Self-similarity In Turbulence Research Articles

Self-similarity in turbulence and its applications.

First, we discuss the non-Gaussian type of self-similar solutions to the Navier-Stokes equations. We revisit a class of self-similar solutions which was studied in Canonne et al. (1996 Commun. Partial. Differ. Equ. 21, 179-193). In order to shed some light on it, we study self-similar solutions to the one-dimensional Burgers equation in detail, completing the most general form of similarity profiles that it can possibly possess. In particular, on top of the well-known source-type solution, we identify a kink-type solution. It is represented by one of the confluent hypergeometric functions, viz. Kummer's function [Formula: see text]. For the two-dimensional Navier-Stokes equations, on top of the celebrated Burgers vortex, we derive yet another solution to the associated Fokker-Planck equation. This can be regarded as a 'conjugate' to the Burgers vortex, just like the kink-type solution above. Some asymptotic properties of this kind of solution have been worked out. Implications for the three-dimensional (3D) Navier-Stokes equations are suggested. Second, we address an application of self-similar solutions to explore more general kind of solutions. In particular, based on the source-type self-similar solution to the 3D Navier-Stokes equations, we consider what we could tell about more general solutions. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Scaling hypothesis leading to generalized extended self-similarity in turbulence.

A scaling hypothesis leading to generalized extended self-similarity (GESS) for velocity structure functions, valid for intermediate scales in isotropic, homogeneous turbulence, is proposed. By introducing an effective scale ŕ, monotonically depending on the physical scale r, with the use of the large deviation theory, the asymptotic forms of the probability densities for the velocity differences u(r) and for the coarse-grained energy-dissipation rate fluctuations epsilon(r), compatible with this GESS, are proposed. The probability density for epsilon(r) is shown to have the form P(r)(epsilon) approximately equal to epsilon(-1)(ŕ/L)(S(ŕ)[z(ŕ)](epsilon))) with z(ŕ)(epsilon)=ln(epsilon/epsilon(L))/ln(L/ŕ), where L and epsilon(L) are the stirring scale and the coarse-grained energy-dissipation rate over the scale L. The concave function S(ŕ)(z), the spectrum, plays the central role of the present approach. Comparing the results with numerical and experimental data, we explicitly obtain the fluctuation spectra S(ŕ)(z).

Mean-field approximation and extended self-similarity in turbulence.

Recent experimental discovery of extended self-similarity (ESS) was one of the most interesting developments, enabling precise determination of the scaling exponents of fully developed turbulence. A sufficient condition for extended self-similarity in a general dynamical system is derived in this paper. It is also shown that if the pressure-gradient contributions are expressed in terms of velocity differences in the mean-field approximation [V. Yakhot, Phys. Rev. E 63, 026307 (2001)], then the ESS is a consequence of the Navier-Stokes equations.

Scaling hypothesis leading to extended self-similarity in turbulence.

A scaling hypothesis leading to extended self-similarity (ESS) for structure functions (the qth order moments of the magnitude of the longitudinal component velocity differences) in isotropic, homogeneous turbulence is proposed. This is done by generalizing the scale variable r to rg(r/L), with a crossover function g. By extending the refined self-similarity, it is shown that the presented scaling also leads to ESS for structure functions of the energy dissipation rate fluctuations, and to ESS bridging relations between velocity and dissipation rate moments. Extended self-similarity on the basis of a universal crossover function g strictly holds toward the outer scale (L) range only. Yet we find at least approximate ESS toward the viscous, inner scale (l) range. Furthermore, the probability densities for the velocity differences and the energy dissipation rate fluctuations which are compatible with this ESS are offered.

Application of extended self-similarity in turbulence

From Navier-Stokes turbulence numerical simulations we show that for the extended self-similarity (ESS) method it is essential to take the third order structure function taken with the modulus and called D*3(r), rather than the standard third order structure function D3(r) itself. If this is done, we find ESS towards scales larger than order ~10 eta , where eta is the Kolmogorov scale. If D3(r) is used, there is no ESS. We also analyze ESS within the Batchelor parametrization of the second and third order longitudinal structure function and focus on the scaling of the transversal structure function. The Re-asymptotic inertial range scaling develops only beyond a Taylor-Reynolds number Re lambda >~ 500.

A Study of Self-Similarity in Turbulence Measured at an Urban Site

Abstract The self-similarity concept has proved useful in describing turbulence for aircraft design purposes. There is an increasing awareness of the interaction of the turbulent planetary boundary layer and buildings and structures. The aeronautical engineering approach could be extended to civil engineering design problems. Wind data from a 118 m tower at an urban site in Melbourne are used to test the appropriateness of the “self-similar” approach. A reasonable model for the site was devised, and it is conjectured that this may form the basis for a “discrete gust” scheme to allow for aerodynamic interactions in the planetary boundary layer.