Comparison of a two-dimensional numerical dust transport model with experimental dust emissions from soil surfaces in a wind tunnel

Near-surface wind-tunnel fugitive dust concentration profiles arising from soil surfaces beds were compared to a finite difference numerical dust transport model. Comparisons of the type shown in this study were previously non-existent in the literature due to the lack of experimental wind-tunnel data for near-surface concentrations over a soil bed. However, in a previous study by the authors, near-surface steady-state concentration profiles were measured in order to obtain fugitive dust emission rates, thus allowing the comparison to models shown in this paper. The novel aspects of the current study include: comparison of concentration profiles of dust obtained experimentally in the wind tunnel with those calculated numerically; comparison of the calculated numerical fetch effect on dust emissions with that obtained in the wind tunnel; and comparison of the emission rates calculated numerically with those obtained experimentally in the wind tunnel. Initial comparisons with the model indicate good agreement implying that the physical mechanism of advection–diffusion is reasonably modeled with the choice of equations for the simple “steady-state” process near the surface. Furthermore, the numerical solutions presented in this paper provide a means to systematically explore the relative impact of varied surface boundary conditions upon the emission process and provide a potential link between wind-tunnel simulations and field scale models.     

A model for contaminant mass flux in capped sediment under consolidation

The paper presents a model for contaminant transport and flux through a consolidating subaqueous sediment and overlying cap. The formulation is based on the effect of consolidation and excess pore pressure dissipation on transient, nonlinear advective component of transport through sediment and the cap. The consolidation is induced by the buoyant weight of the cap when it is placed on the contaminated sediments. One equation is presented for advective-diffusive transport through the sediment that is dependent upon soil/contaminant properties and transient advective velocity, which is calculated from a second equation based on the Terzaghi consolidation theory. A third equation is provided to describe the transport of contaminants in the cap. The parameters, including advective velocity, and boundary conditions used for contaminant transport through the cap are derived from the solution of the first two equations. The finite difference method is used to solve the system of equations for consolidation and contaminant transport. A hypothetical case is analyzed to demonstrate the formulation, and the results show that advection due to consolidation can accelerate breakthrough of contaminant through the cap by orders of magnitude. The derivation and results show that consolidation should be included for cap design, and that reactive caps are essential for delaying and reducing dissolved contaminant flux.     

Coupling geochemistry with a particle tracking transport model

Coupling geochemistry and transport appears unavoidable since it is rare that either of these two phenomena alone can account for the movement of solutes in groundwater. The chemical model is based on thermodynamic equilibrium. The method used is a Gibbs free energy minimization constrained by mass balances. The model calculates the aqueous speciation, the precipitation and the dissolution of pure minerals or solid solutions. The transport equation is solved by the random walk technique which avoids the problem of numerical dispersion for transport, but may be more time consuming than finite differences or elements if a large number of particles are necessary in order to get a sufficiently “smooth” solution. However, when the chemistry deals with a realistic number of elements (e.g., > 10), the cost of the chemistry computation largely dominates that of transport. Special techniques had to be developed in order to solve problems linked to the conditions present in some of the CEC CHEMVAL tests (boundary with fixed concentrations and very low Péclet numbers). The coupling consists of calculating the exchanges of chemical elements between two populations. The first population is sedentary, constituted by a mesh of fixed cells representing the composition of the solid phase. The other population is nomadic, represented by a set of particles which are advected by groundwater flow. A vector of real numbers is associated with each mobile particle. This vector accounts for the mass of each element dissolved in the moving liquid phase. For this reason, the transport equation is only solved once for the whole set of elements. The main assumptions that were necessary to perform the coupling in a simple way are discussed. Two applications are presented: (1) a verification compared to an analytical solution; and (2) the simulation of a percolation experiment through a sandstone core.     

Stepwise superposition approach for the analytical solutions of multi-dimensional contaminant transport in finite- and semi-infinite aquifers

Analytical solutions of contaminant transport in multi-dimensional media are significant for theoretical and practical purposes. However, due to the problems for which the solutions are sought which are complex in most of the cases, most available analytical solutions in multi-dimensional media are not given in their closed forms. Integrals are often included in the solution expressions, which may limit the practitioners to use the solutions. In addition, available multi-dimensional solutions for the third-type sources in bounded media are fairly limited. In this paper, a stepwise superposition approach for obtaining approximate multi-dimensional transport solutions is developed. The approach is based on the condition that the one-dimensional solution along the flow direction is known. The solutions are expressed in their closed forms without integrals. The transport media to the solutions are flexible and can be finite, semi-infinite, or infinite in the transverse directions. The solutions subject to the first- and third-type boundary conditions at the inlet with a distributed source over the domain are obtained. The integrals in some known solutions can also be evaluated by the approach if they can be derived to include known longitudinal integrals with respect to time. The accuracy and efficiency of the solutions proposed in this paper are verified through test problems and calculation examples.     

One-dimensional simulation of solute transfer in saturated-unsaturated porous media using the discontinuous finite elements method

A one-dimensional transport model for simulating water flow and solute transport in homogeneous-heterogeneous, saturated-unsaturated porous media is presented. The model is composed of a combination of accurate numerical algorithms for solving the nonlinear Richard's and advection-dispersion equations (ADE). The mixed form of Richard's equation is solved using a standard finite element method (FEM) with primary variable switching. The transport equation is solved using operator splitting, with the discontinuous finite element method (DFE) for discretization of the advective term. A slope limiting procedure for DFE avoids numerical instabilities but creates very limited numerical dispersion for high Peclet numbers. An implicit finite differences scheme (FD) is used for the dispersive term. The unsaturated flow and transport model (Wamos-T) is applied to a variety of rigorous problems including transient flow, heterogeneous medium and abrupt variations of velocity in magnitude and direction due to time-varying boundary conditions. It produces accurate and mass-conservative solutions for a very large range of grid Peclet numbers. The Wamos-T model is a good and robust alternative for the simulation of mass transport in unsaturated domain.     

One-dimensional simulation of solute transfer in saturated–unsaturated porous media using the discontinuous finite elements method

A one-dimensional transport model for simulating water flow and solute transport in homogeneous–heterogeneous, saturated–unsaturated porous media is presented. The model is composed of a combination of accurate numerical algorithms for solving the nonlinear Richard's and advection–dispersion equations (ADE). The mixed form of Richard's equation is solved using a standard finite element method (FEM) with primary variable switching. The transport equation is solved using operator splitting, with the discontinuous finite element method (DFE) for discretization of the advective term. A slope limiting procedure for DFE avoids numerical instabilities but creates very limited numerical dispersion for high Peclet numbers. An implicit finite differences scheme (FD) is used for the dispersive term.The unsaturated flow and transport model (Wamos-T) is applied to a variety of rigorous problems including transient flow, heterogeneous medium and abrupt variations of velocity in magnitude and direction due to time-varying boundary conditions. It produces accurate and mass-conservative solutions for a very large range of grid Peclet numbers. The Wamos-T model is a good and robust alternative for the simulation of mass transport in unsaturated domain.     

Dimensionless analysis of two analytical solutions for 3-D solute transport in groundwater

Analytical solutions are widely used as screening tools for estimating the potential for contaminant transport in groundwater, or for interpreting tracer tests or groundwater quality data. A solution for three-dimensional solute migration from a plane-source source that is frequently used in practice is the approximate solution of Domenico [J. Hydrol. 91 (1987) 49-58]. A more rigorous solution to the same problem was provided by Sagar [ASCE, J. Hydraul. Div. 108, no. HY1 (1982) 47-62]. A comprehensive and unambiguous comparison between these two solutions is provided using dimensionless analysis. The solutions are first cast in terms of dimensionless parameters and then used to provide type curves covering a wide range of dimensionless parameter values. Results show that while discrepancies between the two solutions are relatively negligible along the plume centreline (for flow regimes dominated by advection and mechanical dispersion), large concentration differences can be observed as lateral distance from the centreline increases, especially in the presence of solute decay.     

Groundwater flow, multicomponent transport and biogeochemistry: development and application of a coupled process model

A research tool for modeling the reactive flow and transport of groundwater contaminants in multiple dimensions is presented. Arbitrarily complex coupled kinetic–equilibrium heterogeneous reaction networks, automatic code generation, transfer-function based solutions, parameter estimation, high-resolution methods for advection, and robust solvers for the mixed kinetic–equilibrium chemistry are some of the features of reactive flow and transport (RAFT) that make it a versatile research tool in the modeling of a wide variety of laboratory and field experiments. The treatment of reactions is quite general so that RAFT can be used to model biological, adsorption/desorption, complexation, and mineral dissolution/precipitation reactions among others. The integrated framework involving automated code generation and parameter estimation allows for the development, characterization, and evaluation of mechanistic process models. The model is described and used to solve a problem in competitive adsorption that illustrates some of these features. The model is also used to study the development of an in situ Fe(II)-zone by encouraging the growth of an iron-reducing bacterium with lactate as the electron donor. Such redox barriers are effective in sequestering groundwater contaminants such as chromate and TCE.     

Kinematic analysis of solute mass flows in rock fractures with spatially random parameters

Field data of physical properties in heterogeneous crystalline bedrock, like porosity and fracture aperture, is associated with uncertainty that can have a significant impact on the analysis of solute transport in rock fractures. Solutions to the central temporal moments of the residence time probability density function (PDF) are derived in a closed form for a solute Dirac pulse. The solutions are based on a model that takes into account advection along the fracture plane, diffusion into the rock matrix and sorption kinetics in the rock matrix. The most relevant rock properties including fracture aperture and several matrix properties as well as flow velocity are assumed to be spatially random along transport pathways. The mass transport is first solved in a general form along one-dimensional pathways, but the results can be extended to multi-dimensional flows simply by substituting the expected travel time for inert water parcels. Based on data obtained with rock samples taken at Asp? Hard Rock Laboratory in Sweden, the solutions indicate that the heterogeneity of the rock properties contributes to increasing significantly both the variance and the skewness of the residence time probability density function for a pulse travelling in a fracture. The Asp? data suggests that the bias introduced in the variance of the residence time PDF by neglecting the effect of heterogeneity of the rock properties on the radionuclide migration is very large for fractures thinner than a few tenths of a millimetre.     

Multidimensional, multicomponent, subsurface reactive transport in nonuniform velocity fields: code verification using an advective reactive streamtube approach

High performance computing has made possible the development of high resolution, multidimensional, multicomponent reactive transport models that can be used to analyze complex geochemical environments. However, as increasingly complex processes are included in these models, the accuracy of the numerical formulation coupling the nonlinear processes becomes difficult to verify. Analytical solutions are not available for realistically complex problems and benchmark solutions are not generally available for specific problems. We present an advective reactive streamtube (ARS) transport technique that efficiently provides accurate solutions of nonlinear multicomponent reactive transport in nonuniform multidimensional velocity fields. These solutions can be compared with results from Eulerian-based advection-dispersion-reaction models to evaluate the accuracy of the numerical formulation used. The ARS technique includes mixed equilibrium and kinetic complexation and precipitation-dissolution reactions subject to the following assumptions: (1) transport is purely advective (i.e., no explicit diffusion or dispersion), and (2) chemistry is described by a canonical system of reactions that evolves with time and is unaffected by position in space. Results from the ARS technique are compared with results from the massively parallel, multicomponent reactive transport model MCTRACKER on a test problem involving irreversible oxidation of organic carbon and reaction of the oxidation products with two immobile mineral phases, gypsum and calcite, and fifteen aqueous complexes. Truncation error, operator splitting error, and the nonlinear transformation of these errors in the high-resolution reactive transport model are identified for this problem.     

Anomalous transport of colloids and solutes in a shear zone

Transport experiments with colloids and radionuclides in a shear zone were conducted during the Colloid and Radionuclide Retardation experiment (CRR) at Nagra's Grimsel Test Site. Breakthrough curves of bentonite colloids and uranine, a non-sorbing solute, were measured in an asymmetric dipole flow field. The colloid breakthrough is earlier than that of uranine. Both breakthrough curves show anomalously long late time tails and the slope of the late time tails for the colloids is slightly higher. Anomalous late time tails are commonly associated with matrix diffusion processes; the diffusive interaction of solutes transported in open channels with the adjacent porous rock matrix or zones of stagnant water. The breakthrough curves for different colloid size classes are very similar and show no signs of fractionation due to their (size-dependent) diffusivity. It is proposed that tailing of the colloids is mainly caused by the structure of the flow field and that for the colloid transport, matrix diffusion is of minor importance. This has consequences for the interpretation of the uranine breakthrough. Comparisons of experimental results with numerical studies and with the evaluation of the colloid breakthrough with continuous time random theory imply that the tailing in the conservative solute breakthrough in this shear zone is not only caused by matrix diffusion. Part of the tailing can be attributed to advective transport in fracture networks and advection in low velocity regions. Models based on the advection-dispersion equation and matrix diffusion do not properly describe the temporal and spatial evolution of colloid and solute transport in such systems with a consistent set of parameters.     

An analytical model for crosswind integrated concentrations released from a continuous source in a finite atmospheric boundary layer

An analytical model for the crosswind integrated concentrations released from a continuous source in a finite atmospheric boundary layer is formulated by considering the wind speed as a power law profile of vertical height above the ground and eddy diffusivity as an explicit function of downwind distance from the source and vertical height. A closed form analytical solution of the resulting advection–diffusion equation for these profiles of wind speed and eddy diffusivity with the physically relevant boundary conditions is derived using the separation of variables technique that leads to a Sturm–Liouville eigen value problem. Various particular cases of the model are deduced.The model is evaluated with the observations obtained from Prairie Grass experiment in various stability classes varying from very unstable to neutral and stable conditions and Hanford diffusion experiment in stable conditions. The agreement is found to be good between the computed and observed concentrations in both the diffusion experiments. For Prairie Grass experiment, the model is predicting 78% cases with in a factor of two and gives a value of NMSE as 0.075. On the other hand, for Hanford observations in stable conditions, it predicts 70% cases with in factor of two. An extensive analysis of statistical measures with the downwind distances from the source reveals that the model is performing well close to the source.     

Modeling tracer diffusion in fractured and unfractured, unsaturated, porous media

A proposed tracer diffusion test for the Exploratory Shaft Facility at Yucca Mountain, NV, is modeled. For the proposed test, a solution containing conservative tracers will be introduced into a borehole in the geologic medium of interest. The tracers will diffuse and advect from the saturated source region into the unsaturated matrix in the surrounding tuff. After some time, the borehole is to be overcored, and tracer concentrations in the fluid will be measured in the core as a function of distance from emplacement. The data will be used to evaluate diffusive behavior and to derive effective diffusion coefficients for the tracers in the specific tuff. Numerical simulations are used to study the effects of effective diffusion coefficient, porosity, saturation, and fracturing on tracer transport. Results are reported for numerical simulations of tests in the Topopah Spring Member and the Tuff of Calico Hills, which have significantly different porosities and saturations. The simulations make the following predictions: The spread of tracer during the test will be sensitive to the effective diffusion coefficient of the tracer. Tracer will diffuse farther in the Topopah Spring Member than in the Tuff of Calico Hills because of the former's lower porosity and saturation. Tracer transport by advection into the Topopah Spring Member will be greater than that into the Tuff of Calico Hills because of capillary effects. While advection will be a significant mechanism for tracer penetration into the Topopah Spring tuff, it will be less significant for tracer penetration into the Calico Hills tuff. The proximity of a single vertical fracture to the source region determines its effects on tracer transport, especially if the fracture diverts fluid flowing from the source region into the matrix.     

Identification of atmospheric boundary layer parameters by inverse problem

This paper deals with the use of inverse analysis techniques for the estimation of micro-meteorological parameters required for the characterization of atmospheric boundary layers. The physical problem is formulated in terms of a transient two-dimensional advection–diffusion equation. Concentration measurements of a tracer are assumed to be available for the inverse analysis. The analysis of the sensitivity coefficients and of the determinant of the information matrix reveals the most appropriate sensor locations and duration of the experiment for the estimation of the unknown parameters. The parameter estimation problem is solved with the Levenberg–Marquardt method of minimization of the least squares norm.     

Numerical validation of a Eulerian hydrochemical code using a 1D multisolute mass transport system involving heterogeneous kinetically controlled reactions

It is demonstrated that at steady state, the 1D thermo-kinetic hydrochemical Eulerian mass balance equations in pure advective mode are indeed identical to the governing mass balance equations of a single reaction path (or geochemical) code in open system mode. Thus, both calculated reaction paths should be theoretically identical whatever the chemical complexity of the water-rock system (i.e., multicomponent, multireaction zones kinetically and equilibrium-controlled). We propose to use this property to numerically test the thermo-kinetic hydrochemical Eulerian codes and we employ it to verify the algorithm of the 1D finite difference code KIRMAT. Compared to the other methods to perform such numerical tests (i.e., comparisons with analytical, semi-analytical solutions, between two Eulerian hydrochemical codes), the advantage of this new method is the absence of constraints on the chemical complexity of the modelled water-rock systems. Moreover, the same thermo-kinetic databases and geochemical functions can be easily and mechanically used in both calculations, when the numerical reference comes from the Eulerian code with no transport terms (u and D = 0) and modify to be consistent with the definition of the open system mode in geochemical modelling. The ability of KIRMAT to treat multicomponent pure advective transport, subjected to several kinetically equilibrium-controlled dissolution and precipitation reactions, and to track their boundaries has been successfully verified with the property of interest. The required numerical validation of the reference calculations is bypassed in developing the Eulerian code from an already checked single reaction path code. A forward time-upstream weighting scheme (a mixing cell scheme) is used in this study. An appropriate choice of grid spacing allows to calculate within the grid size uncertainty the correct mineral reaction zone boundaries, despite the presence of numerical dispersion. Its correction enables us to improve the convergence and to extend the numerical test to mixed advective-dispersive mass transport. However, the skewness factor involves numerical oscillations that prevent to compute different grid spacing. The use of a different chemically controlled time step constraint in both calculations induces some inconsistencies into the validation tests. This numerical validation method may be applied as well as to check a thermo-kinetic hydrochemical finite element based code, from a 1D heterogeneous systems, and 2D-3D systems provided that they are designed so as to be 1D equivalent. A one-step algorithm and the use of a numerical reference coming from the Eulerian code to be tested ensure the potential success (accuracy) of the numerical validation method.     

Stochastic-convective transport with nonlinear reactions and mixing: finite streamtube ensemble formulation for multicomponent reaction systems with intra-streamtube dispersion

An effective streamtube ensemble method is developed to upscale convective-dispersive transport with multicomponent nonlinear reactions in steady nonuniform flow. The transport is cast in terms of a finite ensemble of independent discrete streamtubes that approximate convective transport along macroscopically averaged pathlines and dispersive transport longitudinally as microscopic mixing within streamtubes. The representation of fate and transport via a finite ensemble of effective linear streamtubes, allows the treatment of arbitrarily complex reaction systems involving both homogeneous and heterogeneous reactions, and longitudinal dispersive/diffusive mixing within streamtubes. This allows the use of reactive-transport codes designed to solve such problems in an Eulerian framework, as opposed to reliance on closed-form (convolutional or canonical) expressions for reactive transport in exclusively convective streamtubes. The approach requires both reactive-transport solutions for a representative ensemble of one-dimensional convective-dispersive-reactive streamtubes and the distribution of flux over the streamtube ensemble variants, and it does not allow for lateral mixing between streamtubes. Here, the only ensemble variant is travel time. The discussion details the way that the conventional Eulerian fate and transport model is converted first into an ensemble of transports along three-dimensional streamtubes of unknown geometry, and then to approximate one-dimensional streamtubes that are designed to honor the important global properties of the transport. Conditions under which such an 'equivalent' ensemble of one-dimensional streamtubes are described. The breakthrough curve of a nonreactive tracer in the ensemble is expressed as a combined Volterra-Fredholm integral equation, which serves as the basis for estimation of the distribution of flux over the variant of the ensemble, travel time. Transient convective speed and the effects of errors in flux distributions are described, and the method is applied to a demonstration problem involving nonlinear multicomponent reaction kinetics and strongly nonuniform flow.     

Sensitivity of the solution of the Elder problem to density, velocity and numerical perturbations

In this paper the Elder problem is studied with the purpose of evaluating the inherent instabilities associated with the numerical solution of this problem. Our focus is first on the question of the existence of a unique numerical solution for this problem, and second on the grid density and fluid density requirements necessary for a unique numerical solution. In particular we have investigated the instability issues associated with the numerical solution of the Elder problem from the following perspectives: (i) physical instability issues associated with density differences; (ii) sensitivity of the numerical solution to idealization irregularities; and, (iii) the importance of a precise velocity field calculation and the association of this process with the grid density levels that is necessary to solve the Elder problem accurately. In the study discussed here we have used a finite element Galerkin model we have developed for solving density-dependent flow and transport problems, which will be identified as TechFlow. In our study, the numerical results of Frolkovic and de Schepper [Frolkovic, P. and H. de Schepper, 2001. Numerical modeling of convection dominated transport coupled with density-driven flow in porous media, Adv. Water Resour., 24, 63-72.] were replicated using the grid density employed in their work. We were also successful in duplicating the same result with a less dense grid but with more computational effort based on a global velocity estimation process we have adopted. Our results indicate that the global velocity estimation approach recommended by Yeh [Yeh, G.-T., 1981. On the computation of Darcian velocity and mass balance in finite element modelling of groundwater flow, Water Resour. Res., 17(5), 1529-1534.] allows the use of less dense grids while obtaining the same accuracy that can be achieved with denser grids. We have also observed that the regularity of the elements in the discretization of the solution domain does make a difference in obtaining a unique stationary solution for this problem. The results of our study also indicate that the density differences are critical in the solution of the Elder problem and that high density differences lead to the physical instability that is inherent with this problem. Other than the physical instability associated with the level of density differences used in the Elder problem, the following two points should be considered in solving the Elder problem in a consistent manner: (i) strict attention should be paid to the vertical grid Peclet number in developing the criteria for convergent grid selection; and, (ii) with a globally continuous velocity calculation stable solutions can be obtained at lower grid densities.     

Effects of reduced contaminant loading on downgradient water quality in an idealized two-layer granular porous media

The following explores the issue of how reductions in contaminant loading to plumes will effect downgradient water quality. An idealized scenario of two adjacent layers of uniform geologic media, one transmissive and the other low permeability, is considered. A high concentration source, similar to a thin DNAPL pool, is introduced in the transmissive layer immediately above the low permeability layer. While the source is active, dissolved constituents are driven along the contact by advection and into the low permeability layer by transverse diffusion. Removing the source reverses the concentration gradient between the layers, driving back diffusion of contaminants from the low permeability layer. Laboratory studies involving four contaminants demonstrate that 15 to 44% of the introduced contaminant moves into the low permeability zone (along a distance of 87 cm in a sand tank) over a period of 25 days. The greatest movement of contaminants into the low permeability zone is seen with the contaminants with the greatest sorption coefficients. A unique two-dimensional analytical solution is developed for the two-layer scenario. Processes addressed include advection; transverse dispersion; adsorption and degradation in the transmissive zones; and diffusion, adsorption, and degradation in the low permeability layer. Laboratory data agree favorably with the analytical solutions. Collectively, the laboratory results and analytical solutions provide a basis for testing other modeling approaches that can be applied to more complex problems. A set of field-scale scenarios are considered using the analytical solutions. Results indicate that improvement in water quality associated with source removal diminish with distance downgradient of the source. Furthermore, contaminant degradation and contaminant adsorption in the stagnant zone are shown to be critical factors governing the timing and magnitude of downgradient improvements in water quality. For five of six scenarios considered, observed improvements in water quality 100 m downgradient of the source fall in the range of 1 to 2 orders of magnitude 15 years after complete source removal. The sixth scenario, involving a contaminant half-life of three years and no adsorption, shows greater than three order of magnitude improvements in downgradient water quality within one year of source removal.     

Employing Heisenberg's turbulent spectral transfer theory to parameterize sub-filter scales in LES models

A turbulent subfilter viscosity for large eddy simulation (LES) models is proposed, based on Heisenberg's mechanism of energy transfer. Such viscosity is described in terms of a cutoff wave number, leading to relationships for the grid mesh spacing, for a convective boundary layer (CBL). The limiting wave number represents a sharp filter separating large and small scales of a turbulent flow and, henceforth, Heisenberg's model agrees with the physical foundation of LES models. The comparison between Heisenberg's turbulent viscosity and the classical ones, based on Smagorinsky's parameterization, shows that both procedures lead to similar subgrid exchange coefficients. With this result, the turbulence resolution length scale and the vertical mesh spacing are expressed only in terms of the longitudinal mesh spacing. Through the employment of spectral observational data in the CBL, the mesh spacings, the filter width and the subfilter eddy viscosity are described in terms of the CBL height. The present development shows that Heisenberg's theory naturally establishes a physical criterium that connects the subgrid terms to the large-scale dimensions of the system. The proposed constrain is tested employing a LES code and the results show that it leads to a good representation of the boundary layer variables, without an excessive refinement of the grid mesh.