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 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.

scholarly journals Dissipation-scale fluctuations and mixing transition in turbulent flows

2008 ◽  
Vol 606 ◽  
pp. 325-337 ◽  
Author(s):  
VICTOR YAKHOT
Keyword(s):  
Probability Density ◽  
Turbulent Flows ◽  
Large Scale ◽  
Molecular Diffusion ◽  
Numerical Data ◽  
Reaction Rates ◽  
Diffusion Time ◽  
Necessary Condition ◽  
Scalar Dissipation ◽  
Dissipation Scale

A small separation between reactants, not exceeding 10−8 − 10−7 cm, is the necessary condition for various chemical reactions. It is shown that random advection and stretching by turbulence leads to the formation of scalar-enriched sheets of strongly fluctuating thickness ηc. The molecular-level mixing is achieved by diffusion across these sheets (interfaces) separating the reactigants. Since the diffusion time scale is $\tau_{d}\,{\propto}\,\eta_{c}^{2}$, knowledge of the probability density Q(ηc, Re) is crucial for evaluation of mixing times and chemical reaction rates. According to Kolmogorov–Batchelor phenomenology, the stretching time τeddy ≈ L/urms = O(1) is independent of large-scale Reynolds number Re = urmsL/ν and the diffusion time $\tau_{d}\,{\approx}\,\tau_{\it eddy}/\sqrt{{\it Re}}\,{\ll}\, \tau_{\it eddy}$ is very small. Therefore, in previous studies, molecular diffusion was frequently neglected as being too fast to contribute substantially to the reaction rates. In this paper, taking into account strong intermittent fluctuations of the scalar dissipation scales, this conclusion is re-examined. We derive the probability density Q(ηc, Re, Sc), calculate the mean scalar dissipation scale and predict transition in the reaction rate behaviour from ${\cal R}\,{\propto}\,\sqrt{Re}$ ($Re\,{\leq}\, 10^{3}-10^{4})$ to the high-Re asymptotics ${\cal R}\,{\propto}\, {\it Re}^{0}$. These conclusions are compared with known experimental and numerical data.

The three-dimensional evolution of a plane mixing layer: pairing and transition to turbulence

1993 ◽  
Vol 247 ◽  
pp. 275-320 ◽  
Author(s):  
Robert D. Moser ◽  
Michael M. Rogers
Keyword(s):  
Turbulent Flows ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Linear Stability Theory ◽  
Transition To Turbulence ◽  
Thin Sheets ◽  
Mean Velocity ◽  
Rapid Transition ◽  
The Mean ◽  
Simulated Flow

The evolution of three-dimensional temporally evolving plane mixing layers through as many as three pairings has been simulated numerically. All simulations were begun from a few low-wavenumber disturbances, usually derived from linear stability theory, in addition to the mean velocity. Three-dimensional perturbations were used with amplitudes ranging from infinitesimal to large enough to trigger a rapid transition to turbulence. Pairing is found to inhibit the growth of infinitesimal three-dimensional disturbances, and to trigger the transition to turbulence in highly three-dimensional flows. The mechanisms responsible for the growth of three-dimensionality and onset of transition to turbulence are described. The transition to turbulence is accompanied by the formation of thin sheets of spanwise vorticity, which undergo secondary rollups. The post-transitional simulated flow fields exhibit many properties characteristic of turbulent flows.

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.

scholarly journals A Probabilistic-Based Portfolio Resampling Under the Mean-Variance Criterion

2021 ◽  
Vol 6 (1) ◽  
pp. 45-56
Author(s):  
Anmar Al Wakil
Keyword(s):  
Estimation Error ◽  
Confidence Region ◽  
Three Dimensional ◽  
Efficient Frontier ◽  
Parameter Estimates ◽  
Three Dimensional Representation ◽  
The Mean ◽  
Mean Variance ◽  
Portfolio Return ◽  
Global Mean

Abstract An abundant amount of literature has documented the limitations of traditional unconstrained mean-variance optimization and Efficient Frontier (EF) considered as an estimation-error maximization that magnifies errors in parameter estimates. Originally introduced by Michaud (1998), empirical superiority of portfolio resampling supposedly lies in the addressing of parameter uncertainty by averaging forecasts that are based on a large number of bootstrap replications. Nevertheless, averaging over resampled portfolio weights in order to obtain the unique Resampled Efficient Frontier (REF, U.S. patent number 6,003,018) has been documented as a debated statistical procedure. Alternatively, we propose a probabilistic extension of the Michaud resampling that we introduce as the Probabilistic Resampled Efficient Frontier (PREF). The originality of this work lies in addressing the information loss in the REF by proposing a geometrical three-dimensional representation of the PREF in the mean-variance-probability space. Interestingly, this geometrical representation illustrates a confidence region around the naive EF associated to higher probabilities; in particular for simulated Global-Mean-Variance portfolios. Furthermore, the confidence region becomes wider with portfolio return, as is illustrated by the dispersion of simulated Maximum-Mean portfolios.

Electron Holographic Nano-Characterization of Gold Catalyst at Interface

2002 ◽  
Vol 727 ◽  
Author(s):  
S. Ichikawa ◽  
T. Akita ◽  
M. Okumura ◽  
M. Haruta ◽  
K. Tanaka
Keyword(s):  
Contact Angle ◽  
Titanium Oxide ◽  
Gold Catalyst ◽  
Three Dimensional ◽  
High Resolution Electron Microscopy ◽  
Electron Holography ◽  
Gold Particles ◽  
Oxide Support ◽  
The Mean ◽  
A Charge

AbstractThe catalytic properties of nanostructured gold catalyst are known to depend on the size of the gold particles and to be activated when the size decreases to a few nanometers. We investigated the size dependence of the three-dimensional nanostructure on the mean inner potential of gold catalysts supported on titanium oxide using electron holography and high-resolution electron microscopy (HREM). The contact angle of the gold particles on the titanium oxide tended to be over 90° for gold particles with a size of over 5 nm, and below 90° for a size of below 2 nm. This decreasing change in the contact angle (morphology) acts to increase the perimeter and hence the area of the interface between the gold and titanium oxide support, which is considered to be an active site for CO oxidation. The mean inner potential of the gold particles also changed as their size decreased. The value of the inner potential of gold, which is approximately 25 V in bulk state, rose to over 40 V when the size of the gold particles was less than 2 nm. This phenomenon indicates the existence of a charge transfer at the interface between gold and titanium oxide. The 3-D structure change and the inner potential change should be attributed to the specific electronic structure at the interface, owing to both the “nano size effect” and the “hetero-interface effect.”

The Influence of Blade Wakes on the Performance of Interturbine Diffusers

1996 ◽  
Vol 118 (2) ◽  
pp. 347-352 ◽  
Author(s):  
R. G. Dominy ◽  
D. A. Kirkham
Keyword(s):  
Performance Modeling ◽  
Three Dimensional ◽  
Secondary Flows ◽  
Two Dimensional ◽  
Correct Performance ◽  
Analysis Methods ◽  
Flow Factor ◽  
The Mean ◽  
Lp Turbine

Interturbine diffusers provide continuity between HP and LP turbines while diffusing the flow upstream of the LP turbine. Increasing the mean turbine diameter offers the potential advantage of reducing the flow factor in the following stages, leading to increased efficiency. The flows associated with these interturbine diffusers differ from those in simple annular diffusers both as a consequence of their high-curvature S-shaped geometry and of the presence of wakes created by the upstream turbine. It is shown that even the simplest two-dimensional wakes result in significantly modified flows through such ducts. These introduce strong secondary flows demonstrating that fully three-dimensional, viscous analysis methods are essential for correct performance modeling.

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