σ -Coordinates hydrodynamic numerical model for coastal and ocean three-dimensional circulation

Three-dimensional shallow water system: A relaxation approach

“…The 3-D shallow water system (2)- (5) has been intensively studied and several finite difference (see, e.g., [2,11,12,17,25,34,36,37,45,52]) and finite-volume (see, e.g., [9,13] which is not hyperbolic). In order to address this issue, we propose a relaxation approach inspired by [14,48], where a hyperbolic model of compressible two-phase flow was developed using the pressure relaxation, and [1], where an unconditionally hyperbolic two-layer SWEs were obtained using the relaxation in two auxiliary layer depth variables.…”

“…where U (x, y, σ) is a piecewise linear reconstruction of U at time t, (17) in which ( U x ) i and ( U y ) i are approximations of the horizontal derivatives.…”

“…The 3-D shallow water system, proposed in [6,7], is still substantially simpler than the 3-D Navier-Stokes equations, and unlike the 2-D SWEs it can predict the vertical distribution of primitive variables, information on which is essential in many practical cases. Therefore, the 3-D SWEs have recently attracted a lot of attention as a more accurate alternative to the 2-D SWEs; see, e.g., [2,9,12,13,17,18,25,34,36,37,45,52].…”

A non-horizontal multi-layer numerical model for the flow in natural rivers

“…However, three-dimensional shallow water numerical models (Drago and Iovenitti 2000, Wong and Hon 2002, Zhang and Chan 2003, Jiang and Wai 2005 are scarcely found for the flow simulation in natural rivers, especially in upland rivers, although they have been used extensively in the last 20 years for the prediction of the flow in estuaries and coastal seas. In natural rivers, the free surface displacement compared with the water depth is much larger than that encountered in estuaries and coastal seas; the bed topographies are more irregular, which means that the current models appear to be deficient in dealing with the vertical solution.…”

Numerical and hydraulic simulations of the effect of Density Current Generator in a semi-enclosed tidal bay

“…Apart from two-dimensional (2D) models, there are many three-dimensional (3D) Boussinesq models, e.g. the models of Tabeta and Fujino (1994), Zhang and Gin (2000), Shen et al (2002), and Gross et al (1999), using rectangular z-coordinates (layered model), the Princeton Ocean Model (Blumberg and Mellor, 1987;Mellor, 1996), the models of Drago and Iovenitti (2000) and Jin et al (2000), using sigmacoordinates, the Regional Ocean Model (Song and Haidvogel, 1994), the Delft Hydraulics Model (http://www.wldelft.nl/soft/ d3d/intro), the models of Shankar et al (1997), Wang et al (1992), and Shi et al (2001), using sigma-coordinates in the vertical and curvilinear coordinates in the horizontal directions, respectively, the Telemac-3D (http://www.telemacsystem.com/ gb) and the Dartmouth Circulation Model (Lynch et al, 1996), using sigma-coordinates in the vertical and prismatic elements in the horizontal directions, respectively, in their finite element methods. It should be noted that they utilise the hydrostatic-pressure approximation in 3D coordinates.…”