Multi-block lattice Boltzmann simulations of solute transport in shallow water flows

We report a two-dimensional multi-block lattice Boltzmann model for solute transport in shallow water flows, which is developed based on the advection–diffusion equation for mass transport and the shallow water equations for the flows. A weighting factor is included in the centered scheme for improved accuracy. The model is firstly verified by simulating three benchmark tests: wind-driven circulation in a dish-shaped lake, jet-forced flow in a circular basin, and flow formed by two parallel streams containing different uniform concentrations at the same constant velocity; and then it is applied to a practical wind-induced flow, Baiyangdian Lake, which is characterized by irregular geometries and complex bathymetries. The numerical results have shown that the model is able to produce accurate and detailed results for both water flows and solute transport, which is attractive, especially for flows in narrow zones of practical terrains and certain areas with largely varying pollutant concentrations.     

Lattice Boltzmann simulations of the transient shallow water flows

A two-dimensional lattice Boltzmann model (LBM) is presented for transient shallow water flows. The model is based on the shallow water equations coupled with the large eddy simulation model. In order to obtain accurate results efficiently, a multi-block lattice scheme is applied at the area where a local finer grid is needed for strong change in physical variables. The model is verified by applying to five cases with transient processes: (a) a tidal wave over steps; (b) a perturbation over a submerged hump; (c) partial dam break flow; (d) circular dam break flow; (e) interaction between a dam break surge and four square cylinders. The objectives of this study are to validate the two-dimensional LBM in transient flow simulation and provide the detailed transient processes in shallow water flows.     

Modeling granular soils liquefaction using coupled lattice Boltzmann method and discrete element method

In this paper, a novel coupled pore-scale model of pore-fluid interacting with discrete particles is presented for modeling liquefaction of saturated granular soil. A microscale idealization of the solid phase is achieved using the discrete element method (DEM) while the fluid phase is modeled at a pore-scale using the lattice Boltzmann method (LBM). The fluid forces applied on the particles are calculated based on the momentum exchange between the fluid and particles. The presented model is based on a first principles formulation in which pore-pressure develops due to actual changes in pore space as particles׳ rearrangement occurs during shaking. The proposed approach is used to model the response of a saturated soil deposit subjected to low and large amplitude seismic excitations. Results of conducted simulations show that at low amplitude shaking, the input motion propagates following the theory of wave propagation in elastic solids. The deposit response to the strong input motion indicates that liquefaction took place and it was due to reduction in void space during shaking that led to buildup in pore-fluid pressure. Soil liquefaction was associated with soil stiffness degradation and significant loss of interparticle contacts. Simulation results also indicate that the level of shaking-induced shear strains and associated volumetric strains play a major role in the onset of liquefaction and the rate of pore-pressure buildup.     

用格子玻尔兹曼方法模拟非均匀介质中的电场响应

介绍了用格子玻尔兹曼方法模拟非均匀介质中的电场响应的数值模拟方法. 格子玻尔兹曼方法是从微观领域出发进行数值计算的一种全新的正演模拟方法;从玻尔兹曼碰撞模型出发,利用泰勒展开和Chapman Enskog展开,在基本力学守恒条件和约束条件的限制下,导出了电场响应的扩散方程,得到了局部平衡分布函数的表达式,给出了若干正演模拟的结果;其结果表明,利用这种方法进行非均匀介质中的电场响应正演模拟具有灵活、方便和简单等优点.     

A lattice Boltzmann model coupled with a Large Eddy Simulation model for flows around a groyne

This present paper proposes a two-dimensional lattice Boltzmann model coupled with a Large Eddy Simulation (LES) model and applies it to flows around a non-submerged groyne in a channel. The LES of shallow water equations is efficiently performed using the Lattice Boltzmann Method (LBM) and the turbulence can be taken into account in conjunction with the Smagorinsky Sub-Grid Stress (SGS) model. The bounce-back scheme of the non-equilibrium part of the distribution function is used to determine the unknown distribution functions at inflow boundary, the zero gradient of the distribution function is set normal to outflow boundary to obtain the unknown distribution functions here and the bounce-back scheme, which states that an incoming particle towards the boundary is bounced back into fluid, is applied to the solid wall to ensure non-slip boundary conditions. The initial flow field is defined firstly and then is used to calculate the local equilibrium distributions as initial conditions of the distribution functions. These coupled models successfully predict the flow characteristics, such as circulating flow, velocity and water depth distributions. The comparisons between the simulated results and the experimental data show that the model scheme has the capacity to solve the complex flows in shallow water with reasonable accuracy and reliability.     

Saltwater intrusion modeling in heterogeneous confined aquifers using two-relaxation-time lattice Boltzmann method

This study develops a lattice Boltzmann method (LBM) with a two-relaxation-time collision operator (LTRT) to solve saltwater intrusion problems. A directional-speed-of-sound (DSS) technique is introduced to take into account the hydraulic conductivity heterogeneity and discontinuity, as well as the velocity-dependent dispersion coefficient. The forcing terms in the LTRT model are customized in order to recover the density-dependent groundwater flow and mass transport equations. Using the LTRT with the squared DSS achieves at least second-order accuracy. The LTRT results are verified with Henry’s analytical solution as well as compared with several numerical examples and modified Henry problems that consider heterogeneous hydraulic conductivity and velocity-dependent dispersion. The numerical results show good agreement with the Henry analytical solution and with the numerical solutions obtained by other numerical methods.     

Sensitivity analysis of 2D steady-state shallow water flow. Application to free surface flow model calibration

This paper presents the analytical properties of the solutions of the sensitivity equations for steady-state, two-dimensional shallow water flow. These analytical properties are used to provide guidelines for model calibration and validation. The sensitivity of the water depth/level and that of the longitudinal unit discharge are shown to contain redundant information. Under subcritical conditions, the sensitivities of the flow variables are shown to obey an anisotropic elliptic equation. The main directions of the contour lines for water depth and the longitudinal unit discharge sensitivity are parallel and perpendicular to the flow, while they are diagonal to the flow for the transverse unit discharge sensitivity. Moreover, the sensitivity for all three variables extends farther in the transverse direction than in the longitudinal direction, the anisotropy ratio being a function of the sole Froude number. For supercritical flow, the sensitivity obeys an anisotropic hyperbolic equation. These findings are confirmed by application examples on idealized and real-world simulations. The sensitivities to the geometry, friction coefficient or model boundary conditions are shown to behave in different ways, thus providing different types of information for model calibration and validation.     

Application of the distributed parameter filter to predict simulated tidal induced shallow water flow

Distributed parameter filtering theory is employed for estimating the state variables and associated error covariances of a dynamical distributed system under highly random tidal and meteorological influences. The stochastic-deterministic mathematical model of the physical system under study consists of the shallow water equations described by the momentum and continuity equations in which the external forces such as Coriolis force, wind friction, and atmospheric pressure are considered. White Gaussian noises in the system and measurement equations are used to account for the inherent stochasticity of the system. By using an optimal distributed parameter filter, the information provided by the stochastic dynamical model and the noisy measurements taken from the actual system are combined to obtain an optimal estimate of the state of the system, which in turn is used as the initial condition for the prediction procedure. The approach followed here has numerical approximation carried out at the end, which means that the numerical discretization is performed in the filtering equations, and not in the equations modelling the system. Therefore, the continuous distributed nature of the original system is maintained as long as possible and the propagation of modelling errors in the problem is minimized. The appropriateness of the distributed parameter filter is demonstrated in an application involving the prediction of storm surges in the North Sea. The results confirm excellent filter performance with considerable improvement with respect to the deterministic prediction.     

Application of the distributed parameter filter to predict simulated tidal induced shallow water flow

Distributed parameter filtering theory is employed for estimating the state variables and associated error covariances of a dynamical distributed system under highly random tidal and meteorological influences. The stochastic-deterministic mathematical model of the physical system under study consists of the shallow water equations described by the momentum and continuity equations in which the external forces such as Coriolis force, wind friction, and atmospheric pressure are considered. White Gaussian noises in the system and measurement equations are used to account for the inherent stochasticity of the system. By using an optimal distributed parameter filter, the information provided by the stochastic dynamical model and the noisy measurements taken from the actual system are combined to obtain an optimal estimate of the state of the system, which in turn is used as the initial condition for the prediction procedure. The approach followed here has numerical approximation carried out at the end, which means that the numerical discretization is performed in the filtering equations, and not in the equations modelling the system. Therefore, the continuous distributed nature of the original system is maintained as long as possible and the propagation of modelling errors in the problem is minimized. The appropriateness of the distributed parameter filter is demonstrated in an application involving the prediction of storm surges in the North Sea. The results confirm excellent filter performance with considerable improvement with respect to the deterministic prediction.     

Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations

Shallow water equations with a non-flat bottom topography have been widely used to model flows in rivers and coastal areas. An important difficulty arising in these simulations is the appearance of dry areas where no water is present, as standard numerical methods may fail in the presence of these areas. These equations also have still water steady state solutions in which the flux gradients are nonzero but exactly balanced by the source term. In this paper we propose a high order discontinuous Galerkin method which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. A simple positivity-preserving limiter, valid under suitable CFL condition, will be introduced in one dimension and then extended to two dimensions with rectangular meshes. Numerical tests are performed to verify the positivity-preserving property, well-balanced property, high order accuracy, and good resolution for smooth and discontinuous solutions.     

Monitoring water transport in sandstone using flow propagators: A quantitative comparison of nuclear magnetic resonance measurement with lattice Boltzmann and pore network simulations

A comparison of advective displacement probability distributions (flow propagators) obtained by nuclear magnetic resonance (NMR) experiment with both lattice Boltzmann (LB) and pore network (PN) simulations is presented. Here, we apply all three methods to the exact same sample for the first time: we consider water transport in a Bentheimer sandstone. The LB and PN simulations are based on X-ray micro-tomography (XMT) images of a small rock sample; the NMR experiments are conducted on a much larger rock core-plug from which the small rock sample originated. Despite the limited size of the simulation domains, good agreement is achieved between all three sets of results, verified quantitatively by comparison of the low order moments of the flow propagators. We are concerned primarily with validating the simulations at high liquid flow rates (>10 ml min⁻¹) in high permeability sandstone, ultimately for future application to geological carbon sequestration studies. Under these conditions the LB simulation is found, as expected, to be more robust than the PN model due primarily to the reduced requirement to manually tune the simulation lattice to match the petro-physical properties of the rock.     

A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography

A finite volume MUSCL scheme for the numerical integration of 2D shallow water equations is presented. In the framework of the SLIC scheme, the proposed weighted surface-depth gradient method (WSDGM) computes intercell water depths through a weighted average of DGM and SGM reconstructions, in which the weight function depends on the local Froude number. This combination makes the scheme capable of performing a robust tracking of wet/dry fronts and, together with an unsplit centered discretization of the bed slope source term, of maintaining the static condition on non-flat topographies (C-property). A correction of the numerical fluxes in the computational cells with water depth smaller than a fixed tolerance enables a drastic reduction of the mass error in the presence of wetting and drying fronts. The effectiveness and robustness of the proposed scheme are assessed by comparing numerical results with analytical and reference solutions of a set of test cases. Moreover, to show the capability of the numerical model on field-scale applications, the results of a dam-break scenario are presented.