Mixing of a passive scalar across a thin shearless layer: concentration of intermittency on the sides of the turbulent interface

The advection of a passive scalar through an initial flat interface separating two different isotropic decaying turbulent fields is investigated in two and three dimensions. Simulations have been performed for a range of Taylor’s microscale Reynolds numbers from 45 to 250 and for a Schmidt number equal to 1. Different to the case where the transport involves the momentum and kinetic energy only and one intermittency layer is formed in the low-turbulent energy side of the system, in the passive scalar concentration field two intermittent layers are observed to develop at the sides of the interface. The layers move normally to the interface in opposite directions. The dimensionality produces different time scaling of the passive scalar diffusion, which is much faster in the two-dimensional case. In two dimensions, the propagation of the intermittent layers exhibits a significant asymmetry with respect to the initial position of the interface and is deeper for the layer which moves towards the high kinetic energy side of the system. In three dimensions, the two intermittent layers propagate nearly symmetrically with respect the centre of the mixing region. During the temporal decay, inside the mixing, which is both inhomogeneous and anisotropic but devoid of a mean velocity shear, the passive scalar spectra are computed. In three dimensions, the exponent in the scaling range gets in time a value close to that of the kinetic energy spectrum of isotropic turbulence (?5/3). In two dimensions, instead the exponent settles down to a value that is about one-half of the corresponding isotropic case. By means of an analysis based on simple wavy perturbations of the interface we show that the formation of the double layer of intermittency is a dynamic general feature not specific to the turbulent transport. These results of our numerical study are discussed in the context of experimental results and numerical simulations.     

Torque scaling in turbulent Taylor-Couette flow with co- and counterrotating cylinders

We analyze the global transport properties of turbulent Taylor-Couette flow in the strongly turbulent regime for independently rotating outer and inner cylinders, reaching Reynolds numbers of the inner and outer cylinders of Re(i) = 2×10(6) and Re(o) = ±1.4×10(6), respectively. For all Re(i), Re(o), the dimensionless torque G scales as a function of the Taylor number Ta (which is proportional to the square of the difference between the angular velocities of the inner and outer cylinders) with a universal effective scaling law G ∝ Ta(0.88), corresponding to Nu(ω) ∝ Ta(0.38) for the Nusselt number characterizing the angular velocity transport between the inner and outer cylinders. The exponent 0.38 corresponds to the ultimate regime scaling for the analogous Rayleigh-Bénard system. The transport is most efficient for the counterrotating case along the diagonal in phase space with ω(o) ≈ -0.4ω(i).     

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.     

Generalised scale-by-scale energy-budget equations and large-eddy simulations of anisotropic scalar turbulence at various Schmidt numbers

A necessary condition for the accurate prediction of turbulent flows using large-eddy simulation (LES) is the correct representation of energy transfer between the different scales of turbulence in the LES. For scalar turbulence, transfer of energy between turbulent length scales is described by a transport equation for the second moment of the scalar increment. For homogeneous isotropic turbulence, the underlying equation is the well-known Yaglom equation. In the present work, we study the turbulent mixing of a passive scalar with an imposed mean gradient by homogeneous isotropic turbulence. Both direct numerical simulations (DNS) and LES are performed for this configuration at various Schmidt numbers, ranging from 0.11 to 5.56. As the assumptions made in the derivation of the Yaglom equation are violated for the case considered here, a generalised Yaglom equation accounting for anisotropic effects, induced by the mean gradient, is derived in this work. This equation can be interpreted as a scale-by-scale energy-budget equation, as it relates at a certain scale r terms representing the production, turbulent transport, diffusive transport and dissipation of scalar energy. The equation is evaluated for the conducted DNS, followed by a discussion of physical effects present at different scales for various Schmidt numbers. For an analysis of the energy transfer in LES, a generalised Yaglom equation for the second moment of the filtered scalar increment is derived. In this equation, new terms appear due to the interaction between resolved and unresolved scales. In an a-priori test, this filtered energy-budget equation is evaluated by means of explicitly filtered DNS data. In addition, LES calculations of the same configuration are performed, and the energy budget as well as the different terms are thereby analysed in an a-posteriori test. It is shown that LES using an eddy viscosity model is able to fulfil the generalised filtered Yaglom equation for the present configuration. Further, the dependence of the terms appearing in the filtered energy-budget equation on varying Schmidt numbers is discussed.     

Statistical symmetries and its impact on new decay modes and integral invariants of decaying turbulence

Classical decay laws of isotropic turbulence usually derived from the von Kármán–Howarth equation are essentially based on two paradigms. First, scaling symmetries of space and time, both tracing back to the Navier–Stokes equations in the limit of large Reynolds numbers (or r?η), give rise to a temporal power-law decay for the turbulent kinetic energy and at the same time an algebraic growth of the integral length scale at an exponent that is uniquely coupled to the latter energy decay. Second, global invariants such as Birkhoff or Loitsianskii integrals determine the exponent of both power laws. We presently show that this class of decay laws may be considerably extended considering the entire set of multi-point correlation equations that admit a much wider class of symmetries. It was recently shown that these new symmetries are of paramount importance, e.g. in deriving the logarithmic law of the wall being an analytic solution of the multi-point equations. For the present case, it is particularly an additional scaling group, which we call statistical scaling group, that gives rise to two additional families of ‘canonical’ decay laws including those with an exponential characteristic for both the kinetic energy and the integral length scale. Finally, a second rather generic group admitted by all linear differential equations corresponding to the superposition principle induces an infinite set of scaling laws of rather complex form that may match rather generic initial conditions. All scaling laws are analyzed in the light of the above-mentioned integral invariants that have been further extended in the present contribution to an exponential-type invariant.     

Geometry of intensive scalar dissipation events in turbulence

The maxima of the scalar dissipation rate in turbulence appear in the form of sheets and correspond to the potentially most intensive scalar mixing events. Their cross section extension determines a locally varying diffusion scale of the mixing process and extends the classical Batchelor picture of one mean diffusion scale. The distribution of the local diffusion scales is analyzed for different Reynolds and Schmidt numbers with a fast multiscale technique applied to very high-resolution simulation data. The scales always take values across the whole Batchelor range and beyond. Furthermore, their distribution is traced back to the distribution of the contractive short-time Lyapunov exponent of the flow.     

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.     

On the role of mixing models in the simulation of MILD combustion using finite-rate chemistry combustion models

The present work shows an in-depth analysis about the role of mixing models on the simulation of MILD combustion using a finite-rate combustion model, the Partially Stirred Reactor approach (PaSR). Different approaches of increasing complexity are compared: a simple model based on a fraction of the integral time scale, a fractal-based mixing model and a dynamic mixing model based on the resolution of transport equations for scalar variance and dissipation rate. The approach is validated using detailed experimental data from flames stabilized on the Adelaide Jet-in-Hot Co-flow (JHC) burner at different fuel-jet Reynolds numbers (5k, 10k and 20k) and different co-flow oxygen dilution levels (3%, 6% and 9%). The results indicate the major role of mixing models to correctly handle turbulence/chemistry interactions and clearly indicate the superior performances of the dynamic mixing model over the other tested approaches.     

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.     

Relating Lagrangian passive scalar scaling exponents to Eulerian scaling exponents in turbulence

Intermittency is a basic feature of fully developed turbulence, for both velocity and passive scalars. Intermittency is classically characterized by Eulerian scaling exponent of structure functions. The same approach can be used in a Lagrangian framework to characterize the temporal intermittency of the velocity and passive scalar concentration of a an element of fluid advected by a turbulent intermittent field. Here we focus on Lagrangian passive scalar scaling exponents, and discuss their possible links with Eulerian passive scalar and mixed velocity-passive scalar structure functions. We provide different transformations between these scaling exponents, associated to different transformations linking space and time scales. We obtain four new explicit relations. Experimental data are needed to test these predictions for Lagrangian passive scalar scaling exponents.     

Phenomenology of local scale invariance: from conformal invariance to dynamical scaling

Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent θ or a dynamical exponent z. For a given value of θ (or z), we construct local scale transformations, which can be viewed as scale transformations with a space–time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of θ, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for θ=1 and Schrödinger invariance for θ=2.The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. A particularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber–Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics.     

Intermittency in stochastically perturbed turbulent models

Random dynamical models obtained as a perturbation of the GOY (Gledzer-Ohkitani-Yamada) shell model for three-dimensional turbulence are defined. Both static (time-independent) and dynamical scaling properties of the randomly perturbed model are studied. The random static-inviscid manifold, in contrast to the dynamical evolution, does not show intermittent scaling laws. This behavior is linked to the absence of large deviation in the random-map connecting fluctuations of velocities at different scales. The importance of inviscid conserved quantities on the intermittent statistics is discussed. Different random dynamical perturbations such that only energy is conserved in the inviscid and unforced limit are investigated. Intermittency is weakly affected by random perturbations.     

Stochastic density waves of granular flows: strong-intermittent dissipation fields with self-organization

The quantitative (scaling) results of a recent lattice-gas simulation of granular flows [1] are interpreted in terms of Kolmogorov-Obukhov approach revised for strong space-intermittent systems. Renormalised power spectrum with exponent '-4/3' seems to be an universal spectrum of scalar fluctuations convected by stochastic velocity fields in dissipative systems with inverse energy transfer (some other laboratory and geophysic turbulent flows with this power spectrum as well as an analogy between this phenomenon and turbulent percolation on elastic backbone are pointed out).     

Developing homogeneous isotropic turbulence

We investigate the self-similar evolution of the transient energy spectrum, which precedes the establishment of the Kolmogorov spectrum in homogeneous isotropic turbulence in three dimensions using the EDQNM closure model. The transient evolution exhibits self-similarity of the second kind and has a non-trivial dynamical scaling exponent, which results in the transient spectrum having a scaling that is steeper than the Kolmogorov k^−5/3 spectrum. Attempts to detect a similar phenomenon in DNS data are inconclusive, owing to the limited range of scales available.     

Kinetic Relaxation Models for Energy Transport

Kinetic equations with relaxation collision kernels are considered under the basic assumption of two collision invariants, namely mass and energy. The collision kernels are of BGK-type with a general local Gibbs state, which may be quite different from the Gaussian. By the use of the diffusive length/time scales, energy transport systems consisting of two parabolic equations with the position density and the energy density as unknowns are derived on a formal level. The H theorem for the kinetic model is presented, and the entropy for the energy transport systems, which is inherited from the kinetic model, is derived. The energy transport systems for specific examples of the global Gibbs state, such as a power law with negative exponent, a cut-off power law with positive exponent, the Maxwellian, Bose–Einstein, and Fermi–Dirac distributions, arepresented. MSC classification (2000): Primary: 82C40, 35B40; Secondary: 35K55, 45K05, 82D05, 85A05x     

Local multifractal thermodynamics of 3D turbulence

Using results of a direct numerical simulation (DNS) of 3D turbulence we show that the observed generalized scaling (i.e. scaling moments versus moments of different orders) is consistent with a lognormal-like distribution of turbulent energy dissipation fluctuations with moderate amplitudes for all space scales available in this DNS (beginning from the molecular viscosity scale up to largest ones). Local multifractal thermodynamics has been developed to interpret the data obtained using the generalized scaling, and a new interval of space scales with inverse cascade of generalized energy has been found between dissipative and inertial intervals of scales for sufficiently large values of the Reynolds number. Received 21 July 2000     

Influences of subgrid scale dynamics on resolvable scale statistics in large-eddy simulations

Recently, the -expansion and recursive renormalization group (RNG) theories as well as approximate inertial manifolds (AIM) have been exploited as means of systematically modeling subgrid scales in large-eddy simulations (LES). Although these theoretical approaches are rather complicated mathematically, their key approximations can be investigated using direct numerical simulations (DNS). In fact, the differences among these theories can be traced to whether they retain or neglect interactions between the subgrid-subgrid and subgrid-resolvable scales. In this paper, we focus on the influence of these two interactions on the evolution of the resolvable scales in LES: the effect^A which keeps only the interactions between the small and large scales; and, the effect^B which, on the other hand, keeps only the interactions among the subgrid-subgrid scales. The performance of these models is analyzed using the velocity fields of the direct numerical simulations. Specifically, our comparison is based on the analysis of the energy and enstrophy spectra, as well as higher-order statistics of the velocity and velocity derivatives. We found that the energy spectrum and higher-order statistics for the simulations with the effect^A (referred to, hereafter, as model^A) are in very good agreement with the filtered DNS. The comparison between the computations with effect^B (referred to, hereafter, as model^B) and the filtered DNS, however, is not satisfactory. Moreover, the decorrelation between the filtered DNS and model^A is much slower than that of the filtered DNS and model^B. Therefore, we conclude that the model^A, taking into acciunt the interactions between the subgrid and resolvable scales, is a faithful subgrid model for LES for the range of Reynolds numbers considered.     

Heavy fermion quantum criticality

During the last few years, investigations of rare-earth materials have made clear that heavy fermion quantum criticality exhibits novel physics not fully understood. In this work, we write for the first time the effective action describing the low energy physics of the system. The f fermions are replaced by a dynamical scalar field whose nonzero expected value corresponds to the heavy fermion phase. The effective theory is amenable to numerical studies as it is bosonic, circumventing the fermion sign problem. Via effective action techniques, renormalization group studies, and Callan-Symanzik resummations, we describe the heavy fermion criticality and predict the heavy fermion critical dynamical susceptibility and critical specific heat. The specific heat coefficient exponent we obtain (0.39) is in excellent agreement with the experimental result at low temperatures (0.4).