Adaptation of flux-based solvers to 2D two-layer shallow flows with variable density including numerical treatment of the loss of hyperbolicity and drying/wetting fronts

An important feature of the two-layer shallow flow model is that the resulting system of equations cannot be expressed in conservation-law form. Here, the HLLS and ARoe solvers, derived initially for systems of conservation laws, are reformulated and applied to the two-layer shallow flows in a great variety of problems. Their resulting extension and combination allows us to overcome the loss of the hyperbolic character, ensuring energy or exactly balanced property, guarantees positivity of the solution, and provides a correct drying/wetting advance front without requiring tuning parameters. As a result, in those cases where the rich description of internal and external waves cannot be provided by the ARoe solver, HLLS is applied. Variable density is considered in each layer as a result of a bulk density driven by the mixture of different constituents. A wide variety of test cases is presented confirming the properties of this combination, including exactly balanced scenarios in subcritical and subcritical-transcritical scenarios, dam-break problems over bed variations and wet/dry fronts, non-hyperbolic conditions, transcritical exchange flow with loss of hyperbolicity. Despite the complexity of the test cases presented here, accurate and stable simulations are guaranteed, ensuring positivity of the solution without decreasing the time step.

On Steady Two-dimensional Free-surface Flows with Spatially-varied Discharges

In an open-channel, the transition of a flow from a subcritical to a supercritical state may occur as a result of a lateral inflow or outflow that produces a streamwise discharge variation. Apparently, such a transition cannot be modeled accurately by a conventional hydrostatic pressure approach. In this study, a depth-averaged model that accounts for the effects of a spatially-varied discharge and a non-hydrostatic pressure distribution was developed and applied to simulate the transcritical flow in a lateral-spillway channel and the subcritical flow in a main channel fitted with side weirs. The model results for the axial free-surface profile and variation of discharge in the main channel were compared with the results of a shallow-flow model and experimental data, thereby resulting in a closer match to the measurements than the shallow-flow model. Overall, the investigation results confirmed the efficiency and validity of the non-hydrostatic depth-averaged model in simulating the mean flow characteristics of the subcritical and transcritical free-surface flows with spatially increasing or decreasing discharges, thus demonstrating its potential to be used as a numerical tool in engineering practice.

Adaptive Multi-Scale Shallow Flow Model: a Wavelet-Based Formulation

This work outlines the use of wavelet bases to re-formulate a finite volume (FV) local solution of the shallow water equations (SWEs), so as to achieve mesh adaptivity via local compression and truncation of the numerical solution’s details across successive resolution scales with reference to a single threshold error set by the user. The wavelet bases naturally lead to a scalable FV formulation and how they can readily be exploited to achieve adaptive mesh-resolution selection: up-scaling and/or down- scaling by means of the local solutions’ data (i.e. both flow variables and terrain). Our results show a notable promise in using wavelets as a basis for future flood models to achieve conservative and more autonomous simulation at a wide range of length-scales.

A shallow-flow model for the propagation of tsunamis over complex geometries and mobile beds

A distinguishable feature of overland tsunami propagation is the incorporation of solids within the flow column, either sediment from the natural environment or remains from built infrastructure. This article describes a 2DH (two-dimensional horizontal) mathematical model particularly suited for tsunami propagation over complex and dynamic geometries, such as river and estuarine mobile beds. The discretization scheme is based on a finite-volume method using a flux-splitting technique featuring a reviewed Roe–Riemann solver, with appropriate source-term formulations to ensure full conservativeness. The model is validated with laboratory data and paleo-tsunami evidence. As a forecasting application, it is applied to a tsunami scenario in the Tagus estuary, an effort justified by the numerous catastrophic tsunamis that are known to have struck this location over the past two millennia. The obtained results show that, despite the significant differences in Lisbon's layout and morphology, a 1755-like tsunami would still inflict a devastating impact on this major city.