Scaling is an important measure of multi-scale fluctuation systems. Turbulence as the most remarkable multi-scale system possesses scaling over a wide range of scales. She-Leveque （SL） hierarchical symmetry, since its publication in 1994, has received wide attention. A number of experimental, numerical and theoretical work have been devoted to its verification, extension, and modification. Application to the understanding of magnetohydrodynamic turbulence, motions of cosmic baryon fluids, cosmological supersonic turbulence, natural image, spiral turbulent patterns, DNA anomalous composition, human heart variability are just a few among the most successful examples. A number of modified scaling laws have been derived in the framework of the hierarchical symmetry, and the SL model parameters are found to reveal both the organizational order of the whole system and the properties of the most significant fluctuation structures. A partial set of work related to these studies are reviewed. Particular emphasis is placed on the nature of the hierarchical symmetry. It is suggested that the SL hierarchical symmetry is a new form of the self-organization principle for multi-scale fluctuation systems, and can be employed as a standard analysis tool in the general multi-scale methodology. It is further suggested that the SL hierarchical symmetry implies the existence of a turbulence ensemble. It is speculated that the search for defining the turbulence ensemble might open a new way for deriving statistical closure equations for turbulence and other multi-scale fluctuation systems.
Zhen-Su She Zhi-Xiong Zhang State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, 100871 Beijing, China
Discontinuous Galerkin（DG） method is known to have several advantages for flow simulations,in particular,in fiexible accuracy management and adaptability to mesh refinement. In the present work,the DG method is developed for numerical simulations of both temporally and spatially developing mixing layers. For the temporally developing mixing layer,both the instantaneous fiow field and time evolution of momentum thickness agree very well with the previous results. Shocklets are observed at higher convective Mach numbers and the vortex paring manner is changed for high compressibility. For the spatially developing mixing layer,large-scale coherent structures and self-similar behavior for mean profiles are investigated. The instantaneous fiow field for a three-dimensional compressible mixing layer is also reported,which shows the development of largescale coherent structures in the streamwise direction. All numerical results suggest that the DG method is effective in performing accurate numerical simulations for compressible shear fiows.
Xiao-Tian ShiJun ChenWei-Tao BiChi-Wang ShuZhen-Su She
Using data from direct numerical simulation （DNS） of incompressible and compressible channel flow, we develop a method of sub-ensemble decomposition to investigate the pressure gradient effect on the Karman constant and the additive constant B characterizing the mean velocity profile （MVP）. The sub-ensemble decomposition is defined according to the magnitude of vertical fluctuation velocity, which mimics coherent motions like ejection and sweep. DNS data analysis shows that each sub-ensemble displays a distinct Karman constant, with a variation which mimics effects of pressure gradient. The latter is demonstrated by a relation between sub-ensembles＇ km and Bm similar to empirical data under various pressure gradients. A set of global parameters, k0-pg=0.39 ＆ B0-pg=5.5, are then derived for interpreting two constants observed by Nagib et al.
An experimental study of compressible mixing layers(CMLs)was conducted using planar laser Mie scattering(PLMS)visualizations from condensed ethanol droplets in the flow.Large ensembles of digital images were collected for two flow conditions at convective Mach numbers Mc=0.11 and 0.47.The coherent vortices,braids and eruptions in the mixing zone were observed,interpreted as evidence of multi-scale,three-dimensional structures at a high Reynolds number.The mixing layers with a large visualized range present two stages along the streamwise direction,corresponding to the initial mixing and the well-developed stage.A new method,the gray level ensemble average method(GLEAM),by virtue of the similarity of the mixing layer,was applied to measure the growth rate of the CML thickness.New evidence for a nonlinear growth of CML is reported,providing an interpretation of previous observations of the scattering of the growth rate.
Despite dedicated effort for many decades,statistical description of highly technologically important wall turbulence remains a great challenge.Current models are unfortunately incomplete,or empirical,or qualitative.After a review of the existing theories of wall turbulence,we present a new framework,called the structure ensemble dynamics （SED）,which aims at integrating the turbulence dynamics into a quantitative description of the mean flow.The SED theory naturally evolves from a statistical physics understanding of non-equilibrium open systems,such as fluid turbulence, for which mean quantities are intimately coupled with the fluctuation dynamics.Starting from the ensemble-averaged Navier-Stokes（EANS） equations,the theory postulates the existence of a finite number of statistical states yielding a multi-layer picture for wall turbulence.Then,it uses order functions（ratios of terms in the mean momentum as well as energy equations） to characterize the states and transitions between states.Application of the SED analysis to an incompressible channel flow and a compressible turbulent boundary layer shows that the order functions successfully reveal the multi-layer structure for wall-bounded turbulence, which arises as a quantitative extension of the traditional view in terms of sub-layer,buffer layer,log layer and wake. Furthermore,an idea of using a set of hyperbolic functions for modeling transitions between layers is proposed for a quantitative model of order functions across the entire flow domain.We conclude that the SED provides a theoretical framework for expressing the yet-unknown effects of fluctuation structures on the mean quantities,and offers new methods to analyze experimental and simulation data.Combined with asymptotic analysis,it also offers a way to evaluate convergence of simulations.The SED approach successfully describes the dynamics at both momentum and energy levels, in contrast with all prevalent approaches describing the mean velocity profile only.Moreover,the SED theoretical fr
Semi-periodic structures namely inclined wavy structures （IWS） are experimentally observed in compressible mixing layers at two convective Mach numbers （Mc = 0.11 and 0.47）. Flow structures are visualized by the laserinduced planar laser Mie scattering （PLMS） technique. Two methods are developed to investigate the spatial distribu- tion and geometry of IWS： （1） the dominant mode extrac- tion （DME） method, to extract the dominant modes of IWS from the streamwise gray-level fluctuation, and （2） the phase tracking （PT） method, to identify the shape of IWS. The re- sults suggest that pressure perturbations account for the for- marion of IWS in the initial mixing region and the joint effect of dilatation and coherent vortices enhances IWS in the well- developed region. The large transverse （cross-flow） scale of the IWS and their relation to coherent vortices （CV） indicate that the disturbance originated from CV in the mixing center propagates far into the free streams. The DME and the PT method are shown to be the effective tools to study the geometrical features of wavy structures in compressible shear flows.
A novel notion of turbulent structure the local cascade structure-is introduced to study the convection phenomenon in a turbulent channel flow. A space-time cross-correlation method is used to calculate the convection velocity. It is found that there are two characteristic convection speeds near the wall, one associated with small-scale streaks of a lower speed and another with streamwise vortices and hairpin vortices of a higher speed. The new concept of turbulent structure is powerful to illustrate the dominant role of coherent structures in the near-wall convection, and to reveal also the nature of the convection-the propagation of patterns of velocity fluctuations-which is scale-dependent.
