Abstract
We first develop a new mathematical model for two-fluid interface motion, subjected to the Rayleigh--Taylor (RT) instability in two-dimensional fluid flow, which in its simplest form, is given by $ h_{tt}( \alpha ,t) = A g\, \Lambda h - \frac{ \sigma }{ \rho^++\rho^-} \Lambda^3 h - A \text{\bf\emph{p}}_\alpha(H h_t h_t) $, where $\Lambda = H \text{\bf\emph{p}}_ \alpha $ and $H$ denotes the Hilbert transform. In this so-called $h$-model, $A$ is the Atwood number, $g$ is the acceleration, $ \sigma $ is surface tension, and $\rho^\pm$ denotes the densities of the two fluids. We derive our $h$-model using asymptotic expansions in the Birkhoff--Rott integral-kernel formulation for the evolution of an interface separating two incompressible and irrotational fluids. The resulting $h$-model equation is shown to be locally and globally well-posed in Sobolev spaces when a certain stability condition is satisfied; this stability condition requires that the product of the Atwood number and the initial velocity field ...
Highlights
The instability of a heavy fluid layer supported by a light one is generally known as Rayleigh–Taylor (RT) instability
In the unstable case of a heavier fluid being supported by the lighter fluid, we find good agreement for the growth of the mixing layer with experimental data in the “rocket rig” experiment of Read and Youngs
The Euler equations of inviscid hydrodynamics serve as the basic mathematical model for RT instability and mixing between two fluids
Summary
The instability of a heavy fluid layer supported by a light one is generally known as Rayleigh–Taylor (RT) instability (see Rayleigh [12] and Taylor [17]). The equation does not require a stability condition for a short-time existence theorem We shall investigate this further in future work. Our hmodel equation (24) for the evolution of the height function h(α, t) uses a special parameterization in which the interface Γ(t) is constrained to be the graph (α, h(α, t)) While this model works well in predicting the mixing layer, in the unstable regime and when the RT instability is initiated, the height function h(α, t) can only grow in amplitude. The parameter s determines the order of the operator; for example, for s = 2, we recover the classical Laplace operator, while for s > 2, we can study a variety of hyperviscosity operators We believe that this equation will be an ideal candidate for the C-method artificial viscosity which is localized in both space and time (see [14]), and shall implement this in future work.
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