Probability density function (PDF) calculations are reported for the dispersion from line sources in decaying grid turbulence. The calculations are performed using a modified form of the interaction by exchange with the conditional mean (IECM) mixing model. These flows pose a significant challenge to statistical models because the scalar length scale (of the initial plume) is much smaller than the turbulence integral scale. Consequently, this necessitates incorporating the effects of molecular diffusion in order to model laboratory experiments. Previously, Sawford [Flow Turb. Combust. 72, 133 (2004)] performed PDF calculations in conjunction with the IECM mixing model, modeling the effects of molecular diffusion as a random walk in physical space and using a mixing time scale empirically fit to the experimental data of Warhaft [J. Fluid Mech. 144, 363 (1984)]. The resulting transport equation for the scalar variance contains a spurious production term. In the present work, the effects of molecular diffusion are instead modeled by adding a conditional mean scalar drift term, thus avoiding the spurious production of scalar variance. A laminar wake model is used to obtain an analytic expression for the mixing time scale at small times, and this is used as part of a general specification of the mixing time scale. Based on this modeling, PDF calculations are performed, and comparison is made primarily with the experimental data of Warhaft on single and multiple line sources and with the previous calculations of Sawford. A heated mandoline is also considered with comparison to the experimental data of Warhaft and Lumley [J. Fluid Mech. 88, 659 (1978)]. This establishes the validity of the proposed model and the significant effect of molecular diffusion on the decay of scalar fluctuations. The following are the significant predictions of the model. For the line source, the effect of the source size is limited to early times and can be completely accounted for by simple transformations. The peak centerline ratio of the rms to the mean of the scalar increases with the Reynolds number (approximately as ), whereas this ratio tends to a constant (approximately 0.4) at large times independent of . In addition, the model yields a universal long-time decay exponent for the temperature variance.
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It is to be noted that for the IECM model there exists a mixing rate which yields the correct evolution of the integral scalar variance. But the modified IECM model, in addition to predicting the correct evolution of the integral scalar variance, also yields scalar variance profiles that are in agreement with the laminar thermal wake model at early times (whereas the unmodified IECM model does not).
