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.

Hybrid Eulerian–Lagrangian simulations for polymer–turbulence interactions

2013 ◽  
Vol 717 ◽  
pp. 535-575 ◽  
Author(s):  
Takeshi Watanabe ◽  
Toshiyuki Gotoh
Keyword(s):  
Energy Spectrum ◽  
Energy Dissipation ◽  
Kinetic Energy ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Polymer Concentration ◽  
Hybrid Approach ◽  
Weissenberg Number ◽  
Polymer Additives ◽  
Kinetic Energy Spectrum

AbstractThe effects of polymer additives on decaying isotropic turbulence are numerically investigated using a hybrid approach consisting of Brownian dynamics simulations for an enormous number of dumbbells (of the order of 10 billion,$O(1{0}^{10} )$) and direct numerical simulations of turbulence making full use of large-scale parallel computations. Reduction of the energy dissipation rate and modification of the kinetic energy spectrum in the dissipation range scale were observed when the reaction term due to the polymer additives was incorporated into the equation of motion for the solvent fluid. An increase in the polymer concentration or Weissenberg number${W}_{i} $yielded significant modifications of the turbulence statistics at small scales, such as a suppression of the local energy dissipation fluctuations. A power-law decay of the kinetic energy spectrum$E(k, t)\sim {k}^{- 4. 7} $was observed in the wavenumber range below the Kolmogorov length scale when${W}_{i} = 25$. The generation of intense vortices was suppressed by the polymer additives, consistent with previous studies using the constitutive equations. The field structures of the trace of the polymer stress depended on the intensity of its fluctuation: sheet-like structures were observed for the intermediate intensity region and filamentary structures were observed for the intense region. The results obtained with few polymers and large replicas could approximate those with many polymers and smaller replicas as far as the large-scale statistics were concerned.

The large-scale structure of homogeneous turbulence

1967 ◽  
Vol 27 (3) ◽  
pp. 581-593 ◽  
Author(s):  
P. G. Saffman
Keyword(s):  
Large Scale ◽  
Isotropic Turbulence ◽  
Homogeneous Turbulence ◽  
Wave Number ◽  
Initial Instant ◽  
Force System ◽  
Correlation Tensor ◽  
Number Magnitude ◽  
Small Wave Number ◽  
Conservation Of Momentum

A field of homogeneous turbulence generated at an initial instant by a distribution of random impulsive forces is considered. The statistical properties of the forces are assumed to be such that the integral moments of the cumulants of the force system all exist. The motion generated has the property that at the initial instant\[ E(\kappa) = C\kappa^2 + o(\kappa^2), \]whereE(k) is the energy spectrum function,kis the wave-number magnitude, andCis a positive number which is not in general zero. The corresponding forms of the velocity covariance spectral tensor and correlation tensor are determined. It is found that the terms in the velocity covarianceRij(r) areO(r−3) for large values of the separation magnituder.An argument based on the conservation of momentum is used to show thatCis a dynamical invariant and that the forms of the velocity covariance at large separation and the spectral tensor at small wave number are likewise invariant. For isotropic turbulence, the Loitsianski integral diverges but the integral\[ \int_0^{\infty} r^2R(r)dr = \frac{1}{2}\pi C \]exists and is invariant.

scholarly journals Anisotropic Characteristics of Turbulence Dissipation in Swirling Flow: A Direct Numerical Simulation Study

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Xingtuan Yang ◽  
Nan Gui ◽  
Gongnan Xie ◽  
Jie Yan ◽  
Jiyuan Tu ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Energy Dissipation ◽  
Direct Numerical Simulation ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Vortex Breakdown ◽  
Swirling Flow ◽  
Turbulent Energy ◽  
Swirling Flows ◽  
Small Scale

This study investigates the anisotropic characteristics of turbulent energy dissipation rate in a rotating jet flow via direct numerical simulation. The turbulent energy dissipation tensor, including its eigenvalues in the swirling flows with different rotating velocities, is analyzed to investigate the anisotropic characteristics of turbulence and dissipation. In addition, the probability density function of the eigenvalues of turbulence dissipation tensor is presented. The isotropic subrange of PDF always exists in swirling flows relevant to small-scale vortex structure. Thus, with remarkable large-scale vortex breakdown, the isotropic subrange of PDF is reduced in strongly swirling flows, and anisotropic energy dissipation is proven to exist in the core region of the vortex breakdown. More specifically, strong anisotropic turbulence dissipation occurs concentratively in the vortex breakdown region, whereas nearly isotropic turbulence dissipation occurs dispersively in the peripheral region of the strong swirling flows.

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.

On non-self-similar regimes in homogeneous isotropic turbulence decay

2012 ◽  
Vol 711 ◽  
pp. 364-393 ◽  
Author(s):  
Marcello Meldi ◽  
Pierre Sagaut
Keyword(s):  
Energy Spectrum ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Self Similarity ◽  
Velocity Correlation ◽  
Turbulence Decay ◽  
Long Time ◽  
Self Similar ◽  
Decay Exponent

AbstractBoth theoretical analysis and eddy-damped quasi-normal Markovian (EDQNM) simulations are carried out to investigate the different decay regimes of an initially non-self-similar isotropic turbulence. Breakdown of self-similarity is due to the consideration of a composite three-range energy spectrum, with two different slopes at scales larger than the integral length scale. It is shown that, depending on the initial conditions, the solution can bifurcate towards a true self-similar decay regime, or sustain a non-self-similar state over an arbitrarily long time. It is observed that these non-self-similar regimes cannot be detected, restricting the observation to time exponents of global quantities such as kinetic energy or dissipation. The actual reason is that the decay is controlled by large scales close to the energy spectrum peak. This theoretical prediction is assessed by a detailed analysis of triadic energy transfers, which show that the largest scales have a negligible impact on the total transfers. Therefore, it is concluded that details of the energy spectrum near the peak, which may be related to the turbulence production mechanisms, are important. Since these mechanisms are certainly not universal, this may at least partially explain the significant discrepancies that exist between experimental data and theoretical predictions. Another conclusion is that classical self-similarity theories, which connect the asymptotic behaviour of either the energy spectrum $E(k\ensuremath{\rightarrow} 0)$ or the velocity correlation function $f(r\ensuremath{\rightarrow} + \infty )$ and the turbulence decay exponent, are not particularly relevant when the large-scale spectrum shape exhibits more than one range.

Advances in the Fundamental Aspects of Turbulence: Energy Transfer, Interacting Scales, and Self-Preservation in Isotropic Decay

1998 ◽  
Vol 51 (4) ◽  
pp. 267-301 ◽  
Author(s):  
Ye Zhou ◽  
Charles G. Speziale
Keyword(s):  
Energy Transfer ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Future Research ◽  
Turbulence Energy ◽  
Physical Processes ◽  
Local Isotropy ◽  
Scale Modeling ◽  
Large Eddy ◽  
Subgrid Scale Modeling

The fundamental aspects of isotropic turbulence are reviewed in order to gain a better insight into the physical processes of turbulence. After first reviewing the Kolmogorov energy spectrum and the energy cascade, the Kolmogorov hypothesis of local isotropy is discussed in depth. Then, the detailed physical processes involving energy transfer and interacting scales in isotropic turbulence, including triad interactions, are reviewed. The inertial range and self-similarity are also discussed along with the response of the small scales to large-scale anisotropy and the final stages of the decay process. Results from direct and large-eddy simulations of isotropic turbulence—including a discussion of subgrid scale modeling—are then discussed in detail to illustrate these points. The article closes with a review of self-preservation in isotropic turbulence and a discussion of the prospects for future research. It contains 155 references.

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.

Time-dependent scaling relations and a cascade model of turbulence

1978 ◽  
Vol 88 (2) ◽  
pp. 369-391 ◽  
Author(s):  
Thomas L. Bell ◽  
Mark Nelkin
Keyword(s):  
Energy Spectrum ◽  
Power Law ◽  
Isotropic Turbulence ◽  
Cascade Model ◽  
Time Dependent ◽  
Wave Number ◽  
Grid Turbulence ◽  
A Value ◽  
Model Equations ◽  
Spatial Dimensionality

We study the time-dependent solutions of a nonlinear cascade model for homogeneous isotropic turbulence first introduced by Novikov & Desnyansky. The dynamical variables of the model are the turbulent kinetic energies in discrete wave-number shells of thickness one octave. The model equations contain a parameter C whose size governs the amount of energy cascaded to small wavenumbers relative to the amount cascaded to large wavenumbers. We show that the equations permit scale-similar evolution of the energy spectrum. For 0 ≤ C ≤ 1 and no external force, the freely evolving energy spectrum displays the Kolmogorov k power law, and the total energy decreases in time as a power t−w, where the exponent w depends on the value of C. Grid-turbulence experiments seem to favour a value of C in the range 0·3-0·6. In the presence of an external stirring force acting near a wavenumber k0, the model predicts, in addition to the Kolmogorov k spectrum for k > k0, a scale-similar flow of energy to wavenumbers k < k0. This backward energy flow falls off as a power law in time, and establishes a stationary energy spectrum for k < k0 which is a power law in k less steep than k. We discuss the similarity of the behaviour of the model for C > 1 to the behaviour of turbulent fluid for a spatial dimensionality near 2. The model is shown to approach the Kovasznay and the Leith diffusion approximation equations in the limit in which the thickness of the wavenumber shells approaches zero. However, the cascade model with finite shell thicknesses appears to behave in a more physically reasonable way than the limiting differential equations.

On the large-scale structure of homogeneous two-dimensional turbulence

2007 ◽  
Vol 580 ◽  
pp. 431-450 ◽  
Author(s):  
P. A. DAVIDSON
Keyword(s):  
Energy Spectrum ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Direct Consequence ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Conservation Of Energy ◽  
Dimensional Version ◽  
Spatial Locations

We consider freely decaying two-dimensional isotropic turbulence. It is usually assumed that, in such turbulence, the energy spectrum at small wavenumber, k, takes the form E(k → 0) ∼ Ik3, where I is the two-dimensional version of Loitsyansky's integral. However, a second possibility is E(k → 0) ∼ Lk, where the pre-factor, L, is the two-dimensional analogue of Saffman's integral. We show that, as in three dimensions, L is an invariant and that E ∼ Lk spectra arise whenever the eddies possess a significant amount of linear impulse. The conservation of L is shown to be a direct consequence of the principle of conservation of linear momentum. We also show that isotropic turbulence dominated by a cloud of randomly located monopole vortices has a singular energy spectrum of the form E(k → 0) ∼ Jk−1, where J, like L, is an invariant. However, while E ∼ Jk−1 necessarily implies the existence of a sea of monopoles, the converse need not be true: a sea of monopoles whose spatial locations are not entirely random, but constrained in some way, need not give a E ∼ Jk−1 spectra. The constraint imposed by the conservation of energy is particularly important, ruling out E ∼ Jk−1 spectra for certain classes of initial conditions. Finally, we provide simple explicit examples of random vorticity fields with E ∼ Ik3, E ∼ Lk and E ∼ Jk−1 spectra.

scholarly journals Interaction of Very Large Scale Motion of Coherent Structures with Sediment Particle Exposure

Water ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 248
Author(s):  
Sencer Yücesan ◽  
Daniel Wildt ◽  
Philipp Gmeiner ◽  
Johannes Schobesberger ◽  
Christoph Hauer ◽  
...  
Keyword(s):  
Coherent Structures ◽  
Large Scale ◽  
Numerical Study ◽  
Local Maximum ◽  
Wave Number ◽  
Exposure Level ◽  
Sediment Particle ◽  
High Wave Number ◽  
Large Scale Motion ◽  
Scale Motion

A systematic variation of the exposure level of a spherical particle in an array of multiple spheres in a high Reynolds number turbulent open-channel flow regime was investigated while using the Large Eddy Simulation method. Our numerical study analysed hydrodynamic conditions of a sediment particle based on three different channel configurations, from full exposure to zero exposure level. Premultiplied spectrum analysis revealed that the effect of very-large-scale motion of coherent structures on the lift force on a fully exposed particle resulted in a bi-modal distribution with a weak low wave number and a local maximum of a high wave number. Lower exposure levels were found to exhibit a uni-modal distribution.

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