Phase-field model for the Rayleigh–Taylor instability of immiscible fluids

Abstract

The Rayleigh-Taylor instability of two immiscible fluids in the limit of small Atwood numbers is studied by means of a phase-field description. In this method the sharp fluid interface is replaced by a thin, yet finite, transition layer where the interfacial forces vary smoothly. This is achieved by introducing an order parameter (the phase field) whose variation is continuous across the interfacial layers and is uniform in the bulk region. The phase field model obeys a Cahn-Hilliard equation and is two-way coupled to the standard Navier-Stokes equations. Starting from this system of equations we have first performed a linear analysis from which we have analytically rederived the known gravity-capillary dispersion relation in the limit of vanishing mixing energy density and capillary width. We have performed numerical simulations and identified a region of parameters in which the known properties of the linear phase (both stable and unstable) are reproduced in a very accurate way. This has been done both in the case of negligible viscosity and in the case of nonzero viscosity. In the latter situation only upper and lower bounds for the perturbation growth-rate are known. Finally, we have also investigated the weaklynonlinear stage of the perturbation evolution and identified a regime characterized by a constant terminal velocity of bubbles/spikes. The measured value of the terminal velocity is in perfect agreement with available theoretical prediction. The phase-field approach thus appears to be a valuable tecnhique for the dynamical description of the stages where hydrodynamic turbulence and wave-turbulence enter into play.

Incompressible Rayleigh–Taylor Turbulence

“…We cite surface tension and viscosity (see, e.g., Bellman & Pennington (1954); Mikaelian (1993) ;Chertkov et al (2005); Celani et al (2009) ;Bo↵etta et al (2010c)), magnetic fields (Kruskal & Schwarzschild 1954;Peterson et al 1996), spherical geometries (Plesset 1954;Sakagami & Nishihara 1990), finite-amplitude perturbations (Chang 1959), bubbles (Garabedian 1957;Hecht et al 1994;Goncharov 2002), rotation (Chandrasekhar 1961;Baldwin et al 2015), and compressibility (Newcomb 1983;Scagliarini et al 2010).…”

Linear stability analysis of a horizontal phase boundary separating two miscible liquids

“…[35,37] proved through direct numerical simulations that the classical observations, which were previously obtained on the basis of the Laplace approach, could be successfully reproduced. In [38], the classical dispersion relations for the Rayleigh-Taylor instability were rederived from the phasefield governing equations for immiscible interfaces in the limit of vanishing interface thickness.…”

Lattice Boltzmann methods for thermal flows: Continuum limit and applications to compressible Rayleigh–Taylor systems

“…In particular, it is simple to extend such algorithm to deal with fully 3d systems for ideal, non-ideal and/or even immiscible two fluids systems. High resolution studies of Rayleigh-Taylor systems meant to investigate short wavelengths scaling properties of velocity, density and temperature fields for high Rayleigh, with and without surface tension [39], and using a highly optimized LBM algorithm for the Cell Broadband Engine [75] are under current investigation and will be reported elsewhere [76]. The thermal LBM here proposed still suffers of small spurious oscillations of temperature and perpendicular velocity close to the solid boundaries, making it still not appropriate to study high Rayleigh numbers stationary convection.…”