Multiple anisotropic collisions for advection–diffusion Lattice Boltzmann schemes

This paper develops a symmetrized framework for the analysis of the anisotropic advection–diffusion Lattice Boltzmann schemes. Two main approaches build the anisotropic diffusion coefficients either from the anisotropic anti-symmetric collision matrix or from the anisotropic symmetric equilibrium distribution. We combine and extend existing approaches for all commonly used velocity sets, prescribe most general equilibrium and build the diffusion and numerical-diffusion forms, then derive and compare solvability conditions, examine available anisotropy and stable velocity magnitudes in the presence of advection. Besides the deterioration of accuracy, the numerical diffusion dictates the stable velocity range. Three techniques are proposed for its elimination: (i) velocity-dependent relaxation entries; (ii) their combination with the coordinate-link equilibrium correction; and (iii) equilibrium correction for all links. Two first techniques are also available for the minimal (coordinate) velocity sets. Even then, the two-relaxation-times model with the isotropic rates often gains in effective stability and accuracy. The key point is that the symmetric collision mode does not modify the modeled diffusion tensor but it controls the effective accuracy and stability, via eigenvalue combinations of the opposite parity eigenmodes. We propose to reduce the eigenvalue spectrum by properly combining different anisotropic collision elements. The stability role of the symmetric, multiple-relaxation-times component, is further investigated with the exact von Neumann stability analysis developed in diffusion-dominant limit.     

Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation

We extend lattice Boltzmann (LB) methods to advection and anisotropic-dispersion equations (AADE). LB methods are advocated for the exactness of their conservation laws, the handling of different length and time scales for flow/transport problems, their locality and extreme simplicity. Their extension to anisotropic collision operators (L-model) and anisotropic equilibrium distributions (E-model) allows to apply them to generic diffusion forms. The AADE in a conventional form can be solved by the L-model. Based on a link-type collision operator, the L-model specifies the coefficients of the symmetric diffusion tensor as linear combination of its eigenvalue functions. For any type of collision operator, the E-model constructs the coefficients of the transformed diffusion tensors from linear combinations of the relevant equilibrium projections. The model is able to eliminate the second order tensor of its numerical diffusion. Both models rely on mass conserving equilibrium functions and may enhance the accuracy and stability of the isotropic convection–diffusion LB models.The link basis is introduced as an alternative to a polynomial collision basis. They coincide for one particular eigenvalue configuration, the two-relaxation-time (TRT) collision operator, suitable for both mass and momentum conservation laws. TRT operator is equivalent to the BGK collision in simplicity but the additional collision freedom relates it to multiple-relaxation-times (MRT) models. “Optimal convection” and “optimal diffusion” eigenvalue solutions for the TRT E-model allow to remove next order corrections to AADE. Numerical results confirm the Chapman–Enskog and dispersion analysis.     

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.     

The surface boundary condition in nonhydrostatic ocean models

With increasing resolution in numerical ocean models, nonhydrostatic pressure effects have to be accounted for. In sigma-coordinate mode split ocean models, this pressure may be regarded as a pressure correction. An elliptic equation must be solved for the nonhydrostatic pressure, and the gradients are used to correct the provisional hydrostatic velocity components in each time step. The focus in the present work is on the surface boundary condition for the elliptic equation. In the literature, both Dirichlet and Neumann boundary conditions are suggested and applied. To investigate the sensitivity of the numerical results to the choice of boundary condition, three numerical experiments are performed. The first and second experiments are studies of the propagation and steepening of nonlinear internal waves. The first study is on tank scale and the second experiment is on ocean scale. In the tank-scale experiment, the density and the flow fields are very robust to the choice of boundary condition. In the ocean-scale experiment, the waves produced with a Dirichlet boundary condition become more damped than the waves produced with a Neumann boundary condition. The third study involves a surface buoyant jet. It is shown that well-known characteristics of the plume front are reproduced with a Neumann boundary condition, but the rotating turbulent core of this front is lost with a Dirichlet condition. It is accordingly argued that the appropriate surface boundary condition in mode split nonhydrostatic ocean models is the Neumann condition.     

Thermal diffusivity of seasonal snow determined from temperature profiles

Thermal diffusivity of snow is an important thermodynamic property associated with key hydrological phenomena such as snow melt and heat and water vapor exchange with the atmosphere. Direct determination of snow thermal diffusivity requires coupled point measurements of thermal conductivity and density, which continually change due to snow metamorphism. Traditional methods for determining these two quantities are generally limited by temporal resolution. In this study we present a method to determine the thermal diffusivity of snow with high temporal resolution using snow temperature profile measurements. High resolution (between 2.5 and 10 cm at 1 min) temperature measurements from the seasonal snow pack at the Plaine-Morte glacier in Switzerland are used as initial conditions and Neumann (heat flux) boundary conditions to numerically solve the one-dimensional heat equation and iteratively optimize for thermal diffusivity. The implementation of Neumann boundary conditions and a t-test, ensuring statistical significance between solutions of varied thermal diffusivity, are important to help constrain thermal diffusivity such that spurious high and low values as seen with Dirichlet (temperature) boundary conditions are reduced. The results show that time resolved thermal diffusivity can be determined from temperature measurements of seasonal snow and support density-based empirical parameterizations for thermal conductivity.     

Miscible flow through porous media with dispersion and adsorption

The partial differential nonlinear equation which describes the one-dimensional flow of miscible fluids through porous media with dispersion and Langmuir equilibrium adsorption is numerically solved by finite differences.Local truncation error is determined and von Neumann stability analysis is applied. In order to eliminate either numerical dispersion or unstability, weighting parameters and distance and time increments are conveniently adjusted.Finite differences results are verified with the exact solution for the linear adsorption case. They are obtained for different boundary conditions, whose influence is discussed.Numerical solutions are matched with experimental results from Szabo's¹ polymer flooding tests. Differences between numerical and experimental results are minimized applying optimization techniques to obtain the most suitable physical parameters.     

A new analytical solution solved by triple series equations method for constant-head tests in confined aquifers

The constant-head pumping tests are usually employed to determine the aquifer parameters and they can be performed in fully or partially penetrating wells. Generally, the Dirichlet condition is prescribed along the well screen and the Neumann type no-flow condition is specified over the unscreened part of the test well. The mathematical model describing the aquifer response to a constant-head test performed in a fully penetrating well can be easily solved by the conventional integral transform technique under the uniform Dirichlet-type condition along the rim of wellbore. However, the boundary condition for a test well with partial penetration should be considered as a mixed-type condition. This mixed boundary value problem in a confined aquifer system of infinite radial extent and finite vertical extent is solved by the Laplace and finite Fourier transforms in conjunction with the triple series equations method. This approach provides analytical results for the drawdown in a partially penetrating well for arbitrary location of the well screen in a finite thickness aquifer. The semi-analytical solutions are particularly useful for the practical applications from the computational point of view.     

RATIONAL BASIS FOR SUSPENDED SEDIMENT MODELING

This paper presents a rational basis to model the transport of suspended sediment. The loose-boundary condition for 3D models and the adjustment coefficients for both the depth-integrated 2D and laterally integrated 1D models are treated comprehensively. A combination of Dirichlet and Neumann conditions is proposed as the loose-boundary condition. The adjustment coefficient for 2D models is obtained on the basis of the proposed boundary condition and analytical solutions developed for some simple cases of non-equilibrium transport of sediment in uniform flows. The adjustment coefficient for 1D models for natural rivers is further obtained from lateral integration. Comparisons with analytical solutions and a considerable amount of laboratory and prototype data show that mathematical models developed along the proposed line of attack would well simulate the transport of suspended sediment in practical problems.     

Periodic recharge to finite aquifiers from rectangular areas

Analytical solutions for groundwater flow in unconfined rectangular aquifers are presented for the case of recharge from rectangular areas. The linearised differential equation of the flow is solved using Laplace and finite Fourier transforms. The problem concerns the response of finite aquifers to periodic (seasonal) recharge schemes of variable duration. Results can be obtained for various combinations of Dirichlet and Neumann boundary conditions and can be easily used in a preliminary study of water resources management.     

三维直流电阻率有限元-无限元耦合数值模拟

为解决传统有限元截断边界所引起的问题,本文提出了一种新的三维直流电阻率有限元-无限元耦合数值模拟方法.首先推导了无限元三维单元映射函数,然后提出了一种全新的最优的无限元形函数并与多种其他形函数进行了对比,随后将其与非结构化四面体有限元相结合,取代了传统的混合边界条件,使得电位在无限域内连续并在无限远处衰减为零,最终形成的左端矩阵稀疏对称并与场源位置无关.数值计算表明,该方法可以在近似测区大小的计算范围内得到与混合边界条件相当的计算精度,优于相同计算范围下齐次边界条件的解,有利于减少计算节点数;由于左端矩阵不随场源位置改变,有利于加速反演计算.     

An Analytic Solution to Well-water Level Changes under Barometric Pressure

Under barometric pressure,groundwater flow in well-aquifer systems is a kind of hydro-mechanical coupling problem.Applying the flux boundary conditions on borehole wall and water pressure equilibrium conditions inside and outside the borehole wall under barometric pressure(BP),an analytic solution to well-water level changes has been proposed in this paper.The formulation shows that the BP coefficients increase with time and tend to BP constant.The Change of BP coefficients over time depends only on the ratio of transmissivity(T) to the well radius squared(r2w),and has nothing to do with the change in BP.The BP constant only relates to aquifer loading efficiency(B),and has nothing to do with the aquifer transmissivity and well radius.The BP coefficients change over time in the analytic formulation is consistent with the analysis of measured data from the Nanxi wells.Based on the BP coefficient changes over time,a parameter estimation method is suggested and discussed in its application to the estimation of the aquifer BP constant(or B) and transmissivity by using the Nanxi well data.     

Magnetic equilibria resulting from a helically symmetric kink instability

Abstract

This paper explores magnetic equilibria which could result from the kink instability in a cylindrical magnetic flux tube. We examine a variety of cylindrical magnetic equilibria which are susceptible to the kink, and simulate its evolution in a frictional fluid. We assume that the evolution takes place under conditions of helical symmetry, so the problem becomes effectively two-dimensional. The initial cylindrical equilibrium field is specified in terms of its twist function k(r) = B _θ/(rB_z ) and for a variety of k(r) functions we calculate linear growth rates for the kink instability, assuming that it develops under helical symmetry with pitch τ. We find that the growth rate is sensitive to the value of τ.

We simulate nonlinear evolution of the kink using a Lagrangian frictional code which constrains the field to have helical symmetry of a given pitch τ. Ideal MHD is assumed and the plasma pressure is taken to be small in order to mimic conditions in the solar corona. In some cases the flux tube evolves to a new smooth helically symmetric equilibrium which involves a relatively small change in the maximum electric current. In other cases there is evidence of current-sheet formation.     

Meshless BEM and overlapping Schwarz preconditioners for exterior problems on spheroids

We consider the exterior Neumann problem of the Laplacian with boundary condition on spheroids. We propose to use spherical radial basis functions in the solution of the boundary integral equation arising from the Dirichlet-to-Neumann map. Our meshless approach with radial basis functions is particularly suitable for handling scattered satellite data. We also propose a preconditioning technique based on an overlapping domain decomposition method to deal with ill-conditioned matrices arising from the approximation problem.     

Estimating finite difference block equivalent hydraulic conductivity for numerically solving the Richards' equation

A method is proposed for calculating the equivalent hydraulic conductivity (EHC) within a finite difference block (FDB). Application of the constant‐flux assumption of Darcy's Law, the EHC equals to the integration of effective hydraulic conductivity (K_w) as a function of pressure head (h_w) divided by the head difference at the ends of the FDB. Error analysis show that the constant‐flux (CF) EHC estimates are better than those computed by the commonly used arithmetic‐mean (AM), geometric‐mean (GM), and harmonic‐mean (HM) techniques. CF EHC results are even more superior at larger interblock head difference situations. Simulations of water infiltration experiments show that simulations using the CF EHC or AM or GM weighting technique have only slight difference while applying the Neumann type boundary condition at the ground surface. In case of the Dirichlet type boundary condition, however, the CF EHC is superior to the other two in correctly estimating the depth of infiltration while enlarging the grid size. Therefore, it is recommended to adopt the CF EHC with a larger grid size to the more stable and more efficient results. Copyright © 2007 John Wiley & Sons, Ltd.     

Exact analytical solutions for two-dimensional advection-dispersion equation in cylindrical coordinates subject to third-type inlet boundary condition

Exact analytical solutions for two-dimensional advection-dispersion equation (ADE) in cylindrical coordinates subject to the third-type inlet boundary condition are presented in this study. The finite Hankel transform technique in combination with the Laplace transform method is adopted to solve the two-dimensional ADE in cylindrical coordinates. Solutions are derived for both continuous input and instantaneous slug input. The developed analytical solutions are compared with the solutions for first-type inlet boundary condition to illustrate the influence of the inlet condition on the two-dimensional solute transport in a porous medium system with a radial geometry. Results show significant discrepancies between the breakthrough curves obtained from analytical solutions for the first-type and third-type inlet boundary conditions for large longitudinal dispersion coefficients. The developed solutions conserve the solute mass and are efficient tools for simultaneous determination of the longitudinal and transverse dispersion coefficients from a laboratory-scale radial column experiment or an in situ infiltration test with a tracer.     

Evaluation of Mass Flux to and from Ground Water Using a Vertical Flux Model (VFLUX): Application to the Soil Vacuum Extraction Closure Problem

Site closure for soil vacuum extraction (SVE) application typically requires attainment or specified soil concentration standards based on the premise that mass flux from the vadose zone to ground water not result in levels exceeding maximum contaminant levels (MCLs). Unfortunately, realization of MCLs in ground water may not be attainable at many sites. This results in soil remediation efforts that may be in excess of what is necessary for future protection of ground water and soil remediation goals which often cannot be achieved within a reasonable time period. Soil venting practitioners have attempted to circumvent these problems by basing closure on some predefined percent total mass removal, or an approach to a vapor concentration asymptote. These approaches, however, are subjective and influenced by venting design. We propose an alternative strategy based on evaluation of five components: (1) site characterization, (2) design. (3) performance monitoring, (4) rule-limited vapor transport, and (5) mass flux to and from ground water. Demonstration of closure is dependent on satisfactory assessment of all five components. The focus of this paper is to support mass flux evaluation. We present a plan based on monitoring of three subsurface zones and develop an analytical one-dimensional vertical flux model we term VFLUX. VFLUX is a significant improvement over the well-known numerical one-dimensional model. VLEACH, which is often used for estimation of mass flux to ground water, because it allows for the presence of nonaqueous phase liquids (NAPLs) in soil, degradation, and a lime-dependent boundary condition at the water table inter-face. The time-dependent boundary condition is the center-piece of our mass flux approach because it dynamically links performance of ground water remediation lo SVE closure. Progress or lack of progress in ground water remediation results in either increasingly or decreasingly stringent closure requirements, respectively.     

重力波在中层大气温度波导中的传播模式研究

本文给出了重力波在中层大气温度波导中的导制传播模型,并在此模型的基础上详细讨论了重力波部分导制传播下的对称模式与非对称模式,导出了不同模式下相应的特征函数和色散方程,进一步用离散的方法对两类色散方程进行了求解;同时还利用二维全隐欧拉格式（FICE）对重力波在温度波导中的传播进行了模拟,模拟的结果也成功地展现了对称与非对称两种传播模式.研究表明,下边界的扰动能量在向上传播进入波导区域后被俘获,形成导制传播.不同周期的初始扰动,在波导内均会形成对称与非对称形式两种模式的导制传播,由于两者的行进速度不一致,最终会引起两种不同模式的分离.数值模拟中重力波的水平行进速度与线性模型预测值非常接近.波导中不同模式下重力波的水平波长与初始扰动的水平波长非常一致,然而波导区域内重力波的频率与初始扰动的频率无关,频率不同的初始扰动会激发出相同频率的重力波对称与非对称导制传播模式.这表明在确定的温度波导中,水平波数才是决定重力波传播特性的决定因素.进一步的分析显示,初始扰动的水平波数-频率分布越接近完全导制传播的色散关系时,温度波导中更易于生成以该种模式部分导制传播的重力波.     

Three-dimensional local grid refinement for block-centered finite-difference groundwater models using iteratively coupled shared nodes: a new method of interpolation and analysis of errors

This paper describes work that extends to three dimensions the two-dimensional local-grid refinement method for block-centered finite-difference groundwater models of Mehl and Hill [Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes. Adv Water Resour 2002;25(5):497–511]. In this approach, the (parent) finite-difference grid is discretized more finely within a (child) sub-region. The grid refinement method sequentially solves each grid and uses specified flux (parent) and specified head (child) boundary conditions to couple the grids. Iteration achieves convergence between heads and fluxes of both grids. Of most concern is how to interpolate heads onto the boundary of the child grid such that the physics of the parent-grid flow is retained in three dimensions. We develop a new two-step, “cage-shell” interpolation method based on the solution of the flow equation on the boundary of the child between nodes shared with the parent grid. Error analysis using a test case indicates that the shared-node local grid refinement method with cage-shell boundary head interpolation is accurate and robust, and the resulting code is used to investigate three-dimensional local grid refinement of stream-aquifer interactions. Results reveal that (1) the parent and child grids interact to shift the true head and flux solution to a different solution where the heads and fluxes of both grids are in equilibrium, (2) the locally refined model provided a solution for both heads and fluxes in the region of the refinement that was more accurate than a model without refinement only if iterations are performed so that both heads and fluxes are in equilibrium, and (3) the accuracy of the coupling is limited by the parent-grid size—a coarse parent grid limits correct representation of the hydraulics in the feedback from the child grid.     

Using data assimilation method to calibrate a heterogeneous conductivity field conditioning on transient flow test data

A data assimilation method is developed to calibrate a heterogeneous hydraulic conductivity field conditioning on transient pumping test data. The ensemble Kalman filter (EnKF) approach is used to update model parameters such as hydraulic conductivity and model variables such as hydraulic head using available data. A synthetical two-dimensional flow case is used to assess the capability of the EnKF method to calibrate a heterogeneous conductivity field by assimilating transient flow data from observation wells under different hydraulic boundary conditions. The study results indicate that the EnKF method will significantly improve the estimation of the hydraulic conductivity field by assimilating continuous hydraulic head measurements and the hydraulic boundary condition will significantly affect the simulation results. For our cases, after a few data assimilation steps, the assimilated conductivity field with four Neumann boundaries matches the real field well while the assimilated conductivity field with mixed Dirichlet and Neumann boundaries does not. We found in our cases that the ensemble size should be 300 or larger for the numerical simulation. The number and the locations of the observation wells will significantly affect the hydraulic conductivity field calibration.