湍流两相流动有燃烧颗粒相概率密度函数输运方程理论

由有燃烧的湍流气粒两相流动的瞬态方程和统计力学概率密度函数概念出发,推导了有燃烧颗粒相的质量－动量－能量联合概率密度函数（ＰＤＦ）输运方程,并对方程中条件期望项用梯度模拟概念进行了模拟封闭。封闭后的ＰＤＦ方程可作为建立颗粒拟流体模型方程和封闭二阶矩模型的基础,也可以通过Ｍｏｎｔｅ－Ｃａｒｌｏ 法求解用以直接计算颗粒雷诺应力和湍流动能,以便和二阶 矩模型的结果相对照,改善二阶矩模型。     

A new assessment of the second-order moment of Lagrangian velocity increments in turbulence

The behaviour of the second-order Lagrangian structure functions on state-of-the-art numerical data both in two and three dimensions is studied. On the basis of a phenomenological connection between Eulerian space-fluctuations and the Lagrangian time-fluctuations, it is possible to rephrase the Kolmogorov 4/5-law into a relation predicting the linear (in time) scaling for the second-order Lagrangian structure function. When such a function is directly observed on current experimental or numerical data, it does not clearly display a scaling regime. A parameterisation of the Lagrangian structure functions based on Batchelor model is introduced and tested on data for 3d turbulence, and for 2d turbulence in the inverse cascade regime. Such parameterisation supports the idea, previously suggested, that both Eulerian and Lagrangian data are consistent with a linear scaling plus finite-Reynolds number effects affecting the small- and large timescales. When large-time saturation effects are properly accounted for, compensated plots show a detectable plateau already at the available Reynolds number. Furthermore, this parameterisation allows us to make quantitative predictions on the Reynolds number value for which Lagrangian structure functions are expected to display a scaling region. Finally, we show that this is also sufficient to predict the anomalous dependency of the normalised root mean squared acceleration as a function of the Reynolds number, without fitting parameters.     

Turbulent statistical characteristics associated to the north wind phenomenon in southern Brazil with application to turbulent diffusion

A parameterization for the transport processes in a shear driven planetary boundary layer (PBL) has been established employing turbulent statistical quantities measured during the north wind phenomenon in southern Brazil. Therefore, observed one-dimensional turbulent energy spectra are compared with a spectral model based on the Kolmogorov arguments. The good agreement obtained from this comparison leads to well defined formulations for the turbulent velocity variance, local decorrelation time scale and eddy diffusivity. Furthermore, for vertical regions in which the wind shear forcing is relevant, the eddy diffusivity derived from the north wind data presents a similar profile to those obtained from the non-extensive statistical mechanics theory. Finally, a validation for the present parameterization has been accomplished, using a Lagrangian stochastic dispersion model. The Prairie Grass data set, which presents high mean wind speed, is simulated. The analysis developed in this study shows that the turbulence parameterization constructed from wind data for north wind flow cases is able to describe the diffusion in a high wind speed, shear-dominated PBL.     

Lagrangian dynamics and statistical geometric structure of turbulence

The local statistical and geometric structure of three-dimensional turbulent flow can be described by the properties of the velocity gradient tensor. A stochastic model is developed for the Lagrangian time evolution of this tensor, in which the exact nonlinear self-stretching term accounts for the development of well-known non-Gaussian statistics and geometric alignment trends. The nonlocal pressure and viscous effects are accounted for by a closure that models the material deformation history of fluid elements. The resulting stochastic system reproduces many statistical and geometric trends observed in numerical and experimental 3D turbulent flows, including anomalous relative scaling.     

New sub-grid stochastic acceleration model in LES of high-Reynolds-number flows

The experimental observations of intermittent dynamics of Lagrangian acceleration in a “free” high-Reynolds-number turbulence are shown to be consistent with the Kolmogorov-Oboukhov theory. In line with Kolmogorov-Oboukhov’s predictions, a new sub-grid scale (SGS) model is proposed and is combined with the Smagorinsky model. The new SGS model is focused on simulation of the non-resolved total acceleration vector by two stochastic processes: one for its norm, another for its direction. The norm is simulated by stochastic equation, which was derived from the log-normal stochastic process for turbulent kinetic energy dissipation rate, with the Reynolds number, as the parameter. The direction of the acceleration vector is suggested to be governed by random walk process, with correlation on the Kolmogorov’s timescale. In the framework of this model, a surrogate unfiltered velocity field is emulated by computation of the instantaneous model-equation. The coarse-grid computation of a high-Reynolds-number stationary homogeneous turbulence reproduced qualitatively the main intermittency effects, which were observed in experiment of ENS in Lyon. Contrary to the standard LES with the Smagorinsky eddy-viscosity model, the proposed model provided: (i) non-Gaussianity in the acceleration distribution with stretched tails; (ii) rapid decorrelation of acceleration vector components; (iii) “long memory” in correlation of its norm. The turbulent energy spectra of stationary and decaying homogeneous turbulence are also better predicted by the proposed model.     

Use of Lagrangian statistics for the analysis of the scale separation hypothesis in turbulent channel flow

Turbulence models often involve Reynolds averaging, with a closure providing the Reynolds stress tensor as function of mean velocity gradients, through a turbulence constitutive equation. The main limitation of this linear closure is that it rests on an analogy with kinetic theory. For this analogy to be valid there has to be a scale separation between the mean velocity variations and the turbulent Lagrangian free path whose mean value is the turbulent mixing length. The aim of this work is to better understand this hypothesis from a microscopic point of view. Therefore, fluid elements are tracked in a turbulent channel flow. The flow is resolved by direct numerical simulation (DNS). Statistics on particle trajectories ending on a certain distance y₀ from the wall are computed, leading to estimations of the turbulent mixing length scale and the Knudsen number. Comparing the computed values to the Knudsen number in the case of scale separation, we may know in which region of the flow and to what extent the turbulence constitutive equation is not verified. Finally, a new non-local formulation for predicting the Reynolds stress is proposed.     

Decay and growth laws in homogeneous shear turbulence

Homogeneous anisotropic turbulence has been widely studied in the past decades, both numerically and experimentally. Shear flows have received a particular attention because of the numerous physical phenomena they exhibit. In the present paper, both the decay and growth of anisotropy in homogeneous shear flows at high Reynolds numbers are revisited thanks to a recent eddy-damped quasi-normal Markovian closure adapted to homogeneous anisotropic turbulence. The emphasis is put on several aspects: an asymptotic model for the slow part of the pressure–strain tensor is derived for the return to isotropy process when mean velocity gradients are released. Then, a general decay law for purely anisotropic quantities in Batchelor turbulence is proposed. At last, a discussion is proposed to explain the scattering of global quantities obtained in DNS and experiments in sustained shear flows: the emphasis is put on the exponential growth rate of the kinetic energy and on the shear parameter.     

Closure Approximations for Passive Scalar Turbulence: A Comparative Study on an Exactly Solvable Model with Complex Features

Some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep. This model has a general geometric structure of a jet flow with transverse disturbances, which occur in a number of contexts, and it encompasses a wide variety of possible spatio-temporal statistical structures for the velocity field, including strong long-range correlations. Even though the Eulerian and Lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader-than-Gaussian intermittency, this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations. Our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly. In particular it is shown that the direct interaction approximation (DIA), one of the most sophisticated closure approximations available, fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions, even though the equations it produces are rather complicated and difficult to analyze. Two alternative closure approximations, the Modified DIA (MDIA) and the Renormalized Lagrangian Approximation (RLA), with different levels of sophistication, both are simpler to use than the DIA and perform better. In particular, it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the MDIA is even able to capture intermittency effects.     

Lagrangian velocity statistics in turbulent flows: effects of dissipation

We use the multifractal formalism to describe the effects of dissipation on Lagrangian velocity statistics in turbulent flows. We analyze high Reynolds number experiments and direct numerical simulation data. We show that this approach reproduces the shape evolution of velocity increment probability density functions from Gaussian to stretched exponentials as the time lag decreases from integral to dissipative time scales. A quantitative understanding of the departure from scaling exhibited by the magnitude cumulants, early in the inertial range, is obtained with a free parameter function D(h) which plays the role of the singularity spectrum in the asymptotic limit of infinite Reynolds number. We observe that numerical and experimental data are accurately described by a unique quadratic D(h) spectrum which is found to extend from h(min) approximately 0.18 to h(max) approximately 1.     

Small-scale anisotropy induced by spectral forcing and by rotation in non-helical and helical turbulence

We study the effect of large-scale spectral forcing on the scale-dependent anisotropy of the velocity field in direct numerical simulations of homogeneous turbulence. ABC-type forcing and helical or non-helical Euler-type forcing are considered. We propose a scale-dependent characterisation of anisotropy based on a modal decomposition of the two-point velocity tensor spectrum. This produces direction-dependent spectra of energy, helicity and polarisation. We examine the conditions that allow anisotropy to develop in the small scales due to forcing and we show that the theoretically expected isotropy is not exactly obtained, even in the smallest scales, for ABC and helical Euler forcings. When adding rotation, the anisotropy level in ABC-forced simulations is similar to that of lower Rossby number Euler-forced runs. Moreover, even at low rotation rate, the natural anisotropy induced by the Coriolis force is visible at all scales, and two distinct wavenumber ranges appear from our fine-grained characterisation, not separated by the Zeman scale but by a scale where rotation and dissipation are balanced.     

Lagrangian Markovianized field approximation for turbulence

In a previous communication (W.J.T. Bos and J.-P. Bertoglio 2006, Phys. Fluids, 18, 031706), a self-consistent Markovian triadic closure was presented. The detailed derivation of this closure is given here, relating it to the Direct Interaction Approximation and Quasi-Normal types of closure. The time-scale needed to obtain a self-consistent closure for both the energy spectrum and the scalar variance spectrum is determined by evaluating the correlation between the velocity and an advected displacement vector-field. The relation between this latter correlation and the velocity–scalar correlation is stressed, suggesting a simplified model of the latter. The resulting closed equations are numerically integrated and results for the energy spectrum, scalar fluctuation spectrum and velocity–displacement correlation spectrum are presented for low, unity and high values of the Schmidt number.     

The combined Lagrangian advection method

We present and test a new hybrid numerical method for simulating layerwise-two-dimensional geophysical flows. The method radically extends the original Contour-Advective Semi-Lagrangian (CASL) algorithm [5] by combining three computational elements for the advection of general tracers (e.g. potential vorticity, water vapor, etc.): (1) a pseudo-spectral method for large scales, (2) Lagrangian contours for intermediate to small scales, and (3) Lagrangian particles for the representation of general forcing and dissipation. The pseudo-spectral method is both efficient and highly accurate at large scales, while contour advection is efficient and accurate at small scales, allowing one to simulate extremely fine-scale structure well below the basic grid scale used to represent the velocity field. The particles allow one to efficiently incorporate general forcing and dissipation.     

基于拉氏概率密度的两相湍流二阶矩模型

用颗粒运动的拉氏分析和PDF方法,改进了颗粒相的二阶矩模型。由拉氏两相运动的随机微分方程出发,采用随机过程分析和信号分析法得到湍流两相流动的PDF输运方程,双流体模型方程和两相脉动速度相关的基本模式的封闭式,和用其它方法导出的方程与封闭式的结果一致,对封闭式作了重要的改进,在分析颗粒轨道上的流体湍流作用时间时,全面地引入拉氏分析的轨道穿越效应、惯性效应、连续效应和湍流的各向异性。     

Dynamics and structure of unstably stratified homogeneous turbulence

We characterise the properties of unstably stratified homogeneous turbulence by means of high-resolution direct numerical simulations and a two-point statistical spectral model based on a quasi-normal closure proposed by Burlot et al. Both approaches agree very well regarding the evolution of one- and two-point turbulent statistics, showing that the model is valid at even higher Reynolds numbers than previously considered. From a parametric study with different initial conditions, we confirm that the energy distribution at large scale influences strongly the late time dynamics of the flow. In particular, we assess the existence of backscatter transfer of energy, and evaluate its role in the growth rate of several turbulent quantities. Moreover, thanks to the statistical model, we analyse the scale-by-scale anisotropy of the flow through the decomposition of turbulent spectra in terms of directional anisotropy and polarisation anisotropy, for a refined characterisation of the structure of the flow which is strongly anisotropic in the large scales. This also allows us to study how isotropy is restored in the inertial scales.     

Prandtl number effects in decaying homogeneous isotropic turbulence with a mean scalar gradient

Decaying homogeneous isotropic turbulence with an imposed mean scalar gradient is investigated numerically, thanks to a specific eddy-damped quasi-normal Markovian closure developed recently for passive scalar mixing in homogeneous anisotropic turbulence (BGC). The present modelling is compared successfully with recent direct numerical simulations and other models, for both very large and small Prandtl numbers. First, scalings for the cospectrum and scalar variance spectrum in the inertial range are recovered analytically and numerically. Then, at large Reynolds numbers, the decay and growth laws for the scalar variance and mixed velocity–scalar correlations, respectively, derived in BGC, are shown numerically to remain valid when the Prandtl number strongly departs from unity. Afterwards, the normalised correlation ρ_wθ is found to decrease in magnitude at a fixed Reynolds number when Pr either increases or decreases, in agreement with earlier predictions. Finally, the small scales return to isotropy of the scalar second-order moments is found to depend not only on the Reynolds number, but also on the Prandtl number.     

LES of spatially developing turbulent boundary layer over a concave surface

We revisit the problem of a spatially developing turbulent boundary layer over a concave surface. Unlike previous investigations, we simulate the combined effects of streamline curvature as well as curvature-induced pressure gradients on the turbulence. Our focus is on investigating the response of the turbulent boundary layer to the sudden onset of curvature and the destabilising influence of concave surface in the presence of pressure gradients. This is of interest for evaluating the turbulence closure models. At the beginning of the curve, the momentum thickness Reynolds number is 1520 and the ratio of the boundary layer thickness to the radius of curvature is δ₀/R = 0.055. The radial profiles of the mean velocity and turbulence statistics at different locations along the concave surface are presented. Our recently proposed curvature-corrected Reynolds Averaged Navier-Stokes (RANS) model is assessed in an a posteriori sense and the improvements obtained over the base model are reported. From the large Eddy simulation (LES) results, it was found that the maximum influence of concave curvature is on the wall-normal component of the Reynolds stress. The budgets of wall-normal Reynolds stress also confirmed this observation. At the onset of curvature, the effect of adverse pressure gradient is found to be predominant. This decreases the skin friction levels below that in the flat section.     

Lagrangian Velocity Fluctuations in Fully Developed Turbulence: Scaling, Intermittency, and Dynamics

New aspects of turbulence are uncovered if one considers the flow motion from the perspective of a fluid particle (known as the Lagrangian approach) rather than in terms of a velocity field (the Eulerian viewpoint). Using a new experimental technique, based on the scattering of ultrasound, we have obtained a direct measurement of particle velocities, resolved at all scales, in a fully turbulent flow. We find that the Lagrangian velocity autocorrelation function and the Lagrangian time spectrum are in agreement with the Kolmogorov K41 phenomenology. Intermittency corrections are observed and we give a measurement of the Lagrangian structure function exponents. They are more intermittent than the corresponding Eulerian exponents. We also propose a novel analysis of intermittency in turbulence: our measurement enables us to study it from a dynamical point of view. We thus analyze the Lagrangian velocity fluctuations in the framework of random walks. We find experimentally that the elementary steps in the walk have random uncorrelated directions but a magnitude that displays extremely long-range correlations in time. Theoretically, we study a Langevin equation that incorporates these features and we show that the resulting dynamics accounts for the observed one-point and two-point statistical properties of the Lagrangian velocity fluctuations. Our approach connects the intermittent statistical nature of turbulence to the dynamics of the flow.     

Conserved momenta of a ferromagnetic soliton

Linear and angular momenta of a soliton in a ferromagnet are commonly derived through the application of Noether’s theorem. We show that these quantities exhibit unphysical behavior: they depend on the choice of a gauge potential in the spin Lagrangian and can be made arbitrary. To resolve this problem, we exploit a similarity between the dynamics of a ferromagnetic soliton and that of a charged particle in a magnetic field. For the latter, canonical momentum is also gauge-dependent and thus unphysical; the physical momentum is the generator of magnetic translations, a symmetry combining physical translations with gauge transformations. We use this analogy to unambiguously define conserved momenta for ferromagnetic solitons. General considerations are illustrated on simple models of a domain wall in a ferromagnetic chain and of a vortex in a thin film.     

Multiscale model of gradient evolution in turbulent flows

A multiscale model for the evolution of the velocity gradient tensor in turbulence is proposed. The model couples "restricted Euler" (RE) dynamics describing gradient self-stretching with a cascade model allowing energy exchange between scales. We show that inclusion of the cascade process is sufficient to regularize the finite-time singularity of the RE dynamics. Also, the model retains geometrical features of real turbulence such as preferential alignments of vorticity and joint statistics of gradient tensor invariants. Furthermore, gradient fluctuations are non-Gaussian, skewed in the longitudinal case, and derivative flatness coefficients are in good agreement with experimental data.     

Stretching of polymers in isotropic turbulence: a statistical closure

We present a new closure for the mean rate of stretching of a dissolved polymer by homogeneous isotropic turbulence. The polymer is modeled by a bead-spring-type model (e.g., Oldroyd B, FENE-P, Giesekus) and the analytical closure is obtained assuming the Lagrangian velocity gradient can be modeled as a Gaussian, white-noise stochastic process. The resulting closure for the mean stretching depends upon the ratio of the correlation time for strain and rotation. Additionally, we derived a second-order expression for circumstances when strain and rotation have a finite correlation time. Finally, the base level closure is shown to reproduce results from direct numerical simulations by simply modifying the coefficients.