Fully 3d Rayleigh–taylor Instability in a Boussinesq Fluid

Abstract

Rayleigh-Taylor instability occurs when a heavier fluid overlies a lighter fluid, and the two seek to exchange positions under the effect of gravity. We present linearized theory for arbitrary 3D initial disturbances that grow in time, and calculate the evolution of the interface for early times. A new spectral method is introduced for the fully 3D nonlinear problem in a Boussinesq fluid, where the interface between the light and heavy fluid is approximated with a smooth but rapid density change in the fluid. The results of large-scale numerical calculation are presented in fully 3D geometry, compared and contrasted with the early-time linearized theory.2010 Mathematics subject classification: 76E17: Interfacial stability and instability; 76M22: Spectral methods.

Instability of a dense seepage layer on a sloping boundary

“…For non-zero wall angle, however, there is a nett flow of the dense fluid down the sloping wall, so that K-H-type billows are observed. In the extreme limit θ = π there is a layer of denser fluid above, so that the flow becomes a pure Rayleigh-Taylor instability, and the interface develops rising bubbles of fresh water and downward fingers of dense fluid, in precisely the manner shown by Walters & Forbes (2019). Figure 7 illustrates the temporal evolution of the solution by presenting contour maps of the density S at the four different times t = 10, 14, 18 and 22, and for the three different wall angles θ = π/6, π/4 and π/3.…”

“…This feature is studied in careful detail in figure 12. Here, the billow width for each of the solutions in figure 11 has been plotted as a function of time t. As with figure 10, each solution is presented in figure 12 at 150 values of t, and each of the four cases shown required about 40 hours run time on our parallel-processing computer, details of which are given in Walters & Forbes (2019). Figure 12 thus represents a summary of the results of an enormous amount of computational effort.…”

“…For this, we have again used the routine lgwt written by von Winckel (2004). This large system of equations was solved in parallel using suitable parallelizing hardware and software, as discussed in Walters & Forbes (2019). Using this computer architecture, we are able to use M = 144, N = 288 coefficients, with 720 × 1440 mesh points, so generating results of great accuracy.…”

“…The numerical scheme described here is based on the methods developed by Walters & Forbes (2019) for two-dimensional and three-dimensional unsteady interfacial flows in Boussinesq fluids. They describe in detail their use of a bi-streamfunction approach and the computer hardware they developed for these computations, and they give several examples of the use of these techniques in different applications.…”

“…Nevertheless, a spectral approach to the computation of flows in density-stratified, rotating fluids is also presented by Winters, MacKinnon & Mills (2004), and it, too, makes the Boussinesq approximation. The method of Walters & Forbes (2019) differs from this, in that it eliminates the pressure variable using a vorticity equation (like (4.6) in planar flow) and solves only for density and one streamfunction Ψ (in two dimensions) or two streamfunctions Ψ and χ (in three-dimensional flow). By contrast, Winters et al (2004) retain primitive variables in their formulation, and express each velocity component and pressure as Fourier series in space, with time-dependent coefficients.…”