Multiscale similarity of isotropic homogeneous turbulence

Abstract

A similarity form of energy spectrum involving two lengthscales, the Kolmogorov dissipation lengthscale (η) and the empirical energy-containing lengthscale (£), is proposed for isotropic homogeneous incompressible turbulence. The similarity form is approached by considering the energy spectrum as a summation of a series of ordered functions of variable k£ dominant at small wave numbers and a series of ordered functions of variable kη dominant at large wave numbers. With a corresponding energy transfer spectrum, dynamical equations for these associated expansion functions can be derived from the spectral energy equation. The results of analysis show that the expansion functions may be ordered in powers of \({R_{e\lambda }}^{ - 1/2}\), where Reλ is the Taylor’s Reynolds number and that a power-law energy decay is a necessary condition for similarity to exist. A reasonable Reynolds-number dependence of the skewness of velocity longitudinal derivatives is predicted. Constraints on the expansion functions due to definitions are discussed. An example energy spectrum is constructed and the resulting energy transfer spectrum is investigated.