Strain dynamics for vortex ring mixing process

Simultaneous PIV-PLIF measurements were carried out to investigate the mixing occurring in a laminar vortex ring flow during the formation stage (Re=357-1072). In the first part of the work a control volume analysis was used to determine the variation in time of the scalar concentration mean, variance, and probability density function. In the second part the advection-diffusion differential equations of the scalar, $\xi$, and of its energy, 0.5 $\xi^{2}$, were studied in depth to gain insight into the effect of the strain rate tensor, \textbf{S}, on the local scalar concentration for increasing \textit{Re}. The measurements were obtained with a high spatial resolution (12 $\mu$m for the PLIF) in order to resolve the scalar dissipative scales. Reliable estimates of the scalar dissipation rate ($\nabla \xi \cdot \nabla \xi$), and of the symmetric contraction term ($\nabla \xi \cdot \textbf{S} \cdot \nabla \xi$), shown in equation 1, were obtained. $\nabla \xi \cdot \textbf{S} \cdot \nabla \xi$ accounts for the reduction of scalar dissipation due to the straining component directed as the local scalar gradient (see Southerland et al.\footnote{Southerland K B., Porter III J. R., Dahm, W. J. A., Buch K. A., An experimental study of the molecular mixing process in an axisymmetric laminar vortex ring, Phys. Fluids A 3 (5), May 1991}) Equation 1: $\left( \frac{\partial }{\partial t}+\vec {u}.\nabla +\frac{1}{ReSc}{\nabla ^2} \right)\frac{1}{2}\left( {\nabla \xi .\nabla \xi } \right)=-\left( {\nabla \xi .S.\nabla \xi } \right)-\frac{1}{ReSc}\nabla (\nabla \xi ):\nabla (\nabla \xi$)