Solution Of Advection Research Articles

Self-Similar Solutions for Viscous and Resistive Advection Dominated Accretion Flows

In this paper, self-similar solutions of resistive advection dominated accretion flows (ADAF) in the presence of a pure azimuthal magnetic field are investigated. The mechanism of energy dissipation is assumed to be the viscosity and the magnetic diffusivity due to turbulence in the accretion flow. It is assumed that the magnetic diffusivity and the kinematic viscosity are not constant and vary by position and α-prescription is used for them. In order to solve the integrated equations that govern the behavior of the accretion flow, a self-similar method is used. The solutions show that the structure of accretion flow depends on the magnetic field and the magnetic diffusivity. As the radial infall velocity and the temperature of the flow increase by magnetic diffusivity, the rotational velocity decreases. Also, the rotational velocity for all selected values of magnetic diffusivity and magnetic field is sub-Keplerian. The solutions show that there is a certain amount of magnetic field for which rotational velocity of the flow becomes zero. This amount of the magnetic field depends upon the gas properties of the disc, such as adiabatic index and viscosity, magnetic diffusivity, and advection parameters. The mass accretion rate increases by adding the magnetic diffusivity and the solutions show that in high magnetic pressure, the ratio of the mass accretion rate to the Bondi accretion rate is reduced with an increase in magnetic pressure. Also, the study of Lundquist and magnetic Reynolds numbers based on resistivity indicates that the linear growth of magnetorotational instability (MRI) of the flow reduces by resistivity. This property is qualitatively consistent with resistive magnetohydrodynamics (MHD) simulations.

Isogeometric analysis with Powell–Sabin splines for advection–diffusion–reaction problems

This paper presents the use of Powell–Sabin splines in the context of isogeometric analysis for the numerical solution of advection–diffusion–reaction equations. Powell–Sabin splines are piecewise quadratic C1 functions defined on a given triangulation with a particular macro-structure. We discuss the Galerkin discretization based on a normalized Powell–Sabin B-spline basis. We focus on the accurate detection of internal and boundary layers, and on local refinements. We apply the approach to several test problems, and we illustrate its effectiveness by a comparison with classical finite element and recent isogeometric analysis procedures.

Comment on “Analytical solutions to one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media” by Kumar, A., Jaiswal, D.K., Kumar, N., J. Hydrol., 2010, 380: 330–337

Summary The paper of Kumar et al. (2010) investigated the advection and diffusion of solute in semi-infinite media with variable coefficients. They offered some analytical solutions for several sets of velocities and diffusion coefficients. Despite of its potential interest, the authors mistakenly made use of the transform of partial derivative between two coordinate systems, which made the part of “Temporally dependent dispersion along uniform and steady flow” wrong.

Local Error Analysis of Discontinuous Galerkin Methods for Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems

This paper analyzes the local properties of the symmetric interior penalty upwind discontinuous Galerkin (SIPG) method for the numerical solution of optimal control problems governed by linear reaction-advection-diffusion equations with distributed controls. The theoretical and numerical results presented in this paper show that for advection-dominated problems the convergence properties of the SIPG discretization can be superior to the convergence properties of stabilized finite element discretizations such as the streamline upwind Petrov Galerkin (SUPG) method. For example, we show that for a small diffusion parameter the SIPG method is optimal in the interior of the domain. This is in sharp contrast to SUPG discretizations, for which it is known that the existence of boundary layers can pollute the numerical solution of optimal control problems everywhere even into domains where the solution is smooth and, as a consequence, in general reduces the convergence rates to only first order. In order to prove the nice convergence properties of the SIPG discretization for optimal control problems, we first improve local error estimates of the SIPG discretization for single advection-dominated equations by showing that the size of the numerical boundary layer is controlled not by the mesh size but rather by the size of the diffusion parameter. As a result, for small diffusion, the boundary layers are too “weak” to pollute the SIPG solution into domains of smoothness in optimal control problems. This favorable property of the SIPG method is due to the weak treatment of boundary conditions, which is natural for discontinuous Galerkin methods, while for SUPG methods strong imposition of boundary conditions is more conventional. The importance of the weak treatment of boundary conditions for the solution of advection dominated optimal control problems with distributed controls is also supported by our numerical results.

Diffuse hydrological mass transport through catchments: scenario analysis of coupled physical and biogeochemical uncertainty effects

Abstract. This paper quantifies and maps the effects of coupled physical and biogeochemical variability on diffuse hydrological mass transport through and from catchments. It further develops a scenario analysis approach and investigates its applicability for handling uncertainties about both physical and biogeochemical variability and their different possible cross-correlation. The approach enables identification of conservative assumptions, uncertainty ranges, as well as pollutant/nutrient release locations and situations for which further investigations are most needed in order to reduce the most important uncertainty effects. The present scenario results provide different statistical and geographic distributions of advective travel times for diffuse hydrological mass transport. The geographic mapping can be used to identify potential hotspot areas with large mass loading to downstream surface and coastal waters, as well as their opposite, potential lowest-impact areas within the catchment. Results for alternative travel time distributions show that neglect or underestimation of the physical advection variability, and in particular of those transport pathways with much shorter than average advective solute travel times, can lead to substantial underestimation of pollutant and nutrient loads to downstream surface and coastal waters. This is particularly true for relatively high catchment-characteristic product of average attenuation rate and average advective travel time, for which mass delivery would be near zero under assumed transport homogeneity but can be orders of magnitude higher for variable transport conditions. A scenario of high advection variability, with a significant fraction of relatively short travel times, combined with a relevant average biogeochemical mass attenuation rate, emerges consistently from the present results as a generally reasonable, conservative assumption for estimating maximum diffuse mass loading, when the prevailing physical and biogeochemical variability and cross-correlation are uncertain.

Derivation of the method of characteristics for the fluid dynamic solution of flow advection along porous wall channels

This paper describes in detail a novel formulation of the method of characteristics for its application to solve one-dimensional compressible unsteady non-homentropic flow advected along porous wall channels. In particular, the method is implemented into a wall-flow monolith Diesel particulate filter model whose purpose is the pressure drop prediction. The flow inside the monolith channels is considered to be one-dimensional and the flow through the porous wall treated as a source term agree with the Darcy’s law. The flow dynamic behaviour at internal nodes of the channels is solved by means of shock capturing methods, whereas the end nodes, or boundary conditions, are solved applying the method of characteristics. The derived solution in this study of the Riemann variables and the entropy level includes the variation along the space–time plane due to cross-section area changes, friction and heat transfer as traditionally stated, but also takes into account the key influence on every line of the flow leaving or entering to the channels through the porous walls.

Source areas for a passively diffusing methane gas from South Asian paddy fields using the Flux Footprint Model

The annual rate of methane emissions from paddy fields is uncertain due to the differences in climatic conditions, soil characteristics, duration and patterns of flooding, and other parameters. Thus it is important to identify and estimate the source patches of methane emissions from the paddy fields, and also the magnitude of the flux.The main aim of this work is to identify and quantify the specific source areas for passively diffusing methane gas from a typical South Asian Paddy Field. The SAM2 model is used to characterize each of the source patches corresponding to each of the methane fluxes computed by an indirect method, based on the analytical solution of the Eulerian advection–diffusion equation for vertical diffusion. The results of the model are presented as characteristic dimensions of the source patches: the maximum source location (i.e. the upwind distance of the surface element with the maximum-weight influence), the near and far ends of the source area, and its maximal lateral extension.Micrometeorological parameters like wind direction, wind speed, temperature, relative humidity and concentration of methane gas are measured at certain time intervals. These data are fed into the meteorological pre-processor to generate time dependent atmospheric scaling factors (Richardson Number, Monin–Obukhov Length etc.). These are used directly in SAM2 to generate the elliptic patches with P (the fraction of the total integrated source weight function) varying from 40% (stable) to 99% (unstable). The simulations of SAM2 are related directly to the non-dimensional surface-layer scaling variables by the well-known Levenburg–Marquardt Non-Linear Least Square Method in a parameterized model, which provides a user-friendly estimate of the surface area responsible for the measured methane concentration. The application of SAM2 allows us to draw conclusions about the appropriateness of the spatial representation and localness of methane concentration measurements.

Improved Riemann solvers for complex transport in two-dimensional unsteady shallow flow

The numerical solution of advection–reaction–diffusion transport problems in two-dimensional shallow water flow is split in three subproblems in order to analyze them separately. In the first part, the advection component is solved with the help of an extended Jacobian matrix for the coupled system of flow and advection conservation laws and focusing on the correct definitions of the approximate or weak solutions. Considering that one of the conserved quantities is the solute volume, nonphysical solutions for the solute concentration may appear in complex situations and a solute fix is proposed. This is formulated for first and second order schemes. In the second part of this work, the solution of problems with volumetric reaction terms is studied and the results of single-step as well as multi-step pointwise and upwind approaches are compared in order to establish their relative performance. The upwind treatment is done in 2D cases dividing cell volumes to transform reacting terms in singular source terms. The third part is concerned with the diffusion term. The focus of this part is put on the interference between numerical and physical diffusion. A simple form to estimate the magnitude of the numerical diffusion is proposed and it is shown to improve the accuracy of the results in first and second order approaches.

A Solute Gradient in the Tear Meniscus. I. A Hypothesis to Explain Marx's Line

Marx's line is a line of mucosal staining behind the mucocutaneous junction. It can be demonstrated throughout life in all normal lids by staining with lissamine green and related dyes. Of all the body orifices, only the mucosae of the eye and mouth are directly exposed to the atmosphere. In this paper, we suggest that for the eye, this exposure leads to the formation of Marx's line. The tear meniscus thins progressively toward its apex, where it is pinned at the mucocutaneous junction of the lid. It also thins toward the black line, which segregates the meniscus from the tear film after the blink. We predict that, because of the geometry of the tear meniscus, evaporation generates a solute gradient across the meniscus profile in the anteroposterior plane, which peaks at the meniscus apices at the end of the interblink. One outcome would be to amplify the level of tear molarity at these sites so that they reach hyperosmolar proportions. Preliminary mathematical modeling suggests that dilution of this effect by advection and diffusion of solute away from the meniscus apex at the mucocutaneous junction will be restricted by spatial constraints, the presence of tear and surface mucins at this site, and limited fluid flow. We conclude that evaporative water loss from the tear meniscus may result in a physiological zone of hyperosmolar and related stresses to the occlusal conjunctiva, directly behind the mucocutaneous junction. We hypothesize that this stimulates a high epithelial cell turnover at this site, incomplete epithelial maturation, and a failure to express key molecules such as MUC 16 and galectin-3, which, with the tight junctions between surface epithelial cells, are necessary to seal the ocular surface and prevent penetration of dyes and other molecules into the epithelium. This is proposed as the basis for Marx's line. In Part II of this paper (also published in this issue of The Ocular Surface), we address additional pathophysiological consequences of this mechanism, affecting lid margins.

2‐D radial analytical solutions for solute transport in a dual‐porosity medium

This study presents 2‐D analytical solutions for advective solute transport within a macropore with simultaneous radial diffusion into an unbounded soil matrix. Solutions for three conditions are derived: (1) an instantaneous release of solute into a macropore, (2) a constant concentration of solute at the top of a macropore, and (3) a pulse release of solute into a macropore. A system of two governing equations was solved by the Laplace transform method for solute concentration as a function of space and time. Substituting the asymptotic approximations of the modified Bessel functions, we also obtained approximate solutions for all three cases. For instantaneous and pulse‐type releases of solutes, the solutes initially diffuse into the soil matrix and then reverse direction away from the matrix as they diminish in the macropore. The matrix behaves as a long‐term contaminant source creating long tails in the breakthrough curves. Comparisons between the exact and approximate solutions for all three conditions show that the asymptotic approximations are accurate for relatively short periods of solute movement, with increasing error as time and transport distances increase. The analytical solutions were compared with one set of experimental data and also numerical simulations for contaminant transport in a cylindrical dual‐porosity medium. The analytical solutions for case 3 represented the experimental data reported in the literature well. Comparisons with numerical simulations in a two‐dimensional cylindrical domain that included dispersion in the macropore and advection in the matrix showed that the error caused by neglecting these two processes was minimal when a relatively low permeability matrix was considered for case 2.

Analytical solutions for advective–dispersive solute transport in double‐layered finite porous media

AbstractAnalytical solutions for advection and dispersion of a conservative solute in a one‐dimensional double‐layered finite porous media are presented. The solutions are applicable to five scenarios that have various combinations of fixed concentration, fixed flux and zero concentration gradient conditions at the inlet and outlet boundaries that provide a wide number of options. Arbitrary initial solute concentration distributions throughout the media can be considered via explicit formulations or numerical integration. The analytical solutions presented have been verified against numerical solutions from a finite‐element‐based approach and an existing closed‐form solution for double‐layered media with an excellent correlation being found in both cases. A practical application pertaining to advective transport induced by consolidation of underlying sediment layers on contaminant movement within a capped contaminated sediment system is presented. Comparison of the calculated concentrations and fluxes with alternative approaches clearly illustrates the need to consider advection processes. Consideration of the different features of contaminant transport due to varying pore‐water velocity fields in primary consolidation and secondary consolidation stages is achieved via the use of non‐uniform initial concentration distributions within the proposed analytical solutions. Copyright © 2010 John Wiley & Sons, Ltd.

A fast method for the solution of the Helmholtz equation

In this paper, we consider the numerical solution of the Helmholtz equation, arising from the study of the wave equation in the frequency domain. The approach proposed here differs from those recently considered in the literature, in that it is based on a decomposition that is exact when considered analytically, so the only degradation in computational performance is due to discretization and roundoff errors. In particular, we make use of a multiplicative decomposition of the solution of the Helmholtz equation into an analytical plane wave and a multiplier, which is the solution of a complex-valued advection–diffusion–reaction equation. The use of fast multigrid methods for the solution of this equation is investigated. Numerical results show that this is an efficient solution algorithm for a reasonable range of frequencies.

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.

A fast characteristic finite difference method for fractional advection–diffusion equations

Fractional advection–diffusion equations provide an adequate and accurate description of the movement of solute in an aquifer. However, there are major obstacles that restrict their applications. From a modeling viewpoint, one of the major limitations in the application of fractional advection–diffusion equations to hydrology is the poor predictability of model parameters [27]. From a computational view point, one of the major limitations in numerical solution of fractional advection–diffusion equations in multiple space dimensions is that they generate full coefficient matrices in their numerical approximations, which require O( N 3) of computational cost and O( N 2) storage for a problem of size N. This paper presents a preliminary step towards the efficient numerical solution of fractional advection–diffusion equations. In this paper we develop a fast characteristic finite difference method for the efficient solution of space-fractional transient advection–diffusion equations in one space dimension. This method generates more accurate solutions than standard implicit methods even if much larger time steps and spatial meshes are used, leading to a discrete system with a greatly reduced size. Furthermore, we explore the structure of the coefficient matrix to come up with an efficient iterative solver which requires only O( N) account of storage and roughly O( N log N) account of computational cost. Our preliminary numerical example runs for some simple one dimensional model problems seem to indicate the following observations: to achieve the same accuracy, the new method uses no more than one thousandth of CPU and about one thousandth of the storage used by the standard method. This demonstrates the strong potential of the method.

Numerical simulation of channel segregates during alloy solidification using TVD schemes

PurposeThis paper aims to tackle the problem of thermo‐solutal convection and macrosegregation during ingot solidification of metal alloys. Complex flow structures associated with the development of channels segregate and sharp gradients in the solutal field call for the implementation of accurate methods for numerical modeling of alloy solidification. In particular, the solute transport equation is convection dominated and requires special non‐oscillarity type high‐order schemes to handle the regions of channels segregates.Design/methodology/approachIn the present study, a time‐splitting approach has been adopted to separately handle solute advection and diffusion. This splitting technique allows the application of accurate total variation dimensioning (TVD) schemes for solution of solute advection. Applications of second‐order Lax‐Wendroff TVD SUPERBEE and fifth‐order weighted essentially non‐oscillatory (WENO) schemes are described in the present article. Classical numerical solution of solute transport using hybrid and central‐difference schemes are also employed for the purpose of comparisons. Numerical simulations for solidification of Pb‐18%Sn in a two‐dimensional rectangular cavity have been carried out using different numerical schemes.FindingsNumerical results show the difficulty of obtaining grid‐independent solutions with respect to local details in the region of channels. Grid convergence patterns and numerical uncertainty are found to be dependent on the applied scheme. In general, the first‐order hybrid scheme is diffusive and under predicts the formation of channels. The second‐order central‐difference scheme brings about oscillations with possible non‐physical extremes of solute composition in the region of channel segregates due to sharp gradients in the solutal field. The results obtained using TVD and WENO schemes contain no oscillations and show an excellent capture of channels formation and resolution of the interface between solute‐rich and depleted bands. Different stages of channels formation are followed by analyzing thermo‐solutal convection and macrosegregation at different times during solidification.Research limitations/implicationsAccurate prediction of local variation in the solutal and flow fields in the channels regions requires grid refinement up to scales in the order of microscopic dendrite arm spacing. This imposes limitations in terms of large computational time and applicability of available macroscopic models based on classical volume‐averaging techniques.Practical implicationsThe present study is very useful for numerical simulation of macrosegregation during ingot casting of metal alloys.Originality/valueThe paper provides the methodology and application of TVD schemes to predict channel segregates during columnar solidification of metal alloys. It also demonstrates the limitations of classical schemes for simulation of alloy solidification.

Cement-rock interaction: Infiltration of a high-pH solution into a fractured granite core

Within the framework of the HPF project (Hyperalkaline Plume in Fractured Rock) at the Grimsel Test Site (Switzerland), a small scale core infiltration experiment was performed at the University of Bern. A high-pH solution was continuously injected, under a constant pressure gradient, into a cylindrical core of granite containing a fracture. This high-pH solution was a synthetic version of solutions characteristic of early stages in the degradation of cement. The interaction between the rock and the solutions was reflected by significant changes in the composition of the injected solution, despite the negligible pH-buffering capacity, and a decrease in the permeability of the rock. Changes in the mineralogy and porosity of the fault gouge filling the fracture were only minor. Within the new LCS (Long-Term Cement Studies) project at Grimsel, new one-dimensional reactive transport modeling using CrunchFlow has been used to improve the interpretation of the experimental results. Dispersive and advective solute transport, adsorption processes and mineral reaction kinetics have been taken into account. The evolution of solution composition is mainly controlled by dissolution/precipitation reactions. Adsorption processes (cation exchange, surface complexation) only play a role in the very early stages of the experiment.

On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion

We prove that the Hamilton–Jacobi equation for an arbitrary Hamiltonian H (locally Lipschitz but not necessarily convex) and fractional diffusion of order one (critical) has classical C 1 , α solutions. The proof is achieved using a new Hölder estimate for solutions of advection–diffusion equations of order one with bounded vector fields that are not necessarily divergence free.

ONE-DIMENSIONAL TEMPORALLY DEPENDENT ADVECTION-DISPERSION EQUATION IN POROUS MEDIA: ANALYTICAL SOLUTION

Abstract Analytical solutions of one-dimensional advection–dispersion equation in semi-infinite longitudinal porous domain are obtained in this work. The solute dispersion parameter is considered temporally dependent along uniform flow. The first-order decay term, which is inversely proportional to the dispersion coefficient, is also considered. Initially, the space domain is not solute free. Analytical solutions are obtained for uniform and varying pulse-type input. A new time variable is introduced. The Laplace transform technique is used to get the analytical solutions.

Horizontal pre-asymptotic solute transport in a plane fracture with significant density contrasts

We investigate the dispersion of a finite amount of solute after it has been injected into the laminar flow occurring in a horizontal smooth fracture of constant aperture. When solute buoyancy is negligible, the dispersion process eventually leads to the well-known asymptotic Taylor–Aris dispersion regime, in which the solute progresses along the fracture at the average fluid velocity, according to a one-dimensional longitudinal advection–dispersion process. This paper addresses more realistic configurations for which the solute-induced density contrasts within the fluid play an important role on solute transport, in particular at small and moderate times. Flow and transport are coupled, since the solute distribution impacts the variations in time of the advecting velocity field. Transport is simulated using (i) a mathematical description based on the Boussinesq approximation and (ii) a numerical scheme based on a finite element analysis. This enables complete characterization of the process, in particular at moderate times for which existing analytical models are not valid. At very short times as well as very long times, the overall downward advective solute mass flow is observed to scale as the square of the injected concentration. The asymptotic Taylor–Aris effective dispersion coefficient is reached eventually, but vertical density currents, which are significant at short and moderate times, are responsible for a systematic retardation of the asymptotic mean solute position with respect to the frame moving at the mean fluid velocity, as well as for a time shift in the establishment of the asymptotic dispersion regime. These delays are characterized as functions of the Péclet number and another non-dimensional number which we call advective Archimedes number, and which quantifies the ratio of buoyancy to viscous forces. Depending on the Péclet number, the asymptotic dispersion is measured to be either larger or smaller than what it would be in the absence of buoyancy effects. Breakthrough curves measured at distances larger than the typical distance needed to reach the asymptotic dispersion regime are impacted accordingly. These findings suggest that, under certain conditions, density/buoyancy effects may have to be taken into consideration when interpreting field measurement of solute transport in fractured media. They also allow an estimate of the conditions under which density effects related to fracture wall roughness are likely to be significant.

Analytical solution for one-dimensional advection–dispersion transport equation with distance-dependent coefficients

Summary Mathematical models describing contaminant transport in heterogeneous porous media are often formulated as an advection–dispersion transport equation with distance-dependent transport coefficients. In this work, a general analytical solution is presented for the linear, one-dimensional advection–dispersion equation with distance-dependent coefficients. An integrating factor is employed to obtain a transport equation that has a self-adjoint differential operator, and a solution is found using the generalized integral transform technique (GITT). It is demonstrated that an analytical expression for the integrating factor exists for several transport equation formulations of practical importance in groundwater transport modeling. Unlike nearly all solutions available in the literature, the current solution is developed for a finite spatial domain. As an illustration, solutions for the particular case of a linearly increasing dispersivity are developed in detail and results are compared with solutions from the literature. Among other applications, the current analytical solution will be particularly useful for testing or benchmarking numerical transport codes because of the incorporation of a finite spatial domain.