Theoretical amplitude and period of precursor solition generation in two-layer flow

Abstract

An fKdV equation of two-layer flow and an averaged fKdV equation (AfKdV equation) with respect to phase are derived to determine the theoretical amplitude and period of the precursor solitons in the present paper. In terms of the AfKdV equation derived by the authors, a new theory on the precursor soliton generation based on Lee et al.'s concept is presented. Concepts of asymptotic mean hydraulic fall and level are introduced in our analysis, and the theoretical amplitude and period both depend on the asymptotic mean levels and stratified parameters. From the present theoretical results, it is obtained that when the moving velocity of the topography is at the resonant points, there exist two general relations: (1) amplitude relation Å=2F, (2) period relation$\mathop \tau \limits^ \circ = - 8m_1 m_3^{ - 1} \sqrt {6m_4 m_3^{ - 1} } \mathcal{F}$, in which Å and$\tau$ are the amplitude and period of the precursor solitons at the resonant points respectively,m ₁,m ₃ andm ₄ are coefficients of the fKdV equation, andF is an asymptotic mean half-hydraulic fall at subcritical cutoff points. The theoretical results of this paper are compared with experiments and numerical calculations of two-layer flow over a semicircular topography and all these results are in good agreement. Due to the canonical character of the coefficients of fKdV equations, this theory also holds for any two-dimensional system, which can be reduced to fKdV equations.