Transport Element Method for Axisymmetric Variable-density Flow and Its Application to the Spread and Dispersion of a Dense Cloud

The Lagrangian transport element method for the solution of axisymmetric constant-density ow is extended to variable-density ow. In this method, thin vortex ring-elements of the Rankine type are used to discretize the vorticity eld as well as the density gradient eld. At the axis, since the ring element concept does not apply, several Hill's spherical vortices are used. A simple algebraic expression for determining the density gradient is derived. Using this expression, the rate of baroclinic vorticity generation in the vorticity equation can be easily obtained. The transport element method, free of using Boussinesq approximation and turbulent closure modeling, is used to simulate variable-density ows such as the spread and dispersion of an axisymmetric dense gas cloud following its sudden release. The results show that multiple large scale concentric rings are generated at the cloud surface. Most of the cloud material is associated with the spreading front where the largest and strongest ring is located. The generation of these rings is explained using the baroclinic-vorticity generation mechanism. The entrainment of surrounding air into the dense cloud is shown to occur mainly at the cloud top through the large-scale engulfment by the vortex rings. The results on the cloud dispersion pattern and the rate of the horizontal spreading of the cloud compare well with experimental observations.