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Register a new userA phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows
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A phenomenological theory of the fluctuations of velocity occurring in a fully developed homogeneous and isotropic turbulent flow is presented. The focus is made on the fluctuations of the spatial (Eulerian) and temporal (Lagrangian) velocity increments. The universal nature of the intermittency phenomenon as observed in experimental measurements and numerical simulations is shown to be fully taken into account by the multiscale picture proposed by the multifractal formalism, and its extensions to the dissipative scales and to the Lagrangian framework. The article is devoted to the presentation of these arguments and to their comparisons against empirical data. In particular, explicit predictions of the statistics, such as probability density functions and high order moments, of the velocity gradients and acceleration are derived. In the Eulerian framework, at a given Reynolds number, they are shown to depend on a single parameter function called the singularity spectrum and to a universal constant governing the transition between the inertial and dissipative ranges. The Lagrangian singularity spectrum compares well with its Eulerian counterpart by a transformation based on incompressibility, homogeneity and isotropy and the remaining constant is shown to be difficult to estimate on empirical data. It is finally underlined the limitations of the increment to quantify accurately the singular nature of Lagrangian velocity. This is confirmed using higher order increments unbiased by the presence of linear trends, as they are observed on velocity along a trajectory.
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The Lagrangian (LA) and Eulerian Acceleration (EA) properties of fluid particles in homogeneous turbulence with uniform shear and uniform stable stratification are studied using direct numerical simulations. The Richardson number is varied from $Ri=0$, corresponding to unstratified shear flow, to $Ri=1$, corresponding to strongly stratified shear flow. The probability density functions (pdfs) of both LA and EA have a stretched-exponential shape and they show a strong and similar influence on the Richardson number. The extreme values of the EA are stronger than those observed for the LA. Geometrical statistics explain that the magnitude of the EA is larger than its Lagrangian counterpart due to the mutual cancellation of the Eulerian and convective acceleration, as both vectors statistically show an anti-parallel preference. A wavelet-based scale-dependent decomposition of the LA and EA is performed. The tails of the acceleration pdfs grow heavier for smaller scales of turbulent motion. Hence the flatness increases with decreasing scale, indicating stronger intermittency at smaller scales. The joint pdfs of the LA and EA indicate a trend to stronger correlations with increasing Richardson number and at larger scales of the turbulent motion. A consideration of the terms in the Navier--Stokes equation shows that the LA is mainly determined by the pressure-gradient term, while the EA is dominated by the nonlinear convection term.
We present an investigation of the statistics of velocity gradient related quantities, in particluar energy dissipation rate and enstrophy, along the trajectories of fluid tracers and of heavy/light particles advected by a homogeneous and isotropic turbulent flow. The Refined Similarity Hypothesis (RSH) proposed by Kolmogorov and Oboukhov in 1962 is rephrased in the Lagrangian context and then tested along the particle trajectories. The study is performed on state-of-the-art numerical data resulting from numerical simulations up to Re~400 with 2048^3 collocation points. When particles have small inertia, we show that the Lagrangian formulation of the RSH is well verified for time lags larger than the typical response time of the particle. In contrast, in the large inertia limit when the particle response time approaches the integral-time-scale of the flow, particles behave nearly ballistic, and the Eulerian formulation of RSH holds in the inertial-range.
We revisit the issue of Lagrangian irreversibility in the context of recent results [Xu, et al., PNAS, 111, 7558 (2014)] on flight-crash events in turbulent flows and show how extreme events in the Eulerian dissipation statistics are related to the statistics of power-fluctuations for tracer trajectories. Surprisingly, we find that particle trajectories in intense dissipation zones are dominated by energy gains sharper than energy losses, contrary to flight-crashes, through a pressure-gradient driven take-off phenomenon. Our conclusions are rationalised by analysing data from simulations of three-dimensional intermittent turbulence, as well as from non-intermittent decimated flows. Lagrangian irreversibility is found to persist even in the latter case, wherein fluctuations of the dissipation rate are shown to be relatively mild and to follow probability distribution functions with exponential tails.
We develop a stochastic model for Lagrangian velocity as it is observed in experimental and numerical fully developed turbulent flows. We define it as the unique statistically stationary solution of a causal dynamics, given by a stochastic differential equation. In comparison to previously proposed stochastic models, the obtained process is infinitely differentiable at a given finite Reynolds number, and its second-order statistical properties converge to those of an Ornstein-Uhlenbeck process in the infinite Reynolds number limit. In this limit, it exhibits furthermore intermittent scaling properties, as they can be quantified using higher-order statistics. To achieve this, we begin with generalizing the two-layered embedded stochastic process of Sawford (1991) by considering an infinite number of layers. We then study, both theoretically and numerically, the convergence towards a smooth (i.e. infinitely differentiable) Gaussian process. To include intermittent corrections, we follow similar considerations as for the multifractal random walk of Bacry et al. (2001). We derive in an exact manner the statistical properties of this process, and compare them to those estimated from Lagrangian trajectories extracted from numerically simulated turbulent flows. Key predictions of the multifractal formalism regarding acceleration correlation function and high-order structure functions are also derived. Through these predictions, we understand phenomenologically peculiar behaviours of the fluctuations in the dissipative range, that are not reproduced by our stochastic process. The proposed theoretical method regarding the modelling of infinitely differentiability opens the route to the full stochastic modelling of velocity, including the peculiar action of viscosity on the very fine scales.
We present Lagrangian one-particle statistics from the Risoe PTV experiment of a turbulent flow. We estimate the Lagrangian Kolmogorov constant $C_0$ and find that it is affected by the large scale inhomogeneities of the flow. The pdf of temporal velocity increments are highly non-Gaussian for small times which we interpret as a consequence of intermittency. Using Extended Self-Similarity we manage to quantify the intermittency and find that the deviations from Kolmogorov 1941 similarity scaling is larger in the Lagrangian framework than in the Eulerian. Through the multifractal model we calculate the multifractal dimension spectrum.
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