The axisymmetric equivalent of Kolmogorov's equation

A type of turbulence which is next to local isotropy in order of simplicity, but which corresponds more closely to turbulent flows encountered in practice, is locally axisymmetric turbulence. A representation of the second and third order structure function tensors of homogeneous axisymmetric turbulence is given. The dynamic equation relating the second and third order scalar structure functions is derived. When axisymmetry turns into isotropy, this equation is reduced to the well-known isotropic result: Kolmogorov's equation. The corresponding limiting form is also reduced to the well-known isotropic limiting form of Kolmogorov's equation. The new axisymmetric and theoretical results may have important consequences on several current ideas on the fine structure of turbulence, such as ideas developed by analysis based on the isotropic dissipation rate or such as extended self similarity (ESS) and the scaling laws for the n-order structure functions. Received 20 October 2000 and Received in final form 25 May 2001     

Nonperturbative renormalization group approach to turbulence

We suggest a new, renormalization group (RG) based, nonperturbative method for treating the intermittency problem of fully developed turbulence which also includes the effects of a finite boundary of the turbulent flow. The key idea is not to try to construct an elimination procedure based on some assumed statistical distribution, but to make an ansatz for possible RG transformations and to pose constraints upon those, which guarantee the invariance of the nonlinear term in the Navier-Stokes equation, the invariance of the energy dissipation, and other basic properties of the velocity field. The role of length scales is taken to be inverse to that in the theory of critical phenomena; thus possible intermittency corrections are connected with the outer length scale. Depending on the specific type of flow, we find different sets of admissible transformations with distinct scaling behaviour: for the often considered infinite, isotropic, and homogeneous system K41 scaling is enforced, but for the more realistic plane Couette geometry no restrictions on intermittency exponents were obtained so far. Received: 28 December 1997 / Accepted: 6 August 1998     

Manifestations of drag reduction by polymer additives in decaying, homogeneous, isotropic turbulence

The existence of drag reduction by polymer additives, well established for wall-bounded turbulent flows, is controversial in homogeneous, isotropic turbulence. To settle this controversy, we carry out a high-resolution direct numerical simulation of decaying, homogeneous, isotropic turbulence with polymer additives. Our study reveals clear manifestations of drag-reduction-type phenomena: On the addition of polymers to the turbulent fluid, we obtain a reduction in the energy-dissipation rate, a significant modification of the fluid energy spectrum especially in the deep-dissipation range, a suppression of small-scale intermittency, and a decrease in small-scale vorticity filaments.     

New Sedov-Type Solution of Isotropic Turbulence

The starting point lies in the results obtained by Sedov （1944） for isotropic turbulence with a self-preserving hypothesis. A careful consideration of the mathematical structure of the Karman-Howarth equation leads to an exact analysis of all cases possible and to all admissible solutions of the problem. I study this interesting problem from a new point of view. New solutions are obtained. Based on these exact solutions, some physical significant consequences of recent advances in the theory of self-preserved homogeneous statistical solution of the Navier-Stokes equations are presented.     

Curvature Effect on Compressible Turbulent Flow over a Wavy Wall

A fully developed compressible turbulent flow in a channel with a lower wavy wall and a upper plane wall is studied using large eddy simulation. We mainly attempt to deal with the curvature effect on compressible turbulent flow over the wavy wall. Some typical quantities including the mean turbulence statistics, dilatation and baroclinic terms in the enstrophy equation, turbulent kinetic energy budgets and the near-wall turbulent structures are analysed. The results obtained in this study provide physical insight into the understanding of the effects of curvature and compressibility on wall-bounded compressible turbulent flow.     

Quasi-stationary states in a circular geometry

The quasi-stationary final states of decaying two-dimensional turbulence on a circular domain, with an initial flow field containing either no or a substantial amount of angular momentum, have been investigated numerically. The production of angular momentum is almost absent for these flows on a circular domain with a no-slip wall. Its presence or absence essentially determines the character of the quasi-stationary final state. Based on a minimum-enstrophy principle a diagram is constructed that provides insight into the development of the typical late-time flow patterns on a circular domain with a no-slip wall. The quasi-stationary final states found in the present numerical study can be understood based on the predictions from the minimum-enstrophy principle.     

Axisymmetric form of Kármán-Howarth equation and its limiting forms

Kinematics and dynamics of homogeneous axisymmetric turbulence have been derived with the assumption that the properties of the turbulence are invariant with respect to rotation about a preferred direction . In particular, the “axisymmetric" equivalent of Karman-Howarth “isotropic" equation is derived using Lindborg's representation of the two-point correlation tensors of homogeneous axisymmetric turbulence. When the more constraining assumption of isotropy is made, this equation reduces to the well-known Karman-Howarth equation. There are two interesting limiting forms of the axisymmetric Karman-Howarth equation: the axisymmetric form of the energy balance equation and the axisymmetric form of the vorticity balance equation. Received 1 July 1999 and Received in final form 9 November 1999     

On the use of the discontinuous Galerkin method for numerical simulation of two-dimensional compressible turbulence with shocks

In this paper, the discontinuous Galerkin (DG) method combined with localized artificial diffusivity is investigated in the context of numerical simulation of broadband compressible turbulent flows with shocks for under-resolved cases. Firstly, the spectral property of the DG method is analyzed using the approximate dispersion relation (ADR) method and compared with typical finite difference methods, which reveals quantitatively that significantly less grid points can be used with DG for comparable numerical error. Then several typical test cases relevant to problems of compressible turbulence are simulated, including one-dimensional shock/entropy wave interaction, two-dimensional decaying isotropic turbulence, and two-dimensional temporal mixing layers. Numerical results indicate that higher numerical accuracy can be achieved on the same number of degrees of freedom with DG than high order finite difference schemes. Furthermore, shocks are also well captured using the localized artificial diffusivity method. The results in this work can provide useful guidance for further applications of DG to direct and large eddy simulation of compressible turbulent flows.     

Derivation of a Nonlinear Reynolds Stress Model Using Renormalization Group Analysis and Two-Scale Expansion Technique

Adopting Yoshizawa＇s two-scale expansion technique, the fluctuating field is expanded around the isotropic field. The renormalization group method is applied for calculating the covariance of the fluctuating field at the lower order expansion. A nonlinear Reynolds stress model is derived and the turbulent constants inside are evaluated analytically. Compared with the two-scale direct interaction approximation analysis for turbulent shear flows proposed by Yoshizawa, the calculation is much more simple. The analytical model presented here is close to the Speziale model, which is widely applied in the numerical simulations for the complex turbulent flows.     

Dynamic subgrid scale modeling of turbulent flows using lattice-Boltzmann method

In this paper, we discuss the incorporation of dynamic subgrid scale (SGS) models in the lattice-Boltzmann method (LBM) for large-eddy simulation (LES) of turbulent flows. The use of a dynamic procedure, which involves sampling or test-filtering of super-grid turbulence dynamics and subsequent use of scale-invariance for two levels, circumvents the need for empiricism in determining the magnitude of the model coefficient of the SGS models. We employ the multiple relaxation times (MRT) formulation of LBM with a forcing term, which has improved physical fidelity and numerical stability achieved by proper separation of relaxation time scales of hydrodynamic and non-hydrodynamic modes, for simulation of the grid-filtered dynamics of large-eddies. The dynamic procedure is illustrated for use with the common Smagorinsky eddy-viscosity SGS model, and incorporated in the LBM kinetic approach through effective relaxation time scales. The strain rate tensor in the SGS model is locally computed by means of non-equilibrium moments of the MRT-LBM. We also discuss proper sampling techniques or test-filters that facilitate implementation of dynamic models in the LBM. For accommodating variable resolutions, we employ conservative, locally refined grids in this framework. As examples, we consider the canonical anisotropic and inhomogeneous turbulent flow problem, i.e. fully-developed turbulent channel flow at two different shear Reynolds numbers Re_∗ of 180 and 395. The approach is able to automatically and self-consistently compute the values of the Smagorinsky coefficient, C_S. In particular, the computed value in the outer or bulk flow region, where turbulence is generally more isotropic, is about 0.155 (or the model coefficient ) which is in good agreement with prior data. It is also shown that the model coefficient becomes smaller and approaches towards zero near walls, reflecting the dampening of turbulent length scales near walls. The computed turbulence statistics at these Reynolds numbers are also in good agreement with prior data. The paper also discusses a procedure for incorporation of more general scale-similarity based SGS stress models.     

Turbulence in diffusion replicator equation

Dynamical behaviors in the diffusion replicator equation of three species are numerically studied. We point out the significant role of the heteroclinic cycle in the equation, and analyze the details of the turbulent solution that appeared in this system. Firstly, the bifurcation diagram for a certain parameter setting is drawn. Then it is shown that the turbulence appears with the supercritical Hopf bifurcation of a stationary uniform solution and it disappears under a subcritical-type bifurcation. Secondly, the statistical property of the turbulence near the supercritical Hopf onset point is analyzed precisely. Further, the correlation lengths and correlation times obey some characteristic scaling laws.     

Flame speed and self-similar propagation of expanding turbulent premixed flames

In this Letter we present turbulent flame speeds and their scaling from experimental measurements on constant-pressure, unity Lewis number expanding turbulent flames, propagating in nearly homogeneous isotropic turbulence in a dual-chamber, fan-stirred vessel. It is found that the normalized turbulent flame speed as a function of the average radius scales as a turbulent Reynolds number to the one-half power, where the average radius is the length scale and the thermal diffusivity is the transport property, thus showing self-similar propagation. Utilizing this dependence it is found that the turbulent flame speeds from the present expanding flames and those from the Bunsen geometry in the literature can be unified by a turbulent Reynolds number based on flame length scales using recent theoretical results obtained by spectral closure of the transformed G equation.     

Comparisons of different implementations of turbulence modelling in lattice Boltzmann method

In this paper, we present an alternative approach for the turbulence modelling in the single-relaxation-time lattice Boltzmann method (LBM) framework by treating the turbulence term as an extra forcing term, in addition to the traditional approach of modifying the relaxation time. We compare these two different approaches and their mixture in large-eddy simulation (LES) of three-dimensional decaying isotropic homogenous turbulence using the Smagorinsky model and the mixed similarity model. When the LES was conducted using the Smagorinsky model, where the Boussinesq eddy-viscosity approximation is adopted, the results showed that these three different implementations are equivalent. However, when the mixed similarity model is adopted, which is beyond the Boussinesq eddy-viscosity approximation, our results showed that an equivalent eddy-viscosity will lead to errors, while the forcing approach is more straightforward and accurate. This provides an alternative and more general framework of simulation of turbulence with models in LBM, especially when the Boussinesq eddy-viscosity approximation is invalid.     

Coupling of turbulence on the ignition of multicomponent sprays

This study examines the effect of turbulence on the ignition of multicomponent surrogate fuels and its role in modifying preferential evaporation in multiphase turbulent spray environments. To this end, two zero-dimensional droplet models are considered that are representative of asymptotic conditions of diffusion limit and the distillation limit are considered. The coupling between diffusion, evaporation and combustion is first identified using a scale analysis of 0D homogeneous batch reactor simulations. Subsequently, direct numerical simulations of homogeneously dispersed multicomponent droplets are performed for both droplet models, in decaying isotropic turbulence and at quiescent conditions to examine competing time scale effects arising from evaporation, ignition and turbulence. Results related to intra-droplet transport and effects of turbulence on autoignition and overall combustion are studied using an aviation fuel surrogate. Depending on the characteristic scale, it is shown that turbulence can couple through modulation of evaporation time or defer the ignition phase as a result of droplet cooling or gas-phase homogenization. Both preferential evaporation and turbulence are found to modify the ignition delay time, up to a factor of two. More importantly, identical droplet ignition behavior in homogeneous gas phase can imply fundamentally different combustion modes in heterogeneous environments.     

湍流理论的近代发展

这篇论文的内容是关於不可压缩流体的湍流理论近代发展的综合性介绍舆分析。我们首先评述了根据Reynolds的平均运动方程所建立的混合长度理论。其次,分析关於均匀各向同性湍流的主要理论工作。第三,讨论了运用Reynolds的平均运动方程和根据速度涨落方程求得的速度关联函数的动力学方程来处理具有Reynolds剪应力的普通湍流运动问题。同时说明这个方法虽然能够给出比混合长度理论舆实验较为接近的理论结果并能提出速度涨落平方平均值的理论分布,但是由於在求出的速度关联的动力学方程中出现高次元的速度关联,它继续地导致不封闭的微分方程组因而遇到不易克服的困难。因此,从以上湍流理论发展的回顾和最近关於均匀各向同性湍流在后期衰变运动的涡性结构工作,我们在最后提出了对今后湍流理论研究工作的新看法:湍流运动的基本组成部分是流体粘性作用所引起的涡旋运动;这个涡旋运动的动力学根据是用平均的方法后Navier-Stokes方程所导出的Reyonlds的平均运动方程典带度涨落方程。我们并着重说明Reynolds认识到湍流运动可分作平均运动与涨落运动的重要性。今后的理论工作则在於求这两组动力学方程的涡旋运动解,而这种类型的解并须满足像Колмогоров在高Reynolds数运动的局部各向同性湍流理论中所提出的统计条件,方能使解满足惟一性并可舆实验结果相比较。     

Evaporating droplets in turbulent reacting flows

Three-dimensional direct numerical simulations are carried out to determine the effects of turbulence on the preferential segregation of an evaporating spray and then to study the evolution of the resulting mixture fraction topology and propagating flame. First, the mixing between an initially randomly dispersed phase and the turbulent gaseous carrier phase is studied with non-evaporating particles. According to their inertia and the turbulence properties, the formation of clusters of particles is analyzed (formation delay, cluster characteristic size and density). Once the particles are in dynamical equilibrium with the surrounding turbulent flow, evaporation is considered through the analysis of the mixture fraction evolution. Finally, to mimic ignition, a kernel of burnt gases is generated at the center of the domain and the turbulent flame evolution is described.     

A consistent lattice Boltzmann equation with baroclinic coupling for mixtures

We propose a consistent lattice Boltzmann equation (LBE) with baroclinic coupling between species and mixture dynamics to model the active scalar dynamics in multi-species mixtures. The proposed LBE model is directly derived from the linearized Boltzmann equations for mixtures and it has the following two distinctive features. First, it uses the multiple-relaxation-time collision model so that it has the flexibility of independent Reynolds and Schmidt numbers, and better numerical stability. Second, it satisfies the indifferentiability principle therefore leads to a set of consistent hydrodynamic equations for barycentric velocity for mixtures. The proposed LBE model is validated through simulations of decaying homogeneous isotropic turbulence in three dimensions. We simulate both the active and passive scalar dynamics in decaying turbulence for mixtures. We also compute various statistical quantities and their decay exponents in decaying turbulence. Our results agree well with existing results for both scalar dynamics and decaying turbulence.