International Conference «Mathematical and Information Technologies, MIT-2016»

28 August – 5 September 2016

Vrnjacka Banja, Serbia – Budva, Montenegro

Arkhipov D.G.   Khabakhpashev G.A.  

Mathematical modeling of gravitational waves at two-layer flow interface in a flat horizontal channel

Reporter: Arkhipov D.G.

To date, there are a large number of papers devoted to linear stability of two-layer shear flows in rectangular channels (see for example reviews in monographs [1,2]). An important feature of the two-layer system is the existence of interfacial perturbation mode (in addition to the usual shear modes), which may be unstable in a wide range of problem parameters. In deep layers, when the viscous dissipation may be neglected, the interfacial mode corresponds to internal gravity waves, and the instability is caused by the occurrence of a critical layer. In the two-layer fluid the gravitational waves with a small difference in densities run with small velocity, comparable to the velocity of the steady-state flow. The amplitude of the unstable interfacial perturbations increases rapidly, causing a need to take into account nonlinear effects. The article [3] proposed the weakly nonlinear evolution equation to simulate waves at the interface of the two-layer laminar flow. However, the final equation was derived under the assumption of small ratio of the main flow rate to the wave propagation velocity. The aim of this work is to expand the applicability of the integral-differential equation for larger values of the steady-state flow velocities.

To model the propagation of nonlinear internal waves in a horizontal channel we used the following assumptions: 1) fluids are incompressible and immiscible; 2) moderately long perturbations of small but finite amplitude are to be considered; 3) thicknesses of the viscous boundary layers in a disturbed flow remain thin, i.e. much thinner than the depths of layers; 4) capillary effects are not large; and 5) the perturbation growth rate is small. With these assumptions, using the multiscale method for time, it became possible to build a regular series of perturbation theory to account for the effects of dispersion and nonlinearity of waves, weak dissipation in the viscous boundary layers, as well as perturbation energy excitation by the stationary flow.

The solution in the first order of perturbation theory is a wave moving with constant speed and retaining its initial form. Solving the problem we determine not only the speed of wave propagation, but also the profiles of vertical and horizontal components of perturbed liquid velocity. These data are used in the second order of perturbation theory. On a slow time scale, the initial system of hydrodynamic equations is reduced to a single inhomogeneous differential equation for second order corrections to the vertical velocity perturbation. The right-hand side of this equation includes the previously found velocity profiles. The condition of this equation consistency was found using the adjoint operator method. This condition results in the same integro-differential equation for the interfacial perturbation. Constants of the equation are determined both by main parameters of the problem (ratios of depths, viscosities and densities of liquids) and by vertical profiles of the perturbed flow. The integral in the left-hand side of this equation determines dissipation in unsteady boundary layers, and the right-hand side determines the energy transfer from the main flow into the wave.

The numerical solution of this equation was based on the spectral method with expansion of the unknown function  in spatial Fourier series. Calculations of the evolution of small initial perturbations have shown that the increase of the solitary waves amplitude is accompanied by the corresponding decrease in their length. Eventually, the wave parameters overstep the scope of applicability of the model evolution equations that is looks like a collapse phenomenon.

The study was financially supported by the grant of Russian Science Foundation (project No. 14-22-00174).

1. D. D. Joseph  and  Y. Y. Renardy, Fundamentals of Two-Fluid Dynamics (Springer Verlag, Berlin, 1993).
2. V. K. Andreyev, V. E. Zakhvataev, and  E. A. Ryabitskii, Thermocapillary Instability (Nauka, Novosibirsk, 2000).
3. D. G. Arkhipov  and  G. A. Khabakhpashev, "Modeling of long nonlinear waves on the interface  in a horizontal two-layer viscous channel flow," Fluid Dyn. 40, 126–139 (2005).

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