A method for computing unsteady fully nonlinear interfacial waves

Abstract

We derive a time-stepping method for unsteady fully nonlinear two-dimensional motion of a two-layer fluid. Essential parts of the method are: use of Taylor series expansions of the prognostic equations, application of spatial finite difference formulae of high order, and application of Cauchy's theorem to solve the Laplace equation, where the latter is found to be advantageous in avoiding instability. The method is computationally very efficient. The model is applied to investigate unsteady trans-critical two-layer flow over a bottom topography. We are able to simulate a set of laboratory experiments on this problem described by Melville & Helfrich (1987), finding a very good agreement between the fully nonlinear model and the experiments, where they reported bad agreement with weakly nonlinear Korteweg–de Vries theories for interfacial waves. The unsteady transcritical regime is identified. In this regime, we find that an upstream undular bore is generated when the speed of the body is less than a certain value, which somewhat exceeds the critical speed. In the remaining regime, a train of solitary waves is generated upstream. In both cases a corresponding constant level of the interface behind the body is developed. We also perform a detailed investigation of upstream generation of solitary waves by a moving body, finding that wave trains with amplitude comparable to the thickness of the thinner layer are generated. The results indicate that weakly nonlinear theories in many cases have quite limited applications in modelling unsteady transcritical two-layer flows, and that a fully nonlinear method in general is required.

An intrusion layer in stationary incompressible fluids Part 2: A solitary wave

“…Although (3.24) has a similar form to that of the solitary wave of depression summarized in Grue et al [25], it is nevertheless a different solution, since in the present situation, the lowest fluid is stationary. The result (3.21)-(3.23) thus represents the complete solitary wave solution in the weakly nonlinear analysis.…”

“…This is discussed in detail by Whitham [24], for example. (A summary of the various solitary wave theories is also given in the paper by Grue et al [25].) In terms of the original variables obtained from (3.1) and (3.10), the lower and upper interfaces are predicted by this weakly nonlinear analysis to have the profiles…”

Long Nonlinear Internal Waves

“…In contrast, for the cases in which the cubic nonlinearity became important, the agreement between measurements and the numerical solutions was improved, showing a progression from undular bores upstream, to monotonic bores in the transcritical regime, to steady flow over the topography for supercritical flows (see Figure 6). Grue et al (1997) developed a numerical scheme for fully nonlinear two-layer systems and found very good agreement with the measurements of Melville & Helfrich (1987). They concluded that weakly nonlinear theories may have quite limited application in modeling unsteady transcritical two-layer flows, and that fully nonlinear methods are generally required.…”

Simultaneous synthetic schlieren and PIV measurements for internal solitary waves

“…Here, the interest is in the behaviour of so-called internal solitary waves, which are common in coastal and marginal seas (e.g., Vlasenko et al (2005) and Ostrovsky and Stepanyants (1989)) and the atmospheric boundary layer (e.g., Christie (1989)). Theoretical models (e.g., Grue et al (1997)) have relied heavily on experimental verification (e.g., Grue et al (1999)). However, to date such models have assumed simple two-layer stratifications separated by a sharp interface.…”