Stochastic process model for solute transport and the associated transport equation

A mathematical model of Lagrangian motions of a particle in turbulent flows is developed on the basis of a stochastic differential equation. The model expresses uncertainties involved in turbulence by standard Brownian motion. Because the model does not guarantee smoothness of the path of the particle, local velocity is newly defined so as to be suitable for observation of a velocity time series at a fixed point. Then, it is shown that the newly defined local velocity is governed by a Gaussian distribution. In addition, an estimation method of the turbulent diffusion coefficient involved in the model is proposed by using the local velocity. The estimation method does not require tracer experiments. In order to assess the validity of the proposed local velocity, velocity measurements with three-dimensional acoustic Doppler velocimeters were conducted in agricultural drainage canals. Also, the turbulent diffusion coefficient was estimated by the derived time series of the observed local velocity. Finally, a transport equation of conservative solute is derived by using the linearity of the Kolmogorov forward equation without using gradient-type lows.     

2D numerical simulation of submerged hydraulic jumps

Hydraulic jumps are usually used to dissipate energy in hydraulic engineering. In this paper, the turbulent submerged hydraulic jumps are simulated by solving the unsteady Reynolds averaged Navier–Stokes equations along with the continuity equation and the standard k–? equations for turbulence modeling. The Lagrangian moving grid method is employed for the simulation of the free surface. In the developed model, kinematic free-surface boundary condition is solved simultaneously with the momentum and continuity equations, so that the water elevation can be obtained along with velocity and pressure fields as part of the solution. Computational results are presented for Froude numbers ranging from 3.2 to 8.2 and submergence factors ranging from 0.24 to 0.85. Comparisons with experimental measurements show that numerical model can simulate the velocity field, variation of free surface, maximum velocity, Reynolds shear and normal stresses at various stations with reasonable accuracy.     

Langevin equation of a fluid particle in wall-induced turbulence

We derive the Langevin equation describing the stochastic process of fluid particle motion in wall-induced turbulence (turbulent flow in pipes, channels, and boundary layers including the atmospheric surface layer). The analysis is based on the asymptotic behavior at a large Reynolds number. We use the Lagrangian Kolmogorov theory, recently derived asymptotic expressions for the spatial distribution of turbulent energy dissipation, and also newly derived reciprocity relations analogous to the Onsager relations supplemented with recent measurement results. The long-time limit of the derived Langevin equation yields the diffusion equation for admixture dispersion in wall-induced turbulence.     

Wavelet filtering to study mixing in 2D isotropic turbulence

This paper presents the application of coherent vortex simulation (CVS) filtering, based on an orthogonal wavelet decomposition of vorticity, to study mixing in 2D homogeneous isotropic turbulent flows. The Eulerian and Lagrangian dynamics of the flow are studied by comparing the evolution of a passive scalar and of particles advected by the coherent and incoherent velocity fields, respectively. The former is responsible for strong mixing and produces the same anomalous diffusion as the total flow, due to transport by the coherent vortices, while mixing in the latter is much weaker and corresponds to classical diffusion.     

Statistics of numerically generated turbulence

This paper considers numerically generated turbulence obtained by integrating the complete time-dependent three-dimensional Navier-Stokes equations. The simulated unidirectional turbulent flow, bounded by two parallel planes, is strongly inhomogeneous in the direction normal to the planes but homogeneous in the parallel directions. The resulting flow field, which is considered a numerical realization of fully developed turbulent channel flow, contains detailed information on spatial coherent flow structures as well as on the time-dependency and statistics of the three-dimensional velocity and pressure fields. Focussing here on the statistics of the numerically generated turbulence, second-moments and higher-moments are presented and compared with the most recent PTV and LDV laboratory measurements. It is concluded that direct numerical simulations are an invaluable approach to turbulence which complements field studies and laboratory investigations. Numerical experiments are now becoming a principal source of detailed and reliable information, which play a key role in the deepening of our understanding of turbulent flow phenomena.     

A theoretical model for Reynolds-stress and dissipation-rate budgets in near-wall region

A 3-D wave model for the turbulent coherent structures in near-wall region is proposed. The transport nature of the Reynolds stresses and dissipation rate of the turbulence kinetic energy are shown via computation based on the theoretical model. The mean velocity profile is also computed by using the same theoretical model. The theoretical results are in good agreement with those found from DNS, indicating that the theoretical model proposed can correctly describe the physical mechanism of turbulence in near wall region and it thus possibly opens a new way for turbulence modeling in this region.     

Spin c geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds

From the existence of parallel spinor fields on Calabi-Yau, hyper-Kähler or complex flat manifolds, we deduce the existence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the submanifolds are compact, we obtain sharp estimates on their Betti numbers which generalize those obtained by Smoczyk in [49]. When the ambient manifold is Kähler-Einstein with positive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of Kählerian Killing spinor fields for some particular spin ^c structures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. These results also generalize some theorems by Smoczyk in [50]. Finally, applications on the Morse index of minimal Lagrangian submanifolds are obtained.     

Progress in the Self-Similar Turbulent Flame premixed combustion model

This paper is devoted to premixed combustion modeling in turbulent flow. First, we briefly remind the main features of the Self-Similar Turbulent Flame model that was more extensively developed in a former paper. Then, we carefully describe some improvements of the model. The determination of the turbulent flame velocity is based on the observed self-similarity of the turbulent flame and uses the local flame brush width as a fundamental parameter, which must be retrieved. With respect to the former version, we now derive more rigorously how the density variation has to be taken into account in the width retrieving function. We reformulate the diffusion term as a classical flux divergence term. We enforce the compatibility of the model for the limit of weak turbulence. We include a contracting effect of the source term, thus allowing to give a stationary mono-dimensional asymptotic solution with a finite width. We also include in a preliminary form, a stretch factor, which proves to be useful for controlling the flame behavior close to the flame holder and near the walls. The model implementation in the Star-CD CFD code is then tested on three different flame configurations. Finally, we shortly discuss the model improvements and the simulation results.     

The effect of turbulence on mixing in prototype reaction‐diffusion systems

The effect of turbulence on mixing in prototype reaction‐diffusion systems is analyzed here in the special situation where the turbulence is modeled ideally with two separated scales consisting of a large‐scale mean flow plus a small‐scale spatiotemporal periodic flow. In the limit of fast reaction and slow diffusion, it is rigorously proved that the turbulence does not contribute to the location of the mixing zone in the limit and that this mixing zone location is determined solely by advection of the large‐scale velocity field. This surprising result contrasts strongly with earlier work of the authors that always yields a large‐scale propagation speed enhanced by small‐scale turbulence for propagating fronts. The mathematical reasons for these differences are pointed out. This main theorem rigorously justifies the limit equilibrium approximations utilized in non‐premixed turbulent diffusion flames and condensation‐evaporation modeling in cloud physics in the fast reaction limit. The subtle nature of this result is emphasized by explicit examples presented in the fast reaction and zero‐diffusion limit with a nontrivial effect of turbulence on mixing in the limit. The situation with slow reaction and slow diffusion is also studied in the present work. Here the strong stirring by turbulence before significant reaction occurs necessarily leads to a homogenized limit with the strong mixing effects of turbulence expressed by a rigorous turbulent diffusivity modifying the reaction‐diffusion equations. Physical examples from non‐premixed turbulent combustion and cloud microphysics modeling are utilized throughout the paper to motivate and interpret the mathematical results. © 2000 John Wiley & Sons, Inc.     

The effects of bed form roughness on total suspended load via the Lattice Boltzmann Method

Bed forms in natural rivers and man-made channels provide the dominant contribution to overall flow resistance and hence significantly affect sediment transport rate. Many laboratory experiments and field observations have been conducted on bed forms, and it was found that theoretical flat-bed assumptions do not give the correct estimation for the total suspended load (TSL). In this study, we present a systematic numerical investigation of turbulent open-channel flows over bed forms using the Lattice Boltzmann Method (LBM). A static Smagorinsky model is incorporated into LBM to account for turbulence, and the dynamic interface between fluid and air is captured by a free-surface model. The time-averaged flow velocity, turbulence intensity and Reynolds shear stress in LBM simulations show an excellent agreement with the available experimental data. In addition, the coherent flow structures induced by the bed forms qualitatively agree with previous numerical results from Large Eddy Simulations based the Navier–Stokes equations. We then proceed to investigate the effects of bed form roughness, quantified by the total friction factor f_T, on sediment transport. It is found that the prediction of the TSL based on the theoretical flat-bed assumptions may lead to an overestimation of up to 30%, depending on the bed form roughness. In addition, the normalized TSL is linearly proportional to f_T and nearly inversely proportional to the ratio of downward settling velocity and upward turbulence induced diffusion. Our work proposes a general law linking these quantities to estimate the TSL, which has the potential for a more efficient and accurate engineering design of man-made channels and improved river management.     

湍流边界层中固体小颗粒湍流运动的Lagrangian模型

给出了固体小颗粒在边界层中的Lagrangian运动方程,方程中包括受壁面影响的粘性阻力,Saffman升力及Magus升力等.使用频谱法,得到了颗粒响应流体的Lagrangian能谱的表达式,使用这些结果研究了各种响应特性.本文的结果清楚地表明了固体个颗粒在湍流扩散过程中,其湍流扩散是可能大于流体的.     

Velocity fields with power-law spectra for modeling turbulent flows

We consider a generalization of homogeneous and isotropic Çinlar velocity fields to capture power-law spectra. The random velocity field is non-Gaussian with a representation motivated by Lagrangian and Eulerian observations. A wide range of turbulent flows can be generated by varying the stochastic parameters of the model. The velocity field being a functional version of Poisson shot-noise is constructed as the superposition of eddies randomized through their types and arrival times. We introduce a dependence between the eddy types which are spatial parameters and the decay parameter which is temporal. As a result, long-range correlation in space and a power-law spectrum previously used with Ornstein–Uhlenbeck velocity fields are achieved. We show that a corresponding power-law form for the probability distribution of the eddy diameter is sufficient for this result. The parameters of the probability distribution are further specified in view of Kolmogorov theory of the inertial scales. In particular, ∣k∣^−5/3 scaling of the spectrum is obtained. In the diffusive limit, we show that the parameters governing the decay and the arrival rate, and the speed of rotation of an eddy increase while its diameter decreases. That is, the eddies arrive fast, decay fast, and rotate fast with a small radius for a Brownian limit.     

用局部平均速度结构函数检测湍流边界层多尺度相干结构

测量风洞中平板湍流边界层不同法向位置的流向速度时间序列信号,提出了基于局部多尺度平均意义的湍流多尺度局部平均结构函数的概念,以描述湍流多尺度结构的伸缩变形和相对运动;基于多尺度局部平均结构函数的概念与Harr子波变换的一致性,提出了用湍流多尺度局部平均结构函数的平坦因子检测湍流边界层多尺度相干结构及其间歇性的检测方法,提取近壁区域不同法向位置处多尺度相干结构的条件相位平均波形,以研究多尺度相干结构猝发的动力学过程及其对湍流统计性质的影响．     

A Lagrangian study of eddies in the ocean

A brief review of our results on the application of the Lagrangian approach to study observed and simulated eddies in the ocean is presented. It is shown by a few examples of mesoscale vortex structures in the North Western Pacific how to compute and analyze maps of specific Lagrangian indicators in order to study the birth, formation, evolution, metamorphoses and death of ocean eddies. The examples involve two-dimensional eddies observed in satellitederived velocity fields in the deep ocean and three-dimensional ones simulated in a regional numerical model of circulation with a high resolution.     

扩散速度与组元的质量守恒方程*

在混合物流动中,某组元i的质量迁移速度(绝对速度)等于对流速度(牵连速度)与扩散速度(相对速度)之和.扩散速度——以及扩散系数——依对流速度取法之不同而不同.     

A simple and efficient implicit direct forcing immersed boundary model for simulations of complex flow

This paper presents an implementation of an implicit immersed boundary (IB) method in a flow solver based on the fractional step method and the finite volume method for complex flows involving moving boundaries and complex geometries. In this implementation, a body force caused by the immersed body is first introduced into the N-S equation to model the effect of immersed boundary. However, the body force is not pre-calculated, but implicitly determined in such a way that the velocity at the immersed boundary interpolated from the corrected velocity field accurately satisfies the no-slip and no-penetration conditions. Then, the large-eddy simulation is applied in the solver, where the subgrid-scale stress is determined by the Smagorinsky–Lilly model. Near the immersed boundaries, the subgrid-scale stress is determined by a wall model where the wall shear stress is directly calculated from the Lagrangian force(which represents the action of fluid on solid) on the immersed boundary. Such treatment makes the simulations of high Reynolds number turbulent flows feasible with the IB method. The accuracy and capability of the present method are demonstrated by simulations of a variety of both two- and three-dimensional simulations, including laminar flow past static and oscillating cylinders, rotating hydrofoil and turbulent flow around a three-dimensional circular cylinder and a sphere. It shows that the present implementation provides an easy-to-use, inexpensive and accurate technique for computational fluid dynamics in industrially relevant problems.     

A numerical study on statistical temporal scales in inertia particle dispersion

The statistical temporal scales involved in inertia particle dispersion are analyzed numerically. The numerical method of large eddy simulation, solving a filtered Navier-Stokes equation, is utilized to calculate fully developed turbulent channel flows with Reynolds numbers of 180 and 640, and the particle Lagrangian trajectory method is employed to track inertia particles released into the flow fields. The Lagrangian and Eulerian temporal scales are obtained statistically for fluid tracer particles and three different inertia particles with Stokes numbers of 1, 10 and 100. The Eulerian temporal scales, decreasing with the velocity of advection from the wall to the channel central plane, are smaller than the Lagrangian ones. The Lagrangian temporal scales of inertia particles increase with the particle Stokes number. The Lagrangian temporal scales of the fluid phase ‘seen’ by inertia particles are separate from those of the fluid phase, where inertia particles travel in turbulent vortices, due to the particle inertia and particle trajectory crossing effects. The effects of the Reynolds number on the integral temporal scales are also discussed. The results are worthy of use in examining and developing engineering prediction models of particle dispersion.     

壁湍流多尺度湍涡结构推广的自相似标度律

在水槽中测量了中等雷诺数下平板湍流边界层中的瞬时流向速度的时间序列,验证了Benzi提出的推广的自相似标度律,用子波变换将壁湍流脉动速度分解为多尺度湍涡结构的速度,研究了每一个尺度的湍涡速度结构函数的推广的自相似标度律。主要结论如下:湍流的统计性质是自相似的,这不仅适用于充分发展湍流,而且适用于中等雷诺数和低雷诺数湍流,而且具有相同的标度指数;推广的自相似标度律的适用的尺度范围远远大于惯性子区的范围,可以一直延伸至耗散区的尺度范围;推广的自相似标度律不仅适用于均匀各向同性湍流,也适用于剪切湍流如边界层湍流。     

Direct numerical simulation of turbulent non-Newtonian flow using a spectral element method

A spectral element—Fourier method (SEM) for Direct Numerical Simulation (DNS) of the turbulent flow of non-Newtonian fluids is described and the particular requirements for non-Newtonian rheology are discussed. The method is implemented in parallel using the MPI message passing kernel, and execution times scale somewhat less than linearly with the number of CPUs, however this is more than compensated by the improved simulation turn around times. The method is applied to the case of turbulent pipe flow, where simulation results for a shear-thinning (power law) fluid are compared to those of a yield stress (Herschel–Bulkley) fluid at the same generalised Reynolds number. It is seen that the yield stress significantly dampens turbulence intensities in the core of the flow where the quasi-laminar flow region there co-exists with a transitional wall zone. An additional simulation of the flow of blood in a channel is undertaken using a Carreau–Yasuda rheology model, and results compared to those of the one-equation Spalart-Allmaras RANS (Reynolds-Averaged Navier–Stokes) model. Agreement between the mean flow velocity profile predictions is seen to be good. Use of a DNS technique to study turbulence in non-Newtonian fluids shows great promise in understanding transition and turbulence in shear thinning, non-Newtonian flows.     

VLES turbulence model for an Eulerian–Lagrangian modeling concept for bubble plumes

Traditional Reynolds-averaged Navier–Stokes (RANS) approaches to turbulence modeling, such as the k-ϵ model, have some well-known shortcomings when modeling transient flow phenomena. To mitigate this, a filtered URANS model has been derived where turbulent structures larger than a given filter size (typically grid size) is captured by the flow equations and smaller structures are modeled according to a modified k-ϵ model. This modeling approach is also known as a VLES model (Very Large Eddy Scale model), and provides more details of the transient turbulence than the k-ϵ model at little extra computational cost.In this study a two-phase extension to the VLES model is described. A modeling concept for bubble plumes has been developed in which the bubbles are tracked as particles and the flow of liquid is solved by the Navier–Stokes equations in a traditional mesh based approach. The flow of bubbles and liquid is coupled in an Eulerian–Lagrangian model. Turbulent dispersion of the bubbles is treated by a random walk model. The random walk model depends on an estimation of the eddy life time. The eddy life time for the VLES model differs from a k-ϵ model, and its mathematical expression is derived.The model is applied to ocean plumes emanating from discharge of gas at the ocean floor. Validation with experiments and comparison with k-ϵ model are shown.