Classical Advection-diffusion Equation Research Articles

Reinvestigating the Kinetic Model for the Suspended Sediment Concentration in an Open Channel Flow

The prediction of sediment transport, related to different environmental and engineering problems, requires accurate mathematical models. Most available mathematical models for the concentrations of suspended sediments are based on the classical advection diffusion equation, which remains not efficient enough to describe the complete behavior related to sediment–water two-phase flows and the feedback between the turbulent unsteady flow and suspended sediments. The aim of this paper is to reinvestigate the kinetic model for turbulent two-phase flows, which accounts for both sediment–turbulence interactions and sediment–sediment collisions. The present study provides a detailed and rigorous derivation of the kinetic model equations, clarifications about the mathematical approach and more details about the main assumptions. An explicit link between the kinetic model and the classical advection diffusion equation is provided. Concentration profiles for suspended sediments in open channel flows show that the kinetic model is able to describe the near-bed behavior for coarse sediments.

Suspended Sediments in Environmental Flows: Interpretation of Concentration Profiles Shapes

In environmental flows, field and laboratory measurements of suspended sediments show two kinds of concentration profiles. For coarse sediments, a near-bed upward convex profile is observed beneath the main upward concave profile. In this study, we consider two 1-DV models, namely, the classical advection–diffusion equation (ADE) based on the gradient diffusion model, and the kinetic model. Both need sediment diffusivity, which is related to the eddy viscosity, and an y-dependent β-function (i.e., the inverse of the turbulent Schmidt number). Our study shows that the kinetic model reverts to the classical ADE with an “apparent” settling velocity or sediment diffusivity. For the numerical resolution of the ADE, simple and accurate tools are provided for both the sediment diffusivity and hindered settling. The results for the concentration profiles show good agreement with the experimental data. An interpretation of the concentration profiles is provided by two “criteria” for shapes. The main for steady open-channel flows shows that the shape of the concentration profiles in the Cartesian coordinate depends on the vertical distribution of the derivative of R (the ratio between the sediment diffusivity and the settling velocity of the sediments): dR/dy > −1 for the upward concave concentration profile while dR/dy < −1 for the near-bed upward convex profile. A generalization is proposed for oscillatory flows over sand ripples, where the time-averaged concentration profiles in the semi-log plots are interpreted by a relation between the second derivative of the logarithm of the concentration and the derivative of the product between the sediment diffusivity and an additional parameter related to the convective sediment entrainment process.

A Peridynamic Differential Operator-Based Model for Quantifying Spatial Non-Local Transport Behavior of Pollutants in Heterogeneous Media

Modeling pollutant transport in heterogeneous media is an important task of hydrology. Pollutant transport in a non-homogeneous environment typically exhibits non-local transport dynamics, whose efficient characterization requires a parsimonious model with the non-local feature. This study encapsulates the non-local transport characteristic of pollutants into the peridynamic differential operator (PDDO) and develops a PDDO-based model for quantifying the observed pollutant non-local transport behavior. The simulation results show that the proposed model can describe pollutant non-local transport behavior in various heterogeneous media. The non-local nature of pollutant transport can be adjusted by pre-defined weight function w(|ξ|) and horizon Hx. Applications show that the PDDO-based model can better capture pollutant non-local transport behavior than the classical advection–diffusion equation (ADE) model, especially for quantifying the tail of the experimental data late. Analyses further reveal that the PDDO-based model can characterize both normal (Fickian) and anomalous (Lévy) diffusion regimes.

Generalized finite difference method with irregular mesh for a class of three-dimensional variable-order time-fractional advection-diffusion equations

Fractional advection-diffusion equation, as a generalization of classical advection-diffusion equation, has been always mentioned to simulate anomalous diffusion in porous media. This work introduces a meshless generalized finite difference method (GFDM) to solve a class of three-dimensional variable-order time fractional advection-diffusion equation (TFADE) in finite domains. Three examples with known analytic solutions in different domains are given to demonstrate that the method is accurate and stable. To reduce computational and storage cost, we discretize the time derivative terms of TFADE by a fast finite difference method (FFDM) based on sum-of-exponentials (SOE) approximation. Meanwhile, discretizing space derivative terms, GFDM generates a linear equation set including function values of neighboring nodes with various weight coefficients. Then the partial derivatives of TFADE are indicated as the linear system above. Also, this paper investigates the irregular mesh in the finite spatial domain, which is more closely meets the description of practice problems. Numerical results indicate that models with irregular mesh can also be simulated by GFDM which maintains high accuracy. Furthermore, the method is stable and accurate in solving three-dimensional irregular domain problems, where the relative errors can be less than 0.01%. This paper shows that FFDM based on SOE approximation can improve computational efficiency, and GFDM can flexibly and efficiently solve three-dimensional variable-order and variable-coefficient TFADE.

From micro-scale to macro-scale modeling of solute transport in drying capillary porous media

Drying of capillary porous media initially saturated with saline water is central to many engineering and environmental applications. In order to predict the evolution of solute concentration in a porous medium, the macroscopic continuum models (CMs) are commonly employed. However, the predictive aptitudes of the CMs have been questioned. In this work, we solve the classical advection-diffusion equation for solute transport in an isothermally drying capillary porous medium for the limiting condition of capillary-dominated regime. The solution of the continuum model is compared with pore network simulations. The results of both models are analyzed in terms of local solute concentration profiles for different values of network saturation. On this basis, the ability of the continuum model to reciprocate the pore network results is assessed. The degree of heterogeneity in the liquid phase structure is characterized by performing pore network Monte Carlo simulations distinguishing the percolating liquid cluster from the non-percolating isolated clusters. Based on the statistical analysis of Monte Carlo simulations, the probability of first solid crystals to appear in the respective liquid phase elements is discussed. We observe that solute enrichment is more pronounced in the isolated single liquid throats and isolated clusters due to lack or significant hindrance to back-diffusion as a result of discontinuity in the liquid phase.

Influence of Chemical Osmosis on Solute Transport and Fluid Velocity in Clay Soils

AbstractSolute transport through the clay liner is a significant process in many waste landfills or unmanaged landfills. At present, researchers mainly focus on the test study about semi-membrane property of clay material, however, the influence of chemical osmosis caused by membrane effect on solute transport and fluid velocity is insufficient. In this investigation, based on the classical advection-diffusion equation, a one-dimensional solute transport model for low-permeable clay material has been proposed, in which the coupled fluid velocity related with hydraulic gradient and concentration gradient is introduced, and the semi-membrane effect is embodied in the diffusion mechanism. The influence of chemical osmosis on fluid velocity and solute transport has been analyzed using COMSOL Multiphysics software. The simulated results show that chemical osmosis has a significant retarded action on fluid velocity and pollutant transport. The proposed model can effectively reveal the change in process of coupled fluid velocity under dual gradient and solute transport, which can provide a theoretical guidance for similar fluid movement in engineering.

Ubiquity of anomalous transport in porous media: Numerical evidence, continuous time random walk modelling, and hydrodynamic interpretation

Anomalous transport in porous media is commonly believed to be induced by the highly complex pore space geometry. However, this phenomenon is also observed in porous media with rather simple pore structure. In order to answer how ubiquitous can anomalous transport be in porous media, we in this work systematically investigate the solute transport process in a simple porous medium model with minimal structural randomness. The porosities we consider range widely from 0.30 up to 0.85, and we find by lattice Boltzmann simulations that the solute transport process can be anomalous in all cases at high Péclet numbers. We use the continuous time random walk theory to quantitatively explain the observed scaling relations of the process. A plausible hydrodynamic origin of anomalous transport in simple porous media is proposed as a complement to its widely accepted geometric origin in complex porous media. Our results, together with previous findings, provide evidence that anomalous transport is indeed ubiquitous in porous media. Consequently, attentions should be paid when modelling solute transport by the classical advection-diffusion equation, which could lead to systematic error.

Lattice Boltzmann method for the fractional advection-diffusion equation

Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.

High-order Accurate Numerical Methods for Solving the Space Fractional Advection-dispersion Equation

In this paper, we consider a type of space fractional advection-dispersion equation, which is obtained from the classical advection-diffusion equation by replacing the spatial derivatives with a generalized derivative of fractional order. Firstly, we utilize the modified weighted and shifted Grunwald difference operators to approximate the Riemann-Liouville fractional derivatives and present the finite volume method. Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Secondly, we prove that the scheme is unconditionally stable and convergent with the accuracy of O(τ2 + h2). Furthermore, we apply an extrapolation method to improve the convergence order, which can be O(τ4 + h4). Finally, two numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent agreement with the theoretical analysis.

Numerical solution of fractional advection-diffusion equation with a nonlinear source term

In this paper we use the Jacobi collocation method for solving a special kind of the fractional advection-diffusion equation with a nonlinear source term. This equation is the classical advection-diffusion equation in which the space derivatives are replaced by the Riemann-Liouville derivatives of order 0 < ? ≤ 1 and 1 < μ ≤ 2. Also the stability and convergence of the presented method are shown for this equation. Finally some numerical examples are solved using the presented method.

Stability and convergence of a finite volume method for the space fractional advection–dispersion equation

We consider the space fractional advection–dispersion equation, which is obtained from the classical advection–diffusion equation by replacing the spatial derivatives with a generalised derivative of fractional order. We derive a finite volume method that utilises fractionally-shifted Grünwald formulae for the discretisation of the fractional derivative, to numerically solve the equation on a finite domain with homogeneous Dirichlet boundary conditions. We prove that the method is stable and convergent when coupled with an implicit timestepping strategy. Results of numerical experiments are presented that support the theoretical analysis.

하천 오염물질의 모의를 위한 프랙탈 이송확산방정식의 해석적 유도

프랙탈 이송확산방정식은 정수 차수의 미분연산자로 구성된 고전적인 이송확산방정식과 비교하여 프랙탈 차수의 미분연산자로 구성된 보다 상위개념의 방정식으로써 정의된다. 지금까지의 프랙탈 이송확산방정식은 추계학적인 기법을 동원하여 푸리에-라플라스 공간에서 주로 해석되었으나, 본 연구에서는 실제 공간에서 유한차분개념을 도입하여 보다 직접적으로 하천에서의 오염물 이송확산에 관한 지배방정식을 유도하였다. 이러한 개념의 유도방법은 프랙탈 차수 및 관련 확산계수의 물리적인 추정에 관한 실마리를 제공할 수 있다. 고전적인 이송확산방정식과는 달리 프랙탈 이송확산방정식은 실제 하천에서 관측되는 오염물의 시간-농도 분포곡선의 왜곡현상과 분포곡선의 전후방부 농도를 보다 실제에 가깝게 모의할 수 있을 것으로 기대되어진다. The fractal advection-diffusion equation (ADE) is a generalization of the classical AdE in which the second-order derivative is replaced with a fractal order derivative. While the fractal ADE have been analyzed with a stochastic process In the Fourier and Laplace space so far, in this study a fractal ADE for describing solute transport in rivers is derived with a finite difference scheme in the real space. This derivation with a finite difference scheme gives the hint how the fractal derivative order and fractal diffusion coefficient can be estimated physically In contrast to the classical ADE, the fractal ADE is expected to be able to provide solutions that resemble the highly skewed and heavy-tailed time-concentration distribution curves of contaminant plumes observed in rivers.

Longitudinal dispersion in natural channels: 2. The roles of shear flow dispersion and dead zones in the River Severn, U.K.

Abstract. The classical one-dimensional advection-diffusion equation (ADE) gives an inadequate description of tracer cloud evolution in the River Severn, U.K. A solute transport model incorporating the effects of tracer storage in dead zones is presented in which the channel is conceived as being divided into two parallel regions. The bulk flow region occurs in the central part. Its longitudinal dispersive properties are described by the ADE. Adjacent to this, an additional cross-sectional area is defined in which tracer can be stored temporarily in regions of slowly moving water called dead zones. Exchange between the two regions follows a first order rate equation. Applying the model to the River Severn shows that a dispersing cloud’s evolution occurs in two distinct stages with a rapid transitional phase. Initially, shear-dispersion is dominant while the tracer particles mix fully over the bulk flow. Once this has occurred, dead zone storage accounts well for the non-Fickian evolution of the cloud. After the transitional phase the dead zone storage mechanism clearly dominates over shear-dispersion. Overall, the combined shear flow dispersion – dead zone model (D-DZM) provides a much better, physically consistent description of the tracer cloud’s evolution than the simple classical ADE approach can do alone. Keywords: Channels; dispersion; dead zones; tracers; River Severn

A fully coupled heat and mass transfer model incorporating contaminant gas transfer in an unsaturated porous medium

Abstract This paper presents a fully coupled heat and mass transfer numerical model describing the migration of a contaminant gas through an engineered clay liner of a sanitary landfill site. The governing system of non-linear partial differential equations are derived from a mechanistic approach where the mass and energy conservation laws are defined for a particular phase into which Darcy’s law and Fick’s law are substituted. The model treats the migration of liquid water, heat, air and contaminant gas separately with independent system variables of capillary potential, temperature, pore air pressure and molar concentration of the contaminant gas. A numerical solution is presented, which is compared against the classical advection–diffusion equation, to highlight the effect of a temperature gradient on the contaminant gas concentration.