMs possess a form similar to roll cells centred above the logarithmic layer and that they appear to play an important role in organizing the flow, while themselves contributing only a minor fraction of the flow turbulent kinetic energy. The roll cell motions play an important role with the smaller scales of motion that are necessary to create the strong streamwise streaks of low-velocity fluctuation that characterize the flow.
... The helical form would be gradually approached with many data samples (snapshots) owing to the homogeneity in x and θ (as the eigenvalue spectra of simpler systems suggest (e.g. Duggleby & Paul 2010)). For the set of data in Hellström et al. (2011), the two most energetic modes contain long regions of u that are streamwise-aligned (helix angle of zero). ...
Regions of negative velocity fluctuation are extracted from a turbulent
pipe simulation at Reτ=685. The lengths of these regions
are characterized for a range of thresholds. The organizational patterns
of these structures are analyzed using a linear stochastic estimate of a
negative velocity fluctuation event. Instantaneous examples of these
events are compared with the patterns in the estimate to understand the
relationship between structure organization and the two-point spatial
correlation. Close examination of the very long motions reveal distinct
forms of smaller motions with different characteristic scales. Studying
these structural elements provides insight into how the smaller motions
compose the very long motions. Cross-sections in various planes are
compared to features observed in experimental investigations, including
vortical motions. Three-dimensional vortical structures are extracted
using several methods, and their geometries and relationships with the
velocity motions are examined.
We probe the effectiveness of using topological defects to characterize the leading Lyapunov vector for a high-dimensional chaotic convective flow field. This is accomplished using large-scale parallel numerical simulations of Rayleigh–Bénard convection for experimentally accessible conditions. We quantify the statistical correlations between the spatiotemporal dynamics of the leading Lyapunov vector and different measures of the flow field pattern’s topology and dynamics. We use a range of pattern diagnostics to describe the flow field structures which includes many of the traditional diagnostics used to describe convection as well as some diagnostics tailored to capture the dynamics of the patterns. We use the ideas of precision and recall to build a statistical description of each pattern diagnostic’s ability to describe the spatial variation of the leading Lyapunov vector. The precision of a diagnostic indicates the probability that it will locate a region where the Lyapunov vector is larger than a threshold value. The recall of a diagnostic indicates its ability to locate all of the possible spatial regions where the Lyapunov vector is above threshold. By varying the threshold used for the Lyapunov vector magnitude, we generate precision-recall curves which we use to quantify the complex relationship between the pattern diagnostics and the spatiotemporally varying magnitude of the leading Lyapunov vector. We find that pattern diagnostics which include information regarding the flow history outperform pattern diagnostics that do not. In particular, an emerging target defect has the highest precision of all of the pattern diagnostics we have explored.
Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive definite kernel is also the covariance kernel of a Gaussian process. Given a fixed sigma-finite measure $\mu$, we consider positive definite kernels defined on the subset of the sigma algebra having finite $\mu$ measure. We show that then the corresponding Hilbert factorizations consist of signed measures, finitely additive, but not automatically sigma-additive. We give a necessary and sufficient condition for when the measures in the RKHS, and the Hilbert factorizations, are sigma-additive. Our emphasis is the case when $\mu$ is assumed non-atomic. By contrast, when $\mu$ is known to be atomic, our setting is shown to generalize that of Shannon-interpolation. Our RKHS-approach further leads to new insight into the associated Gaussian processes, their Ito calculus and diffusion. Examples include fractional Brownian motion, and time-change processes.
Particle Image Velocimetry from the centerline of a 3 x 5 scaled down array of wind turbines in a wind tunnel was analyzed to gain further understanding of how turbulent transport brings mean kinetic energy (MKE) into the array from the neutrally stable Atmospheric Boundary Layer (ABL) above. Vertical fluxes of MKE due to the Reynolds stresses were computed i.e. -< U > < u'v'> A modal expansion for < U >< u'v'> was constructed based on the Proper Orthogonal Decomposition (POD). By determining each mode's fractional contribution to the total entrainment it was shown that a small number of modes (the first 25) account for 75% of the entrainment. The remaining 25% is achieved asymptotically as the remainder of the modes are included in the representation. Based on this behavior the labels "idiosyncratic" and "asymptotic" were applied to the different mode types. A characteristic wavelength for each mode was defined as the length of a mode's longest positive contribution to the energy entrainment. By this definition it was shown that idiosyncratic and asymptotic modes are characterized by wavelengths greater than and less than D (rotor diameter) respectively so that large percentages of the energy brought into the wind farm are done at scales greater than D. Physical reasoning indicates the idiosyncratic modes are associated with larger scale coherent motions whereas the asymptotic modes are associated with small scale turbulent fluctuations. The analysis was repeated for PIV data without turbines. It was shown that the idiosyncratic modes represent the scales which are affected by the presence of the turbines. This further established that the idiosyncratic modes were connected with the larger scales of turbulent motion.
A blind Large-Eddy Simulation (LES) of film-cooling heat transfer is performed on a canonical cylindrical cooling hole geometry using a massively-parallel, geometrically-flexible, open-source spectral element solver NEK5000. The simulation is for a blowing-ratio of 1.0, density-ratio of 1.5, and Reynolds-number Reθ = 4,300 based on boundary layer momentum thickness and ReD = 32,000 based on hole diameter. A low-Mach ideal gas formulation is used to match the density ratio. A spectral-damping LES subgrid model is used which does not restrict time-stepping, allowing CFL numbers of 5–10 through characteristics time-integration. The numerical mesh resolves the boundary layer and coarsens to acceptable LES sizing in the free stream, resulting in 88 million grid points (410,464 elements at 5th order polynomial). For this blowing ratio, the coolant hole Mach number is too large for the low-Mach formulation (> 0.3). This results in faster hole velocities as opposed to fluid compression, effectively changing the momentum ratio leading to coolant lift-off as compared to experiment. The film-cooling effectiveness along centerline and spanwise locations of x/D = 2 and 8 are lower than experiment. Ideal parallel scaling is shown up to 256 processors and estimated to continue at ideal scaling to 2048 processors.
Current supercomputing systems have tens or hundreds of thousands of cores and are trending to GPU and co-compute platforms that deliver thousands of cores per node. Modern computational fluid dynamics codes must be designed to take advantage of these developments in order to further their use in the design cycle. Furthermore, these codes must be highly accurate, stable, and geometrically flexible. NEK5000 is a massively-parallel spectral element code that exhibits these characteristics but currently only for incompressible and low-Mach flows. Adding capabilities for NEK5000 to solve the fully compressible Navier-Stokes equations will extend its usefulness to aerospace applications. As a first step the following work extends NEK5000’s capabilities to solve the 2D compressible Euler equations. Using the conservative formulation, the equations are discretized using a non-staggered spectral element mesh, and the state variables are advanced using 1st order explicit Euler time stepping. A channel with a 10% bump is used as a test case for the modification. The modified NEK5000 code performs very well despite not being optimized for use in hyperbolic equations.
Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-Bénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading-order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading-order Lyapunov exponent, and we quantify the details of their response to the dynamics of defects. The leading-order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary-dominated dynamics to bulk-dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics.
Spatiotemporally chaotic dynamics in laboratory experiments on convection are characterized using a new dimension, D(CH), determined from computational homology. Over a large range of system sizes, D(CH) scales in the same manner as D(KLD), a dimension determined from experimental data using Karhuenen-Loéve decomposition. Moreover, finite-size effects (the presence of boundaries in the experiment) lead to deviations from scaling that are similar for both D(CH) and D(KLD). In the absence of symmetry, D(CH) can be determined more rapidly than D(KLD).
A study is made of the number of dimensions needed to specify chaotic Rayleigh–Bénard convection, over a range of Rayleigh numbers (γ = Ra/Rac < 102). This is based on the calculation of Lyapunov dimension over the range, as well as the notion of Karhunen–Loéve dimension. An argument suggesting a universal relation between these estimates and supporting numerical evidence is presented. Numerical evidence is also presented that the reciprocal of the largest Lyapunov exponent and the correlation time are of the same order of magnitude. Several other universal features are suggested. In particular it is suggested that the intrinsic attractor dimension is $O(Ra^{\frac{2}{3}})$, which is sharper than previous results.
A parametric study is made of chaotic Rayleigh–Bénard convection over moderate Rayleigh numbers. As a basis for comparison over the Rayleigh number (Ra) range we consider mean quantities, r.m.s. fluctuations, Reynolds number, probability distributions and power spectra. As a further means of investigating the flow we use the Karhunen–Loéve procedure (empirical eigenfunctions, proper orthogonal decomposition). Thus, we also examine the variation in eigenfunctions with.Ra. This in turn provides an analytical basis for describing the manner in which the chaos is enriched both temporarily and spatially as Ra increases. As Ra decreases, the significant mode count decreases but, in addition, the eigenfunctions tend more nearly to the eigenfunctions of linearized theory. As part of this parametric study a variety of scaling properties are investigated. For example it is found that the empirical eigenfunctions themselves show a simple scaling in Ra.
For three-dimensional flows with one inhomogeneous spatial coordinate and two periodic directions, the Karhunen–Loeve procedure is typically formulated as a spatial eigenvalue problem. This is normally referred to as the direct method (DM). Here we derive an equivalent formulation in which the eigenvalue problem is formulated in the temporal coordinate. It is shown that this so-called method of snapshots (MOS) has some numerical advantages when compared to the DM. In particular, the MOS can be formulated purely as a matrix composed of scalars, thus avoiding the need to construct a matrix of matrices as in the DM. In addition, the MOS avoids the need for so-called weight functions, which emerge in the DM as a result of the non-uniform grid typically employed in the inhomogeneous direction. The avoidance of such weight functions, which may exhibit singular behaviour, guarantees satisfaction of the boundary conditions. The MOS is applied to data sets recently obtained from the direct simulation of turbulence in a channel in which viscoelasticity is imparted to the fluid using a Giesekus model. The analysis reveals a steep drop in the dimensionality of the turbulence as viscoelasticity is increased. This is consistent with the results that have been obtained with other viscoelastic models, thus revealing an essential generic feature of polymer-induced drag reduced turbulent flows. Published in 2006 by John Wiley & Sons, Ltd.
Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.
We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow.
Using large-scale numerical calculations, we explore the proper orthogonal decomposition of low Reynolds number turbulent pipe flow, using both the translational invariant (Fourier) method and the method of snapshots. Each method has benefits and drawbacks, making the 'best' choice dependent on the purpose of the analysis. Owing to its construction, the Fourier method includes all the flow fields that are translational invariants of the simulated flow fields. Thus, the Fourier method converges to an estimate of the dimension of the chaotic attractor in less total simulation time than the method of snapshots. The converse is that for a given simulation, the method of snapshots yields a basis set that is more optimal because it does not include all of the translational invariants that were not a part of the simulation. Using the Fourier method yields smooth structures with definable subclasses based upon Fourier wavenumber pairs, and results in a new dynamical systems insight into turbulent pipe flow. These subclasses include a set of modes that propagate with a nearly constant phase speed, act together as a wave packet and transfer energy from streamwise rolls. It is these interactions that are responsible for bursting events and Reynolds stress generation. These structures and dynamics are similar to those found in turbulent channel flow. A comparison of structures and dynamics in turbulent pipe and channel flows is reported to emphasize the similarities and differences.
A method is presented for the representation of (pictures of) faces. Within a specified framework the representation is ideal. This results in the characterization of a face, to within an error bound, by a relatively low-dimensional vector. The method is illustrated in detail by the use of an ensemble of pictures taken for this purpose.
We report experiments on convection patterns in a cylindrical cell with a large aspect ratio. The fluid had a Prandtl number of approximately 1. We observed a chaotic pattern consisting of many rotating spirals and other defects in the parameter range where theory predicts that steady straight rolls should be stable. The correlation length of the pattern decreased rapidly with increasing control parameter so that the size of a correlated area became much smaller than the area of the cell. This suggests that the chaotic behavior is intrinsic to large aspect ratio geometries. Comment: Preprint of experimental paper submitted to Phys. Rev. Lett. May 12 1993. Text is preceeded by many TeX macros. Figures 1 and 2 are rather long
Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on the frequency with which a typical trajectory visits different regions of the attractor. Both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value, which we call the “dimension of the natural measure”, and all of the metric dimensions take on a common value, which we call the “fractal dimension”. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.
In this work we study, using the results of direct numerical simulations [Housiadas and Beris, “Polymer-induced drag reduction: Viscoelastic and inertia effects of the variations in viscoelasticity and inertia,” Phys. Fluids 15, 2369 (2003)], the effects of changes in the flow viscoelasticity and the friction Reynolds number on several higher order statistics of turbulence, such as the Reynolds stress, the enstrophy, the averaged equations for the conformation tensor, as well as on the coherent structures through a Karhunen–Loeve (K-L) analysis and selected flow and conformation visualizations. In particular, it is shown that, as the zero friction Weissenberg number WeΤ0 increases (for a constant zero friction Reynolds number ReΤ0) dramatic reductions take place in many terms in the averaged equations for the Reynolds stresses and in all terms of the averaged enstrophy equations. From a Karhunen–Loeve analysis of the eigenmodes of the flow we saw that the presence of viscoelasticity increases significantly the coherence and energy content of the first few modes. The K-L dimension of the flow at WeΤ0=125 is fully one order of magnitude lower than its Newtonian counterpart. As far as the effect of viscoelasticity is concerned, it is manifested primarily by changes in the boundary layer, which are mostly accomplished by WeΤ0=50–62.5. In comparison, increasing the ReΤ0, from 125 to 590, induces significant changes in various terms in the budgets, despite the fact that the drag reduction remains practically the same over that range. However, the near the wall region seems to change significantly only up to ReΤ0=395, with few changes observed upon a further increase to ReΤ0=590.
By analyzing solutions of the 1D Kuramoto-Sivashinsky and the 2D Miller-Huse models in domains of varying size L, we examine how the Karhunen-Loève decomposition (KLD) can be used to quantify extensive chaos. For sufficiently large L, the KLD energies of both the system and of its various subsystems scale extensively, yielding the same KLD energy densities which, in turn, have a similar parametric dependence compared to that of the Lyapunov dimension density. We show that the KLD energy density is a local quantity that can be used to characterize inhomogeneous spatiotemporal chaos.
This review summarizes results for Rayleigh-Bénard convection that have been obtained over the past decade or so. It concentrates on convection in compressed gases and gas mixtures with Prandtl numbers near one and smaller. In addition to the classical problem of a horizontal stationary fluid layer heated from below, it also briefly covers convection in such a layer with rotation about a vertical axis, with inclination, and with modulation of the vertical acceleration.
The aerodynamic performance of a wing deteriorates considerably as the Reynolds number decreases from 106 to 104. In particular, flow separation can result in substantial change in effective airfoil shape and cause reduced aerodynamic performance. Lately, there has been growing interest in developing suitable techniques for sustained and robust flight of micro air vehicles (MAVs) with a wingspan of 15cm or smaller, flight speed around 10m/s, and a corresponding Reynolds number of 104–105. This paper reviews the aerodynamics of membrane and corresponding rigid wings under the MAV flight conditions. The membrane wing is observed to yield desirable characteristics in delaying stall as well as adapting to the unsteady flight environment, which is intrinsic to the designated flight speed. Flow structures associated with the low Reynolds number and low aspect ratio wing, such as pressure distribution, separation bubble and tip vortex are reviewed. Structural dynamics in response to the surrounding flow field is presented to highlight the multiple time-scale phenomena. Based on the computational capabilities for treating moving boundary problems, wing shape optimization can be conducted in automated manners. To enhance the lift, the effect of endplates is evaluated. The proper orthogonal decomposition method is also discussed as an economic tool to describe the flow structure around a wing and to facilitate flow and vehicle control.
The paper considers the dynamics of coherent structures in the wall region of a turbulent channel flow. The Karhunen-Loeve eigenfunctions and Galerkin procedure are employed to derive the dynamical description. A well-posed Hermitian theory is developed and convergence questions do not arise. No exterior pressure is required by this theory. It is shown that the behavior of the resulting model equations include intermittency, quasi-periodic, and chaotic solutions. Three-dimensional effects are introduced into the dynamics in order to produce a physically more realistic dynamical theory. It is argued that the bursting and ejection events in turbulent boundary layers are explained more satisfactorily within this framework.
Low-temperature techniques were applied to the study of a hydrodynamic instability. High-precision results for the Nusselt number N as a function of the Rayleigh number R for liquid and gaseous helium revealed no singularities in N(R), except at the convective threshold Rc. For R>2.19Rc, a new turbulent state was found and characterized by measuring the frequency spectrum and the amplitude of N(R).
Spatially extended dynamical systems exhibit complex behaviour in both
space and time—spatiotemporal chaos1, 2. Analysis of dynamical
quantities (such as fractal dimensions and Lyapunov exponents3)
has provided insights into low-dimensional systems; but it has proven more
difficult to understand spatiotemporal chaos in high-dimensional systems,
despite abundant data describing its statistical properties1, 4, 5.
Initial attempts have been made to extend the dynamical approach to higher-dimensional
systems, demonstrating numerically that the spatiotemporal chaos in several
simple models is extensive6, 7, 8 (the number of dynamical degrees
of freedom scales with the system volume). Here we report a computational
investigation of a phenomenon found in nature, ‘spiral defect’
chaos5, 9 in Rayleigh–Bénard convection, in which
we find that the spatiotemporal chaos in this state is extensive and characterized
by about a hundred dynamical degrees of freedom. By studying the detailed
space–time evolution of the dynamical degrees of freedom, we find that
the mechanism for the generation of chaotic disorder is spatially and temporally
localized to events associated with the creation and annihilation of defects.
The results of a comparative analysis based upon a Karhunen–Loève expansion of turbulent pipe flow and drag reduced turbulent pipe flow by spanwise wall oscillation are presented. The turbulent flow is generated by a direct numerical simulation at a Reynolds number Re = 150. The spanwise wall oscillation is imposed as a velocity boundary condition with an amplitude of A + = 20 and a period of T + = 50. The wall oscillation results in a 27% mean velocity increase when the flow is driven by a constant pressure gradient. The peaks of the Reynolds stress and root-mean-squared velocities shift away from the wall and the Karhunen–Loève dimension of the turbulent attractor is reduced from 2763 to 1080. The coherent vorticity structures are pushed away from the wall into higher speed flow, causing an increase of their advection speed of 34% as determined by a normal speed locus. This increase in advection speed gives the propagating waves less time to interact with the roll modes. This leads to less energy transfer and a shorter lifespan of the propagating structures, and thus less Reynolds stress production which results in drag reduction. © 2007 American Institute of Physics.
The dynamical equations for the energy in a turbulent channel flow have been developed by using the Karhunen-Loéve modes to represent the velocity field. The energy balance equations show that all the energy in the flow originates from the applied pressure gradient acting on the mean flow. Energy redistribution occurs through triad interactions, which is basic to understanding the dynamics. Each triad interaction determines the rate of energy transport between source and sink modes via a catalyst mode. The importance of the proposed method stems from the fact that it can be used to determine both the rate of energy transport between modes as well as the direction of energy flow. The effectiveness of the method in determining the mechanisms by which the turbulence sustains itself is demonstrated by performing a detailed analysis of triad interactions occurring during a turbulent burst in a minimal channel flow. The impact on flow modification is discussed. Copyright © 2002 John Wiley & Sons, Ltd.
The Karhunen-Loeve (K-L) procedure for generating the empirical eigenfunctions is used to analyze two turbulent channel flow simulations of different geometry and origins. In one instance the Reynolds number ReT based on wall shear velocity, u r, and channel halfwidth, δ, is 80 and in the second instance ReT = 125. The latter case showed a well defined log-layer, while this was absent in the former case. According to accepted convention, the turbulence is said to be continuous when Rer = 80, and fully developed when Rer = 125. In both instances the empirical eigenfunctions reveal the presence of propagating plane wave structures. Thus the contrasts in the two cases is of some importance since they virtually eliminate the possibility that the waves are artifacts.
In addition to the propagating structures the K-L analysis reveals a complimentary set of modes (non-propagating) which can be identified with the streaks that are found in wall bounded turbulence. The energy contained in the propagating structures is relatively small compared to that contained in the remaining streaky modes. Nevertheless the waves are shown to play an essential role in the local production of turbulence via bursting and sweeping.
A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency 0 of the instability.
Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those depend only on metric properties, and those that depend on the frequency with which a typical trajectory visits different regions of the attractor. Both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value, which we call the “dimension of the natural measure”, and all of the metric dimensions take on a common value, which we call the “fractal dimension”. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.
A variety of chaotic flows evolving in relatively high-dimensional spaces are considered. It is shown through the use of an optimal choice of basis functions, which are consequence of the Karhunen-Loeve procedure, that an accurate description can be given in a relatively low-dimensional space. Particular examples of this procedure, which are presented, are the Ginzburg-Landau equation, turbulent convection in an unbounded domain and turbulent convection in a bounded domain.
Efficient solution of the Navier–Stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems. For unsteady incompressible flows, the pressure operator is the leading contributor to stiffness, as the characteristic propagation speed is infinite. In the context of operator splitting formulations, it is the pressure solve which is the most computationally challenging, despite its elliptic origins. We examine several preconditioners for the consistent 2Poisson operator arising in the N− N−2spectral element formulation of the incompressible Navier–Stokes equations. We develop a finite element-based additive Schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior Gauss points. For large two-dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation.
The proper orthogonal decomposition technique is applied in the frequency domain to obtain a reduced-order model of the flow in a turbomachinery cascade. The flow is described by an inviscid–viscous interaction model where the inviscid part is described by the full potential equation and the viscous part is described by an integral boundary layer model. The fully nonlinear steady flow is computed and the unsteady flow is linearized about the steady solution. A frequency-domain model is constructed and validated, showing to provide similar results when compared with previous computational and experimental data presented in the literature. The full model is used to obtain a reduced-order model in the frequency domain. A cascade of airfoils forming a slightly modified Tenth Standard Configuration is investigated to show that the reduced-order model with only 25 degrees of freedom accurately predicts the unsteady response of the full system with approximately 10 000 degrees of freedom.
The threshold of the transition to turbulence of low Prandtl number convective flows occurs much closer to the convective threshold in an extended cylindrical cell, than one could infer from a straight roll stability analysis. Convection then involves non local effects together with a closure of the spatial scales. We solve these problems by constructing an explicit analytical solution of the phase fields and of their mean flow fields, which is valid at the dominant orders of pattern distortions. We hence provide an understanding of the low value of the threshold of turbulence at low Prandtl numbers in a cylindrical cell and of the mechanisms that lead to it. Le seuil de transition à la turbulence des écoulements convectifs de bas nombre de Prandtl se situe bien plus près du seuil de convection dans une boîte cylindrique étendue, que ne le laisserait penser l'analyse de stabilité de rouleaux droits. La convection met alors en jeu des effets non locaux et une fermeture des échelles spatiales. Nous résolvons ces problèmes en construisant une solution analytique explicite des champs de phase et de leurs écoulements moyens, valide aux ordres dominants de distorsion des structures. Nous aboutissons ainsi à une explication de la faible valeur du seuil de turbulence à bas nombre de Prandtl en géométrie cylindrique ainsi que des mécanismes qui y conduisent.
We have measured, in a Rayleigh-Benard experiment in liquid helium, the time dependent effects occurring above the onset of convection, using a local probe. Results are very dependent on the aspect ratio Γ, ratio of the cylinder cell radius to its height. For a small aspect ratio, Γ = 2 and Γ = 2.5, well defined sharp oscillations are present for a large range of Rayleigh numbers. For Γ = 6, a low frequency noise appears for R/Rc = 2, where Rc is the critical Rayleigh number for convection. No well-defined oscillations are present but the noise peaks at a dimensionless frequency around ω = 14, when R/Rc > 3.5. This effect exists also for Γ = 4 and Γ = 12. Γ = 3 is thus a transition point between the two regimes. Sur une expérience de Rayleigh-Benard dans l'hélium liquide, nous avons étudié les effets dépendants du temps au-dessus du seuil de convection. Les résultats dépendent fortement du facteur de forme Γ, rapport du rayon de la cellule cylindrique à sa hauteur. Pour Γ = 2 et Γ = 2,5, nous observons des oscillations bien définies pour une grande gamme de nombres de Rayleigh. Pour Γ = 6, un spectre de bruit basse fréquence apparaît à R/Rc = 2. Pour R/Rc > 3,5, le spectre de bruit présente un maximum local autour de ω ∼ 14. (ω fréquence sans dimension, Rc nombre de Rayleigh critique d'établissement de la convection.) Cet effet persiste pour Γ = 4 et Γ = 12. Γ = 3 représente donc le point de transition entre ces deux régimes.
Available in film copy from University Microfilms International. Computer-produced copy. Thesis (Ph. D.)--Brown University, 1991. Vita. Includes bibliographical references (leaf 128).
The results of an analysis of low-Reynolds-number turbulent channel flow based on the Karhunen-Loeve (K-L) expansion are presented. The turbulent flow field is generated by a direct numerical simulation of the Navier-Stokes equations at a Reynolds number Re(tau) = 80 (based on the wall shear velocity and channel half-width). The K-L procedure is then applied to determine the eigenvalues and eigenfunctions for this flow. The random coefficients of the K-L expansion are subsequently found by projecting the numerical flow field onto these eigenfunctions. The resulting expansion captures 90 percent of the turbulent energy with significantly fewer modes than the original trigonometric expansion. The eigenfunctions, which appear either as rolls or shearing motions, possess viscous boundary layers at the walls and are much richer in harmonics than the original basis functions.
A direct simulation of turbulent flow in a channel is analyzed by the method of empirical eigenfunctions (Karhunen-Loeve procedure, proper orthogonal decomposition). This analysis reveals the presence of propagating plane waves in the turbulent flow. The velocity of propagation is determined by the flow velocity at the location of maximal Reynolds stress. The analysis further suggests that the interaction of these waves appears to be essential to the local production of turbulence via bursting or sweeping events in the turbulent boundary layer, with the additional suggestion that the fast acting plane waves act as triggers.
The origin of the power-law decay measured in the power spectra of low Prandtl number Rayleigh-Bénard convection near the onset of chaos is addressed using long time numerical simulations of the three-dimensional Boussinesq equations in cylindrical domains. The power law is found to arise from quasidiscontinuous changes in the slope of the time series of the heat transport associated with the nucleation of dislocation pairs and roll pinch-off events. For larger frequencies, the power spectra decay exponentially as expected for time continuous deterministic dynamics.
Using large-scale numerical calculations we explore spatiotemporal chaos in Rayleigh-Bénard convection for experimentally relevant conditions. We calculate the spectrum of Lyapunov exponents and the Lyapunov dimension describing the chaotic dynamics of the convective fluid layer at constant thermal driving over a range of finite system sizes. Our results reveal that the dynamics of fluid convection is truly chaotic for experimental conditions as illustrated by a positive leading-order Lyapunov exponent. We also find the chaos to be extensive over the range of finite-sized systems investigated as indicated by a linear scaling between the Lyapunov dimension of the chaotic attractor and the system size.
We describe the development and implementation of an efficient spectral element code for multimillion gridpoint simulations of incompressible flows in general two- and three-dimensional domains. Key to this effort has been the development of scalable solvers for elliptic problems and a stabilization scheme that admits full use of the methods high-order accuracy. We review these and other recently developed algorithmic underpinnings that have resulted in good parallel and vector performance on a broad range of architectures and that, with sustained performance of 319 GFLOPS on 2048 nodes of the Intel ASCI-Red machine at Sandia, readies us for the multithousand node terascale computing systems now coming on line at the DOE labs.
The results of an analysis of turbulent pipe flow based on a Karhunen-Lo`eve decomposition are presented. The turbulent flow is generated by a direct numerical simulation of the Navier-Stokes equations using a spectral element algorithm at a Reynolds number Re_\tau=150. This simulation yields a set of basis functions that captures 90% of the energy after 2,453 modes. The eigenfunctions are categorised into two classes and six subclasses based on their wavenumber and coherent vorticity structure. Of the total energy, 81% is in the propagating class, characterised by constant phase speeds; the remaining energy is found in the non propagating subclasses, the shear and roll modes. The four subclasses of the propagating modes are the wall, lift, asymmetric, and ring modes. The wall modes display coherent vorticity structures near the wall, the lift modes display coherent vorticity structures that lift away from the wall, the asymmetric modes break the symmetry about the axis, and the ring modes display rings of coherent vorticity. Together, the propagating modes form a wave packet, as found from a circular normal speed locus. The energy transfer mechanism in the flow is a four step process. The process begins with energy being transferred from mean flow to the shear modes, then to the roll modes. Energy is then transfer ed from the roll modes to the wall modes, and then eventually to the lift modes. The ring and asymmetric modes act as catalysts that aid in this four step energy transfer. Physically, this mechanism shows how the energy in the flow starts at the wall and then propagates into the outer layer. Comment: 28 pages, 20 figures. Updated with reviewer's comments / suggestions
Rayleigh-B\'{e}nard convection is studied and quantitative comparisons are made, where possible, between theory and experiment by performing numerical simulations of the Boussinesq equations for a variety of experimentally realistic situations. Rectangular and cylindrical geometries of varying aspect ratios for experimental boundary conditions, including fins and spatial ramps in plate separation, are examined with particular attention paid to the role of the mean flow. A small cylindrical convection layer bounded laterally either by a rigid wall, fin, or a ramp is investigated and our results suggest that the mean flow plays an important role in the observed wavenumber. Analytical results are developed quantifying the mean flow sources, generated by amplitude gradients, and its effect on the pattern wavenumber for a large-aspect-ratio cylinder with a ramped boundary. Numerical results are found to agree well with these analytical predictions. We gain further insight into the role of mean flow in pattern dynamics by employing a novel method of quenching the mean flow numerically. Simulations of a spiral defect chaos state where the mean flow is suddenly quenched is found to remove the time dependence, increase the wavenumber and make the pattern more angular in nature. Comment: 9 pages, 10 figures
By analyzing large-aspect-ratio spiral-defect-chaos (SDC) convection images, we show that the Karhunen-Lo\`eve decomposition (KLD) scales extensively for subsystem-sizes larger than 4d (d is the fluid depth), which strongly suggests that SDC is extensively chaotic. From this extensive scaling, the intensive length \xi_KLD is computed and found to have a different dependence on the Rayleigh number than the two-point correlation length \xi_2. Local computations of \xi_KLD reveal a substantial spatial nonuniformity of SDC that extends over radii 18d< r < 45d in a \Gamma=109 aspect-ratio cell. Comment: 10 pages single-spaced (total), 3 figues, 2 tables
Terascale implementations
- Hm Tufo
- Fischerpf
Tufo HM,FischerPF.Terascale implementations. In: Proceedings of the ACM/IEEE SC99 Conference on High Performance Networking and Computing. IEEE Computer Society; 1999 [Gordon Bell Prize paper].
Ergodic theory of chaos and strange attractors
- Eckmann