Symmetry and Scaling of Turbulent Mixing

We presently show that the infinite set of multi-point correlation equations, which are direct statistical consequences of the Navier-Stokes equations, admit a rather large set of Lie symmetry groups. This set is considerable extended compared to the set of groups which are implied from the original set of equations of fluid mechanics. Specifically a new scaling group and translational groups of the correlation vectors and all independent variables have been discovered. These new statistical groups have important consequences on our understanding of turbulent scaling laws to be exemplarily revealed by two examples. Firstly, one of the key foundations of statistical turbulence theory is the universal law of the wall with its essential ingredient is the logarithmic law. We demonstrate that the log-law fundamentally relies on one of the new translational groups. Furthermore, we consider a rotating channel flow, whose scaling behavior can only be described using the new statistical symmetries. It can be seen that the direction of rotation axes plays an important role, because different axes result in very different scaling laws.

Fluid mechanics and its applications, 1998

Mixing of a passive scalar, e, by a turbulent flow is a problem of obvious practical importance, and provides an interesting way to test our understanding of turbulence itself [1]. Theoretical considerations, inspired by the Kolmogorov K41 theory [2], suggest that the second order structure function behaves as r2/3, as indeed found experimentally [3]. Higher order structure functions show significant departures from the predictions of the K41 theory. A particularly strong effect has been observed in a weakly heated turbulent boundary layers. The temperature signal shows a ramp like structure [4] resulting in a strong assymetry in the scalar derivative, and in a skewness of order 1 of (8x e), x being the coordinate along the downstream direction. The third order structure function, < (e(x)e(0))3 > behaves as X A 3, A3 ~ 1, whereas considerations based on the K41 phenomenology would rather suggest a value of order 5/3. A similar anisotropy of the scalar derivatives has been observed experimentally [5] and numerically [6, 7] when a large scale gradient is imposed. The numerical results suggest that the effect does not crucially depend on the statistical properties of the velocity field. This suggests that much can be understood with the help of simplified models. Mixing by a random, Gaussian, white in time velocity field, scaling in space with a power law: < va(r', t)Vb(T', t') >= C~b(Tr'o(tt'), with: a b D!b(T) == (C~b(O)-C~b(T)) = Do«d + 1-€)Oab-(2-€) 1~2)1T1 2-E (1) (d is the space dimension) has been introduced by Kraichnan [8, 9]. In this problem, the steady state N-poiJ;lt correlation function obeys a closed (Hopf) equation of the form [10] : 577

Physical Review Letters, 2005

We analyze multi-point correlation functions of a tracer in an incompressible flow at scales far exceeding the scale L at which fluctuations are generated (quasi-equilibrium domain) and compare them with the correlation functions at scales smaller than L (turbulence domain). We demonstrate that the scale invariance can be broken in the equilibrium domain and trace this breakdown to the statistical integrals of motion (zero modes) as has been done before for turbulence. Employing

Europhysics Letters (EPL), 2002

Physics of Fluids, 2005

We present direct numerical simulations of the mixing of the passive scalar at modest Taylor microscale ͑10ഛ R ഛ 42͒ and Schmidt numbers larger than unity ͑2 ഛ Scഛ 32͒. The simulations resolve below the Batchelor scale up to a factor of 4. The advecting turbulence is homogeneous and isotropic, and is maintained stationary by stochastic forcing at low wave numbers. The passive scalar is rendered stationary by a mean scalar gradient in one direction. The relation between geometrical and statistical properties of scalar field and its gradients is examined. The Reynolds numbers and Schmidt numbers are not large enough for either the Kolmogorov scaling or the Batchelor scaling to develop and, not surprisingly, we find no fractal scaling of scalar level sets, or isosurfaces, in the intermediate viscous range. The area-to-volume ratio of isosurfaces reflects the nearly Gaussian statistics of the scalar fluctuations. The scalar flux across the isosurfaces, which is determined by the conditional probability density function ͑PDF͒ of the scalar gradient magnitude, has a stretched exponential distribution towards the tails. The PDF of the scalar dissipation departs distinctly, for both small and large amplitudes, from the log-normal distribution for all cases considered. The joint statistics of the scalar and its dissipation rate, and the mean conditional moment of the scalar dissipation, are studied as well. We examine the effects of coarse-graining on the probability density to simulate the effects of poor probe-resolution in measurements.

Physical review, 1999

A recently proposed evolution equation ͓Vaienti et al., Physica D 85, 405 ͑1994͔͒ for the probability density functions ͑PDF's͒ of turbulent passive scalar increments obtained under the assumptions of fully threedimensional homogeneity and isotropy is submitted to validation using direct numerical simulation ͑DNS͒ results of the mixing of a passive scalar with a nonzero mean gradient by a homogeneous and isotropic turbulent velocity field. It is shown that this approach leads to a quantitatively correct balance between the different terms of the equation, in a plane perpendicular to the mean gradient, at small scales and at large Péclet number. A weaker assumption of homogeneity and isotropy restricted to the plane normal to the mean gradient is then considered to derive an equation describing the evolution of the PDF's as a function of the spatial scale and the scalar increments. A very good agreement between the theory and the DNS data is obtained at all scales. As a particular case of the theory, we derive a generalized form for the well-known Yaglom equation ͑the isotropic relation between the second-order moments for temperature increments and the third-order velocity-temperature mixed moments͒. This approach allows us to determine quantitatively how the integral scale properties influence the properties of mixing throughout the whole range of scales. In the simple configuration considered here, the PDF's of the scalar increments perpendicular to the mean gradient can be theoretically described once the sources of inhomogeneity and anisotropy at large scales are correctly taken into account. ͓S1063-651X͑99͒02108-X͔

Physical Review Letters, 1999

We address scaling in inhomogeneous and anisotropic turbulent flows by decomposing structure functions into their irreducible representation of the SO(3) symmetry group which are designated by j, m indices. Employing simulations of channel flows with Re λ ≈ 70 we demonstrate that different components characterized by different j display different scaling exponents, but for a given j these remain the same at different distances from the wall. The j = 0 exponent agrees extremely well with high Re measurements of the scaling exponents, demonstrating the vitality of the SO(3) decomposition.

Theories of Turbulence, 2002

First a short introduction to the notion of symmetries of differential equations is given including infinitesimal transformations, invariant functions and invariant solutions. Then it is shown that the symmetry properties i.e. invariant transformations of the Navier-Stokes equations are pivotal to understand the physics of fluid flow. We demonstrate that all common symmetries "transfer" to the statistical equations such as the Reynolds stress transport equations or the multi-point correlation equations. From the knowledge of the symmetries we derive from the latter equations a broad variety of invariant solutions (scaling laws) using only first principles. These solutions comprise classical results such as the logarithmic-law-of-the-wall and other wall bounded shear flows. Also homogeneous and inhomogeneous time-dependent flows are analyzed and solutions are discussed. Since the symmetries of fluid motion are admitted by all statistical quantities of turbulent flows we give necessary conditions on turbulence models such that they "capture" the proper physics i.e. the symmetries and their corresponding invariant solutions. Particularly we will investigate two-equation models such as the kmodel as well as Reynolds stress transport models with respect to their symmetry properties. Finally we give conditions for the sub-grid scale model in large-eddy simulation of turbulence to obey the proper symmetries. For all of the latter turbulence models it is demonstrated that symmetry violation gives rather disadvantageous prediction capabilities of the model under investigation.

Physical Review Letters, 2001

We present the first measurements of anisotropic statistical fluctuations in perfectly homogeneous turbulent flows. We address both problems of intermittency in anisotropic sectors and hierarchical ordering of anisotropies on a direct numerical simulation of a three dimensional random Kolmogorov flow. We achieved an homogeneous and anisotropic statistical ensemble by randomly shifting the forcing phases. We observe high intermittency as a function of the order of the velocity correlation within each fixed anisotropic sector and a hierarchical organization of scaling exponents at fixed order of the velocity correlation at changing the anisotropic sector. 47.27.Nz, 47.27.Ak At the basis of Kolmogorov 1941 theory there is the idea of restoring of universality and isotropy at small scales in turbulent flows. Memory of large scale anisotropic forcing and/or boundary conditions should be quickly lost during the process of energy transfer toward small scales. The overall result being a local recovering of isotropy and universality for turbulent fluctuations at small enough scales and large enough Reynolds numbers.

2014

A detailed theoretical investigation is given which demonstrates that a recently proposed statistical scaling symmetry is physically void. Although this scaling is mathematically admitted as a unique symmetry transformation by the underlying statistical equations for incompressible Navier-Stokes turbulence on the level of the functional Hopf equation, by closer inspection, however, it leads to physical inconsistencies and erroneous conclusions in the theory of turbulence. The new statistical symmetry is thus misleading in so far as it forms within an unmodelled theory an analytical result which at the same time lacks physical consistency. Our investigation will expose this inconsistency on different levels of statistical description, where on each level we will gain new insights for its non-physical transformation behavior. With a view to generate invariant turbulent scaling laws, the consequences will be finally discussed when trying to analytically exploit such a symmetry. In fact, a mismatch between theory and numerical experiment is conclusively quantified. We ultimately propose a general strategy on how to not only track unphysical statistical symmetries, but also on how to avoid generating such misleading invariance results from the outset. All the more so as this specific study on a physically inconsistent scaling symmetry only serves as a representative example within the broader context of statistical invariance analysis. In this sense our investigation is applicable to all areas of statistical physics in which symmetries get determined in order to either characterize complex dynamical systems, or in order to extract physically useful and meaningful information from the underlying dynamical process itself.