Rayleigh–Taylor Flow with Two Interfaces: The Completed Boussinesq Approximation

A system of three horizontal fluid layers is considered, with two interfaces separating them. When the upper fluids are of higher density, the system is unstable and Rayleigh–Taylor instabilities occur, as interfacial disturbances grow with time and the fluids overturn. A linearized solution is presented for the corresponding inviscid problem. It reveals a neutrally stable situation when the fluid densities decrease with height. However, whenever a high density fluid lies above a less dense one, the linearized solution predicts exponential growth of the interface between them. With two interfaces present, several different flow scenarios are possible, depending on the two density ratios between the three fluids The interfacial waves can occur either in a sinuous or a varicose formation. A semi-numerical spectral method is used to obtain nonlinear solutions for three-layer viscous fluids, using a recently-published “Completed Boussinesq Approximation”. These nonlinear results are compared with the linearized inviscid solution and also with interface shapes obtained from an SPH algorithm. Results are shown for sinuous and varicose solution types, and inversion layer flows are discussed.