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RAYLEIGH−TAYLOR INSTABILITY IN TWO-FLUID AND STRATIFIED MEDIA

Sergey N. Yakovenko
Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Institutskaya Street, 4/1, Novosibirsk 630090, Russia

Abstract

Two-fluid interface evolution is studied by direct simulations of Navier−Stokes and volume-fraction equations. To capture surface tension, the continuum surface force model is used where the mollified volume-fraction function varies smoothly across the interface due to convolving the original function with the eight-order polynomial smooth kernel. Tests of Rayleigh−Taylor instability show that the average of spike and bubble amplitudes has an initial exponential growth, corresponding to a linear instability stage with the constant growth rate. Evolution of this rate shows that both viscosity and surface tension effects damp the instability development in agreement with theory and measurements data. For real fluids (water-air) good prediction is obtained for both linear instability and nonlinear stages. If density difference across the interface is not so large, the nonlinear stage shows Kelvin−Helmholtz instability effects leading to typical mushroom-like structures. For large density difference (e.g. for water-air interface), the heavier fluid penetrates deeply into the lighter one and forms high columns. Surface tension omission gives spurious distortion of the interface, then its fragmentation. The vortex-sheet method yields under-estimation of the growth rate. Application of the continuum surface force model leads to the correct interface evolution within the experiments data scatter. The similar convective structures as for two-fluid flows with moderate density difference are observed in direct numerical simulations of a single-phase stably stratified flow above the obstacle (when overturning/breaking internal gravity waves produce unstable layers with strong density gradients) and provide a source of quasi-steady turbulent patches. To define the buoyancy terms in the Navier−Stokes equations with the Boussinesq approximation, the density-deviation equation is used. The overall behavior and the wavelength of dominant structures evaluated from the simulation results correspond to the second (non-linear) stage of the scenarios for the Rayleigh−Taylor instability evolution when memory of initial conditions is lost.

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