Probability density function integrals and small-scale mixing in turbulent plumes

We analyse cross-wind-integrated statistics of the concentration field of a conserved scalar for point and line sources in grid turbulence. In particular, using wind-tunnel measurements we calculate the cross-wind integrated probability density function (pdf) for the scalar concentration. We then use that quantity in the exact evolution equation for the pdf to calculate the cross-wind integrated mean of the rate of dissipation of scalar variance, conditional on the scalar concentration. Much of the variation of these statistics with distance downstream is accounted for by scaling with concentration, length and time scales based on the development of the mean plume. This scaling thus suggests some simple practical parameterisations of these statistics in terms of mean-field quantities. One of the motivations for this work is to find a simple parameterisation for the scalar dissipation that can be used for modelling chemical reactions in plumes. We also consider the cross-wind integral of the first few moments of the concentration field and show that the integration greatly simplifies the budgets for these moments. Thus the first moment is just the downstream flux of the scalar, which is constant. The second moment budget provides a check on the mean dissipation estimated directly from the pdf evolution equation.