What is the entrainment coefficient of a pure turbulent line plume?

Abstract

Despite its pivotal role in the classic theory of turbulent line plumes, there has been no consensus on the value of the entrainment coefficient $\alpha$ suitable for a pure plume. Reported measurements vary by 100 %, from $\alpha =0.1$ to $\alpha =0.2$, hindering the predictive capabilities of plume theory. Following our theoretical developments, measurements of plume entrainment using a new approach and a rigorous assessment of reported values for $\alpha$, we conclude that ${\alpha =0.11\pm 15\,\%}$ should be adopted as the consensus value. Our theoretical framework demonstrates how $\alpha$ is determined from underlying plume measurements, and places an emphasis on the link between measurement uncertainty and uncertainty in $\alpha$. This framework inspired our experimental design, intentionally conceived to precisely determine $\alpha$. From measurements of the plume scalar width and the entrainment velocity outside the plume, we determine that $\alpha =0.108\pm 2\,\%$ ($95\,\%$ confidence interval). Complementing our experiments is an evaluation of the historical data which, after we explain why some reported values of $\alpha$ are erroneous, supports the range $0.095\lesssim \alpha \lesssim 0.13$. The proposed consensus value thus represents both our precisely determined value and the variation in the published data. The significance of a consensus value for $\alpha$ can be summarised as follows: (i) it enhances confidence in the application of plume theory to practical situations and (ii) it permits more detailed comparison of entrainment between pure line plumes and related turbulent flows, including forced and lazy line plumes and wall plumes.