The present work studies the isotropic and homogeneous turbulence for incompressible fluids through a specific Lyapunov analysis, assuming that the turbulence is due to the bifurcations associated to the velocity field.
The analysis consists in the study of the mechanism of the energy cascade from large to small scales through the Lyapunov analysis of the relative motion between two particles and in the calculation of the velocity fluctuation through the Lyapunov analysis of the local deformation and the Navier-Stokes equations.
The analysis provides an explanation for the mechanism of the energy cascade, leads to the closure of the von Kármán-Howarth equation, and describes the statistics of the velocity difference.
Several tests and numerical results are presented.
In this work, we calculate the self-similar longitudinal velocity correlation function and the statistical properties of velocity difference using the results of the Lyapunov analysis of the fully developed isotropic homogeneous turbulence just presented by the author in a previous work (arXiv:0911.1463). There, a closure of the von Karman-Howarth equation is proposed and the statistics of velocity difference is determined through a specific analysis of the Fourier-transformed Navier-Stokes equations. The correlation functions correspond to steady-state solutions of the von Karman-Howarth equa…
The present work studies the isotropic and homogeneous turbulence for incompressible fluids through a specific Lyapunov analysis, assuming that the turbulence is due to the bifurcations associated to the velocity field. The analysis consists in the calculation of the velocity fluctuation through the Lyapunov analysis of the local deformation and the Navier-Stokes equations and in the study of the mechanism of the energy cascade from large to small scales through the finite scale Lyapunov analysis of the relative motion between two particles. The analysis provides an explanation for the mechani…
We calculate the self-similar longitudinal velocity correlation function, the energy spectrum and the corresponding other properties using a theory on the isotropic homogeneous turbulence just presented by the author in a previous work. The correlation functions correspond to steady-state solutions of the evolution equation under the self-similarity hypothesis introduced by von Kármán. These solutions are numerically calculated and the results adequately describe several properties of the isotropic turbulence.
This work presents a review of previous articles dealing with an original turbulence theory proposed by the author, and provides new theoretical insights into some related issues. The new theoretical procedures and methodological approaches confirm and corroborate the previous results. These articles study the regime of homogeneous isotropic turbulence for incompressible fluids and propose theoretical approaches based on a specific Lyapunov theory for determining the closures of the von Kármán-Howarth and Corrsin equations, and the statistics of velocity and temperature difference. Furthermore…
Lyapunov exponents measure the average exponential growth rate of typical linear perturbations in a chaotic system, and the inverse of the largest exponent is a measure of the time horizon over which the evolution of the system can be predicted. Here, Lyapunov exponents are determined in forced homogeneous isotropic turbulence for a range of Reynolds numbers. Results show that the maximum exponent increases with Reynolds number faster than the inverse Kolmogorov time scale, suggesting that the instability processes may be acting on length and time scales smaller than Kolmogorov scales. Analysi…
The present work analyzes the distribution function of the finite scale local Lyapunov exponent of a pair fluid particles trajectories in fully developed incompressible homogeneous isotropic turbulence. According to the hypothesis of fully developed chaos, this PDF is reasonably estimated by maximizing the entropy associated to such distribution, resulting to be an uniform distribution function in a proper interval of variation of the local Lyapunov exponents. From this PDF, we determine the relationship between the average and maximum Lyapunov exponents and the longitudinal velocity correlati…
In the context of the analysis of the chaotic properties of homogeneous and isotropic turbulence, direct numerical simulations are used to study the fluctuations of the finite time Lyapunov exponent (FTLE) and its relation to Reynolds number, lattice size and the choice of the steptime used to compute the Lyapunov exponents. The results show that using the FTLE method produces Lyapunov exponents that are remarkably stable under the variation of the steptime and lattice size. Furthermore, it reaches such stability faster than other characteristic quantities such as energy and dissipation rate. …
In order to better understand deviations from equilibrium in turbulent flows, it is meaningful to characterize the dynamics rather than the statistics of turbulence. To this end, the Lyapunov theory provides a useful description of turbulence through the study of the perturbation dynamics. In this work, the Lyapunov spectrum of forced homogeneous isotropic turbulent flows is computed. Using the Lyapunov exponents of a flow at different Reynolds numbers, the scaling of the dimension of the chaotic attractor for a three-dimensional homogeneous isotropic flow (HIT) is obtained for the first time …
The present work proposes a theory of isotropic and homogeneous turbulence for incompressible fluids, which assumes that the turbulence is due to the bifurcations associated to the velocity field. The theory is formulated using a representation of the fluid motion which is more general than the classical Navier-Stokes equations, where the fluid state variables are expressed in terms of the referential coordinates. The theory is developed according to the following four items: 1) Study of the route toward the turbulence through the bifurcations analysis of the kinematic equations. 2) Referentia…
Based on the Karman-Howarth equation in 3D incompressible fluid, a new isotropic turbulence scale evolution equation and its related theory progress. The present results indicate that the energy cascading process has remarkable similarities with the determinisitic construction rules of the isotonic oscillators in quantum mechanics.
Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as 'elastic turbulence' when the Weissenberg number exceeds a critical value. The two-dimensional Oldroyd-B model is the simplest constitutive model that reproduces this phenomenon. To make a practical estimate of the resolution scale of the dynamics requires an assumption that an attractor of the Oldroyd-B model exists : numerical simulations show that the quantities on which this assumption is based are bounded. We estimate the Lyapunov dimension of this assumed attractor as a function of the Weissenberg number by combin…
This article accompanies the submission of a fluid dynamics video (entry 102248) of inertial cloud droplet clustering in isotropic turbulence for the Gallery of Fluid Motion of the 66th Annual Meeting of the American Physical Society, Division of Fluid Dynamics.
A new form of the well known Lin equation for the energy balance in isotropic turbulence is given which takes account of the filtered-partitioned transfer spectra which were used to resolve the scale paradox. It is argued that this new formulation provides a new starting point for both theoretical and numerical methods which may lead to a better understanding of the underlying energy processes in turbulence.
We conduct a careful analysis of the data provided by Krogstad & Davidson (2011) and show that their data do not support their conclusions. According to their published data, their decaying approximately homogeneous isotropic turbulent flows are, invariably, clearly different from Saffman turbulence; and very clearly marked differences exist between the far downstream turbulence behaviors generated by their conventional grid and by their multiscale cross grids.
It is shown that the correlation integrals (invariants) play an important, and at certain conditions a dominant, role in helical isotropic homogeneous turbulence both in the inertial and near-dissipation ranges. Results of direct numerical simulations and recent laboratory experiments with multiscale grids have been used for this purpose. A possibility of spontaneous breaking of reflection symmetry has been also briefly discussed in this context.
The effects of Reynolds number and Stokes number on particle-pair relative velocity (RV) were investigated systematically using a recently developed planar four-frame particle tracking technique in a novel homogeneous and isotropic turbulence chamber.
Using fluctuating hydrodynamics we investigate the effect of thermal fluctuations in the dissipation range of homogeneous, isotropic turbulence. Simulations confirm theoretical predictions that the energy spectrum is dominated by these fluctuations at length scales comparable to the Kolmogorov length. We also find that the extreme intermittency in the far-dissipation range predicted by Kraichnan is replaced by Gaussian thermal equipartition.
The present work analyzes the statistics of finite scale local Lyapunov exponents of pairs of fluid particles trajectories in fully developed incompressible homogeneous isotropic turbulence. According to the hypothesis of fully developed chaos, this statistics is here analyzed assuming that the entropy associated to the fluid kinematic state is maximum. The distribution of the local Lyapunov exponents results to be an unsymmetrical uniform function in a proper interval of variation. From this PDF, we determine the relationship between average and maximum Lyapunov exponents, and the longitudina…
We demonstrate an unexpected connection between isotropic turbulence and wall-bounded shear flows. We perform direct numerical simulations of isotropic turbulence forced at large scales at moderate Reynolds numbers and observe sudden transitions from a chaotic dynamics to a spatially simple flow, analogous to the laminar state in wall bounded shear flows. We find that the survival probabilities of turbulence are exponential and the typical lifetimes increase superexponentially with the Reynolds number. Our results suggest that both isotropic turbulence and wall-bounded shear flows qualitativel…
The Navier-Stokes equations generate an infinite set of generalized Lyapunov exponents defined by different ways of measuring the distance between exponentially diverging perturbed and unperturbed solutions. This set is demonstrated to be similar, yet different, from the generalized Lyapunov exponent that provides moments of distance between two fluid particles below the Kolmogorov scale. We derive rigorous upper bounds on dimensionless Lyapunov exponent of the fluid particles that demonstrate the exponent's decay with Reynolds number $Re$ in accord with previous studies. In contrast, terms of…
It is shown that the distributed chaos in the simple Hamiltonian (conservative) dynamical systems, such as the Nose-Hoover oscillator and double oscillator, can mimic the distributed chaos in the isotropic homogeneous turbulence. Direct numerical simulations with the classic Toda lattice and with the nonlinear Schrödinger equation (soliton turbulence) under random initial conditions have been also discussed in this context. These properties of the distributed chaos have been related to analytical properties of the Hamiltonian systems. Decrease of the smoothness results in the power-law spectra…
Temporal decorrelations in compressible isotropic turbulence are studied using the space-time correlation theory and direct numerical simulation. A swept-wave model is developed for dilatational components while the classic random sweeping model is proposed for solenoidal components. The swept-wave model shows that the temporal decorrelations in dilatational fluctuations are dominated by two physical processes: random sweeping and wave propagation. These models are supported by the direct numerical simulation of compressible isotropic turbulence, in the sense of that all curves of normalized t…
We study the statistical properties of helicity in direct numerical simulations of fully developed homogeneous and isotropic turbulence and in a class of turbulence shell models. We consider correlation functions based on combinations of vorticity and velocity increments that are not invariant under mirror symmetry. We also study the scaling properties of high-order structure functions based on the moments of the velocity increments projected on a subset of modes with either positive or negative helicity (chirality). We show that mirror symmetry is recovered at small-scales, i.e., chiral terms…
The streamwise and transverse velocities are measured simultaneously in grid turbulence at Reynolds numbers 100-300. We extract typical intermittency patterns, which are consistent with velocity profiles of Burgers and Lamb-Oseen vortices. The radii of these vortex tubes are estimated to be several of the Kolmogorov length.
We consider equilibrium statistics for high Reynolds number isotropic turbulence in an incompressible flow driven by steady forcing at the largest scale. Motivated by shell model observations, we develop a similarity theory for the inertial range from clearly stated assumptions. In the right variables, the theory is scaling invariant, but in traditional variables it shows anomalous scaling. We obtain the underlying probability density function, the scaling exponents, and the coefficients for the structure functions. An inertial range length scale also emerges.
Using direct numerical simulation we study the behavior of the maximal Lyapunov exponent in thin-layer turbulence, where one dimension of the system is constrained geometrically. Such systems are known to exhibit transitions from fully three dimensional turbulence through a mixed two and three dimensional phenomenology state and then onto fully two dimensional dynamics. We find a discontinuous jump in the Lyapunov exponent at this second transition implying the predictability of such systems can change dramatically. Such transitions are seen in a number of different turbulent systems, for exam…
The assumption of similarity and self-preservation, which permits an analytical determination of the energy decay in isotropic turbulence, has played an important role in the development of turbulence theory for more than half a century. Sedov (1944), who first found an ingenious way to obtain two equations from one. Nonethless, it appears that this problem has never been reinvestigated in depth subsequent to this earlier work. In the present paper, such an analysis is carried out, yielding a much more complete picture of self-preservation isotropic turbulence. Based on these exact solutions, …
We discuss the irreversibility, nonlocality, and fluctuations, as well as the Lyapunov and hydrodynamic instabilities characterizing atomistic, smooth-particle, and finite-difference solutions of the two-dimensional Rayleigh-Bénard problem. To speed up the numerical analysis we control the time-dependence of the Rayleigh number so as to include many distinct flow morphologies in a single simulation. The relatively simple nature of these computational algorithms and the richness of the results they can yield make such studies and their interpretation particularly well suited to graduate-level r…
The purpose of this brief comunication is to improve a hypothesis of the previous work of the author (de Divitiis, Theor Comput Fluid Dyn, doi:10.1007/s00162-010-0211-9) dealing with the finite--scale Lyapunov analysis of isotropic turbulence. There, the analytical expression of the structure function of the longitudinal velocity difference $Δu_r$ is derived through a statistical analysis of the Fourier transformed Navier-Stokes equations, and by means of considerations regarding the scales of the velocity fluctuations, which arise from the Kolmogorov theory. Due to these latter considerations…
In every turbulent flow with non-zero viscosity, heat is generated by viscous friction. This heat is then mixed by the velocity field. We consider how heat fluctuations generated this way are injected and distributed over length scales in isotropic turbulence. A triadic closure is derived and numerically integrated. It is shown how the heat fluctuation spectrum depends on the Reynolds and Prandtl numbers.
This paper is the forth part of our series of work, is devoted to the analysis on the multiscales and cascade aspects of the statistical theory of isotropic turbulence based on the new Sedov-type solution. In this paper, we use the explicit map method to analyse the nonlinear dynamical behaviour for cascade in isotorpic turbulence. This deductive scale analysis is shown to provide the first visual evidence of the celebrated Richardson cascade, and reveals in partcular its multiscale character. The results also indicate that the energy cascading process has remarkable similarities with the dete…
The radial relative velocity between particles suspended in turbulent flow plays a critical role in droplet collision and growth. We present a simple and accurate approach to RV measurement in isotropic turbulence - planar 4-frame particle tracking velocimetry - using routine PIV hardware. This study demonstrates the feasibility of accurately measuring RV using routine hardware and verifies, for the first time, the path-history and inertial filtering effects on particle-pair RV at large particle separations experimentally.
The recent development of a statistical model for incompressible Navier-Stokes (NS) fluids based on inverse kinetic theory (IKT, 2004-2008) poses the problem of searching for particular realizations of the theory which may be relevant for \textit{the statistical description of turbulence} and in particular for the so-called \textit{homogeneous, isotropic}and \textit{stationary} turbulence (HIST). Here the problem is set in terms of the $1-$point velocity probability density function (PDF) which determines a complete IKT-statistical model for NS fluids. This raises the interesting question of i…
This study analyzes the temperature fluctuations in incompressible homogeneous isotropic turbulence through the finite scale Lyapunov analysis of the relative motion between two fluid particles. The analysis provides an explanation of the mechanism of the thermal energy cascade, leads to the closure of the Corrsin equation, and describes the statistics of the longitudinal temperature derivative through the Lyapunov theory of the local deformation and the thermal energy equation. The results here obtained show that, in the case of self-similarity, the temperature spectrum exhibits the scaling l…
Direct numerical simulation is used to investigate the decay exponent of isotropic homogeneous turbulence over a range of Reynolds numbers sufficient to display both high and low Re number decay behavior. The initial turbulence is generated by the stirring action of the flow past many small randomly placed cubes. Stirring occurs at 1/30th of the simulation domain size so that the low-wavenumber and large scale behavior of the turbulent spectrum is generated by the fluid and is not imposed. It is shown that the decay exponent in the resulting turbulence closely matches the theoretical predictio…
In this paper, we present an extensive study of linearly forced isotropic turbulence. By using an analytical method, we identified two parametric choices that are new to our knowledge. We proved that the underlying nonlinear dynamical system for linearly forced isotropic turbulence is the general case of a cubic Lienard equation with linear damping (Dumortier and Rousseau 1990).
We use the recently developed Macroscopic Forcing Method [Mani and Park, Physical Review Fluids, 6:054607, 2021] to compute the scale-dependent eddy diffusivity characterizing ensemble-averaged scalar and momentum transport in incompressible homogeneous isotropic turbulence. For scales larger than the energy containing eddies, eddy diffusivity is found to be constant and consistent with the Boussinesq approximation. However for small scales eddy diffusivity is found to vanish inversely proportional to the wavenumber. Behavior at all scales is reasonably captured by a non-local eddy diffusivity…
An improved understanding of turbulence is essential for the effective modelling and control of industrial and geophysical processes. Homogeneous, isotropic turbulence (HIT) is the archetypal field for developing turbulence physics theory. Based on the Schur transform, we introduce an additive decomposition of the velocity gradient tensor into a normal part (containing the eigenvalues) and a non-normal or shear-related tensor. We re-interrogate some key properties of HIT and show that the the tendency of the flow to form disc-like structures is not a property of the normal tensor; it emerges f…
Small scale characteristics of turbulence such as velocity gradients and vorticity fluctuate rapidly in magnitude and oscillate in sign. Much work exists on the characterization of magnitude variations, but far less on sign oscillations. While averages performed on large scales tend to zero because of the oscillatory character, those performed on increasingly smaller scales will vary with the averaging scale in some characteristic way. This characteristic variation at high Reynolds numbers is captured by the so-called cancellation exponent, which measures how local averages tend to cancel out …
We present the most extensive direct numerical simulations, attempted so far, of statistically steady, homogeneous, isotropic turbulence in two-dimensional fluid films with air-drag-induced friction and with polymer additives. Our study reveals that the polymers (a) reduce the total fluid energy, enstrophy, and palinstrophy, (b) modify the fluid energy spectrum both in inverse- and forward-cascade regimes, (c) reduce small-scale intermittency, (d) suppress regions of large vorticity and strain rate, and (e) stretch in strain-dominated regions. We compare our results with earlier experimental s…
We show by direct numerical simulation (DNS) that the Lagrangian cross correlation of velocity gradients in homogeneous isotropic turbulence increases at short times, whereas its auto-correlation decreases. Kinematic considerations allow to show that two invariants of the turbulent velocity field determine the short-time velocity gradient correlations. In order to get a more intuitive understanding of the dynamics for longer times, heuristic models are proposed involving the combined action of local shear and rotation. These models quantitatively reproduce the effects and disentangle the diffe…
Power spectrum of the distributed chaos can be represented by a weighted superposition of the exponential functions which is converged to a stretched exponential $\propto \exp-(k/k_β)^{β}$. An asymptotic theory has been developed in order to estimate the value of $β$ for the isotropic turbulence. This value has been found to be $β=3/4$. Excellent agreement has been established between this theory and the data of direct numerical simulations not only for the velocity field but also for the passive scalar, energy dissipation rate, and magnetic fields. One can conclude that the isotropic turbulen…
A dynamical model is proposed for isotropic turbulence driven by steady forcing that yields a viscosity independent dynamics for the small-scale (inertial) regime. This reproduces the Kolmogorov spectrum for the two-point velocity correlation function in the fully developed (stationary) stage, while predicting intermittency corrections for higher order moments. The model also yields a transient stage with a power-law time evolution. The crossover time to fully developed turbulence scales with the turbulent system size as $\sim L^{11/3}$. The physical origin of the transient behavior is explain…
We investigate the conditional vorticity budget of fully developed three-dimensional homogeneous isotropic turbulence with respect to coherent and incoherent flow contributions. The Coherent Vorticity Extraction based on orthogonal wavelets allows to decompose the vorticity field into coherent and incoherent contributions, of which the latter are noise-like. The impact of the vortex structures observed in fully developed turbulence on statistical balance equations is quantified considering the conditional vorticity budget. The connection between the basic structures present in the flow and the…
Using time-resolved measurements of the orientation of rigid inertial fibers in a turbulence-tank, we investigate the autocorrelation of their tumbling rate. The correlation time ($τ_d$) is well predicted by Kolmogorov inertial-range scaling based on the fiber length (L) when the fiber inertia can be neglected. For inertial fibers, we propose a simple model considering fiber inertia (measured by a tumbling Stokes number) and a viscous torque which accurately predicts both the correlation time and the variance of the tumbling rate.
We investigate the single-point probability density function of the velocity in three-dimensional stationary and decaying homogeneous isotropic turbulence. To this end we apply the statistical framework of the Lundgren-Monin-Novikov hierarchy combined with conditional averaging, identifying the quantities that determine the shape of the probability density function. In this framework the conditional averages of the rate of energy dissipation, the velocity diffusion and the pressure gradient with respect to velocity play a key role. Direct numerical simulations of the Navier-Stokes equation are…
We study the rotational dynamics of {\it inertial} disks and rods in three-dimensional, homogeneous isotropic turbulence. In particular, we show how the alignment and the decorrelation time-scales of such spheroids depend, critically, on both the level of inertia and the aspect ratio of these particles. These results illustrate the effect of inertia---which leads to a preferential sampling of the local flow geometry---on the statistics of both disks and rods in a turbulent flow. Our results are important for a variety of natural and industrial settings where the turbulent transport of asymmetr…
We study the effect of the Basset history force on the dynamics of small particles transported in homogeneous and isotropic turbulence and show that this term, often neglected in previous numerical studies, reduces the small-scale clustering typical of inertial particles. The contribution of this force to the total particle acceleration is, on average, responsible for about 10% of the total acceleration and particularly relevant during rare strong events. At moderate density ratios, i.e. sand or metal powder in water, its presence alters the balance of forces determining the particle accelerat…
We study the competition between domain coarsening in a symmetric binary mixtures below the critical temperature and turbulent fluctuations. We find that the coarsening process is arrested in presence of turbulence. The physics of the process shares remarkable similarities with the behaviour of diluted turbulent emulsions and the arrest length scale can be estimated with an argument similar to the one proposed by Kolmogorov and Hinze for the maximal stability diameter of droplets in turbulence. Although in the absence of flow the microscopic diffusion constant is negative, turbulence does effe…
Inertial-range features of turbulence are investigated using data from experimental measurements of grid turbulence and direct numerical simulations of isotropic turbulence simulated in a periodic box, both at the Taylor-scale Reynolds number $R_λ\sim 1000$. In particular, oscillations modulating the power-law scaling in the inertial range are examined for structure functions up to sixth order moments. The oscillations in exponent ratios decrease with increasing sample size in simulations though, in experiments, they survive at a low value of $4$ parts in $1000$ even after massive averaging. T…
We present developments of the Hamiltonian approach to problems of the freely decay of isotropic turbulence, and also consider specific applications of the modified Prelle-Singer procedure to isotropic turbulence. It demonstrates that a nonlinear second order ordinary differential equation is intimately related to the self-preserving solution of Karman-Howarth equation, admitting time-dependent first integrals and also proving the nonstandard Hamiltonian structure, as well as the Liouville sense of integrability.
In this letter, we present an extensive study of the linearly forced isotropic turbulence. By using analytical method, we identify two parametric choices, of which they seem to be new as far as our knowledge goes. We prove that the underlying nonlinear dynamical system for linearly forced isotropic turbulence is the general case of a cubic Lienard equation with linear damping. We also discuss a Fokker-Planck approach to this new dynamical system,which is bistable and exhibits two asymmetric and asymptotically stable stationary probability densities.
We generalize Batchelor's parameterization of the autocorrelation functions of isotropic turbulence in a form involving a product expansion with multiple small scales. The richer small scale structure acquired this way, compared to the usual Batchelor function, is necessary so that the associated energy spectrum approximate well actual spectra in the universal equilibrium range. We propose that the generalized function provides an approximation of arbitrary accuracy for actual spectra of isotropic turbulence over the universal equilibrium range. The degree of accuracy depends on the number of …
