Modelling solute transport in shallow water with the lattice Boltzmann method

The lattice Boltzmann method is used to investigate the solute transport in shallow water flows. Shallow water equations are solved using the lattice Boltzmann equation on a D2Q9 lattice with multiple-relaxation-time (MRT-LBM) and Bhatnagar–Gross–Krook (BGK-LBM) terms separately, and the advection–diffusion equation is also solved with a LBM-BGK on a D2Q5 lattice. Three cases: open channel flow with side discharge, shallow recirculation flow and flow in a harbour are simulated to verify the described methods. Agreements between predictions and experiments are satisfactory. In side discharge flow, the reattachment length for different ratios of side discharge velocity to main channel velocity has been studied in detail. Furthermore, the performance of MRT-LBM and BGK-LBM for these three cases has been investigated. It is found that LBM-MRT has better stability and is able to satisfactorily simulate flows with higher Reynolds number. The study shows that the lattice Boltzmann method is simple and accurate for simulating solute transport in shallow water flows, and hence it can be applied to a wide range of environmental flow problems.     

A Hybrid Method to Solve Shallow Water Flows with Horizontal Density Gradients

In this work, a model for shallow water flows that accounts for the effects of horizontal density fluctuations is presented and derived. While the density is advected by the flow, a two-way feedback between the density gradients and the time evolution of the fluid is ensured through the pressure and source terms in the momentum equations. The model can be derived by vertically averaging the Euler equations while still allowing for density fluctuations in horizontal directions. The approach differs from multi-layer shallow water flows where two or more layers are considered, each of them having their own depth, velocity and constant density. A Roe-type upwind scheme is developed and the Roe matrices are computed systematically by going from the conservative to the quasi-linear form at a discrete level. Properties of the model are analyzed. The system is hyperbolic with two shock-wave families and a contact discontinuity associated to interfaces of regions with density jumps. This new field is degenerate with pressure and velocity as the corresponding Riemann invariants. We show that in some parameter regimes numerically recognizing such invariants across contact discontinuities is important to correctly compute the flow near those interfaces. We present a numerical algorithm that correctly captures all waves with a hybrid strategy. The method integrates the Riemann invariants near contact discontinuities and switches back to the conserved variables away from it to properly resolve shock waves. This strategy can be applied to any numerical scheme. Numerical solutions for a variety of tests in one and two dimensions are shown to illustrate the advantages of the strategy and the merits of the scheme.     

Application of mesh-adaptation for pollutant transport by water flow

An adaptive finite volume method is proposed for the numerical solution of pollutant transport by water flows. The shallow water equations with eddy viscosity, bottom friction forces and wind shear stresses are used for modelling the water flow whereas, a transport-diffusion equation is used for modelling the advection and dispersion of pollutant concentration. The adaptive finite volume method uses simple centred-type discretization for the source terms, can handle complex topography using unstructured grids and satisfies the conservation property. The adaptation criteria are based on monitoring the pollutant concentration in the computational domain during its dispersion process. The emphasis in this paper is on the application of the proposed method for numerical simulation of pollution dispersion in the Strait of Gibraltar. Results are presented using different tidal conditions and wind-induced flow fields in the Strait.     

Application of mesh-adaptation for pollutant transport by water flow

An adaptive finite volume method is proposed for the numerical solution of pollutant transport by water flows. The shallow water equations with eddy viscosity, bottom friction forces and wind shear stresses are used for modelling the water flow whereas, a transport-diffusion equation is used for modelling the advection and dispersion of pollutant concentration. The adaptive finite volume method uses simple centred-type discretization for the source terms, can handle complex topography using unstructured grids and satisfies the conservation property. The adaptation criteria are based on monitoring the pollutant concentration in the computational domain during its dispersion process. The emphasis in this paper is on the application of the proposed method for numerical simulation of pollution dispersion in the Strait of Gibraltar. Results are presented using different tidal conditions and wind-induced flow fields in the Strait.     

A mimetic mass, momentum and energy conserving discretization for the shallow water equations

A spatial semi-discretization is developed for the two-dimensional depth-averaged shallow water equations on a non-equidistant structured and staggered grid. The vector identities required for energy conservation in the continuous case are identified. Discrete analogues are developed, which lead to a finite-volume semi-discretisation which conserves mass, momentum, and energy simultaneously. The key to discrete energy conservation for the shallow water equations is to numerically distinguish storage of momentum from advective transport of momentum. Simulation of a large-amplitude wave in a basin confirms the conservative properties of the new scheme, and demonstrates the enhanced robustness resulting from the compatibility of continuity and momentum equations. The scheme can be used as a building block for constructing fully conservative curvilinear, higher order, variable density, and non-hydrostatic discretizations.     

The collection of mathematical models of Well-4 for the calculation of flows in steam-water geothermal wells

The mathematical models for the calculation of flows in steam-water geothermal wells are described. The models cover the full range of the possible flow conditions: a vertical well with invariable mass flow, a vertical well with variable mass flow, a slant well with invariable mass flow, and a slant well with variable mass flow.     

Finite-Volume-Particle Methods for Models of Transport of Pollutant in Shallow Water

We present a new hybrid numerical method for computing the transport of a passive pollutant by a flow. The flow is modeled by the Saint-Venant system of shallow water equations and the pollutant propagation is described by a transport equation. The idea behind the new finite-volume-particle (FVP) method is to use different schemes for the flow and the pollution computations: the shallow water equations are numerically integrated using a finite-volume scheme, while the transport equation is solved by a particle method. This way the specific advantages of each scheme are utilized at the right place. This results in a significantly enhanced resolution of the computed solution     

Coupled model of surface water flow, sediment transport and morphological evolution

This paper presents a mathematical model coupling water flow and sediment transport dynamics that enables calculating the changing surface morphology through time and space. The model is based on the shallow water equations for flow, conservation of sediment concentration, and empirical functions for bed friction, substrate erosion and deposition. The sediment transport model is a non-capacity formulation whereby erosion and deposition are treated independently and influence the sediment flux by exchanging mass across the bottom boundary of the flow. The resulting hyperbolic system is solved using a finite volume, Godunov-type method with a first-order approximate Riemann solver. The model can be applied both to short time scales, where the flow, sediment transport and morphological evolution are strongly coupled and the rate of bed evolution is comparable to the rate of flow evolution, or to relatively long time scales, where the time scale of bed evolution associated with erosion and/or deposition is slow relative to the response of the flow to the changing surface and, therefore, the classical quasi-steady approximation can be invoked. The model is verified by comparing computed results with documented solutions. The developed model can be used to investigate a variety of problems involving coupled flow and sediment transport including channel initiation and drainage basin evolution associated with overland flow and morphological changes induced by extreme events such as tsunami.     

Particle mixing and reactive front motions in chaotic but closed shallow flows

A numerical study is presented of wind-induced active mixing and transport processes in closed shallow flows that are able to support chaotic advection. The wind-induced non-linear shallow water flow field is predicted using a quadtree grid based Godunov-type finite volume solver. Particles are tracked by numerically integrating the advection equations using velocity information interpolated from the predicted flow field. In complex oscillating flows, storage of all the necessary velocity information becomes problematical. Instead, we utilize the mean field and the first few dominant unsteady contributions as determined using Singular Value Decomposition. The advected particles are assumed to support autocatalytic reaction defined as A + B → 2B. Wind-induced reactive particle advection is considered in a realistic mine tailings pond with somewhat idealized bed topography. The reactive process reaches a stationary stage where reaction products occupy the whole closed flow domain. However, in the transient stage, particles undergo active advection and trace out filamentary structures that are similar to those in open flows. Because of the impossibility of particle escape and the global fine-scale chaotic mixing, the initial stages of chaotic mixing in closed flows are more efficient than in open flows. The results qualitatively validate a surface reaction theory derived by Károlyi and Tél [Károlyi G, Tél T. Chemical transients in closed chaotic flows: the role of effective dimensions. Phys Rev Lett 2005;95:264501-1-4] for closed systems.     

A coupled flow and transport model for simulation of groundwater reservoirs

As in many cases the quality of water, rather than the available amount restricts water use, a management of groundwater reservoirs has to include the investigation of the transport of dissolved solids in many cases. A special problem of coastal aquifers is a density dependent groundwater flow, caused by different saltwater concentration. For the efficient simulation of aquifer reaction a model is described, which permits the simultaneous computation of groundwater flow and mass transport.     

twodee-2: A shallow layer model for dense gas dispersion on complex topography

twodee-2 is a Fortran 90 code based on previous code (twodee). It is designed to solve the shallow water equations for fluid depth, depth-averaged horizontal velocities and depth-averaged fluid density. The shallow layer approach used by twodee-2 is a compromise between the complexity of CFD models and the simpler integral models. It can be used for forecasting gas dispersion near the ground and/or for hazard assessment over complex terrains. The inputs to the model are topography, terrain roughness, wind measurements from meteorological stations and gas flow rate from the ground sources. Optionally the model can be coupled with the output of a meteorological processor which generates a zero-divergence wind field incorporating terrain effects. Model outputs are gas concentration, depth-averaged velocity, averaged cloud thickness and dose. The model can be a useful tool for gas hazard assessment by evaluating where and when lethal concentrations for humans and animals can be reached.     

A method to generate pressure gradients for molecular simulation of pressure-driven flows in nanochannels

One of the difficulties in molecular simulation of pressure-driven fluid flow in nanochannels is to find an appropriate pressure control method. When periodic boundary conditions (PBCs) are applied, a gravity-like field has been widely used to replace actual pressure gradients. The gravity-fed method is not only artificial, but not adequate for studying properties of fluid systems which are essentially inhomogeneous in the flow direction. In this paper, a method is proposed which can generate any desired pressure difference to drive the fluid flow by attaching a ??pump?? to the nanofluidic system, while the model is still compatible with PBCs. The molecular dynamics model based on the proposed method is applied to incompressible flows in smooth nanochannels, and the predicted velocity profiles are identical to those by the gravity-fed method, as expected. For compressible flows, the proposed model successfully predicts the changes of fluid density and velocity profile in the flow direction, while the gravity-fed method can only predict constant fluid properties. For fluid flows in nanochannels with a variable cross-sectional area, the proposed model predicts higher mass flow rates as compared to the gravity-fed method and possible reasons for the difference are discussed.     

Particle mixing and reactive front motion in unsteady open shallow flow - Modelled using singular value decomposition

Passive and active tracers are used to examine particle mixing and reactive front dynamics in an open shallow flow of water past a circular cylinder. A quadtree grid based Godunov-type shallow water equation solver predicts the unsteady flow hydrodynamics of the wake behind the cylinder. The resulting periodic flow field consisting of a von Kármán vortex street is decomposed and stored over one oscillatory period using Singular Value Decomposition (SVD). Particles are advected according to the reconstructed flow field from the SVD modes, with continuous spatial velocity information obtained via bilinear interpolation. Passive particle dynamics driven by different SVD flow modes is investigated, and it is found that the flow field recovered from the mean flow and the first pair of time varying modes is adequate to represent the complicated dynamical properties induced by the original flow field. Active autocatalytic reaction, A + B → 2B, is incorporated into the particle advection model, assuming surface reaction. Active particles are found to trace out an expanded version of the unstable manifold of the chaotic saddle in the wake, in qualitative agreement with published analytical results. The numerical model is applicable to mixing and transport processes in more complicated shallow environmental flows.     

Addition of simultaneous heat and solute transport and variable fluid viscosity to SEAWAT

SEAWAT is a finite-difference computer code designed to simulate coupled variable-density ground water flow and solute transport. This paper describes a new version of SEAWAT that adds the ability to simultaneously model energy and solute transport. This is necessary for simulating the transport of heat and salinity in coastal aquifers for example. This work extends the equation of state for fluid density to vary as a function of temperature and/or solute concentration. The program has also been modified to represent the effects of variable fluid viscosity as a function of temperature and/or concentration. The viscosity mechanism is verified against an analytical solution, and a test of temperature-dependent viscosity is provided. Finally, the classic Henry–Hilleke problem is solved with the new code.     

位移法浅水波方程的解及其特性

不同于传统流体力学,在Lagrange坐标下推导浅水波方程.若将水平位移作为基本变量,则推导出的浅水波数学模型可描述为固体力学的非线性大位移问题.运用不可压缩条件,通过变分原理推导出位移法浅水波方程,给出椭圆函数形式的行波解,并分析孤波解产生的条件.该基础研究建立了在分析结构力学中分析浅水波问题的理论基础,有利于进一步开展水动力学的研究.     

“Law of the nano-wall” in nano-channel gas flows

Molecular dynamics simulations of force-driven nano-channel gas flows show two distinct flow regions. While the bulk flow region can be determined using kinetic theory, transport in the near-wall region is dominated by gas–wall interactions. This duality enables definition of an inner-layer scaling, \(y^{*}\), based on the molecular dimensions. For gas–wall interactions determined by Lennard–Jones potential, the velocity distribution for \(y^{*} \le 3\) exhibits a universal behavior as a function of the local Knudsen number and gas–wall interaction parameters, which can be interpreted as the “law of the nano-wall.” Knowing the velocity and density distributions within this region and using the bulk flow velocity profiles from Beskok–Karniadakis model (Beskok and Karniadakis in Microscale Thermophys Eng 3(1):43–77, 1999), we outline a procedure that can correct kinetic-theory-based mass flow rate predictions in the literature for various nano-channel gas flows.     

A high-resolution method for the depth-integrated solute transport equation based on an unstructured mesh

This paper presents a high-resolution numerical method for solving mass transport problems involving advection and anisotropic diffusion in shallow water based on unstructured mesh. An alternating operator-splitting technique is adopted to advance the numerical solution with advection and diffusion terms solved separately in two steps. By introducing a new r-factor into the Total Variation Diminishing (TVD) limiter, an improved finite-volume method is developed to solve the advection term with significant reduction of numerical diffusion and oscillation errors. In addition, a coordinate transformation is introduced to simplify the diffusion term with the Green-Gauss theorem used to deal with the anisotropic effect based on unstructured mesh. The new scheme is validated against three benchmark cases with separated and combined advection and diffusion transport processes involved. Results show that the scheme performs better than existing methods in predicting the advective transport, particularly when a sharp concentration front is in presence. The model also provides a sound solution for the anisotropic diffusion phenomenon. Anisotropic diffusion has been largely neglected by existing flow models based on unstructured mesh, which usually treat the diffusion process as being isotropic for simplicity. Based on the flow field provided by the ELCIRC model, the developed transport model was successfully applied to simulate the transport of a hypothetical conservative tracer in a bay under the influence of tides.     

Surface charge-dependent transport of water in graphene nano-channels

Deionized water flow through positively charged graphene nano-channels is investigated using molecular dynamics simulations as a function of the surface charge density. Due to the net electric charge, Ewald summation algorithm cannot be used for modeling long-range Coulomb interactions. Instead, the cutoff distance used for Coulomb forces is systematically increased until the density distribution and orientation of water atoms converged to a unified profile. Liquid density near the walls increases with increased surface charge density, and the water molecules reorient their dipoles with oxygen atoms facing the positively charged surfaces. This effect weakens away from the charged surfaces. Force-driven water flows in graphene nano-channels exhibit slip lengths over 60 nm, which result in plug-like velocity profiles in sufficiently small nano-channels. With increased surface charge density, the slip length decreases and the apparent viscosity of water increases, leading to parabolic velocity profiles and decreased flow rates. Results of this study are relevant for water desalination applications, where optimization of the surface charge for ion removal with maximum flow rate is desired.     

Dynamics and solutions to some control problems for water-tanksystems

We consider a tank containing a fluid. The tank is subjected to directly controlled translations and rotations. The fluid motion is described by linearized wave equations under shallow water approximations. For irrotational flows, a new variational formulation of Saint-Venant equations is proposed. This provides a simple method to establish the equations when the tank is moving. Several control configurations are studied: one and two horizontal dimensions; tank geometries (straight and nonstraight bottom, rectangular and circular shapes), tank motions (horizontal translations with and without rotations). For each configuration, we prove that the linear approximation is steady-state controllable and provide a simple and flatness-based algorithm for computing the steering open-loop control. These algorithms rely on operational calculus. They lead to second order equations in space variables whose fundamental solutions define delay operators corresponding to convolutions with compact support kernels. For each configuration, several controllability open-problems are proposed and motivated     

Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography

Many natural terrains have complicated surface topography. The simulation of steep-fronted flows that occur after heavy rainfall flash floods or as inundation from dyke breaches is usually based on the non-linear shallow water equations in hyperbolic conservation form. Particular challenges to numerical modellers are posed by the need to balance correctly the flux gradient and source terms in Godunov-type finite volume shock-capturing schemes and by the moving wet-dry boundary as the flood rises or falls. This paper presents a Godunov-type shallow flow solver on adaptive quadtree grids aimed at simulating flood flows as they travel over natural terrain. By choosing the stage and discharge as dependent variables in the hyperbolic non-linear shallow water equations, a new deviatoric formulation is derived that mathematically balances the flux gradient and source terms in cases where there are wet-dry fronts. The new formulation is more general in application than previous a priori approaches. Three benchmark tests are used to validate the solver, and include steady flow over a submerged hump, flow disturbances propagating over an elliptical-shaped hump, and free surface sloshing motions in a vessel with a parabolic bed. The model is also used to simulate the propagation of a flood due to a dam break over an initially dry floodplain containing three humps.