A multiscale model for Rayleigh-Taylor and Richtmyer-Meshkov instabilities

Abstract

We develop a novel multiscale model of interface motion for the Rayleigh-Taylor instability (RTI) and Richtmyer-Meshkov instability (RMI) for two-dimensional, inviscid, compressible flows with vorticity, which yields a fast-running numerical algorithm that produces both qualitatively and quantitatively similar results to a resolved gas dynamics code, while running approximately two orders of magnitude (in time) faster. Our multiscale model is founded upon a new compressible-incompressible decomposition of the velocity field u = v + w. The incompressible component w of the velocity is also irrotational and is solved using a new asymptotic model of the Birkhoff-Rott singular integral formulation of the incompressible Euler equations, which reduces the problem to one spatial dimension. This asymptotic model, called the higher-order z-model, is derived using small nonlocality as the asymptotic parameter, allows for interface turn-over and roll-up, and yields a significant simplification for the equation describing the evolution of the amplitude of vorticity. This incompressible component w of the velocity controls the small scale structures of the interface and can be solved efficiently on fine grids. Meanwhile, the compressible component of the velocity v remains continuous near contact discontinuities and can be computed on relatively coarse grids, while receiving subgrid scale information from w. We first validate the incompressible higher-order z-model by comparison with classical RTI experiments as well as full point vortex simulations. We then consider both the RTI and the RMI problems for our multiscale model of compressible flow with vorticity, and show excellent agreement with our high-resolution gas dynamics solutions. 4 Numerical implementation of the z-model 15

A Comparison of Interface Growth Models Applied to Rayleigh–Taylor and Richtmyer–Meshkov Instabilities

“…This is a limitation of the pseudo-spectral method, rather than a deficiency of the Z17 model. Recent work [21] has addressed improving the interface roll-up of the Z17 model. Figure 6 shows the half-mix widths, measured from bubble tip to spike tip for the constant mean acceleration, G ¼ 0:74 reported in W01.…”

“…11 Time series of the maximum G02 (red) and Z17 (blue) model amplitudes compared to the experiments of V14 (black dots) Fig. 12 The Z17 model bubble (blue) and spike (black) comparison with the case 1 experiment of V14 (red dots), showing spread of data between modeled bubbles and spikes New generations of the z-model have recently been derived, for example in the study of Ramani and Shkoller [21]. In Ref.…”

“…[21], a "higher-order" z-model has been developed that extends the amount of interface roll-up that can be modeled and has been shown to give better agreement with W01 than Z17 does. The work of Shkoller [21] extends the z-model to a hybrid z-model/ compressible Euler approach, where the velocity field of the Euler equations has been decomposed into its compressible and incompressible parts. Future efforts will involve investigating the hybrid approach of Shkoller [21].…”

“…The work of Shkoller [21] extends the z-model to a hybrid z-model/ compressible Euler approach, where the velocity field of the Euler equations has been decomposed into its compressible and incompressible parts. Future efforts will involve investigating the hybrid approach of Shkoller [21]. Also, extending the z-model to higher dimensions is of interest.…”

Models for Damped Water Waves

“…and, furthermore, Longuet-Higgins [29] stated that "For certain applications, however, viscous damping of the waves is important, and it would be highly convenient to have equations and boundary conditions of comparable simplicity as for undamped waves." The purpose of this paper is to derive new weakly nonlinear asymptotic models (in the spirit of [6,16,17,18,30,31,32,36]) describing damped water waves and, at the same time, keeping the features of potential flows. We observe that, at first sight, the idea of viscous damping of potential flows is somehow paradoxical since the hypothesis of irrotational velocity implies that the viscous term in the Navier-Stokes equations vanishes.…”