A central-upwind scheme for two-layer shallow-water flows with friction and entrainment along channels

We present a new high-resolution, non-oscillatory semi-discrete central-upwind scheme for one-dimensional two-layer shallow-water flows with friction and entrainment along channels with arbitrary cross sections and bottom topography. These flows are described by a conditionally hyperbolic balance law with non-conservative products. A detailed description of the properties of the model is provided, including entropy inequalities and asymptotic approximations of the eigenvalues of the corresponding coefficient matrix. The scheme extends existing central-upwind semi-discrete numerical methods for hyperbolic conservation and balance laws and it satisfies two properties crucial for the accurate simulation of shallow-water flows: it {\it preserves the positivity} of the water depth for each layer, and it is {\it well balanced}, {\it i.e.}, the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. Along with the description of the scheme and proofs of these two properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

In this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

Energy estimation of resonant waves in channels with lateral cavities

<p>Macro-roughness elements, such as lateral cavities and embayments, are usually built in the banks of rivers for different purposes. They can be used to create harbors, or to promote morphological diversity that enhance habitat suitability in an attempt to restore the sediment cycle in channelized rivers. In presence of lateral cavities, shallow water flows may exhibit a rhythmic water surface oscillation, called seiche. The formation of the seiche is triggered by the partially bounded in-cavity water body which leads to the generation of a standing wave. Amplitude and periodicity of the seiche is jointly controlled by the dominant eigenmodes of the standing wave and by the turbulent shear layer structures created at the opening of the cavity. Seiches have been studied in the past decades putting the focus on their impact on river hydrodynamics and morphodynamics. However, the study of the seiches from an energy harvesting perspective is still unexplored. Seiche waves could represent a distributed hydropower source with a low environmental impact, being energy extraction possibly integrated with river restoration works. In this work, we use an in-house  numerical simulation model to reproduce the water surface oscillation in a channel with multiple lateral cavities and study their wave energy potential. The interaction of multiple cavities has an additional effect in the propagation and formation of multiple standing waves, ultimately leading to two-dimensional and multi-modal seiche waves. Therefore, a detailed analysis of the seiche amplitude and energy spatial distribution is presented.</p>