Models and methods for two-layer shallow water flows

Abstract

Two-layer shallow water models present at least two fundamental difficulties that are addressed in the present contribution. The first one is related to the lack of hyperbolicity of most existing models. By considering weak compressibility of the phases, a strictly hyperbolic formulation with pressure relaxation is obtained. It is shown to tend to the conventional two-layer model in the stiff pressure relaxation limit. The second issue is related to the non-conservative terms in the momentum equations. Analyzing the Riemann problem structure, local constants appear precisely at locations where the non-conservative products need definition. Thanks to these local constants, a locally conservative formulation of the equations is obtained, simplifying the Riemann problem resolution through a HLL-type Riemann solver. The method is compared to literature data, showing accurate and oscillation free solutions. Additional numerical experiments show robustness and accuracy of the method.

Re-evaluating efficiency of first-order numerical schemes for two-layer shallow water systems by considering different eigenvalue solutions

“…Although analytical expressions given by Eqs. (18) and (19) This algorithm is not a subject of the current study, but its description and details are available in [25].…”

“…These equations are challenging to solve numerically because of the layer coupling and non-conservative source terms accounting for the variable geometry, friction, or entrainment. Over the last two decades, a numerical resolution of two-layer SWE has been an object of intense research [15,14,26,16,31,6,21,28,8,18].…”

A consistent reduction of the two-layer shallow-water equations to an accurate one-layer spreading model

“…This form was originally described by Ovsyannikov,22 and is referred to in more recent works as "the conventional two-layer shallow-water model". 32…”

“…29 Attempts to amend the non-hyperbolicity of the systems include adding numerical (non-physical) friction forces, 30 operator-splitting approaches, 31 and introduction of an artificial compressibility. 32 Due to their comparative simplicity, the one-layer shallow-water equations (1LSWE) have often been used to model two-layer phenomena like liquid-on-liquid spreading and gravity currents where one assumes that the layers are in a buoyant equilibrium. In this case, a forced constant Froude-number boundary condition at the leading edge of a spreading liquid is used to account for the effect of the missing layer.…”

Extended Noble–Abel Stiffened-Gas Equation of State for Sub-and-Supercritical Liquid-Gas Systems Far from the Critical Point

“…Material interface problems [15], chemical reactions [16], phase change [17], surface tension [18], solid-fluid [19], plastic transformation [20], dense and dilute flows [21], and shallow water flows [22] can be cited for instance. In these flow models, compressibility of each phase is responsible for the hyperbolic character of the equations and an appropriate and convex EOS is required for each fluid.…”