Effects of turbulence structure, molecular diffusion and source size on scalar fluctuations in homogeneous turbulence

A Lagrangian stochastic model of two-point displacements which includes explicitly the effects of molecular diffusion and viscosity is developed from the marked-particle model of Durbin (1980) and used to study the influence of these molecular processes on scalar fluctuations in stationary homogeneous turbulence. It is shown that for the homogeneous scalar field resulting from a uniform-gradient source distribution or for a cloud produced by a source large compared with the Kolmogorov microscale, the variance of scalar fluctuations $\overline{\theta^{\prime 2}}$ is independent of the molecular diffusivity for large Reynolds number Re provided that the Prandtl number Pr is finite. In these circumstances $\overline{\theta^{\prime 2}}$ can be calculated from marked-particle-pair statistics. For source sizes that are not large compared with the Kolmogorov microscale, $\overline{\theta^{\prime 2}}$ depends explicitly on the effect of molecular diffusion under the action of the straining motion on the Kolmogorov microscale. The model is consistent with Saffman's (1960) calculation for the mean concentration near such a small source and with Townsend's (1951) measurements on small heat spots. For the fluctuation field in stationary homogeneous turbulence these small-scale processes remain important far from the source. It is shown that dissipation of $\overline{\theta^{\prime 2}}$ is intimately connected to the process of relative dispersion and that the non-dissipative model for $\overline{\theta^{\prime 2}}$ discussed by Corrsin (1952) and Chatwin & Sullivan (1979) corresponds to the point-sample, infinite- Pr limit of the two-point theory. The two-point model is also used to examine the effect of instrumental averaging on $\overline{\theta^{\prime 2}}$ . For finite Pr , the reduction in $\overline{\theta^{\prime 2}}$ due to a fixed sampling volume is eventually negligible because the lengthscale of $\overline{\theta^{\prime 2}}$ grows with time. For infinite Pr , the linear-strain field of the turbulence on the Kolmogorov microscale generates an increasingly fine-scale structure in the scalar field. Then a finite sampling volume reduces $\overline{\theta^{\prime 2}}$ but not as much as the effect of reducing Pr to a finite value. A finite sampling volume and infinite Pr is not necessarily equivalent to a finite value of Pr and a point sample. Timescales for important stages in the development of the scalar field in a cloud or downwind of a continuous source (for example, the onset of dissipation or the stage at which fluctuations are dominated by internal structure or ‘streakiness’ within the cloud rather than bulk motion or ‘meandering’) have been estimated. For small sources and large Re these timescales are significantly less than the integral timescale t L . Many real flows evolve on the timescale t L , so that the present results for stationary homogeneous turbulence should apply to such flows for small sources and large Re , a situation typical of the atmosphere.