Effect of Reynolds-number-dependent turbulence structure on dispersion of fluid and particles

The influence of the spatio-temporal structure of turbulence on the dispersions of fluid elements and particles with inertia is investigated. The spatial structure is represented by an extended von Karman energy spectrum model which is significantly more realistic than the Gaussian spectrum assumed in previous work and which allows evaluation of the effects of Reynolds number Reλ based on Taylor micro scale. Dispersion of fluid is analyzed using four different models for the Eulerian auto-correlation function D(i:): i) a Gaussian; ii) a simple exponential; iii) a composite form of Gaussian and exponential; and iv) the model proposed by Hunt et al. (1987). The first three models for D(i:) predict that the fluid diffusivity normalized by the integral length scale L 11 and the root-mean-squared turbulent velocity uo decreases as Reλ increases. The parameter cE=Touo/L11, in which To is the Eulerian integral time scale, has a significant effect on the fluid diffusivity. An approximate, empirical expression for cE as a function of Reλ is obtained by matching the predicted Lagrangian time scale (based on the composite form of D(i:)) with the experimental measurement of Sato & Yamamoto (hereinafter referred as SY, 1987) for low Re,._ turbulence and with the prediction based on renormalization group analysis of Yakhot & Orszag (hereinafter referred as YO, 1986) for high Reλ turbulence. The predicted fluid Lagrangian correlations are not exactly similar in shape for different values of Reλ; but they may be approximated by exponential function exp(-,;/TL) for simplicity. Particle dispersion with finite inertia and settling velocity is A analyzed. Four regions in the (β, λ) parameter space are distinguished, where β is the particle inertial parameter normalized by u0 (2π)112/L11 and 1.. is the ratio of the particle settling velocity VT to u0. They are: i) large settling rate; ii) small particle inertia and small settling rate; iii) large particle inertia and small settling rate; and iv) order-one values of β and λ. Simple asymptotic results are obtained in the first three regions and numerical integration is carried out for order-one β and λ. For cE given by the empirical expression, both particle intensity (a=l, 2) and diffusivity are influenced strongly by the turbulence structure. It is found that the ratio of particle diffusivity to fluid diffusivity is weakly dependent on Reλ for β~1 and β» 1. For β«l, the diffusivity ratio depends strongly on Reλ and can be as high as 2.22 for massive particles in high Reλ turbulence. The intensity ratio