scholarly journals Growth Rate Exponents of Richtmyer-Meshkov Mixing Layers

2005 ◽  
Vol 73 (3) ◽  
pp. 461-468 ◽  
Author(s):  
Timothy T. Clark ◽  
Ye Zhou
Keyword(s):  
Flow Field ◽  
Power Law ◽  
Mixing Layer ◽  
Power Laws ◽  
Mixing Layers ◽  
Spatial Gradients ◽  
Power Law Exponent ◽  
Virtual Time ◽  
Time Origin ◽  
Density Ratios

The Richtmyer-Meshkov mixing layer is initiated by the passing of a shock over an interface between fluid of differing densities. The energy deposited during the shock passage undergoes a relaxation process during which the fluctuational energy in the flow field decays and the spatial gradients of the flow field decrease in time. This late stage of Richtmyer-Meshkov mixing layers is studied from the viewpoint of self-similarity. Analogies with weakly anisotropic turbulence suggest that both the bubble-side and spike-side widths of the mixing layer should evolve as power-laws in time, with the same power-law exponent and virtual time origin for both sides. The analogy also bounds the power-law exponent between 2∕7 and 1∕2. It is then shown that the assumption of identical power-law exponents for bubbles and spikes yields fits that are in good agreement with experiment at modest density ratios.

Organized large structure in the post-transition mixing layer. Part 1. Experimental evidence

2013 ◽  
Vol 737 ◽  
pp. 466-498 ◽  
Author(s):  
A. D’Ovidio ◽  
C. M. Coats
Keyword(s):  
Large Scale ◽  
Linear Growth ◽  
Initial Conditions ◽  
Mixing Layer ◽  
Mixing Layers ◽  
Continuous Linear ◽  
Transition Flow ◽  
Large Structures ◽  
The Individual ◽  
Density Ratios

AbstractNew flow-visualization experiments on mixing layers of various velocity and density ratios are reported. It is shown that, in mixing layers developing from laminar initial conditions, the familiar mechanism of growth by vortex amalgamation is replaced at the mixing transition by a previously unrecognized mechanism in which the spanwise-coherent large structures individually undergo continuous linear growth. In the organized post-transition flow it is this continuous linear growth of the individual structures that produces the self-similar growth of the mixing-layer thickness, with the occasional interactions between neighbouring structures occurring as a consequence of their growth, not its cause. It is also observed that periods during which the post-transition mixing layer comprises orderly processions of large structures alternate with periods during which no large-scale organization is apparent downstream of the transition location. These two fully turbulent flow states are characterized by different growth rates, entrainment ratios and orientations of the mixing layer relative to the free streams. The implications of these findings are discussed.

scholarly journals Half Way There: Theoretical Considerations for Power Laws and Sticks in Diffusion MRI for Tissue Microstructure

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1871
Author(s):  
Matt G. Hall ◽  
Carson Ingo
Keyword(s):  
Power Law ◽  
Power Laws ◽  
Tissue Microstructure ◽  
Confluent Hypergeometric Functions ◽  
A Value ◽  
Power Law Exponent ◽  
Weighted Magnetic Resonance Imaging ◽  
Diffusion Weighting ◽  
Theoretical Considerations ◽  
Insight Into

In this article, we consider how differing approaches that characterize biological microstructure with diffusion weighted magnetic resonance imaging intersect. Without geometrical boundary assumptions, there are techniques that make use of power law behavior which can be derived from a generalized diffusion equation or intuited heuristically as a time dependent diffusion process. Alternatively, by treating biological microstructure (e.g., myelinated axons) as an amalgam of stick-like geometrical entities, there are approaches that can be derived utilizing convolution-based methods, such as the spherical means technique. Since data acquisition requires that multiple diffusion weighting sensitization conditions or b-values are sampled, this suggests that implicit mutual information may be contained within each technique. The information intersection becomes most apparent when the power law exponent approaches a value of 12, whereby the functional form of the power law converges with the explicit stick-like geometric structure by way of confluent hypergeometric functions. While a value of 12 is useful for the case of solely impermeable fibers, values that diverge from 12 may also reveal deep connections between approaches, and potentially provide insight into the presence of compartmentation, exchange, and permeability within heterogeneous biological microstructures. All together, these disparate approaches provide a unique opportunity to more completely characterize the biological origins of observed changes to the diffusion attenuated signal.

scholarly journals Fame, Obscurity and Power Laws in Music History

2020 ◽  
Vol 14 (3-4) ◽  
pp. 186
Author(s):  
Andrew Gustar
Keyword(s):  
Power Law ◽  
Power Laws ◽  
Music History ◽  
Average Value ◽  
Musical Activity ◽  
Power Law Exponent ◽  
Fundamental Process ◽  
The Common ◽  
Individual Importance ◽  
Power Law Distributions

This paper investigates the processes leading to musical fame or obscurity, whether for composers, performers, or works themselves. It starts from the observation that the patterns of success, across many historical music datasets, follow a similar mathematical relationship known as a power law, often with an exponent approximately equal to two. It presents several simple models which can produce power law distributions. An examination of these models' transience characteristics suggests parallels with some historical music examples, giving clues to the ways that success and obscurity might emerge in practice and the extent to which success might be influenced by inherent musical quality. These models can be seen as manifestations of a more fundamental process resulting from the law of maximum entropy, subject to a constraint on the average value of the logarithm of the success measure. This implies that musical success is a multiplicative quality, and suggests that musical markets operate to strike a balance between familiarity (socio-cultural importance) and novelty (individual importance). The common power law exponent of two is seen to emerge as a consequence of the tendency for musical activity to be spread evenly across the log-success bands.

scholarly journals Power law statistics of force and acoustic emission from a slowly penetrated granular bed

2014 ◽  
Vol 21 (1) ◽  
pp. 1-8 ◽  
Author(s):  
K. Matsuyama ◽  
H. Katsuragi
Keyword(s):  
Acoustic Emission ◽  
Power Law ◽  
Glass Bead ◽  
Power Laws ◽  
Solid Sphere ◽  
Granular Bed ◽  
Power Law Distribution ◽  
Energy Scaling ◽  
Power Law Exponent ◽  
Ae Signal

Abstract. Penetration-resistant force and acoustic emission (AE) from a plunged granular bed are experimentally investigated through their power law distribution forms. An AE sensor is buried in a glass bead bed. Then, the bed is slowly penetrated by a solid sphere. During the penetration, the resistant force exerted on the sphere and the AE signal are measured. The resistant force shows power law relation to the penetration depth. The power law exponent is independent of the penetration speed, while it seems to depend on the container's size. For the AE signal, we find that the size distribution of AE events obeys power laws. The power law exponent depends on grain size. Using the energy scaling, the experimentally observed power law exponents are discussed and compared to the Gutenberg–Richter (GR) law.

The Simulation of Mixing Layers Driven by Compound Buoyancy and Shear

1996 ◽  
Vol 118 (2) ◽  
pp. 370-376 ◽  
Author(s):  
D. M. Snider ◽  
M. J. Andrews
Keyword(s):  
Growth Rate ◽  
Flow Field ◽  
Numerical Solution ◽  
Mixing Layer ◽  
Measured Data ◽  
Mixing Layers ◽  
Two Dimensional ◽  
One Dimensional ◽  
Self Similar ◽  
Accurate Numerical Solution

Fully developed compound shear and buoyancy driven mixing layers are predicted using a k-ε turbulence model. Such mixing layers present an exchange of equilibrium in mixing flows. The k-ε buoyancy constant Cε3 = 0.91, defined in this study for buoyancy unstable mixing layers, is based on an approximate self-similar analysis and an accurate numerical solution. One-dimensional transient and two-dimensional steady calculations are presented for buoyancy driven mixing in a uniform flow field. Two-dimensional steady calculations are presented for compound shear and buoyancy driven mixing. The computed results for buoyancy alone and compound shear and buoyancy mixing compare well with measured data. Adding shear to an unstable buoyancy mixing layer does not increase the mixing growth rate compared with that from buoyancy alone. The nonmechanistic k-ε model which balances energy generation and dissipation using constants from canonical shear and buoyancy studies predicts the suppression of the compound mixing width. Experimental observations suggest that a reduction in growth rate results from unequal stream velocities that skew and stretch the normally vertical buoyancy plumes producing a reduced mixing envelope width.

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

Regional variation of recession flow power-law exponent

2018 ◽  
Vol 32 (7) ◽  
pp. 866-872 ◽  
Author(s):  
Swagat Patnaik ◽  
Basudev Biswal ◽  
Dasika Nagesh Kumar ◽  
Bellie Sivakumar
Keyword(s):  
Power Law ◽  
Regional Variation ◽  
Power Law Exponent ◽  
Flow Power

scholarly journals Power Laws in Economics: An Introduction

2016 ◽  
Vol 30 (1) ◽  
pp. 185-206 ◽  
Author(s):  
Xavier Gabaix
Keyword(s):  
Stock Market ◽  
Power Law ◽  
Power Laws ◽  
Economic Fluctuations ◽  
Formal Definition ◽  
Empirical Power

Many of the insights of economics seem to be qualitative, with many fewer reliable quantitative laws. However a series of power laws in economics do count as true and nontrivial quantitative laws—and they are not only established empirically, but also understood theoretically. I will start by providing several illustrations of empirical power laws having to do with patterns involving cities, firms, and the stock market. I summarize some of the theoretical explanations that have been proposed. I suggest that power laws help us explain many economic phenomena, including aggregate economic fluctuations. I hope to clarify why power laws are so special, and to demonstrate their utility. In conclusion, I list some power-law-related economic enigmas that demand further exploration. A formal definition may be useful.

scholarly journals Free Convective MHD Flow Past a Vertical Cone with Variable Heat and Mass Flux

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
J. Prakash ◽  
S. Gouse Mohiddin ◽  
S. Vijaya Kumar Varma
Keyword(s):  
Boundary Layer ◽  
Power Law ◽  
Mass Flux ◽  
Numerical Study ◽  
Convection Boundary Layer ◽  
Vertical Cone ◽  
Local Skin ◽  
Surface Mass ◽  
Flow Past ◽  
Power Law Exponent

A numerical study of buoyancy-driven unsteady natural convection boundary layer flow past a vertical cone embedded in a non-Darcian isotropic porous regime with transverse magnetic field applied normal to the surface is considered. The heat and mass flux at the surface of the cone is modeled as a power law according to qwx=xm and qw*(x)=xm, respectively, where x denotes the coordinate along the slant face of the cone. Both Darcian drag and Forchheimer quadratic porous impedance are incorporated into the two-dimensional viscous flow model. The transient boundary layer equations are then nondimensionalized and solved by the Crank-Nicolson implicit difference method. The velocity, temperature, and concentration fields have been studied for the effect of Grashof number, Darcy number, Forchheimer number, Prandtl number, surface heat flux power-law exponent (m), surface mass flux power-law exponent (n), Schmidt number, buoyancy ratio parameter, and semivertical angle of the cone. Present results for selected variables for the purely fluid regime are compared with the published results and are found to be in excellent agreement. The local skin friction, Nusselt number, and Sherwood number are also analyzed graphically. The study finds important applications in geophysical heat transfer, industrial manufacturing processes, and hybrid solar energy systems.

The mixing layer over a deep cavity at high-subsonic speed

2003 ◽  
Vol 475 ◽  
pp. 101-145 ◽  
Author(s):  
NICOLAS FORESTIER ◽  
LAURENT JACQUIN ◽  
PHILIPPE GEFFROY
Keyword(s):  
High Speed ◽  
Mixing Layer ◽  
Interaction Mechanism ◽  
Boundary Layer Separation ◽  
Mixing Layers ◽  
Deep Cavity ◽  
Collective Interaction ◽  
Flow Over A Cavity ◽  
Conditional Statistics ◽  
Forced Mixing

The flow over a cavity at a Mach number 0.8 is considered. The cavity is deep with an aspect ratio (length over depth) L/D = 0.42. This deep cavity flow exhibits several features that makes it different from shallower cavities. It is subjected to very regular self-sustained oscillations with a highly two-dimensional and periodic organization of the mixing layer over the cavity. This is revealed by means of a high-speed schlieren technique. Analysis of pressure signals shows that the first tone mode is the strongest, the others being close to harmonics. This departs from shallower cavity flows where the tones are usually predicted well by the standard Rossiter’s model. A two-component laser-Doppler velocimetry system is also used to characterize the phase-averaged properties of the flow. It is shown that the formation of coherent vortices in the region close to the boundary layer separation has some resemblance to the ‘collective interaction mechanism’ introduced by Ho & Huang (1982) to describe mixing layers subjected to strong sub-harmonic forcing. Otherwise, the conditional statistics show close similarities with those found in classical forced mixing layers except for the production of random perturbations, which reaches a maximum in the structure centres, not in the hyperbolic regions with which turbulence production is usually associated. An attempt is made to relate this difference to the elliptic instability that may be observed here thanks to the particularly well-organized nature of the flow.

Sign in / Sign up

Export Citation Format

Share Document