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Applied Math Seminar A nonlinear evolution of an unstable fluid interface |
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The Rayleigh-Taylor instability (RTI) occurs at the interface between two fluids, when a light fluid accelerates a heavy fluid. The turbulent mixing produced by the instability is of extreme importance in inertial confinement fusion, plasmas, astrophysics, flames, and many other applications. The dynamics of RTI is governing by a system of conservation laws, which are nonlinear partial differential equations. The singular aspects of the interface evolution cause theoretical difficulties and preclude elementary methods of solution. The evolution of the large-scale coherent structure, the dynamics of small-scale structures, and the cascades of energy are the fundamental issues to be understood. We suggest a new theoretical approach to the problem. It is based on group theory and applies a separation of scales in the governing equations. This approach allows one to account for non-local properties of the flow that has singularities, and to obtain regular asymptotic solutions describing the nonlinear coherent motion. The analysis determines the number of key properties of the RT flows (universality of the three-dimensional dynamics, discontinuity of the dimensional crossover, dependence of the motion on the symmetry of the flow as well as on the acceleration type), and it is applicable for a one and two-fluid systems. Our results show that a balance between the inverse and direct cascades is required to keep isotropy of the coherent structure. The theory explains existing observations, finds reliable diagnostic parameters for simulations and experiments, and predicts a new type of the interface evolution. |