The Initial Stage of Dam-Break Flow of Two Immiscible Fluids. Linear Analysis of Global Flow

Abstract

Two-dimensional impulsive flow of two immiscible fluids is studied within the potential flow theory. Initially the fluids of different depths and different densities are at rest and separated with a thin vertical plate. The plate is withdrawn suddenly and gravity-driven flow of the fluids starts. During the early stage the flow is described by the linear potential theory. Attention is paid to the motion of the interface between the fluids and the singular behaviour of the velocity field at the triple point, where the free surfaces of the fluids and the interface meet each other. The linear problem is solved by the Fourier series method. Local analysis of the flow field close to the triple point reveals that the singularity of the flow depends on the ratio of the fluid densities with a coefficient dependent on both the density ratio and the shape of the flow region. The flow velocity is also log-singular at the point where the interface meets the bottom. The intensity of this singularity depends on the density ratio. The latter singularity disappears when the densities of the fluids are equal. The Fourier series solution supplemented by the singularity analysis at the corner points resolves these initial singularities. Comparisons with solutions obtained through the boundary element method are established for validation purposes. The numerical analysis of the problem by the boundary element method is carried out and it compares quite well with the Fourier series solution. The singular flow field which leads to the jet formation at the initial instant has been observed by both methods. The problem of dam-break flow for the wet-bed case corresponds to the present problem with equal densities of the fluids. Comparisons with data available in literature are established in the case of fluids with the same density.