The Entrainment Rate for Buoyant Plumes in a Crossflow

Abstract

We consider large-eddy simulations (LES) of buoyant plumes from a circular source with initial buoyancy flux F ₀ released into a stratified environment with constant buoyancy frequency N and a uniform crossflow with velocity U. We make a systematic comparison of the LES results with the mathematical theory of plumes in a crossflow. We pay particular attention to the limits ${\tilde{U}\ll1}$ and ${\tilde{U}\gg 1}$, where ${\tilde{U}=U/(F_0 N)^{1/4}}$, for which analytical results are possible. For ${\tilde{U}\gg 1}$, the LES results show good agreement with the well-known two-thirds law for the rise in height of the plume. Sufficiently far above the source, the centreline vertical velocity of the LES plumes is consistent with the analytical z ^−1/3 and z ^−1/2 scalings for respectively ${\tilde{U}\ll 1}$ and ${\tilde{U}\gg 1}$. In the general case, where the entrainment is assumed to be the sum of the contributions from the horizontal and vertical velocity components, we find that the discrepancy between the LES data and numerical solutions of the plume equations is largest for ${\tilde{U}=O(1)}$. We propose a modified additive entrainment assumption in which the contributions from the horizontal and vertical velocity components are not equally weighted. We test this against observations of the plume generated by the Buncefield fire in the U.K. in December 2005 and find that the results compare favourably. We also show that the oscillations of the plume as it settles down to its final rise height may be attenuated by the radiation of gravity waves. For ${\tilde{U}\ll 1}$ the oscillations decay rapidly due to the transport of energy away from the plume by gravity waves. For ${\tilde{U}>rsim 1}$ the gravity waves travel in the same direction and at the same speed as the flow. In this case, the oscillations of the plume do not decay greatly by radiation of gravity waves.