Experimental assessment of fractal scale similarity in turbulent flows. Part 2. Higher-dimensional intersections and non-fractal inclusions

1997 ◽  
Vol 338 ◽  
pp. 89-126 ◽  
Author(s):  
RICHARD D. FREDERIKSEN ◽  
WERNER J. A. DAHM ◽  
DAVID R. DOWLING
Keyword(s):  
Turbulent Flows ◽  
Three Dimensional ◽  
Local Scale ◽  
Scalar Dissipation ◽  
Experimental Assessment ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Scale Similarity ◽  
Data Points ◽  
Simple Average

Results from an earlier experimental assessment of fractal scale similarity in one-dimensional spatial and temporal intersections in turbulent flows are here extended to two- and three-dimensional spatial intersections. Over 25000 two-dimensional (2562) intersections and nearly 40 three-dimensional (2563) intersections, collectively representing more than 2.3 billion data points, were analysed using objective statistical methods to determine which intersections were as fractal as stochastically scale-similar fractal gauge sets having the same record length. Results for the geometry of Sc [Gt ]1 scalar isosurfaces and the scalar dissipation support span the range of lengthscales between the scalar and viscous diffusion scales λD and λν. The present study finds clear evidence for stochastic fractal scale similarity in the dissipation support. With increasing intersection dimension n, the data show a decrease in the fraction of intersections satisfying the criteria for fractal scale similarity, consistent with the presence of localized non-fractal inclusions. Local scale similarity analyses on three-dimensional (643) intersections directly show such intermittent non-fractal inclusions with characteristic lengthscale comparable to λν. These inclusions lead to failure of the relation among codimensions Dn≡D−(3−n) when applied to simple average dimensions, which has formed the basis for most previous assessments of fractal scale-similarity. Unlike the dissipation support geometry, scalar isosurface geometries from the same data were found not to be as fractal as fractional Brownian motion gauge sets over the range of scales examined.

Experimental assessment of fractal scale-similarity in turbulent flows. Part 1. One-dimensional intersections

1996 ◽  
Vol 327 ◽  
pp. 35-72 ◽  
Author(s):  
Richard D. Frederiksen ◽  
Werner J. A. Dahm ◽  
David R. Dowling
Keyword(s):  
Turbulent Flows ◽  
Temporal Structure ◽  
Three Dimensional ◽  
Scalar Fields ◽  
Energy Dissipation Rate ◽  
Random Sets ◽  
Scalar Dissipation ◽  
Temporal Scales ◽  
Scale Similarity ◽  
Spatio Temporal

Results are presented from an assessment of the applicability of fractal scale-similarity in the spatio–temporal structure of Sc [Gt ] 1 conserved scalar fields ζ(x, t) and scalar energy dissipation rate fields ∇(x, t) in turbulent flows. Over 2 million spatial and temporal intersections were analysed from fully resolved three-dimensional (256) spatial measurements as well as fully resolved four-dimensional spatio–temporal measurements containing up to 3 million points. Statistical criteria were used to assess both deterministic and stochastic fractal scale-similarity and to differentiate between fractal and random sets. Results span the range of spatio–temporal scales from the scalar diffusion scales (ΛD, TD) to the viscous diffusion scales (Λv, Tv) and to the outer scales (δ, Tδ). Over this entire range of scales, slightly over 99.0% of all intersections with the scalar dissipation support geometry showed scale-similarity as fractal as stochastically self-similar fBm sets having the same record length. Dissipation values above the mean were found to have support dimension D = 0.66. The dissipation support dimension decreased sharply with increasing dissipation values. Virtually no intersections showed scaling as random as a random set with the same relative cover. In contrast, intersections with scalar isosurfaces showed scaling only approximately as fractal as a corresponding fBm set and only over the range of spatio–temporal scales between (ΛD, TD) and (Λv, Tv). On these inner scales the isosurface dimension was D = 0.48 and was largely independent of the isoscalar value. At larger scales, scalar isosurfaces showed no fractal scale-similarity. In contrast, isoscalar level crossing sets showed no fractal scale-similarity over any range of scales, even though the scalar dissipation support geometry for the same data is clearly fractal. These results were found to be unaffected by noise.

Experimental assessment of fractal scale similarity in turbulent flows. Part 4. Effects of Reynolds and Schmidt numbers

1998 ◽  
Vol 377 ◽  
pp. 169-187 ◽  
Author(s):  
RICHARD D. FREDERIKSEN ◽  
WERNER J. A. DAHM ◽  
DAVID R. DOWLING
Keyword(s):  
Turbulent Flows ◽  
Dissipation Rate ◽  
Single Point ◽  
Energy Dissipation Rate ◽  
Scalar Dissipation ◽  
Turbulent Shear ◽  
Turbulent Shear Flow ◽  
Outer Scale ◽  
Scale Invariant ◽  
Scale Similarity

Experimental results are presented for the influence of Reynolds number on multifractal scale similarity in turbulent flows. These are obtained from single-point measurements of a dynamically passive Sc≈1 conserved scalar quantity ζ(t) in a turbulent shear flow at outer-scale Reynolds numbers of 14000[les ]Reδ[les ]110000. Statistical criteria based on the maximum allowable scale-to-scale variation L1(ε) in multiplier distributions P(Mε) from multifractal gauge sets allow accurate discrimination between multifractal and non-multifractal scaling. Results show that the surrogate scalar energy dissipation rate χs(t)≡(dζ/dt)2is found to display a scale-invariant similarity consistent with a random multiplicative cascade characterized by a bilinear multiplier distribution P(Mε) over a range of scales extending downward from the outer scaleTδ. For a range of scales extending upward from the inner (diffusive) scale TD, the dissipation rate displays a different scale-invariant similarity characterized by a uniform multiplier distribution. The former scale-invariance becomes evident in the present Sc≈1 data only when Reδ is sufficiently large. Comparisons with results from Sc 1 data indicate that this scale-invariant similarity applies when the outer-to-inner scale ratio Tδ/TD≈0.09 Re3/4δSc1/2 is greater than about 400. In contrast to the scalar dissipation rate field, the scalar field is found to lack any multifractal scale similarity.

Coordination Polymers with a Bulky Perylene-Based Tetracarboxylate Ligand: Syntheses, Crystal Structures, and Luminescent Properties

2010 ◽  
Vol 63 (3) ◽  
pp. 463 ◽  
Author(s):  
Chun-Sen Liu ◽  
Min Hu ◽  
Song-Tao Ma ◽  
Qiang Zhang ◽  
Li-Ming Zhou ◽  
...  
Keyword(s):  
Coordination Polymers ◽  
Three Dimensional ◽  
Luminescent Properties ◽  
Supramolecular Interactions ◽  
Polymeric Chain ◽  
Stacking Interactions ◽  
One Dimensional ◽  
Π Stacking ◽  
Higher Dimensional ◽  
Interchain Interactions

To explore the coordination possibilities of perylene-based ligands with a larger conjugated π-system, four ZnII, MnII, and CoII coordination polymers with perylene-3,4,9,10-tetracarboxylate (ptc) and the chelating 1,10-phenanthroline (phen) ligands were synthesized and characterized: {[Zn2(ptc)(phen)2](H2O)10}∞ (1), {[Zn3(ptc)(OH)2(phen)2](H2O)3}∞ (2), {[Mn(ptc)0.5(phen)(H2O)2](H2O)1.5}∞ (3), and {[Co(ptc)0.5(phen)(H2O)2](H2O)2.5}∞ (4). Structural analysis reveals that complexes 1 and 2 both take one-dimensional polymeric chain structures with dinuclear and trinuclear units as nodes, respectively, which are further extended via the accessorial secondary interchain interactions, such as C–H···O H-bonding or aromatic π···π stacking interactions, to give rise to the relevant higher-dimensional frameworks. Compound 3 has a two-dimensional sheet structure that is further assembled to form a three-dimensional framework by interlayer π···π stacking interactions. Complex 4 is a one-dimensional ribbon-like array structure that is interlinked by the co-effects of intermolecular π···π stacking and C–H···π supramolecular interactions, resulting in a higher-dimensional framework from the different crystallographic directions. Moreover, complexes 1–4 exhibit strong solid-state luminescence emissions at room temperature, which mainly originate from intraligand π→π* transitions of ptc.

scholarly journals Synthetic Weyl Points of the Shear Horizontal Guided Waves in One-Dimensional Phononic Crystal Plates

2021 ◽  
Vol 12 (1) ◽  
pp. 167
Author(s):  
Hongbo Zhang ◽  
Shaobo Zhang ◽  
Jiang Liu ◽  
Bilong Liu
Keyword(s):  
Guided Waves ◽  
Dimensional Space ◽  
Phononic Crystal ◽  
Phase Interface ◽  
Three Dimensional ◽  
Interface States ◽  
Geometrical Parameters ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Shear Horizontal

Weyl physics in acoustic and elastic systems has drawn extensive attention. In this paper, Weyl points of shear horizontal guided waves are realized by one-dimensional phononic crystal plates, in which one physical dimension plus two geometrical parameters constitute a synthetic three-dimensional space. Based on the finite element method, we have not only observed the synthetic Weyl points but also explored the Weyl interface states and the reflection phase vortices, which have further proved the topological phase interface states. As the first realization of three-dimensional topological phases through one-dimensional phononic crystal plates in the synthetic dimension, this research demonstrates the great potential of applicable one-dimensional plate structural systems in detecting higher-dimensional topological phenomena.

CORRELATION AND SPECTRAL PROPERTIES OF MULTIDIMENSIONAL THUE–MORSE SEQUENCES

2007 ◽  
Vol 17 (04) ◽  
pp. 1265-1303 ◽  
Author(s):  
A. BARBÉ ◽  
F. VON HAESELER
Keyword(s):  
Spectral Properties ◽  
Three Dimensional ◽  
Number System ◽  
Dimensional Case ◽  
Binary Number ◽  
Singular Continuous Spectrum ◽  
One Dimensional ◽  
Number Systems ◽  
Higher Dimensional ◽  
The One

This paper considers higher-dimensional generalizations of the classical one-dimensional two-automatic Thue–Morse sequence on ℕ. This is done by taking the same automaton-structure as in the one-dimensional case, but using binary number systems in ℤm instead of in ℕ. It is shown that the corresponding ±1-valued Thue–Morse sequences are either periodic or have a singular continuous spectrum, dependent on the binary number system. Specific results are given for dimensions up to six, with extensive illustrations for the one-, two- and three-dimensional case.

Numerical Modeling of Some Parameters for Performance Prediction of Centrifugal Impellers

2000 ◽  
Author(s):  
JongSik Oh ◽  
KoonSup Oh
Keyword(s):  
Turbulent Flows ◽  
Performance Prediction ◽  
Effective Means ◽  
Three Dimensional ◽  
Cfd Analysis ◽  
One Dimensional ◽  
Engineering Models ◽  
Time Marching ◽  
And Diffusion ◽  
Marching Method

The numerical results of a CFD analysis for 5 impellers are presented and discussed to generate simple correlations for the slip factors and the aerodynamic exit blockages of centrifugal compressors. The purpose of the analysis and modeling is to supply an effective means of estimating both parameters used in the meanline performance prediction stage, only in the agile engineering sense. A finite volume time marching method was used in the analysis of three dimensional compressible turbulent flows. To generate one dimensional representative values from the three dimensional results, a mass-averaged concept was used on each impeller exit plane. The Wiesner’s slip factor was found to fail to predict accurate level of values and also the trend of variation, when the flow rate was changed, especially in case of backswept impellers. Aerodynamic blockage at the impeller exit was also found to vary with the flow rates, the blade exit angle and diffusion ratio. Some useful engineering models of both parameters were suggested to improve the current level of prediction for the impeller exit performance.

scholarly journals Bounding the scalar dissipation scale for mixing flows in the presence of sources

2011 ◽  
Vol 688 ◽  
pp. 443-460 ◽  
Author(s):  
A. Alexakis ◽  
A. Tzella
Keyword(s):  
Turbulent Flows ◽  
Molecular Diffusion ◽  
Three Dimensional ◽  
Homogenization Theory ◽  
Scalar Dissipation ◽  
Mixing Properties ◽  
The Mean ◽  
Source Sink ◽  
Mean Variance ◽  
Large Peclet Number

AbstractWe investigate the mixing properties of scalars stirred by spatially smooth, divergence-free flows and maintained by a steady source–sink distribution. We focus on the spatial variation of the scalar field, described by the dissipation wavenumber, ${k}_{d} $, that we define as a function of the mean variance of the scalar and its gradient. We derive a set of upper bounds that for large Péclet number ($\mathit{Pe}\gg 1$) yield four distinct regimes for the scaling behaviour of ${k}_{d} $, one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of $\mathit{Pe}$ and the ratio $\rho = {\ell }_{u} / {\ell }_{s} $, where ${\ell }_{u} $ and ${\ell }_{s} $ are, respectively, the characteristic length scales of the velocity and source fields. A fifth regime is revealed by homogenization theory. These regimes reflect the balance between different processes: scalar injection, molecular diffusion, stirring and bulk transport from the sources to the sinks. We verify the relevance of these bounds by numerical simulations for a two-dimensional, chaotically mixing example flow and discuss their relation to previous bounds. Finally, we note some implications for three-dimensional turbulent flows.

Experimental assessment of fractal scale similarity in turbulent flows. Part 3. Multifractal scaling

1997 ◽  
Vol 338 ◽  
pp. 127-155 ◽  
Author(s):  
RICHARD D. FREDERIKSEN ◽  
WERNER J. A. DAHM ◽  
DAVID R. DOWLING
Keyword(s):  
Scalar Field ◽  
Turbulent Flows ◽  
Finite Length ◽  
Scalar Dissipation ◽  
Statistical Criterion ◽  
Cascade Process ◽  
Scale Invariant ◽  
Scale Similarity ◽  
Multifractal Scaling ◽  
Multiplicative Cascade

Earlier experimental assessments of fractal scale similarity in geometric properties of turbulent flows are extended to assess the applicability of multifractal scale-similarity in the conserved scalar field ζ(x, t) and in the true scalar energy dissipation rate field ∇ζ·∇ζ(x, t). Fully resolved four-dimensional spatio-temporal measurements from a turbulent flow at Reλ≈41 and Reδ≈3000 are analysed. The utility of various classical constructs for identifying multifractal scale similarity in data records of finite length is examined. An objective statistical criterion based on the maximum allowable scale-to-scale variation L1(ε) in multiplier distributions 〈P(Mε)〉 obtained from multifractal gauge fields is developed to allow accurate discrimination between multifractal and non-multifractal scaling in finite-length experimental data records. Results from analyses of temporal intersections show that for scales greater than 0.03 λν/u, corresponding to 1.4 λD/u, the scalar dissipation field clearly demonstrates a scale-invariant similarity consistent with a multiplicative cascade process that can be modelled with a bilinear multiplier distribution. However, the conserved scalar field from precisely the same data does not follow any scale similarity consistent with a multiplicative cascade at scales below 0.5 λν/u. At larger scales, there are indications of a possible scale-invariant similarity in the scalar field, but with a fundamentally different multiplier distribution.

Structure and function of neural networks in the retina

1988 ◽  
Vol 46 ◽  
pp. 26-27
Author(s):  
Peter Sterling
Keyword(s):  
Ganglion Cells ◽  
Three Dimensional ◽  
Structure And Function ◽  
Synaptic Connections ◽  
Visual Axis ◽  
One Dimensional ◽  
Cat Retina ◽  
Ultrathin Sections ◽  
And Function ◽  
Insight Into

The synaptic connections in cat retina that link photoreceptors to ganglion cells have been analyzed quantitatively. Our approach has been to prepare serial, ultrathin sections and photograph en montage at low magnification (˜2000X) in the electron microscope. Six series, 100-300 sections long, have been prepared over the last decade. They derive from different cats but always from the same region of retina, about one degree from the center of the visual axis. The material has been analyzed by reconstructing adjacent neurons in each array and then identifying systematically the synaptic connections between arrays. Most reconstructions were done manually by tracing the outlines of processes in successive sections onto acetate sheets aligned on a cartoonist's jig. The tracings were then digitized, stacked by computer, and printed with the hidden lines removed. The results have provided rather than the usual one-dimensional account of pathways, a three-dimensional account of circuits. From this has emerged insight into the functional architecture.

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