An Efficient Two-Layer Non-hydrostatic Approach for Dispersive Water Waves

Abstract

In this paper, we propose a two-layer depth-integrated non-hydrostatic system with improved dispersion relations. This improvement is obtained through three free parameters: two of them related to the representation of the pressure at the interface and a third one that controls the relative position of the interface concerning the total height. These parameters are then optimized to improve the dispersive properties of the resulting system. The optimized model shows good linear wave characteristics up to kH ≈ 10, that can be improved for long waves. The system is solved using an efficient formally second-order well-balanced and positive preserving hybrid finite volume/difference numerical scheme. The scheme consists of a two-step algorithm based on a projection-correction type scheme. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite-volume method. Second, the dispersive terms are solved using finite differences. The method has been applied to idealized and challenging physical situations that involve nearshore breaking. Agreement with laboratory data is excellent. This technique results in an accurate and efficient method.

A General Non-hydrostatic Hyperbolic Formulation for Boussinesq Dispersive Shallow Flows and Its Numerical Approximation

“…As it will be seen in the numerical tests shown in the next section, the PDE systems considered in this paper cannot describe this process without an additional term that allows the model to dissipate the required amount of energy in such situations. In this work, we adopt a simplified version of the breaking mechanism introduced in [17], [22] and later adopted for approximated hyperbolic dispersive systems in [18] where a friction term is added to the right hand side of the equations:…”

“…The general formulation (22) has two main advantages. On the one hand, as it has already been said, it covers all the classical and well known dispersive systems.…”

Hyperbolic model for free surface shallow water flows with effects of dispersion, vorticity and topography

“…The non-negative constants σ and κ are the empirical parameters responsible for the mixing and energy dissipation. According to [38,3,24] these parameters are as follows σ ≈ 0.15 and κ ∈ [2,6]. A mild slope approximation means that the dimensionless bottom variation is weak [32,11]:…”

“…Both models (27) and (12) include the empirical parameters σ and κ, which may differ slightly from the fixed above values. In particular, the following range of these parameters were used in [24,21,10] for layered hydrostatic and dispersive models: σ ∈ [0.15, 0.20] and κ ∈ [2,6]. A change in the empirical parameters in the specified range does not lead to a significant change in the solution.…”

“…A hierarchy of non-hydrostatic shallow water-type models approximating the incompressible Euler and Navier-Stokes systems with free surface was presented in [4,9]. Suggested in [6] a two-layer dispersive system provides an efficient and accurate approach to model the propagation of waves near coastal areas and intermediate waters.…”