Chaotic mixing in a torus map

The advection and diffusion of a passive scalar is investigated for a map of the 2-torus. The map is chaotic, and the limit of almost-uniform stretching is considered. This allows an analytic understanding of the transition from a phase of constant scalar variance (for short times) to exponential decay (for long times). This transition is embodied in a short superexponential phase of decay. The asymptotic state in the exponential phase is an eigenfunction of the advection-diffusion operator, in which most of the scalar variance is concentrated at small scales, even though a large-scale mode sets the decay rate. The duration of the superexponential phase is proportional to the logarithm of the exponential decay rate; if the decay is slow enough then there is no superexponential phase at all.     

Large-scale structure of passive scalar turbulence

We investigate the large-scale statistics of a passive scalar transported by a turbulent velocity field by means of direct numerical simulations. We focus on scales larger than the characteristic length scale of scalar injection, yet smaller than the correlation length of the velocity. We show the existence of nontrivial long-range correlations in the form of new power laws for the decay of high-order coarse-grained scalar cumulants. This result contradicts the classical scenario of Gibbs equilibrium statistics that should hold in the absence of scalar flux. The breakdown of "thermal equilibrium" at large scales is traced back to the statistical geometry of turbulent dispersion of two scalar blobs. The numerical values obtained for the scaling exponents of the coarse-grained scalar cumulants are in agreement with recent theoretical results.     

雷诺数对圆柱尾流中被动标量场的影响

本文通过实验研究雷诺数对加热圆柱尾流中温度场的影响.实验中雷诺数Re(≡U∞d/v,其中U∞为来流速度、d为圆柱直径、v为流体黏度)的取值范围为1200-8600.实验中温度是由直径为0.63 μm的冷线探针测量的.实验结果表明,一般而言,雷诺数对整个尾流的标量混合特性有着显著的影响.随着雷诺数的增加,平均标量场向外的扩散速度加快、标量脉动强度增加了但衰减也加快.本文还发现:尾流中似乎存在两个区域,一个位于卡门涡街下游靠后,另一个就是传统的远场自相似区;在这两个区域,某些描述标量和动量的相似关系式近似成立.     

Passive scalar transport in peripheral regions of random flows

We investigate statistical properties of the passive scalar mixing in random (turbulent) flows assuming its diffusion to be weak. Then at advanced stages of the passive scalar decay, its unmixed residue is primarily concentrated in a narrow diffusive layer near the wall and its transport to the bulk goes through the peripheral region (laminar sublayer of the flow). We conducted Lagrangian numerical simulations of the process for different space dimensions d and revealed structures responsible for the transport, which are passive scalar tongues pulled from the diffusive boundary layer to the bulk. We investigated statistical properties of the passive scalar and of the passive scalar integrated along the wall. Moments of both objects demonstrate scaling behavior outside the diffusive boundary layer. We propose an analytic scheme for the passive scalar statistics, explaining the features observed numerically.     

Conserved scalar mixing in a confined-opposed-jets flow

The focus of this paper is on the mixing of a conserved passive scalar for Sc = 1 (Sc is the Schmidt number) in axisymmetric turbulence for which the initial injections of turbulent kinetic energy and scalar variance are similar. Two confined-opposed-jets (COJ) are experimentally studied through simultaneous PIV (particle image velocimetry) and PLIF (planar laser induced fluorescence) measurements, for different flow regimes. One-point transport equation for the scalar variance is assessed through experimental data, along the common axis of the two opposed jets, and different physical phenomena are revealed (production, diffusion, dissipation). The production of scalar variance is equilibrated by the diffusion term (～75%) and the mean dissipation of the scalar variance (～25%). To further assess the scalar behaviour at each scale in this anisotropic, but axisymmetric, flow, a scale-by-scale scalar variance budget equation is derived for axisymmetric turbulence. This equation reduces to Yaglom's 4/3 law, under additional restrictions. The equation is assessed through experimental data, in the impingement region between the two COJ. In particular, the anisotropic energy transfer along different directions is quantified. It is shown that for scales smaller than the size of the central region, Δ, the cascade of the scalar variance is completely inhibited, independently of the particular direction. For scales larger than Δ, the apparent aspect of the energy transfer is that of an inverse cascade, with positive values of the scalar variance transfer. Nonetheless, inhomogeneity of the flow and mixing at those scales is directly responsible for these positive values.     

Large-scale anisotropy in scalar turbulence

The effect of anisotropy on the statistics of a passive tracer transported by a turbulent flow is investigated. We show that under broad conditions an arbitrarily small amount of anisotropy propagates to the large scales where it eventually dominates the structure of the concentration field. This result is obtained analytically in the framework of an exactly solvable model and confirmed by numerical simulations of scalar transport in two-dimensional turbulence.     

Numerical integration of stochastic differential equations

The stability of incompressible turbulent fluids with respect to weak mean flow perturbations is discussed. It is shown that for a statistically homogeneous, isotropic, and stationary model such perturbations will decay. This is in marked contrast to the compressible case.     

How does flow in a pipe become turbulent?

The transition to turbulence in pipe flow does not follow the scenario familiar from Rayleigh-Benard or Taylor-Couette flow since the laminar profile is stable against infinitesimal perturbations for all Reynolds numbers. Moreover, even when the flow speed is high enough and the perturbation sufficiently strong such that turbulent flow is established, it can return to the laminar state without any indication of the imminent decay. In this parameter range, the lifetimes of perturbations show a sensitive dependence on initial conditions and an exponential distribution. The turbulence seems to be supported by threedimensional travelling waves which appear transiently in the flow field. The boundary between laminar and turbulent dynamics is formed by the stable manifold of an invariant chaotic state. We also discuss the relation between observations in short, periodically continued domains, and the dynamics in fully extended puffs.     

Effect of mean flow shear on cross phase and transport reconsidered

We reconsider the important question of the effect of a strong mean shear flow on the transport of a passive scalar field. By incorporating the effect of resonance, we show that the flux scales with the mean shear Omega as Omega(-1). The results also indicate that the scaling of the flux and cross phase with shear is rather weak and that the cross phase is not always more heavily suppressed than the amplitude of the turbulence. Furthermore, we show that the scalings of flux and cross phase with Omega depend on the statistics of the turbulent flow.     

Decaying turbulence: What happens when the correlation length varies spatially in two adjacent zones

We have imagined a numerical experiment to explore the onset of turbulent intermittency associated with a spatial perturbation of the correlation length. We place two isotropic regions, with different integral scales, inside a volume where the turbulent kinetic energy is initially uniform and leave them to interact and evolve in time. The different length scales produce different decay rates in the two regions. Since the smaller-scale region decays faster, a transient turbulent energy gradient is generated at the interface between the two regions. The transient is characterized by three phases in which the kinetic energy gradient across the interface grows, peaks and then slowly decays. The transient lifetime is almost proportional to the initial ratio of the correlation lengths. The direct numerical simulations also show that the interface width grows in time. The velocity moments inside this interaction zone are seen to depart from their initial isotropic values and, with a certain lag, the anisotropy is seen to spread to small scales. The longitudinal derivative moments also become anisotropic after a few eddy turnover times. This anisotropic behaviour is different from that observed in sheared homogeneous turbulent flows, where high transverse derivative moments are generated, but longitudinal moments almost maintain the isotropic turbulence values. Apart from the behaviour of the energy gradient transients, the results also show the timescaling of the interface diffusion width, and data on the anisotropy of the large and small scales, observed through one-point statistics determined inside the intermittency sublayer, which is associated with the interaction zone.     

Modelling of lifted turbulent diffusion flames in a channel mixing layer by the flame hole dynamics

The partial quenching structure of turbulent diffusion flames in a turbulent mixing layer is investigated by the method of flame hole dynamics as an effort to develop a prediction model for the turbulent flame lift off. The essence of the flame hole dynamics is derivation of the random walk mapping, from the flame-edge theory, which governs expansion or contraction of the quenching holes initially created by the local quenching events. The numerical simulation for the flame hole dynamics is carried out in two stages. First, a direct numerical simulation is performed for a constant-density fuel–air channel mixing layer to obtain the background turbulent flow and mixing fields, from which a time series of two-dimensional scalar-dissipation-rate array is extracted. Subsequently, a Lagrangian simulation of the flame hole random walk mapping, projected to the scalar dissipation rate array, yields a temporally evolving turbulent extinction process and its statistics on partial quenching characteristics. In particular, the probability of encountering the reacting state, while conditioned with the instantaneous scalar dissipation rate, is examined to reveal that the conditional probability has a sharp transition across the crossover scalar dissipation rate, at which the flame edge changes its direction of propagation. This statistical characteristic implies that the flame edge propagation instead of the local quenching event is the main mechanism controlling the partial quenching events in turbulent flames. In addition, the conditional probability can be approximated by a heavyside function across the crossover scalar dissipation rate.     

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通过使用一排16根冷线探头排在多个空间点同时测量微加热圆柱的尾流温度场, 用小波分析技术对瞬时温度场的时间序列信号进行多尺度分析, 目的是研究不同尺度脉动温度对总体温度场的贡献.直径为d = 12.7 mm 的圆柱产生了被测的尾流, 对应的雷诺数为5500, 测量区域位于下游距离为2d 和 20d 之间. 基于小波多尺度分辨技术, 尾流温度场被分解为不同温度脉动特征尺度的小波分量. 通过分析这些小波分量的瞬时温度等值线图, 能够直接观测到不同特征尺度的涡结构运动特征和湍流间歇过程. 特别地, 我们在近场区从原始信号分解获得的高频区域中发现了K-H涡的存在. 不同尺度的温度方差沿流向的变化表明, 在下游距离为x=3d和 20d之间, 中等尺度的结构比大尺度和小尺度结构对总的温度均方根的贡献更大. 不同尺度的自相关函数表明, 大尺度和中等尺度的结构显示出较大的相关性, 而高频的小波分量则更快地失去了原有的拟序性. 关键词： 湍流尾流 被动标量 小波分析     

Hierarchical Parcel-Swapping Representation of Turbulent Mixing. Part 1. Formulation and Scaling Properties

An economical representation of effects of turbulence on the time-evolving structure of diffusive scalar fields is obtained by introducing a hierarchical (tree) network connecting fluid parcels, with effects of turbulent advection represented by swapping pairs of sub-trees at rates determined by turbulence time scales associated with the sub-trees. The fluid parcels reside at the base of the tree. The tree structure partitions the fluid parcels into adjacent pairs (or more generally, p-tuples). Adjacent parcels intermix at rates governed by diffusion time scales based on molecular diffusivities and parcel sizes. This simple procedure efficiently accomplishes long-standing objectives of turbulent mixing model development, such as generating physically based time histories of fluid-parcel nearest-neighbor encounters and the associated spatial structure of turbulent scalar fields. Correspondences between features of the hierarchical formulation and turbulent mixing phenomenology, both generic and case-specific, are noted.     

Revisiting the boundary layer structure used in Craig and Gordon's model of isotope fractionation in evaporation

Craig and Gordon's (CG) model of isotope fractionation in evaporation was derived more than 40 years ago and was based on the turbulent boundary layer structure model acceptable at that time. That view suggested that turbulent flows consist of eddies with a wide range of length scales moving randomly in the flow domain. There is evidence that some parameters in CG model do not fully correspond to data in the literature. Owing to advances in fluid dynamics research techniques, it has been shown in recent decades that the apparent chaotic flow in turbulent boundary layers is in fact governed by few well-organised structures. This article reviews the major characteristics of these coherent structures based on available results from experimental, numerical and theoretical studies of turbulent and laminar boundary layers. The review points on some differences between past and present views of the boundary layer structure and on their relation and possible influence on power laws in CG model.     

The effect of subgrid-scale models on the entrainment of a passive scalar in a turbulent planar jet

Classical large-eddy simulation (LES) modelling assumes that the passive subgrid-scale (SGS) models do not influence large-scale quantities, even though there is now ample evidence of this in many flows. In this work, direct numerical simulation (DNS) and large-eddy simulations of turbulent planar jets at Reynolds number Re_H = 6000 including a passive scalar with Schmidt number Sc = 0.7 are used to study the effect of several SGS models on the flow integral quantities e.g. velocity and scalar jet spreading rates. The models analysed are theSmagorinsky, dynamic Smagorinsky, shear-improved Smagorinsky and the Vreman. Detailed analysis of the thin layer bounding the turbulent and non-turbulent regions – the so-called turbulent/non-turbulent interface (TNTI) – shows that this region raises new challenges for classical SGS models. The small scales are far from equilibrium and contain a high fraction of the total kinetic energy and scalar variance, but the situation is worse for the scalar than for the velocity field. Both a-priori and a-posteriori (LES) tests show that the dynamic Smagorinsky and shear-improved models give the best results because they are able to accurately capture the correct statistics of the velocity and passive scalar fluctuations near the TNTI. The results also suggest the existence of a critical resolution Δ_x, of the order of the Taylor scale λ, which is needed for the scalar field. Coarser passive scalar LES i.e. Δ_x ≥ λ results in dramatic changes in the integral quantities. This fact is explained by the dynamics of the small scales near the jet interface.     

Quasiadiabatic decay of capillary turbulence on the charged surface of liquid hydrogen

We study the free decay of capillary turbulence on the charged surface of liquid hydrogen. We find that decay begins from the high frequency end of the spectral range, while most of the energy remains localized at low frequencies. The apparent discrepancy with the self-similar theory of nonstationary wave turbulent processes is accounted for in terms of a quasiadiabatic decay wherein fast nonlinear wave interactions redistribute energy between frequency scales in the presence of finite damping at all frequencies. Numerical calculations based on this idea agree well with experimental data.     

标量湍流的能量传输特性

应用直接数值模拟数据,从标量湍流传输的三波关系出发,进行湍流及标量湍流传输谱的多尺度分析,研究不同尺度间的能量传输性质,证实标量能量的传输与湍动能传输具有不同性质,大尺度速度脉动对标量传输有较大贡献,尤其是与标量小尺度脉动的相互作用,使标量模拟需要有比速度场更高的网格分辨率;并发现标量湍流的能量传输具有明显的非局部性;另外,定义了能量传输系数,发现在相同的Re数和Pe数条件下,标量湍流的对流惯性较速度脉动的惯性子区宽.     

Dynamical ensembles in stationary states

We propose, as a generalization of an idea of Ruelle's to describe turbulent fluid flow, a chaotic hypothesis for reversible dissipative many-particle systems in nonequilibrium stationary states in general. This implies an extension of the zeroth law of thermodynamics to nonequilibrium states and it leads to the identification of a unique distribution describing the asymptotic properties of the time evolution of the system for initial data randomly chosen with respect to a uniform distribution on phase space. For conservative systems in thermal equilibrium the chaotic hypothesis implies the ergodic hypothesis. We outline a procedure to obtain the distribution : it leads to a new unifying point of view for the phase space behavior of dissipative and conservative systems. The chaotic hypothesis is confirmed in a nontrivial, parameter-free, way by a recent computer experiment on the entropy production fluctuations in a shearing fluid far from equilibrium. Similar applications to other models are proposed, in particular to a model for the Kolmogorov-Obuchov theory for turbulent flow.     

Evolution and scaling of the peak flame surface density in spherical turbulent premixed flames subjected to decaying isotropic turbulence

The peak flame surface density within the turbulent flame brush is central to turbulent premixed combustion models in the flamelet regime. This work investigates the evolution of the peak surface density in spherically expanding turbulent premixed flames with the help of direct numerical simulations at various values of the Reynolds and Karlovitz number. The flames propagate in decaying isotropic turbulence inside a closed vessel. The effects of turbulent transport, transport due to mean velocity gradient, and flame stretch on the peak surface density are identified and characterized with an analysis based on the transport equation for the flame surface density function. The three mechanisms are governed by distinct flow time scales; turbulent transport by the eddy turnover time, mean transport by a time scale related to the pressure rise in the closed chamber, and flame stretch by the Kolmogorov time scale. Appropriate scaling of the terms is proposed and shown to collapse the data despite variations in the dimensionless groups. Overall, the transport terms lead to a reduction in the peak value of the surface density, while flame stretch has the opposite effect. In the present configuration, a small imbalance between the two leads to an exponential decay of the peak surface density in time. The dimensionless decay rate is found to be consistent with the evolution of the wrinkling scale as defined in the Bray-Moss-Libby model.