A turbulent diffusion model of Rayleigh-Taylor mixing

A simple turbulent diffusion model is described for predicting late term mixing due to Rayleigh-Taylor instability at an interface that initially separates two semi-infinite regions of fluid of different densities. The one-dimensional, transient species conservation equation is combined with an available vertical dispersion equation which relates the turbulent diffusivity to the local density gradient and a characteristic mixing length. The similarity solution of the species conservation equation yields the well-known long time result for the vertical extent of the mixing region and shows that the dimensionless growth constant that appears in the mixing thickness equation is simply a measure of the size of the characteristic mixing length. A simple polynomial relation is derived for the heavy fluid volume fraction profile within the mixing region which is in close agreement with the profile obtained from a previously reported numerical simulation of Rayleigh-Taylor mixing. It is demonstrated how the model can be readily applied to predict Rayleigh-Taylor mixing in a finite fluid region between top and bottom boundaries.