Susan Kurien

Hyperviscosity, Galerkin truncation and bottlenecks in turbulence

It is shown that the use of a high power alpha of the Laplacian in the dissipative term of hydrodynamical equations leads asymptotically to truncated inviscid conservative dynamics with a finite range of spatial Fourier modes. Those at large wave numbers thermalize, whereas modes at small wave numbers obey ordinary viscous dynamics [C. Cichowlas et al., Phys. Rev. Lett. 95, 264502 (2005)10.1103/Phys. Rev. Lett. 95.264502]. The energy bottleneck observed for finite alpha may be interpreted as incomplete thermalization. Artifacts arising from models with alpha>1 are discussed.

Recovering isotropic statistics in turbulence simulations: The Kolmogorov 4/5th law

One of the main benchmarks in direct numerical simulations of three-dimensional turbulence is the Kolmogorov prediction for third-order structure functions with homogeneous and isotropic statistics in the infinite Reynolds number limit. Previous direct numerical simulations (DNS) techniques to obtain isotropic statistics have relied on time-averaging structure functions in a few directions over many eddy-turnover times, using forcing schemes carefully constructed to generate isotropic data. Motivated by recent theoretical work, which removes isotropy requirements by spherically averaging the structure functions over all directions, we will present results which supplement long-time averaging by angle-averaging over up to 73 directions from a single flow snapshot. The directions are among those natural to a square computational grid, and are weighted to approximate the spherical average. We use this angle-averaging procedure to compare the statistically steady flows generated by two different forcing schemes in a periodic box. Our results show that despite the apparent differences in the two flows, their isotropic components, as measured by the Kolmogorov laws, are essentially identical. This procedure may be used to investigate the isotropic part of the small-scale statistics of any quantity of interest. The averaging process is inexpensive, and for the Kolmogorov 4/5 law, reasonable results can be obtained from a single snapshot of data. This implies consistency with the recently derived local versions of the Kolmogorov laws, which do not require long-time averages.

Measures of Anisotropy and the Universal Properties of Turbulence

Local isotropy, or the statistical isotropy of small scales, is one of the basic assumptions underlying Kolmogorov’s theory of universality of small-scale turbulent motion. The literature is replete with studies purporting to examine its validity and limitations. While, until the mid-seventies or so, local isotropy was accepted as a plausible approximation at high enough Reynolds numbers, various empirical observations that have accumulated since then suggest that local isotropy may not obtain at any Reynolds number. This throws doubt on the existence of universal aspects of turbulence. Part of the problem in refining this loose statement is the absence until now of serious efforts to separate the isotropic component of any statistical object from its anisotropic components. These notes examine in some detail the isotropic and anisotropic contributions to structure functions by considering their SO(3) decomposition. After an initial discussion of the status of local isotropy (Sect. 1) and the theoretical background for the SO(3) decomposition (Sect. 2), we provide an account of the experimental data (Sect. 3) and their analysis (Sects. 4–6). Viewed in terms of the relative importance of the isotropic part to the anisotropic parts of structure functions, the basic conclusion is that the isotropic part dominates the small scales at least up to order 6. This follows from the fact that, at least up to that order, there exists a hierarchy of increasingly larger power-law exponents, corresponding to increasingly higher-order anisotropic sectors of the SO(3) decomposition. The numerical values of the exponents deduced from experiment suggest that the anisotropic parts in each order roll off less sharply than previously thought by dimensional considerations, but they do so nevertheless.

Anisotropy of small-scale scalar turbulence

The anisotropy of small-scale temperature fluctuations in shear flows is analysed by making measurements in high-Reynolds-number atmospheric surface layers. A spherical harmonics representation of the moments of scalar increments is proposed, such that the isotropic part corresponds to the index j = 0 and increasing degrees of anisotropy correspond to increasing j . The parity and angular dependence of the odd moments of the scalar increments show that the moments cannot contain any isotropic part ( j = 0), but can be satisfactorily represented by the lowest-order anisotropic term corresponding to j = 1. Thus, the skewnesses of scalar increments (and derivatives) are inherently anisotropic quantities, and are not suitable indicators of the tendency towards isotropy.

Generation of anisotropy in turbulent flows subjected to rapid distortion

A computational tool for the anisotropic time-evolution of the spectral velocity correlation tensor is presented. We operate in the linear, rapid distortion limit of the mean-field-coupled equations. Each term of the equations is written in the form of an expansion to arbitrary order in the basis of irreducible representations of the SO(3) symmetry group. The computational algorithm for this calculation solves a system of coupled equations for the scalar weights of each generated anisotropic mode. The analysis demonstrates that rapid distortion rapidly but systematically generates higher-order anisotropic modes. To maintain a tractable computation, the maximum number of rotational modes to be used in a given calculation is specified a priori. The computed Reynolds stress converges to the theoretical result derived by Batchelor and Proudman [Quart. J. Mech. Appl. Math. 7, 83 (1954)QJMMAV0033-561410.1093/qjmam/7.1.83] if a sufficiently large maximum number of rotational modes is utilized; more modes are required to recover the solution at later times. The emergence and evolution of the underlying multidimensional space of functions is presented here using a 64-mode calculation. Alternative implications for modeling strategies are discussed.