Relative Behaviour of Velocity and Scalar Structure Functions in Turbulent Flows

Abstract

This chapter reviews in a critical manner the existing analytical framework for describing the behaviour of velocity and scalar structure functions in turbulent flows. The assumptions which underpin this framework are only likely to be validated at very large Reynolds numbers and for relatively homogeneous and isotropic flows. These conditions are unlikely to apply in the laboratory. The major emphasis is on the likely dependence of second-order structure functions (or equivalently spectra) on both the Taylor micro-scale Reynolds number R_λ and other parameters, such as the large scale anisotropy or the dissipation timescale ratio or, more generally, the initial conditions of the flow. Measurements strongly indicate that the influence of R_λ and of the other parameters cannot be ignored. The retention of the non-homogeneity of the flow in the Navier-Stokes and heat transport equations provides a better idea of how large the magnitude of R_λ should be before the “asymptotic” results of Kolmogorov and Yaglom may be attained. Special attention is given to a suitable framework which allows velocity and scalar fluctuations to be compared meaningfully. The analogy between scalar and energy structure functions (or spectra) appears to work well for flows with a continuous injection of turbulent energy and scalar variance.