Similarity and self-preservation in isotropic turbulence

1951 ◽  
Vol 243 (867) ◽  
pp. 359-386 ◽  
Keyword(s):  
Energy Spectrum ◽  
Isotropic Turbulence ◽  
Initial Period ◽  
Reynolds Numbers ◽  
Wave Number ◽  
Inertial Subrange ◽  
Local Similarity ◽  
Mean Flow ◽  
Spectrum Function ◽  
Energy Spectrum Function

Measurements of the double and triple velocity correlation functions and of the energy spectrum function have been made in the uniform mean flow behind turbulence-producing grids of several shapes at mesh Reynolds numbers between 2000 and 100000. These results have been used to assess the validity of the various theories which postulate greater or less degrees of similarity or self-preservation between decaying fields of isotropic turbulence. It is shown that the conditions for the existence of the local similarity considered by Kolmogoroff and others are only fulfilled for extremely small eddies at ordinary Reynolds numbers, and that the inertial subrange in which the spectrum function varies as k -35 ( k is the wave-number) is non-existent under laboratory conditions. Within the range of local similarity, the spectrum function is best represented by an empirical function such as k -a log k , and it is concluded that all suggested forms for the inertial transfer term in the spectrum equation are in error. Similarity of the large scale structure of flows of differing Reynolds numbers at corresponding times of decay has been confirmed, and approximate measurements of the Loitsianski invariant in the initial period have been made. Its value, expressed non-dimensionally, decreases slowly with grid Reynolds number within the range of observation. Turbulence-producing grids of widely different shapes are found to produce flows identical in energy decay and in structure of the smaller eddies. The largest eddies depend markedly on the grid shape and are, in general, significantly anisotropic. Within the initial period of decay, the greater part of the energy spectrum function is self-preserving, and this part has a shape independent of the shape of the turbulence-producing grid. The part that is not self-preserving contains at least one-third of the total energy, and it is concluded that theories postulating quasi-equilibrium during decay must be considered with great caution.

The multiple-scale cumulant expansion for isotropic turbulence

1978 ◽  
Vol 85 (1) ◽  
pp. 97-142 ◽  
Author(s):  
Tomomasa Tatsumi ◽  
Shigeo Kida ◽  
Jiro Mizushima
Keyword(s):  
Reynolds Number ◽  
Energy Spectrum ◽  
Isotropic Turbulence ◽  
Numerical Data ◽  
Multiple Scale ◽  
Reynolds Numbers ◽  
Inertial Subrange ◽  
Universal Spectrum ◽  
Velocity Derivative ◽  
Wavenumber Region

A method of multiple-scale expansion is applied to the theory of incompressible isotropic turbulence in order to close the infinite system of cumulant equations. The dynamical equation for the energy spectrum derived from this method is found to give positive-definite solutions at all Reynolds numbers. At large Reynolds numbers the spectrum takes the form of Kolmogorov's$-\frac{5}{3}$power spectrum in the inertial subrange, whose extent increases indefinitely with Reynolds number. The spectrum in the energy-containing range satisfies an inviscid similarity law, so that the rate of energy decay or of viscous dissipation is also independent of the viscosity. In the higher wavenumber region beyond the inertial subrange the spectrum takes a universal form which is independent of its structure at lower wavenumbers. The universal spectrum is composed of three different subspectra, which are, in order of increasing wavenumber, the$k^{-\frac{5}{3}}$spectrum, thek−1spectrum and the exp [−σk1·5] spectrum, σ being a constant. Various statistical quantities such as the energy, the skewness of the velocity derivative, the microscale and the microscale Reynolds number are calculated from the numerical data for the energy spectrum. Theoretical results are discussed in detail in comparison with experimental results.

The modified cuinulant expansion for two-dimensional isotropic turbulence

1981 ◽  
Vol 110 ◽  
pp. 475-496 ◽  
Author(s):  
Tomomasa Tatsumi ◽  
Shinichiro Yanase
Keyword(s):  
Numerical Solutions ◽  
Isotropic Turbulence ◽  
Initial Period ◽  
Three Dimensional ◽  
Critical Time ◽  
Reynolds Numbers ◽  
Inviscid Limit ◽  
Inertial Subrange ◽  
Quasi Equilibrium ◽  
Two Dimensional

The two-dimensional isotropic turbulence in an incompressible fluid is investigated using the modified zero fourth-order cumulant approximation. The dynamical equation for the energy spectrum obtained under this approximation is solved numerically and the similarity laws governing the solution in the energy-containing and enstrophy-dissipation ranges are derived analytically. At large Reynolds numbers the numerical solutions yield the k−3 inertial subrange spectrum which was predicted by Kraichnan (1967), Leith (1968) and Batchelor (1969) assuming a finite enstrophy dissipation in the inviscid limit. The energy-containing range is found to satisfy an inviscid similarity while the enstrophy-dissipation range is governed by the quasi-equilibrium similarity with respect to the enstrophy dissipation as proposed by Batchelor (1969). There exists a critical time tc which separates the initial period (t < tc) and the similarity period (t > tc) in which the enstrophy dissipation vanishes and remains non-zero respectively in the inviscid limit. Unlike the case of three-dimensional turbulence, tc is not fixed but increases indefinitely as the viscosity tends to zero.

The theory of decay process of incompressible isotropic turbulence

1957 ◽  
Vol 239 (1216) ◽  
pp. 16-45 ◽  
Keyword(s):  
Energy Spectrum ◽  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Initial Period ◽  
Wave Number ◽  
Power Series Solution ◽  
Initial State ◽  
Dynamical Equations ◽  
Present Solution ◽  
Initial Spectrum

One of the most serious difficulties in the theory of homogeneous turbulence is the indeterminacy of the equation for the velocity correlation function of any order, each involving the correlation of higher-by-one order. In the present paper this difficulty is resolved by treating the two dynamical equations for the second- and third-order velocity correlations, and by introducing the assumption of the zero fourth-order cumulant of the velocity field which yields a relationship between the fourth- and second-order velocity correlations. Actual calculation, however, is carried out in the wave-number space, and a pair of simultaneous equations for the energy spectrum function are derived in part I. Another difficulty of the subject arises from the present lack of knowledge about the initial state of turbulence. In part II, some probable initial conditions for the energy spectrum are examined, among which the initial spectrum of single-line type is chosen as the most suitable for the present problem and its dynamical consequences are fully discussed. The power-series solution for the initial spectrum as well as the energy decay law due to it are computed and compared with experimental data. It is found that the solution, in so far as the approximate expression calculated in the present paper is concerned, corresponds to the earlier initial period of decay. A solution which would be essentially in agreement with experiments is expected to be given by extending the present solution to the further developed stage of decay.

Some Aspects of Gas-Solid Suspension Turbulence

1971 ◽  
Vol 93 (4) ◽  
pp. 631-635 ◽  
Author(s):  
P. S. H. Baw ◽  
R. L. Peskin
Keyword(s):  
Energy Spectrum ◽  
Continuous Medium ◽  
Solid Phase ◽  
Particle Energy ◽  
Solid Particles ◽  
Suspension Flow ◽  
Wave Number ◽  
Spectrum Function ◽  
Solid Suspension ◽  
Energy Spectrum Function

An analysis is given for the apparent particle energy spectrum function in addition to the analysis of the effect of particles on the fluid energy spectrum in a turbulent gas-solid suspension flow. The analysis assumes that the problem of the motion of a continuous medium containing solid particles can be treated as two interacting continuous media, namely, the gas—and the solid—phase. The results obtained show that upon introduction of the particles the energy density of the fluid decreases more rapidly than for the case of pure fluid as wave number increases.

RESEARCH OF EXTERNAL MASS TRANSFER PROCESSES FOR ADSORPTION FROM SOLUTIONS IN A APPARATUS WITH STIRRING

2021 ◽  
pp. 11-20
Author(s):  
V. Solovej ◽  
K. Gorbunov ◽  
V. Vereshchak ◽  
O. Gorbunova
Keyword(s):  
Mass Transfer ◽  
Energy Spectrum ◽  
Energy Dissipation ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Wave Number ◽  
Wave Form ◽  
Local Isotropy ◽  
Stirred Vessel ◽  
Almost All

A study has been mode of transport-controlled mass transfer-controlled to particles suspended in a stirred vessel. The motion of particle in a fluid was examined and a method of predicting relative velocities in terms of Kolmogoroff’s theory of local isotropic turbulence for mass transfer was outlined. To provide a more concrete visualization of complex wave form of turbulence, the concepts of eddies, of eddy velocity, scale (or wave number) and energy spectrum, have proved convenient. Large scale motions of scale contain almost all of the energy and they are directly responsible for energy diffusion throughout the stirring vessel by kinetic and pressure energies. However, almost no energy is dissipated by the large-scale energy-containing eddies. A scale of motion less than is responsible for convective energy transfer to even smaller eddy sires. At still smaller eddy scales, close to a characteristic microscale, both viscous energy dissipation and convection are the rule. The last range of eddies has been termed the universal equilibrium range. It has been further divided into a low eddy size region, the viscous dissipation subrange, and a larger eddy size region, the inertial convection subrange. Measurements of energy spectrum in mixing vessel are shown that there is a range, where the so called -(5/3) power law is effective. Accordingly, the theory of local isotropy of Kolmogoroff can be applied because existence of the internal subrange. As the integrated value of local energy dissipation rate agrees with the power per unit mass of liquid from the impeller, almost all energy from the impeller is viscous dissipated in eddies of microscale. The correlation for mass transfer to particles suspended in a stirred vessel is recommended. The results of experimental study are approximately 12 % above the predicted values.

scholarly journals The role of big eddies in homogeneous turbulence

1949 ◽  
Vol 195 (1043) ◽  
pp. 513-532 ◽  
Keyword(s):  
Fourier Transforms ◽  
Stokes Equations ◽  
Initial Period ◽  
Rate Of Change ◽  
Homogeneous Turbulence ◽  
Wave Number ◽  
Small Range ◽  
Large Wave ◽  
Correlation Tensor ◽  
Spectrum Function

Recent experimental work on the decay of isotropic turbulence has shown that big eddies play an important part in the motion. There is a range of eddy sizes which, during the initial period of decay, contains a negligible proportion of the total energy and is excluded from the similarity possessed by the smaller eddies. This paper examines the motion associated with this small range of large wave-lengths in the more general case of homogeneous turbulence. For this purpose it is convenient to introduce a spectrum tensor, defined as the three-dimensional Fourier transform of the double-velocity correlation tensor. This spectrum function is also suitable for the application of similarity hypotheses, unlike the conventional one-dimensional spectrum function. The properties of the spectrum as a function of the wave-number vector k, are discussed with particular reference to small values of the magnitude k . When k is small the energy per unit interval of wave-number magnitude varies as k 4 . The rate of change of the spectrum function is obtained from the Navier-Stokes equations in terms of Fourier transforms of the triple-velocity and pressure-velocity mean values. After taking into account the continuity condition it is found that the terms of the first and second degree in the expansion of the spectrum function in powers of components of k are constant throughout the decay. The biggest eddies of the turbulence are therefore permanent, being determined wholly by the initial conditions, and are dominant in the final period when the smaller eddies have decayed. The action of smaller eddies on the invariant big eddies is equivalent to that of a turbulent viscosity, the value of which may vary with direction. The implications of the analysis for similarity hypotheses are discussed briefly.

scholarly journals SOME ADDITIONAL SOLUTIONS OF CONFORMAL TURBULENCE

1993 ◽  
Vol 08 (07) ◽  
pp. 619-623 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Energy Spectrum ◽  
Careful Study ◽  
2D Turbulence ◽  
Spectrum Function ◽  
Energy Spectrum Function

We made a careful study of Polyakov’s Diofantian equations for 2D turbulence and found several additional CFTs which meet his criterion. This fact implies that we need further conditions for CFT in order to determine the exponent of the energy spectrum function.

scholarly journals The decay of isotropic magnetohydrodynamics turbulence and the effects of cross-helicity

2018 ◽  
Vol 84 (1) ◽  
Author(s):  
Antoine Briard ◽  
Thomas Gomez
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Energy Spectrum ◽  
Kinetic Energy ◽  
Total Energy ◽  
Isotropic Turbulence ◽  
Reynolds Numbers ◽  
Inertial Range ◽  
Mhd Turbulence ◽  
Previous Prediction

Decaying homogeneous and isotropic magnetohydrodynamics (MHD) turbulence is investigated numerically at large Reynolds numbers thanks to the eddy-damped quasi-normal Markovian (EDQNM) approximation. Without any background mean magnetic field, the total energy spectrum $E$ scales as $k^{-3/2}$ in the inertial range as a consequence of the modelling. Moreover, the total energy is shown, both analytically and numerically, to decay at the same rate as kinetic energy in hydrodynamic isotropic turbulence: this differs from a previous prediction, and thus physical arguments are proposed to reconcile both results. Afterwards, the MHD turbulence is made imbalanced by an initial non-zero cross-helicity. A spectral modelling is developed for the velocity–magnetic correlation in a general homogeneous framework, which reveals that cross-helicity can contain subtle anisotropic effects. In the inertial range, as the Reynolds number increases, the slope of the cross-helical spectrum becomes closer to $k^{-5/3}$ than $k^{-2}$. Furthermore, the Elsässer spectra deviate from $k^{-3/2}$ with cross-helicity at large Reynolds numbers. Regarding the pressure spectrum $E_{P}$, its kinetic and magnetic parts are found to scale with $k^{-2}$ in the inertial range, whereas the part due to cross-helicity rather scales in $k^{-7/3}$. Finally, the two $4/3$rd laws for the total energy and cross-helicity are assessed numerically at large Reynolds numbers.

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