Two-layer Shallow Water Flows Research Articles

Weakly Compressible Two-Layer Shallow-Water Flows Along Channels

In this paper, we formulate a model for weakly compressible two-layer shallow water flows with friction in general channels. The formulated model is non-conservative, and in contrast to the incompressible limit, our system is strictly hyperbolic. The generalized Rankine–Hugoniot conditions are provided for the present system with non-conservative products to define weak solutions. We write the Riemann invariants along each characteristic field for channels with constant width in an appendix. A robust well-balanced path-conservative semi-discrete central-upwind scheme is proposed and implemented to validate exact solutions to the Riemann problem. We also present numerical tests in general channels to show the merits of the scheme.

A modified Rusanov method for simulating two-layer shallow water flows with irregular topography

A modified Rusanov method for simulating two-layer shallow water flows with irregular topography

A Nonintrusive Reduced-Order Model for Uncertainty Quantification in Numerical Solution of One-Dimensional Free-Surface Water Flows Over Stochastic Beds

Free-surface water flows over stochastic beds are complex due to the uncertainties in topography profiles being highly heterogeneous and imprecisely measured. In this study, the propagation and influence of several uncertainty parameters are quantified in a class of numerical methods for one-dimensional free-surface flows. The governing equations consist of both single-layer and two-layer shallow water equations on either flat or nonflat topography. For this purpose, the free-surface profiles are computed for different realizations of the random variables when the bed is excited with sources whose statistics are well defined. Many research studies have been dedicated to the development of numerical methods to achieve some order of accuracy in free-surface flows. However, little concern was given to examine the performance of these numerical methods in the presence of uncertainty. This work addresses this specific area in computational hydraulics with regards to the uncertainty generated from bathymetric forces. As numerical solvers for the one-dimensional shallow water equations, we implement four finite volume methods. To reduce the required number of samples for uncertainty quantification, we combine the proper orthogonal decomposition method with the polynomial chaos expansions for efficient uncertainty quantifications of complex hydraulic problems with large number of random variables. Numerical results are shown for several test examples including dam-break problems for single-layer and two-layer shallow water flows. The problem of flow exchange through the Strait of Gibraltar is also solved in this study. The obtained results demonstrate that in some hydraulic applications, a highly accurate numerical method yields an increase in its uncertainty and makes it very demanding to use in an operational manner with measured data from the field. On the other hand, when the complexity of physics increases, these highly accurate numerical methods display less uncertainty compared to the low accurate methods.

A well-balanced numerical model for depth-averaged two-layer shallow water flows

A well-balanced numerical model for depth-averaged two-layer shallow water flows

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 preserves the positivity of the water depth for each layer, and it is well balanced, 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.

A new well-balanced finite-volume scheme on unstructured triangular grids for two-dimensional two-layer shallow water flows with wet-dry fronts

In this study, a novel two-dimensional finite-volume method on unstructured triangular meshes for two-layer shallow water flows is developed. First, the one-dimensional relaxation approach developed in Abgrall and Karni (2009) [1] is extended to the currently studied two-dimensional two-layer shallow water equations to build an unconditionally hyperbolic system. Next, a two-dimensional finite-volume HLL scheme on triangular grids for such a non-conservative hyperbolic system is constructed. Special attention is paid to guaranteeing the well-balanced property even in the presence of the wet-dry fronts. To this end, techniques such as spatial reconstruction for the secondary variables, special cell classifications for the two-layer fluid system, and special local reconstruction of auxiliary surface variables are proposed. These special techniques are further combined with new discrete formulas for the nonconservative products and the bed-slope terms, and it thus leads to an exactly well-balanced numerical scheme for the studied two-layer system even in the presence of wet-dry fronts. The developed numerical model is second order accurate and preserves the nonnegative layer depths. The performances of the proposed numerical model are investigated by several numerical experiments.

Nonlinear stability of two-layer shallow water flows with a free surface.

The problem of two layers of immiscible fluid, bordered above by an unbounded layer of passive fluid and below by a flat bed, is formulated and discussed. The resulting equations are given by a first-order, four-dimensional system of PDEs of mixed-type. The relevant physical parameters in the problem are presented and used to write the equations in a non-dimensional form. The conservation laws for the problem, which are known to be only six, are explicitly written and discussed in both non-Boussinesq and Boussinesq cases. Both dynamics and nonlinear stability of the Cauchy problem are discussed, with focus on the case where the upper unbounded passive layer has zero density, also called the free surface case. We prove that the stability of a solution depends only on two 'baroclinic' parameters (the shear and the difference of layer thickness, the former being the most important one) and give a precise criterion for the system to be well-posed. It is also numerically shown that the system is nonlinearly unstable, as hyperbolic initial data evolves into the elliptic region before the formation of shocks. We also discuss the use of simple waves as a tool to bound solutions and preventing a hyperbolic initial data to become elliptic and use this idea to give a mathematical proof for the nonlinear instability.

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

The efficiency of several first-order numerical schemes for two-layer shallow water equations (SWE) are evaluated here by considering different eigenvalue solutions. This study is a continuation of our previous work (Krvavica et al., 2018) in which we have proposed an efficient implementation of a Roe solver for two-layer SWE based on analytical expressions for eigenvalues and eigenvectors. In this work, the accuracy and computational cost of numerical, analytical, and approximated eigenvalue solvers are compared when implemented in Roe, Intermediate Field CaPturing (IFCP) and Polynomial Viscosity Matrix (PVM) schemes. Several numerical tests are performed to examine the overall efficiency of numerical schemes with different eigenvalue solvers when computing two-layer shallow-water flows. The results confirm that analytical eigenvalue solutions are much faster than numerical solvers, with a computational cost closer to approximate expressions. Consequently, the Roe scheme with analytical solutions to the eigenstructure is equally efficient as the IFCP scheme. On the other hand, IFCP and PVM schemes with analytical solutions to eigenvalues are found to be equally efficient as those with approximated expressions. Analytical eigenvalues show slightly better results when dealing with larger density differences between the layers.

Wetting and drying of two-layer shallow-water flow in a moving vessel via semi-Lagrangian simulation

The sloshing motion of two inviscid, incompressible, immiscible, shallowwater fluid layers in a rectangular vessel with a rigid lid is considered. The vessel is forced to oscillate in a horizontal rectilinear motion, at the lowest resonant frequency of the system, such that the lower fluid hits the rigid lid causing a wetting-drying scenario. The two-layer shallow-water equations are solved using a conservative quasi-monotone semi-Lagrangian scheme which conserves the mass of the system. Two examples are considered: a low-fill example where the lower fluid does not interact with the upper-rigid lid, in order to validate the code against an existing Lagrangian particle path scheme, and a high-fill example where there is wetting and drying of the upper rigid lid. In both examples, results of the semi-Lagrangian simulations are compared to existing experimental data and good agreement is obtained.

Models and methods for two-layer shallow water flows

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.

Salt-Wedge Response to Variable River Flow and Sea-Level Rise in the Microtidal Rječina River Estuary, Croatia

ABSTRACT Krvavica, N.; Travas, V., and Ožanic, N., 2017. Salt-wedge response to variable river flow and sea-level rise in the microtidal Rjecina River Estuary, Croatia. A finite-volume model for two-layer shallow-water flow is presented and applied to study the dynamic response of a salt wedge in a microtidal estuary to changes in river flow rate and sea-level rise (SLR). First, the shape of the arrested salt wedge was computed for different hydrographic conditions. Next, the response of the salt wedge to highly variable river flow was investigated. Finally, this model was applied to predict the impacts of the SLR on salinity intrusions in the Rjecina River Estuary. To assess the model performance and to examine the salinity structure in the estuary, a field-sampling campaign was conducted during 2014 and 2015. Field observations revealed negligible longitudinal density variations in both freshwater and saltwater layers and highly stratified conditions for all considered river flow rates and sea levels. F...

Numerical modelling of two-layer shallow water flow in microtidal salt-wedge estuaries: Finite volume solver and field validation

Abstract A finite volume model for two-layer shallow water flow in microtidal salt-wedge estuaries is presented in this work. The governing equations are a coupled system of shallow water equations with source terms accounting for irregular channel geometry and shear stress at the bed and interface between the layers. To solve this system we applied the Q-scheme of Roe with suitable treatment of source terms, coupling terms, and wet-dry fronts. The proposed numerical model is explicit in time, shock-capturing and it satisfies the extended conservation property for water at rest. The model was validated by comparing the steady-state solutions against a known arrested salt-wedge model and by comparing both steady-state and time-dependant solutions against field observations in Rječina Estuary in Croatia. When the interfacial friction factor λi was chosen correctly, the agreement between numerical results and field observations was satisfactory.

A discontinuous Galerkin method for two-layer shallow water equations

In this paper, we study a discontinuous Galerkin method to approximate solutions of the two-layer shallow water equations on non-flat topography. The layers can be formed in the shallow water model based on the vertical variation of water density which in general depends on the water temperature and salinity. For a water body with equal density the model reduces to the canonical single-layer shallow water equations. Thus, for a model with equal density on flat bottom, the method is equivalent to the discontinuous Galerkin method for conservation laws. The considered method is a stable, highly accurate and locally conservative finite element method whose approximate solutions are discontinuous across inter-element boundaries; this property renders the method ideally suited for the hp-adaptivity. Several numerical results illustrate the performance of the method and confirm its capability to solve two-layer shallow water flows including tidal conditions on the water free-surface and bed frictions on the bottom topography.

Conservation law modelling of entrainment in layered hydrostatic flows

A methodology is developed for modelling entrainment in two-layer shallow water flows using non-standard conserved quantities, replacing layerwise mass conservation by global energy conservation. Thus, the energy that the standard model would regularly dissipate at internal shocks is instead available to exchange fluid between the layers. Two models are considered for the upper boundary of the flow: a rigid lid and a free surface. The latter provides a selection principle for choosing physically relevant conservation laws among the infinitely many that the former possesses, when the ratio between the baroclinic and barotropic speeds tends to zero. Solutions of the equations are studied analytically and numerically, applied to the lock-exchange problem, and compared with other closures.

A new composite scheme for two-layer shallow water flows with shocks

This paper is devoted to solve the system of partial differential equations governing the flow of two superposed immiscible layers of shallow water flows. The system contains source terms due to bottom topography, wind stresses, and nonconservative products describing momentum exchange between the layers. The presence of these terms in the flow model forms a nonconservative system which is only conditionally hyperbolic. In addition, two-layer shallow water flows are often accompanied with moving discontinuities and shocks. Developing stable numerical methods for this class of problems presents a challenge in the field of computational hydraulics. To overcome these difficulties, a new composite scheme is proposed. The scheme consists of a time-splitting operator where in the first step the homogeneous system of the governing equations is solved using an approximate Riemann solver. In the second step a finite volume method is used to update the solution. To remove the non-physical oscillations in the vicinity of shocks a nonlinear filter is applied. The method is well-balanced, non-oscillatory and it is suitable for both low and high values of the density ratio between the two layers. Several standard test examples for two-layer shallow water flows are used to verify high accuracy and good resolution properties for smooth and discontinuous solutions.

A Non-oscillatory Central Scheme for One-Dimensional Two-Layer Shallow Water Flows along Channels with Varying Width

We present a new high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional two-layer shallow-water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and it enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, 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 a detailed description of the scheme and proofs of these two properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

Well-posedness of two-layer shallow-water flow between two horizontal rigid plates

This paper deals with well-posedness of the system modelling two-layer shallow-water flow between two horizontal rigid plates. More precisely, we generalize results obtained by Guyenne (2010 Well-posedness of the Cauchy problem for models of large amplitude internal waves Nonlinearity 23 237–75) to the non-irrotational case. The main idea is an iterative construction with a hyperbolic equation for the bottom depth h assuming the following condition on the data: , where h0 is the initial depth of the bottom layer, and denote, respectively, the initial velocity of the bottom and top layer, and ρ = ρ1/ρ2.

Instabilities of two-layer shallow-water flows with vertical shear in the rotating annulus

Being motivated by the recent experiments on instabilities of the two-layer flows in the rotating annulus with super-rotating top, we perform a full stability analysis for such system in the shallow-water approximation. We use the collocation method which is benchmarked by comparison with analytically solvable one-layer shallow-water equations linearized about a state of cyclogeostrophic equilibrium. We describe different kinds of instabilities of the cyclogeostrophically balanced state of solid-body rotation of each layer (baroclinic, Rossby–Kelvin (RK) and Kelvin–Helmholtz (KH) instabilities), and give the corresponding growth rates and the structure of the unstable modes. We obtain the full stability diagram in the space of parameters of the problem and demonstrate the existence of crossover regions where baroclinic and RK, and RK and KH instabilities, respectively, compete having similar growth rates.

Observations of downslope winds and rotors in the Falkland Islands

A field campaign aimed at observing the near-surface flow field across and downwind of a mountain range on the Falkland Islands, South Atlantic, is described. The objective was to understand and eventually predict orographically generated turbulence. The instrumentation was based primarily on an array of automatic weather stations (AWSs), which recorded 30 s mean surface pressure, wind speed and direction (at 2 m), temperature and relative humidity for approximately one year. These measurements were supported by twice-daily radiosonde releases. The densest part of the AWS array was located to the south of the Wickham mountain range, across Mount Pleasant Airfield (MPA). In northerly flow the array provides a detailed study of the flow downwind of the mountain range. The dataset contains several episodes in which the flow downwind of the mountains is accelerated relative to the upwind flow. During some of these episodes short-lived (typically ∼1 hour) periods of unsteady flow separation are observed and these are associated with the formation of rotors aloft. Such events present a significant hazard to aviation at MPA. Examination of radiosonde profiles suggests that the presence of a strong temperature inversion at a height similar to the mountain height is a necessary condition for both downwind acceleration and the formation of rotors. The data are used to show that the downwind fractional speed-up is proportional to the non-dimensional mountain height (based on upstream near-surface winds and a depth-averaged Brunt-Vaisala frequency diagnosed from radiosonde data). Similarly, a relationship is established between a quantity that describes the spatial variability of the flow downwind of the mountains and the upstream wind and depth-averaged Brunt-Vaisala frequency. The dependence of the flow behaviour on the Froude number (defined in the usual way for two-layer shallow-water flow) and ratio of mountain height to inversion height is presented in terms of a flow regime diagram.

Numerical simulation of two-layer shallow water flows through channels with irregular geometry

This paper deals with the numerical simulation of flows of stratified fluids through channels with irregular geometry. Channel cross-sections are supposed to be symmetric but not necessarily rectangular. The fluid is supposed to be composed of two shallow layers of immiscible fluids of constant densities, and the flow is assumed to be one-dimensional. Therefore, the equations to be solved are a coupled system composed of two Shallow Water models with source terms involving depth and breadth functions. Extensions of the Q-schemes of van Leer and Roe are proposed where a suitable treatment of the coupling and source terms is performed by adapting the techniques developed in [J. Comput. Phys. 148 (1999) 497; Comput. Fluids 29(8) (2000) 17; Math. Model. Numer. Anal. 35(1) (2001) 107]. An enhanced consistency condition, the so-called C-property, introduced in [Comput. Fluids 23(8) (1994) 1049] is extended to this case and a general result providing sufficient conditions to ensure this property is shown. Then, some numerical tests to validate the resulting schemes are presented. First we verify that, in practice, the numerical schemes satisfy the C-property, even for extremely irregular channels. Then, in order to validate the schemes, we compare some approximate steady solutions obtained with the generalized Q-scheme of Van Leer with those obtained by using the asymptotic techniques developed by Armi and Farmer for channels with simplified geometries. Finally we apply the numerical scheme to the simulation of the flow through the Strait of Gibraltar. Real bathymetric and coast-line data are considered to include in the model the main features of the abrupt geometry of this natural strait connecting the Atlantic Ocean and the Mediterranean Sea. A steady-state solution is obtained from lock-exchange initial conditions. This solution is then used as initial condition to simulate the main semidiurnal and diurnal tidal waves in the Strait of Gibraltar through the imposition of suitable boundary conditions obtained from observed tidal data. Comparisons between numerical results and observed data are also presented.