Solution Of Advection Research Articles

Generalized semi-analytical solution of advection–diffusion–reaction in finite and semi-infinite cylindrical ducts

Abstract A semi-analytical solution for the transient advection–diffusion–reaction problem within finite and semi-infinite ducts is derived. The solution allows for general radial- and time-dependent inlet/outlet conditions, complex boundary conditions on the duct wall including adsorption and decay, and arbitrary velocity profiles of the transporting fluid. The only numerical step of the solution is the inverse Laplace transform in the time variable. Therefore, the approach also produces fully analytical steady-state solutions. The solution is verified against computational fluid dynamics (CFD) simulations under various boundary conditions and velocity profiles (Newtonian and power-law), and in all cases good agreement is obtained. Although theoretically applicable to all regimes, the solution is computationally difficult at very high Peclet numbers and very early times due to numerical instabilities as a result of finite precision arithmetic of computers. A convergence analysis is conducted to delineate the boundaries of this limit for two important cases. The solution was derived using a new approach for solving two-dimensional partial differential equations (PDEs) with non-constant coefficients which parallels the Frobenius and power series methods for solving ordinary differential equations (ODEs). The approach reduces the original PDE to a single infinite-order ODE with constant coefficients. The approach is suspected to provide solutions to a large class of PDEs of this type. The solution may find applications in a number of engineering and/or biomedical fields, it can be used to verify numerical simulators, and serve as a simple and easy-to-implement alternative where access to numerical simulators is not available.

Analytical solutions for one-dimensional advection–dispersion equation of the pollutant concentration

We present simple analytical solutions for the unsteady advection–dispersion equations describing the pollutant concentration C(x, t) in one dimension. The solutions are obtained by using Laplace transformation technique. In this study we divided the river into two regions x ≤ 0 and x≥0 and the origin at x = 0. The variation of C(x, t) with the time t from t = 0 up to t → ∞ (the steady state case) is taken into account in our study. The special case for which the dispersion coefficient D = 0 is studied in detail. The parameters controlling the pollutant concentration along the river are determined.

Cumulative distribution function solutions of advection-reaction equations with uncertain parameters.

We derive deterministic cumulative distribution function (CDF) equations that govern the evolution of CDFs of state variables whose dynamics are described by the first-order hyperbolic conservation laws with uncertain coefficients that parametrize the advective flux and reactive terms. The CDF equations are subjected to uniquely specified boundary conditions in the phase space, thus obviating one of the major challenges encountered by more commonly used probability density function equations. The computational burden of solving CDF equations is insensitive to the magnitude of the correlation lengths of random input parameters. This is in contrast to both Monte Carlo simulations (MCSs) and direct numerical algorithms, whose computational cost increases as correlation lengths of the input parameters decrease. The CDF equations are, however, not exact because they require a closure approximation. To verify the accuracy and robustness of the large-eddy-diffusivity closure, we conduct a set of numerical experiments which compare the CDFs computed with the CDF equations with those obtained via MCSs. This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties of the two input parameters, such as their correlation lengths and variance of the coefficient that parametrizes the advective flux.

Time periodic traveling wave solutions for periodic advection–reaction–diffusion systems

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a class of periodic advection–reaction–diffusion systems. Under certain conditions, we prove that there exists a maximal wave speed c⁎ such that for each wave speed c≤c⁎, there is a time periodic traveling wave connecting two periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c≤c⁎ are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves with speed c>c⁎.

Optimized strong stability preserving IMEX Runge–Kutta methods

We construct and analyze robust strong stability preserving IMplicit–EXplicit Runge–Kutta (IMEX RK) methods for models of flow with diffusion as they appear in astrophysics, and in many other fields where equations with similar structure arise. It turns out that besides the optimization of the region of absolute monotonicity, some other properties of the methods are crucial for the success of such simulations. In particular, the models in our focus dictate to also take into account the step size limits associated with dissipativity, positivity of the stiff parabolic terms which represent transport by diffusion, the uniform convergence with respect to different stiffness properties of those same terms, etc. Furthermore, in the literature, some other properties, like the inclusion of a part of the imaginary axis in the stability region, have been argued to be relevant.In this paper, we construct several new IMEX RK methods which differ from each other by taking various or even all of these constraints simultaneously into account. It is demonstrated for some simple examples as well as for the problem of double-diffusive convection, that the newly constructed schemes provide a significant computational advantage over other methods from the literature. Due to their accumulation of different stability properties, the optimized IMEX RK methods obtained in this paper are robust schemes that may also be useful for general models which involve the solution of advection–diffusion equations, or other transport equations with similar stability requirements.

Pollutant Spreading in a Small Stream: A Case Study in Mala Nitra Canal in Slovakia

The Water Framework Directive requires as an obligatory goal to achieve and to keep “good water quality” status within the defined period (for Slovakia—up to the year 2015). For surface waters, the main criterion is the ecological and chemical status of the water. Mathematical and numerical modelling allows evaluating various situations of contaminants spreading in rivers (from everyday wastewater disposal through fatal accidents and discharges of the toxic substances) without immediate destructive impact to the environment. Determination of longitudinal and transverse dispersion coefficient values, as the main hydrodynamic characteristics of the dispersion, has the highest extent of uncertainty for hydrodynamic models simulating pollutant transport in streams. This paper deals with the determination of dispersion coefficients based on field tracer experiments performed in a small modified stream (basic hydrodynamic parameters during the experiments were: discharge Q = 0.138–0.553 m3.s−1, depth h = 0.29–0.48 m, width B = 5.2–5.9 m). During the experiments, various conditions and situations were taken into account, e.g., continuous and instantaneous pollution source, as well as various positions of pollution source along the river width, among others. Field measurements were evaluated using three different methods for dispersion coefficient determination: based on statistical evaluation, based on analytical solutions of advection–dispersion equation, and using numerical models. The dimensionless dispersion coefficients values were determined, which can be used for numerical simulation of pollutant transport in similar types of streams.

Book Review: “Effective Parameters of Hydrogeological Models” May Lead Readers Safely Through the Deep Waters of Uncertainty

Hydrogeological models are useful tools for understanding complex interactions in hydrogeological systems. Obtaining quantitative hydrogeological predictions allows one to make data-driven decisions. Models have become a widely used tool to deal with various practical and theoretical problems in hydrogeology; however, the difficulty of measuring aquifer properties and the complexity of hydrogeological systems indicate that data measurements and model predictions are subject to significant uncertainty. The second edition of “Effective Parameters of Hydrogeological Models” by Vikenti Gorokhovski (2014) (Springer Hydrogeology Series, Hardback ISBN 978-3-319-03568-0, eBook ISBN 978-3-319-03569-7) is an excellent introduction to hydrogeological models and the variety of methods to treat uncertainty surrounding models. The book contains 11 chapters, 68 illustrations with 35 in color. This book is part of the Springer Hydrogeology Series, which seeks to publish a broad portfolio of scientific books for researchers, students, and others interested in hydrogeology. The book is an updated version of the first edition (2012), also published by Springer as part of the Springer Briefs in Earth Sciences Series. The second edition contains a new chapter on advective solute transport through porous media, which provides a more comprehensive view of the topic. As its title indicates, the book introduces methods to evaluate model parameters that are critical to effective hydrogeological simulations. It is an excellent addition to the existing literature related to model uncertainty analysis, and will be of great value to hydrogeologists, modelers and hydrogeological students, and potentially useful for decision makers, project managers and practitioners. The second edition advances the field of hydrogeological uncertainty analysis in terms of both theory and methodology. The fact that hydrogeological model predictions are highly uncertain requires critical evaluation of model validity and model performance and this has attracted a great deal of research attention. As Carrera et al. (1993) and Wu and Zeng (2013) have pointed out, three types of uncertainty are involved in hydrogeological modeling: (1) uncertainty associated with Environ. Process. (2014) 1:187–192 DOI 10.1007/s40710-014-0008-8

Phoretic self-propulsion at finite Péclet numbers

AbstractPhoretic self-propulsion is a unique example of force- and torque-free motion on small scales. The classical framework describing the flow field around a particle swimming by self-diffusiophoresis neglects the advection of the solute field by the flow and assumes that the chemical interaction layer is thin compared to the particle size. In this paper we quantify and characterize the effect of solute advection on the phoretic swimming of a sphere. We first rigorously derive the regime of validity of the thin-interaction-layer assumption at finite values of the Péclet number (${Pe}$). Under this assumption, we solve computationally the flow around Janus phoretic particles and examine the impact of solute advection on propulsion and the flow created by the particle. We demonstrate that although advection always leads to a decrease of the swimming speed and flow stresslet at high values of the Péclet number, an increase can be obtained at intermediate values of${Pe}$. This possible enhancement of swimming depends critically on the nature of the chemical interactions between the solute and the surface. We then derive an asymptotic analysis of the problem at small${Pe}$which allows us to rationalize our computational results. Our computational and theoretical analysis is accompanied by a parallel study of the influence of reactive effects at the surface of the particle (Damköhler number) on swimming.

Stochastic Partial Differential Equation Based Modelling of Large Space–Time Data Sets

SummaryIncreasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data. We show that the solution of a stochastic advection–diffusion partial differential equation provides a flexible model class for spatiotemporal processes which is computationally feasible also for large data sets. The Gaussian process defined through the stochastic partial differential equation has, in general, a non-separable covariance structure. Its parameters can be physically interpreted as explicitly modelling phenomena such as transport and diffusion that occur in many natural processes in diverse fields ranging from environmental sciences to ecology. To obtain computationally efficient statistical algorithms, we use spectral methods to solve the stochastic partial differential equation. This has the advantage that approximation errors do not accumulate over time, and that in the spectral space the computational cost grows linearly with the dimension, the total computational cost of Bayesian or frequentist inference being dominated by the fast Fourier transform. The model proposed is applied to post-processing of precipitation forecasts from a numerical weather prediction model for northern Switzerland. In contrast with the raw forecasts from the numerical model, the post-processed forecasts are calibrated and quantify prediction uncertainty. Moreover, they outperform the raw forecasts, in the sense that they have a lower mean absolute error.

SURCOS: A software tool to simulate irrigation and fertigation in isolated furrows and furrow networks

A software tool useful for the numerical computation of surface irrigation and fertigation in furrows and furrow networks was developed. The model solves the complete one-dimensional St-Venant equations together with the transport equation of a passive solute. The flow equations and the solute advection are solved with a high resolution TVD explicit Eulerian scheme. The solute dispersion is solved with a centered implicit Eulerian scheme to avoid further restriction in the allowable time step. The computational speed of the model is high in isolated furrows. In cases of large furrow networks over extended irrigation times the model is slower but affordable computational speed is achieved. The computational model has been designed to be robust, intuitive and able to supply useful visual results. Both the executable and the source code, as well as the examples presented can be downloaded, edited and distributed under a BSD type license.

A finite volume method with linearisation in time for the solution of advection–reaction–diffusion systems

The numerical solution in one space dimension of advection–reaction–diffusion systems with nonlinear source terms may invoke a high computational cost when the presently available methods are used. Numerous examples of finite volume schemes with high order spatial discretisations together with various techniques for the approximation of the advection term can be found in the literature.Almost all such techniques result in a nonlinear system of equations as a consequence of the finite volume discretisation especially when there are nonlinear source terms in the associated partial differential equation models.This work introduces a new technique that avoids having such nonlinear systems of equations generated by the spatial discretisation process when nonlinear source terms in the model equations can be expanded in positive powers of the dependent function of interest.The basis of this method is a new linearisation technique for the temporal integration of the nonlinear source terms as a supplementation of a more typical finite volume method. The resulting linear system of equations is shown to be both accurate and significantly faster than methods that necessitate the use of solvers for nonlinear system of equations.

Contaminant Transport in Capped Deformable Partially Saturated Sediments

ABSTRACT Zhang, H.; Yang, H., and Jeng, D.-S., 2015. Contaminant transport in capped deformable partially saturated sediments. Contaminated sediment tends to consolidate in response to the loading imposed by an overlying cap layer. The form of transient solute advection through sediments and caps is consequently changed. Therefore, the contaminant discharge into the overlying water is expected to increase and the effectiveness of the in situ capping system will be strongly affected. In this study, a one-dimensional sediment–cap model coupling consolidation and advection–dispersion equation was established. A simplified model was proposed through the sensitivity analysis, and then a parametric study was conducted. Numerical examples concluded that the absorption, degree of sediment saturation, cap thickness, and sediment depth significantly affect the contaminant discharge from the top of the cap. Therefore, these parameters should be considered carefully during cap design.

Approximate solutions for the fractional advection–dispersion equation using Legendre pseudo-spectral method

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. Fractional advection–dispersion equation (FADE) is used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium and for modeling transport at the Earth surface. In this paper, an efficient numerical method for solving FADE is considered. The fractional derivative is described in the Caputo sense. The method is based on Legendre approximations. The properties of Legendre polynomials are utilized to reduce FADE to a system of ODEs, which is solved using the finite difference method. Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of FADE are presented and the results are compared with the exact solution.

An analytical model for solute transport through a water-saturated single fracture and permeable rock matrix

An analytical model for solute transport through a water-saturated single fracture and permeable rock matrix J. E. Houseworth, 1 D. Asahina, 1 and J. T. Birkholzer 1 The problem of solute transport through a water-saturated single fracture in a permeable rock matrix is examined using an analytical modeling approach. A closed-form analytical solution is obtained that accounts for transverse and longitudinal advective transport in the fracture and matrix and transverse diffusion in the matrix. The solution also accounts for both diffusive and advective solute exchange between the fracture and matrix and a general solute source position in either the fracture or matrix. The novel features are the incorporation of advective transport in the matrix and a general source position into a closed-form solution for the solute-transport problem. Examples of the solution behavior are presented, which demonstrate the effects of matrix advection in combination with advection along the fracture, transverse diffusion in the matrix for solute release in the fracture and matrix. A semianalytical solution in the form of a superposition integral is also derived that includes these transport features, plus independent levels of longitudinal diffusion and dispersion in the matrix and fracture, respectively. Examples are presented that include advective transport in the fracture and matrix, longitudinal and transverse diffusion in the matrix, longitudinal dispersion in the fracture, as well as solute release from the fracture and matrix. An approximate criterion is proposed to evaluate the significance of longitudinal diffusion and dispersion relative to longitudinal spreading caused by fracture- matrix interaction. Flow through fractures displays highly heterogeneous and complex flow patterns controlled by small-scale fracture features. Furthermore, transport processes at small scales that govern fracture-matrix exchange can dramatically influence larger-scale transport behavior [Grisak and Pick- ens, 1980]. Transport through saturated fractured rock was first investigated in an analytical model by Neretnieks [1980]. Tang et al. [1981] extended this work to include the effects of longitudinal dispersion in the fracture, and Sudicky and Frind [1982] and Maloszewski and Zuber [1985] further generalized the transport problem from a single fracture to a system of parallel fractures with interceding rock matrix. Sharifi Haddad et al. [2012] developed a semianalytical model of solute transport in a system of parallel fractures for a radially symmetric flow field associated with well injection. Maloszewski and Zuber [1990] considered the effects of linear kinetic interactions between solute and the rock for a single fracture and rock matrix. All of these mod- els were restricted to advective transport in the fracture, transverse diffusion in the matrix, and diffusive fracture- matrix solute exchange. A review of these modeling approaches among others is provided by Bodin et al. [2003a, 2003b]. Cihan and Tyner [2011] developed exact analytical solutions for advective transport through cylin- drical macropores and diffusive exchange with a soil ma- trix, for an instantaneous release of solute into a macropore, a constant concentration of solute at the top of a macropore, and a pulse release of solute into a macropore. 1 Introduction Solute transport in groundwater flow through frac- tured rock is a subject that has been investigated for nuclear waste disposal and other environmental groundwater con- tamination problems [National Research Council, 1996]. Fractures are a common feature of consolidated rock sys- tems and typically present much higher permeability than unfractured rock matrix, such that flow through fractures often dominates overall flow behavior. Matrix, on the other hand, typically dominates the overall pore volume of a fractured rock. These attributes often lead to much higher solute-transport velocities through fractures than in unfrac- tured rock or unconsolidated soils [Berkowitz, 2002]. This behavior often makes fracture flow and transport critical characteristics to examine for any geologic site where sol- ute transport is a concern. However, despite their impor- tance, fractures remain a difficult feature to represent accurately in mathematical models for groundwater flow and transport [Matth€ai et al., 2009 ; Wu and Pruess, 2000]. Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA. Corresponding author : J. E. Houseworth, Earth Sciences Division, Law- rence Berkeley National Laboratory, Berkeley, CA 94720, USA. (jehouseworth@lbl.gov)

Closed Form Solution and Equivalent Equation Approximation of Linear Advection by a Non Dissipative Second Order Scheme for Step Initial Conditions

We analyse the solution of the linear advection equation on a uniform mesh by a non dissipative second order scheme for discontinuous initial condition. These schemes are known to generate parasitic oscillations in the vicinity of the discontinuity. An approximate way to predict these oscillations is provided by the equivalent equation method. More specifically, we focus on the case of advection of a step function by the leapfrog scheme. Numerical experiments show that the equivalent equation method fails to reproduce the oscillations generated by the scheme far from the discontinuity. Thus, we derive closed form exact and approximate solutions for the scheme that accurately predict these oscillations. We study the relationship between equivalent equation approximation and exact solution for the scheme, to determine its range of validity.

On some energy inequalities and supnorm estimates for advection–diffusion equations in [formula omitted

We show how some suitable standard energy inequalities can be used for a simple short derivation of the fundamental supnorm estimate ‖u(⋅,t)‖L∞(Rn)≤K(n,p)‖u(⋅,0)‖Lp(Rn)t−n2p∀t>0 for solutions of advection–diffusion equations of the form ut+divf(u)=Δu (and some generalizations) in the whole space Rn, where n≥1 is arbitrary. Our approach can be used for other parabolic equations as well, including degenerate cases like porous medium type equations and p-Laplacian evolution equations. Some open problems and related results of interest are also given.

Diffusiophoresis of colloidal particles in neutral solute gradients at finite Péclet number

AbstractThe role of neutral solute advection on the diffusiophoretic motion of colloidal particles is quantified. Theoretical analyses of this phenomenon usually assume that the solute concentration evolves solely via diffusion; that is, the Péclet number ($\mathit{Pe}$) for solute transport is identically zero. This leads to the conclusion that the translational diffusiophoretic velocity of a colloid is independent of its size, shape, and orientation with respect to the imposed solute gradient, provided that the colloid has uniform surface properties and that the length scale of interaction between the solute and the particle surface is much smaller than the particle size (Morrison, J. Colloid Interface Sci. vol. 34, 1970, p. 210). For a single spherical colloid, we show that the particle velocity decreases monotonically with increasing $\mathit{Pe}$. Moreover, the solute concentration and fluid flow around the colloid become markedly fore–aft asymmetric as $\mathit{Pe}$ is increased. Next, an asymptotic expansion at small $\mathit{Pe}$ predicts that solute advection leads to relative phoretic motion between two identical spherical colloids, which ultimately align in a plane normal to the imposed gradient (there is no relative motion at $\mathit{Pe}= 0$). Finally, asymptotic analysis of the diffusiophoretic motion of a slightly non-spherical colloid at small $\mathit{Pe}$ demonstrates that advection leads to a shape- and orientation-dependent particle velocity, in contrast to the insensitivity of the velocity to shape and orientation at $\mathit{Pe}= 0$.

An analytical model for solute transport through a GCL-based two-layered liner considering biodegradation

An analytical model for solute advection and dispersion in a two-layered liner consisting of a geosynthetic clay liner (GCL) and a soil liner (SL) considering the effect of biodegradation was proposed. The analytical solution was derived by Laplace transformation and was validated over a range of parameters using the finite-layer method based software Pollute v7.0. Results show that if the half-life of the solute in GCL is larger than 1year, the degradation in GCL can be neglected for solute transport in GCL/SL. When the half-life of GCL is less than 1year, neglecting the effect of degradation in GCL on solute migration will result in a large difference of relative base concentration of GCL/SL (e.g., 32% for the case with half-life of 0.01year). The 100-year solute base concentration can be reduced by a factor of 2.2 when the hydraulic conductivity of the SL was reduced by an order of magnitude. The 100-year base concentration was reduced by a factor of 155 when the half life of the contaminant in the SL was reduced by an order of magnitude. The effect of degradation is more important in approving the groundwater protection level than the hydraulic conductivity. The analytical solution can be used for experimental data fitting, verification of complicated numerical models and preliminary design of landfill liner systems.

The solution of two-dimensional advection–diffusion equations via operational matrices

In this paper we describe a spectrally accurate, unconditionally stable, efficient method using operational matrices to solve numerically two-dimensional advection–diffusion equations on a rectangular domain.The novelty of this paper is to relate for the first time evolution partial differential equations and Sylvester-type equations, avoiding Kronecker tensor products. Furthermore, to reach large times, the calculation of just two matrix exponentials is required, for which we compare different techniques based on Padéʼs approximations, matrix decompositions and Krylov spaces, as well as a new technique which avoids the computation of matrix exponentials. We also illustrate how to take advantage of multiple precision arithmetic.Finally, possible generalizations to non-linear problems and higher-dimensional problems, as well as to unbounded domains, are considered.

Spontaneous autophoretic motion of isotropic particles

Suspended colloidal particles interacting chemically with a solute can self-propel by autophoretic motion when they are asymmetrically patterned (Janus colloids). Here we demonstrate theoretically that such anisotropy is not necessary for locomotion and that the nonlinear interplay between surface osmotic flows and solute advection can produce spontaneous and self-sustained motion of isotropic particles. Solving the classical autophoretic framework for isotropic particles, we show that, for given material properties, there exists a critical particle size (or Péclet number) above which spontaneous symmetry-breaking and autophoretic motion occur. A hierarchy of instabilities is further identified for quantized critical Péclet numbers.