Deterministic and stochastic dynamics of Rayleigh–Taylor mixing with a power-law time-dependent acceleration

Abstract

Rayleigh–Taylor (RT) mixing occurs in a variety of natural and man-made phenomena in fluids, plasmas and materials, from celestial event to atoms. In many circumstances, RT flows are driven by variable acceleration, whereas majority of existing studies have considered only sustained acceleration. In this work we perform detailed analytical and numerical study of RT mixing with a power-law time-dependent acceleration. A set of deterministic nonlinear non-homogeneous ordinary differential equations and nonlinear stochastic differential equations with multiplicative noise are derived on the basis of momentum model. For a broad range of parameters, self-similar asymptotic solutions are found analytically, and their statistical properties are studied numerically. We identify two sub-regimes of RT mixing dynamics depending on the acceleration exponent—the acceleration-driven mixing and dissipation-driven mixing. Transition between the sub-regimes is studied, and it is found that each sub-regime has its own characteristic dimensionless invariant quantity.

Supernova, nuclear synthesis, fluid instabilities, and interfacial mixing

“…For solution ðh d , v d Þ, the exponent is set by the drag, ð2 + a cr Þ = ð1 + CÞ −1 , and the prefactor B d is set by the initial conditions. This mixing is RM type; it is driven by the drag (the dissipation, since μ = «=v) (35). For a > a cr the mixing is RT type.…”

“…For RT type, the acceleration sets the timescale at early stage and defines the nonlinear dynamics and the interfacial mixing at later stages. For RM type, the initial growth rate sets the timescale at early stage; at late stages, the drag defines the nonlinear dynamics and the interfacial mixing, and the initially supplied energy gradually dissipates (33,35). The critical values of the exponent at which the transition occurs from RT-to RM-type dynamics are distinct for the linear, nonlinear, and mixing regimes.…”

“…The critical values of the exponent at which the transition occurs from RT-to RM-type dynamics are distinct for the linear, nonlinear, and mixing regimes. Particularly for blast wave-induced accelerations, the linear and nonlinear dynamics are RT type, and the mixing is RM type, but RM-type mixing develops quicker than the acceleration prescribes (33,35). While for subdiffusive mixing dynamics, superdiffusive canonical turbulence may be a challenge to occur, other mechanisms are possible for energy accumulation at small scales.…”

“…For a < a cr , the mixing is RM type. At a ∼ a cr , a transition occurs from RT-to RM-type mixing (33,35).…”

“…Per observations, at a = 0, the imbalance is small, ðμ − μÞ=μ << 1 (51-53). Hence, RT mixing may develop when the amplitude h is the scale for energy dissipation, L ∼ h (14,33,35). It may also develop due to the growth of period λ ∼ Gt a+2 in the nonlinear regime.…”