The Kelvin-Helmholtz Instabilities in Two-Fluids Shallow Water Models

Abstract

The goal of this paper is to describe the formation of Kelvin-Helmholtz instabilities at the interface of two fluids of different densities and the ability of various shallow water models to reproduce correctly the formation of these instabilities. Working first in the so called rigid lid case, we derive by a simple linear analysis an explicit condition for the stability of the low frequency modes of the interface perturbation, an expression for the critical wave number above which Kelvin-Helmholtz instabilities appear, and a condition for the stability of all modes when surface tension is present. Similar conditions are derived for several shallow water asymptotic models and compared with the values obtained for the full Euler equations. Noting the inability of these models to reproduce correctly the scenario of formation of Kelvin-Helmholtz instabilities, we derive new models that provide a perfect matching. A comparisons with experimental data is also provided. Moreover, we briefly discuss the more complex case where the rigid lid is replaced by a free surface. In this configuration, it appears that some frequency modes are stable when the velocity jump at the interface is large enough; we explain why such stable modes do not appear in the rigid lid case.

Long ring waves in a stratified fluid over a shear flow

“…The graph also shows the onset of the K-H instability for the long waves at U 1 ≈ 0.5, and stabilisation for values of the shear flow exceeding U 1 ≈ 2, in agreement with the results of Ovsyannikov (1979). We note that the stabilisation persists within the scope of the full equations of motion (Ovsyannikov 1985;Lannes & Ming 2015).…”

“…show that long waves are stable both for small shears, as in the rigid lid case, but also for sufficiently large shears (Ovsyannikov 1979(Ovsyannikov , 1985; see also Barros & Choi (2014) and Lannes & Ming (2015).…”

“…Among such stabilising mechanisms not present in the simplified model are the continuity of the actual density and shear flow profiles, with a thin intermediate layer in between the two main layers, and surface tension (see Turner (1973), Boonkasame & Milewski (2011), Lannes & Ming (2015) and references therein). The two-layer model is an abstraction of the actual continuous density and shear flow profiles, suitable for the theoretical study of long internal waves.…”

“…In the rigid lid approximation, the condition for the stability of the long waves in a two-layer fluid of finite depth (see figure 3) in the non-dimensional variables used in our paper is given by (Ovsyannikov 1979(Ovsyannikov , 1985; see also Bontozoglou (1991), Boonkasame & Milewski (2011) and Lannes & Ming (2015) and references therein. The free surface results…”

A New Class of Two‐Layer Green–Naghdi Systems with Improved Frequency Dispersion

“…Many attempts have been made to "regularize" the Green-Naghdi model, that is proposing new models with formally the same precision as the original model, but which are not subject to high-frequency Kelvin-Helmholtz instabilities, even without surface tension [11][12][13][14][15]. The strategies adopted in these works rely on change of unknowns and/or Benjamin-Bona-Mahony type tricks; see [16, section 5.2] for a thorough presentation of such methods in the free-surface setting.…”