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.

scholarly journals A stochastic view of isotropic turbulence decay

2011 ◽  
Vol 668 ◽  
pp. 351-362 ◽  
Author(s):  
MARCELLO MELDI ◽  
PIERRE SAGAUT ◽  
DIDIER LUCOR
Keyword(s):  
Reynolds Number ◽  
High Reynolds Number ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Response Surfaces ◽  
Turbulence Decay ◽  
Initial Spectrum ◽  
Long Time ◽  
High Reynolds

A stochastic eddy-damped quasi-normal Markovian (EDQNM) approach is used to investigate self-similar decaying isotropic turbulence at a high Reynolds number (400 ≤ Reλ ≤ 104). The realistic energy spectrum functional form recently proposed by Meyers & Menevau (Phys. Fluids, vol. 20, 2008, p. 065109) is generalized by considering some of the model constants as random parameters, since they escape measure in most experimental set-ups. The induced uncertainty on the solution is investigated, building response surfaces for decay power-law exponents of usual physical quantities. Large-scale uncertainties are considered, the emphasis being put on Saffman and Batchelor turbulences. The sensitivity of the solution to initial spectrum uncertainties is quantified through probability density functions of the decay exponents. It is observed that the initial spectrum shape at very large scales governs the long-time evolution, even at a high Reynolds number, a parameter which is not explicitly taken into account in many theoretical works. Therefore, a universal asymptotic behaviour in which kinetic energy decays as t−1 is not detected. However, this decay law is observed at finite Reynolds numbers with low probability for some initial conditions.

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.

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.

The structure of weakly compressible grid-generated turbulence

2001 ◽  
Vol 432 ◽  
pp. 219-283 ◽  
Author(s):  
G. BRIASSULIS ◽  
J. H. AGUI ◽  
Y. ANDREOPOULOS
Keyword(s):  
Mach Number ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Three Dimensional ◽  
High Amplitude ◽  
Time Dependent ◽  
Wide Range ◽  
Dynamical Flow ◽  
Set Up ◽  
Decay Exponent

A decaying compressible nearly homogeneous and nearly isotropic grid-generated turbulent flow has been set up in a large scale shock tube research facility. Experiments have been performed using instrumentation with spatial resolution of the order of 7 to 26 Kolmogorov viscous length scales. A variety of turbulence-generating grids provided a wide range of turbulence scales with bulk flow Mach numbers ranging from 0.3 to 0.6 and turbulent Reynolds numbers up to 700. The decay of Mach number fluctuations was found to follow a power law similar to that describing the decay of incompressible isotropic turbulence. It was also found that the decay coefficient and the decay exponent decrease with increasing Mach number while the virtual origin increases with increasing Mach number. A possible mechanism responsible for these effects appears to be the inherently low growth rate of compressible shear layers emanating from the cylindrical rods of the grid. Measurements of the time-dependent, three dimensional vorticity vectors were attempted for the first time with a 12-wire miniature probe. This also allowed estimates of dilatation, compressible dissipation and dilatational stretching to be obtained. It was found that the fluctuations of these quantities increase with increasing mean Mach number of the flow. The time-dependent signals of enstrophy, vortex stretching/tilting vector and dilatational stretching vector were found to exhibit a rather strong intermittent behaviour which is characterized by high-amplitude bursts with values up to 8 times their r.m.s. within periods of less violent and longer lived events. Several of these bursts are evident in all the signals, suggesting the existence of a dynamical flow phenomenon as a common cause.

Turbulent energy density in scale space for inhomogeneous turbulence

2018 ◽  
Vol 842 ◽  
pp. 532-553 ◽  
Author(s):  
Fujihiro Hamba
Keyword(s):  
Energy Spectrum ◽  
Energy Density ◽  
Isotropic Turbulence ◽  
Scale Space ◽  
Scale Dependence ◽  
Energy Cascade ◽  
Homogeneous Isotropic Turbulence ◽  
Velocity Correlation ◽  
Inhomogeneous Turbulence ◽  
Point Velocity

The energy spectrum is commonly used to describe the scale dependence of the turbulent fluctuations in homogeneous isotropic turbulence. In contrast, one-point statistical quantities, such as the turbulent kinetic energy, are employed for inhomogeneous turbulence modelling. To obtain a better understanding of inhomogeneous turbulence, some attempts have been made to describe its scale dependence by using the second-order structure function and the two-point velocity correlation. However, previous expressions for the energy density in the scale space do not satisfy the requirement that it should be non-negative. In this work, a new expression for the energy density in the scale space is proposed on the basis of the two-point velocity correlation; the integral with a filter function is introduced to satisfy the non-negativity of the energy density. Direct numerical simulation (DNS) data of homogeneous isotropic turbulence were first used to assess the role of the energy density by comparing it with the energy spectrum. DNS data of a turbulent channel flow were then used to investigate the energy density and its transport equation in inhomogeneous turbulence. It was shown that the new energy density is positive in the scale space of the homogeneous direction. The energy transfer was successfully examined in the scale space both in the homogeneous and inhomogeneous directions. The energy cascade from large to small scales was clearly observed. Moreover, the inverse energy cascade from large to very large scales was observed in the scale space of the spanwise direction.

scholarly journals A Lagrangian direct-interaction approximation for homogeneous isotropic turbulence

1997 ◽  
Vol 345 ◽  
pp. 307-345 ◽  
Author(s):  
SHIGEO KIDA ◽  
SUSUMU GOTO
Keyword(s):  
Energy Spectrum ◽  
Differential Equations ◽  
Isotropic Turbulence ◽  
Longitudinal Velocity ◽  
Direct Interaction ◽  
Homogeneous Isotropic Turbulence ◽  
Velocity Correlation ◽  
Stationary Turbulence ◽  
Direct Interaction Approximation ◽  
Velocity Derivative

A set of integro-differential equations in the Lagrangian renormalized approximation (Kaneda 1981) is rederived by applying a perturbation method developed by Kraichnan (1959), which is based upon an extraction of direct interactions among Fourier modes of a velocity field and was applied to the Eulerian velocity correlation and response functions, to the Lagrangian ones for homogeneous isotropic turbulence. The resultant set of integro-differential equations for these functions has no adjustable free parameters. The shape of the energy spectrum function is determined numerically in the universal range for stationary turbulence, and in the whole wavenumber range in a similarly evolving form for the freely decaying case. The energy spectrum in the universal range takes the same shape in both cases, which also agrees excellently with many measurements of various kinds of real turbulence as well as numerical results obtained by Gotoh et al. (1988) for a decaying case as an initial value problem. The skewness factor of the longitudinal velocity derivative is calculated to be −0.66 for stationary turbulence. The wavenumber dependence of the eddy viscosity is also determined.

A consequence of the zero fourth cumulant approximation

1962 ◽  
Vol 13 (3) ◽  
pp. 369-382 ◽  
Author(s):  
Edward E. O'Brien ◽  
George C. Francis
Keyword(s):  
Energy Spectrum ◽  
Numerical Computation ◽  
Root Mean Square ◽  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Passive Scalar ◽  
Turbulent Energy ◽  
Length Scale ◽  
Mean Square ◽  
Stationary Turbulence

Recent investigations by Kraichnan (1961) and Ogura (1961) have raised doubts concerning the usefulness of the zero fourth cumulant approximation in turbulence dynamics. It appears extremely tedious to examine, by numerical computation, the consequences of this approximation on the turbulent energy spectrum although the appropriate equations have been established by Proudman & Reid (1954) and Tatsumi (1957). It has proved possible, however, to compute numerically the sequences of an analogous assumption when applied to an isotropic passive scalar in isotropic turbulence.The result of such computation, for specific initial conditions described herein, and for stationary turbulence, is that the scalar spectrum does develop negative values after a time approximately $2 \Lambda | {\overline {(u^2)}} ^{\frac {1}{2}}$, Where Λ is a length scale typical of the energy-containing components of both the turbulent and scalar spectra and $\overline {(u^2)}^{\frac {1}{2}}$ is the root mean square turbulent velocity.

THE ENERGY CASCADE OF TURBULENCE WITH QUASI-SELF-SIMILARITY

2009 ◽  
Vol 23 (03) ◽  
pp. 513-516 ◽  
Author(s):  
HAO ZHU ◽  
KEMING CHENG
Keyword(s):  
Energy Spectrum ◽  
Turbulent Flows ◽  
Iterated Function System ◽  
Three Dimensional ◽  
Energy Cascade ◽  
Function System ◽  
Self Similarity ◽  
Iterated Function ◽  
Self Similar ◽  
Nonlinear Disturbance

In this article, we investigate the energy cascade of three-dimensional turbulent flows, in which the break-up process of eddy is quasi-self-similar. Mathematically this kind of turbulence with quasi-self-similar structure eddies can be regarded as cookie-cutter system, and can be generated by self-similar iterated function system (IFS) with added nonlinear disturbance. Using Bowen's result, we can calculate the exponent of dissipative correlated function, dissipated velocity, energy spectrum supported on cookie-cutter system. The present results show that the β-model is feasible for this kind of quasi-self-similar turbulence.

scholarly journals The fate of random initial vorticity distributions for two-dimensional Euler equations on a sphere

2015 ◽  
Vol 786 ◽  
pp. 1-4 ◽  
Author(s):  
Paul K. Newton
Keyword(s):  
Numerical Simulations ◽  
Euler Equations ◽  
Large Scale ◽  
Initial Conditions ◽  
Infinite Family ◽  
Small Scale ◽  
Two Dimensional ◽  
Random Initial Conditions ◽  
Long Time ◽  
Initial Vorticity

The paper by Dritschel et al. (J. Fluid Mech., vol. 783, 2015, pp. 1–22) describes the long-time behaviour of inviscid two-dimensional fluid dynamics on the surface of a sphere. At issue is whether the flow settles down to an equilibrium or whether, for generic (random) initial conditions, the long-time solution is periodic, quasi-periodic or chaotic. While it might be surprising that this issue is not settled in the literature, it is important to keep in mind that the Euler equations form a dissipationless Hamiltonian system, hence the set of equations only redistributes the initial vorticity, generating smaller and smaller scales, while keeping kinetic energy, angular impulse and an infinite family of vorticity moments (Casimirs) intact. While special solutions that never settle down to an equilibrium state can be constructed using point vortices, vortex patches and other distributions, the fate of random initial conditions is a trickier problem. Previous statistical theories indicate that the long-time state should be a stationary large-scale distribution of vorticity. By carrying out careful numerical simulations using two different methods, the authors make a compelling case that the generic long-time state resembles a large-scale oscillating quadrupolar vorticity field, surrounded by persistent small-scale vortices. While numerical simulations can never conclusively settle this issue, the results might help guide future theories that seek to prove the existence of such an interesting dynamical long-time state.

Transform/oblique rift system : what have we learned from numerical modelling and what's next ?

2020 ◽  
Author(s):  
Laetitia Le Pourhiet ◽  
Anthony Jourdon
Keyword(s):  
Numerical Modelling ◽  
Large Scale ◽  
Initial Conditions ◽  
Strike Slip ◽  
Rift System ◽  
New Concepts ◽  
Long Time ◽  
3D Numerical Modelling ◽  
Transform Margin ◽  
Conjugate Margins

<p>For very long time, transform margins have been treated and described  based on oceanic transform fault concepts. Their was no change in kinematics nor structures with time and thermally speaking, it was hypothesed that the margin was reheated as the mid-oceanic ridge translated passively along the margin.  In the last 10 years, 3D numerical modelling has been made available and numbers of studies have challenged this view. It is time to review the concepts that have emerge. Interrestingly, many modelling contributions have tackled the obliquity at very different scales, with initial conditions varying from simple flat layered homogeneous lithosphere to subduction of opposite vergence. Moreover some contributions have focus on rheological aspect and other on inheritance at different scale and different physical coupling have been used. Some models were targeting at reproducing the oceanic transform concepts, other at exploring how large scale structure can emerge.  I will therefore try to review  the state of art in numerical modelling of transform margin and oblique extensional system based on my own work and literature review. I will try to emphize the important differences and similarities used in the different modelling. Using different models with different boundary conditions and scale I will try to introduce a new conceptual model of transform margin which captures important characteristics like the delay in continental break-up highlighted by the tracing of sediments and water-depth as well as the obliquity between syn-rift and post-rift subsidence.  Some models of oblique extension have also been producing new type of strike slip ocean continent transition which somehow could be interpreted as steep transform margins but appears to be mainly strike slip and have no conjugate margins. To conclude, all these 3D numerical modelling  allow us today to present a very different view of transform margins than 10 years ago. Some of the new concepts that have emerged  mendate to re assess our interpretation of exisiting datasets.</p>

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