Landau Damping and Entropy

1967 ◽  
Vol 22 (11) ◽  
pp. 1682-1689
Author(s):  
F. Winterberg
Keyword(s):  
Velocity Fluctuation ◽  
Natural Generalization ◽  
Landau Damping ◽  
Mean Square ◽  
Velocity Fluctuations ◽  
Equilibrium Configurations ◽  
The Mean ◽  
Definition Of ◽  
Non Equilibrium

It is shown that the well-known LANDAU-damping entropy paradox can be resolved by considering a macroscopic entropy as defined by CLAUSIUS instead of the microscopic entropy defined by BOLTZMΛNN. Although both entropy definitions are identical for equilibrium configurations, the same in general is not true for non-equilibrium configurations. To extend the entropy definition of CLAUSIUS to non-equilibrium configurations requires a generalization of the concept of temperature to nonequilibrium configurations. A natural generalization of the concept of temperature to non-equilibrium configurations is to put the temperature proportional to the mean square velocity fluctuation. In this way a macroscopic entropy can be defined which is then proportional to the logarithm of the mean square velocity fluctuation. In the case of LANDAU damping it can be shown that two states having the same statistical permutability and thus the same microscopic entropy may have different mean square velocity fluctuations. One should therefore consider, besides the microscopic disorder defined by BOLTZMANN, which is proportional to the logarithm of the permutability, a macroscopic disorder which is proportional to the logarithm of the mean square velocity fluctuation.In the case of LANDAU damping the microscopic entropy does not change with time; the macroscopic entropy, however, increases steadily with time.

Larger-Eddy Simulation of Velocity Fluctuation in a Mixing T-Junction

2012 ◽  
Vol 152-154 ◽  
pp. 1313-1318
Author(s):  
Tao Lu ◽  
Su Mei Liu ◽  
Ping Wang ◽  
Wei Yyu Zhu
Keyword(s):  
Experimental Data ◽  
Velocity Fluctuation ◽  
Flow Model ◽  
Numerical Results ◽  
Mean Square ◽  
Main Duct ◽  
Velocity Fluctuations ◽  
Eddy Simulation ◽  
Large Eddy ◽  
Les Model

Velocity fluctuations in a mixing T-junction were simulated in FLUENT using large-eddy simulation (LES) turbulent flow model with sub-grid scale (SGS) Smagorinsky–Lilly (SL) model. The normalized mean and root mean square velocities are used to describe the time-averaged velocities and the velocities fluctuation intensities. Comparison of the numerical results with experimental data shows that the LES model is valid for predicting the flow of mixing in a T-junction junction. The numerical results reveal the velocity distributions and fluctuations are basically symmetrical and the fluctuation at the upstream of the downstream of the main duct is stronger than that at the downstream of the downstream of the main duct.

scholarly journals A Comparison Theorem for Stability of Linear Stochastic Implicit Difference Equations of Index-1

2020 ◽  
Vol 36 (3) ◽  
Author(s):  
Nguyen Hong Son
Keyword(s):  
Comparison Theorem ◽  
Difference Equations ◽  
Mean Square ◽  
Mean Square Stability ◽  
Method Of Solution ◽  
The Mean ◽  
Definition Of

In this paper we study linear stochastic implicit difference equations (LSIDEs for short) of index-1. We give a definition of solution and introduce an index-1 concept for these equations. The mean square stability of LSIDEs is studied by using the method of solution evaluation. An example is given to illustrate the obtained results.

Comparison of Acoustic Velocity Perturbation Measurements Using PIV vs. Two-Microphone Technique

2013 ◽  
Author(s):  
Saravanan Balusamy ◽  
Simone Hochgreb
Keyword(s):  
Velocity Fluctuation ◽  
Transfer Functions ◽  
Reference Value ◽  
Acoustic Velocity ◽  
Temperature Phase ◽  
Velocity Perturbation ◽  
Mean Velocity ◽  
Velocity Fluctuations ◽  
Piv Measurements ◽  
The Mean

Understanding combustion instabilities requires accurate measurements of the acoustic velocity perturbation into injectors. This is often accomplished via the use of the two microphone technique, as this only requires two pressure transducers. However, measurements of the actual velocities emerging from the injectors are not often taken, leaving questions regarding the assumptions about the acoustic velocity. A comparison of velocity measured at downstream of the injector with that of two-microphone technique can show the accuracy and limitations of two-microphone technique. In this paper, velocity measurements are taken using both particle image velocimetry (PIV) and the two-microphone technique in a high pressure facility designed for aeroengine injector measurements. The flow is excited using an area modulation device installed on the choked end of the combustion chamber, with PIV measurements enabled by optical access downstream of the injector through a quartz tube and windows. Acoustic velocity perturbations at the injector are determined by considering the Fourier transformed pressure fluctuations for two microphones installed at a known distance upstream of the injector. PIV measurements are realized by seeding the air flow with micrometric water particles under 2.5 bar pressure at ambient temperature. Phase locked velocity fields are realized by synchronizing the acquisition of PIV images with the revolution of the acoustic modulator using the pressure signal measured at the face of injector. The mean velocity fluctuation is calculated as the difference between maximum and minimum velocities, normalized by the mean velocity of the unforced case. Those values are compared with the peak-to-peak velocity fluctuation amplitude calculated by the two-microphone technique. Although the ranges of velocity fluctuations for both techniques are similar, the variation of fluctuation with forcing frequencies diverges significantly with frequency. The differences can be attributed to several limitations associated with of both techniques, such as the quality of the signal, the signal/noise ratio, the accuracy of PIV measurements and the assumption of isentropic flow of the particle velocity from the plenum through the injector. We conclude that two-microphone methods can be used as a reference value for the velocity fluctuation in low order applications such as flame transfer functions, but not for drawing conclusions regarding the absolute velocity fluctuations in the injector.

Material’s Chemical Reaction Progression Variables Fluctuating and Detailed Equilibrium Principle

2012 ◽  
Vol 424-425 ◽  
pp. 861-864
Author(s):  
Qing Hua Zou
Keyword(s):  
Chemical Reaction ◽  
Equilibrium Equation ◽  
Internal Variable ◽  
Local System ◽  
Mean Square ◽  
Mean Square Deviation ◽  
Variable Chemical ◽  
The Mean ◽  
The Stability ◽  
Non Equilibrium

The paper derived and gave the mean square deviation formula relating to the internal variable (chemical reaction progression variable) fluctuating of the linear non–equilibrium local system, and also rendered the condition of the stability. Furthermore, It discussed the related relations among the corresponding mean square deviation, the detailed equilibrium equation, the relax time and other thermal mechanics variables

Particle dynamics in the channel flow of a turbulent particle–gas suspension at high Stokes number. Part 2. Comparison of fluctuating force simulations and experiments

2011 ◽  
Vol 687 ◽  
pp. 41-71 ◽  
Author(s):  
Partha S. Goswami ◽  
V. Kumaran
Keyword(s):  
Relaxation Time ◽  
Particle Velocity ◽  
Volume Fraction ◽  
Quantitative Agreement ◽  
Mean Square ◽  
Mass Loading ◽  
Mean Velocity ◽  
Velocity Fluctuations ◽  
The Mean ◽  
Viscous Relaxation

AbstractThe particle and fluid velocity fluctuations in a turbulent gas–particle suspension are studied experimentally using two-dimensional particle image velocimetry with the objective of comparing the experiments with the predictions of fluctuating force simulations. Since the fluctuating force simulations employ force distributions which do not incorporate the modification of fluid turbulence due to the particles, it is of importance to quantify the turbulence modification in the experiments. For experiments carried out at a low volume fraction of $9. 15\ensuremath{\times} 1{0}^{\ensuremath{-} 5} $ (mass loading is 0.19), where the viscous relaxation time is small compared with the time between collisions, it is found that the gas-phase turbulence is not significantly modified by the presence of particles. Owing to this, quantitative agreement is obtained between the results of experiments and fluctuating force simulations for the mean velocity and the root mean square of the fluctuating velocity, provided that the polydispersity in the particle size is incorporated in the simulations. This is because the polydispersity results in a variation in the terminal velocity of the particles which could induce collisions and generate fluctuations; this mechanism is absent if all of the particles are of equal size. It is found that there is some variation in the particle mean velocity very close to the wall depending on the wall-collision model used in the simulations, and agreement with experiments is obtained only when the tangential wall–particle coefficient of restitution is 0.7. The mean particle velocity is in quantitative agreement for locations more than 10 wall units from the wall of the channel. However, there are systematic differences between the simulations and theory for the particle concentrations, possibly due to inadequate control over the particle feeding at the entrance. The particle velocity distributions are compared both at the centre of the channel and near the wall, and the shape of the distribution function near the wall obtained in experiments is accurately predicted by the simulations. At the centre, there is some discrepancy between simulations and experiment for the distribution of the fluctuating velocity in the flow direction, where the simulations predict a bi-modal distribution whereas only a single maximum is observed in the experiments, although both distributions are skewed towards negative fluctuating velocities. At a much higher particle mass loading of 1.7, where the time between collisions is smaller than the viscous relaxation time, there is a significant increase in the turbulent velocity fluctuations by ${\ensuremath{\sim} }1$–2 orders of magnitude. Therefore, it becomes necessary to incorporate the modified fluid-phase intensity in the fluctuating force simulation; with this modification, the mean and mean-square fluctuating velocities are within 20–30 % of the experimental values.

scholarly journals ASSESSMENT OF INFLUENCE OF RANGE FINDERS WITH THE IMPROVED CHARACTERISTICS ON THE ACCURACY OF GEOMETRICAL PARAMETERS CONTROL OF VEHICLES

2018 ◽  
Vol 15 (4) ◽  
pp. 526-537
Author(s):  
A. M. Askhabov ◽  
E. S. Voevodin ◽  
V. A. Zeer ◽  
A. S. Kashura
Keyword(s):  
Mean Square Error ◽  
Geometrical Parameters ◽  
Control Points ◽  
Mean Square ◽  
Measuring Systems ◽  
Laser Range ◽  
Laser Range Finders ◽  
The Mean ◽  
Definition Of ◽  
Error Of Measurement

Introduction. The purpose of the research is the determination of parameters and characteristics of the laser measuring systems providing decrease in errors of measurement of geometrical parameters of vehicles.The goal is achieved at the expense of the solution of the relevant task connected with the parameters research of the laser equipment and definition of requirements to accuracy (a passport error) of the laser range finders applied at geometrical parameters control of vehicles.Methods and materials. The algorithm of calculation of spatial coordinates of the car points and errors of their definition has been created and used for the research. In this case calculations of spatial provision of the point and its error have been reduced to definition and to the solution of the equations system of the second order in the MathCAD program environment. As the controlled and measured parameters at coordinates’ calculation of distances from laser measuring instruments to the studied car points were accepted.Results. As aresult, the mean square error of measurement of distance between control points significantly decreases at reduction of the passport error of laser range finders. At the size of the passport error of laser range finders equals 0,8 mm, the mean square error of measurement of distance between control points (from 0 to 3000 mm) equals 2,2-2,9 mm, and an error of definition of the control point is 1,5-1,9 mm.Discussion and conclusion. The conducted research demonstrates that it is necessary to compare the reached accuracy parameters at the set configuration of laser measuring system with the shown standard restrictions for an error of measurements. Therefore, further improvement of characteristics of accuracy of measuring systems is possible due to progressive decrease in an error of laser range finders as a result of their technical improvement.

scholarly journals Space-time fractional stochastic equations on regular bounded open domains

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Vo V. Anh ◽  
Nikolai N. Leonenko ◽  
María D. Ruiz-Medina
Keyword(s):  
Evolution Equations ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Weak Sense ◽  
Laplacian Operator ◽  
Mean Square ◽  
Fractional Polynomial ◽  
The Mean ◽  
Negative Laplacian ◽  
Definition Of

AbstractFractional (in time and in space) evolution equations defined on Dirichlet regular bounded open domains, driven by fractional integrated in time Gaussian spatiotemporal white noise, are considered here. Sufficient conditions for the definition of a weak-sense Gaussian solution, in the mean-square sense, are derived. The temporal, spatial and spatiotemporal Hölder continuity, in the mean-square sense, of the formulated solution is obtained, under suitable conditions, from the asymptotic properties of the Mittag-Leffler function, and the asymptotic order of the eigenvalues of a fractional polynomial of the Dirichlet negative Laplacian operator on such bounded open domains.

Fluctuating of Chemical Reaction Progression Variable

2011 ◽  
Vol 347-353 ◽  
pp. 927-930
Author(s):  
Qing Hua Zou
Keyword(s):  
Chemical Reaction ◽  
Equilibrium Equation ◽  
Internal Variable ◽  
Local System ◽  
Mean Square ◽  
Mean Square Deviation ◽  
Variable Chemical ◽  
The Mean ◽  
The Stability ◽  
Non Equilibrium

Abstract: The paper derived and gave the mean square deviation formula relating to the internal variable (chemical reaction progression variable) fluctuating of the linear non–equilibrium local system, and also rendered the condition of the stability. Furthermore, It discussed the related relations among the corresponding mean square deviation, the detailed equilibrium equation, the relax time and other thermal mechanics variables.

Turbulent Heat Transport in Circular Ducts With Circumferentially Varying Heat Flux

1984 ◽  
Vol 106 (1) ◽  
pp. 64-70 ◽  
Author(s):  
J. W. Baughn ◽  
M. A. Hoffman ◽  
B. E. Launder ◽  
R. K. Takahashi
Keyword(s):  
Heat Flux ◽  
Heat Transport ◽  
Numerical Solutions ◽  
Experimental Results ◽  
Turbulent Heat ◽  
Mean Square ◽  
Velocity Fluctuations ◽  
The Mean ◽  
Turbulent Air Flow ◽  
Energy Equations

Experiments have been performed with turbulent air flow in a tube with uniform and with top-half heating; the flow was hydrodynamically fully developed at the start of the heated section, and there were negligible buoyancy effects. Numerical solutions of the mean momentum and energy equations which use the suggestion [1] that the ratio of radial-to-circumferential turbulent diffusivities is equal to the corresponding ratio of mean square velocity fluctuations in these directions agree well with the experimental results.

The Bordeaux Automatic Meridian Circle

1978 ◽  
Vol 48 ◽  
pp. 227-228
Author(s):  
Y. Requième
Keyword(s):  
Mean Square Error ◽  
Mean Square ◽  
Meridian Circle ◽  
The Mean ◽  
Full Automation

In spite of important delays in the initial planning, the full automation of the Bordeaux meridian circle is progressing well and will be ready for regular observations by the middle of the next year. It is expected that the mean square error for one observation will be about ±0.”10 in the two coordinates for declinations up to 87°.

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