Advection numerical solution

The advection—diffusion equation with a source term can be solved by CFD algorithms in general. Patankar provided an excellent introduction to numerical fluid flow and heat transfer. Oran and Boris discussed numerical solutions of diffusion—convection problems with chemical reactions. Since fuel cells feature an aspect ratio of the order of 100, 0(100), the upwind scheme for the flow-field solution is applicable and proves to be very effective. Unstructured meshes are commonly employed in commercial CFD codes. 

To understand the behavior of the movement of the contaminant in ground-water, people solve Eq. (1) forward in time. In solving this equation forward in time, one assumes that the plume is originated from somewhere and will travel through the porous media due to advection and dispersion. The conventional procedure to solve Eq. (1) is to use finite difference or finite element methods. For simple cases, closed-form solutions exist. Quantitative descriptions of the processes forward in time are well understood. Multidimensional models of these processes have been used successfully in practice [50]. Numerical solute transport models were first developed about 25 years ago. When properly applied, these models can provide useful information about transport processes and can assist in the design of remedial programs. 

The numerical solution to the advection-dispersion equation and associated adsorption equations can be performed using finite difference schemes, either in their implicit and/or explicit form. In the one-dimensional MRTM model (Selim et al., 1990), the Crank-Nicholson algorithm was applied to solve the governing equations of the chemical transport and retention in soils. The web-based simulation system for the one-dimensional MRTM model is detailed in Zeng et al. (2002). The alternating direction-implicit (ADI) method is used here to solve the three-dimensional models. 

These numerical solutions account for advective transport of the reactant gases, as well as both ordinary and Knudsen diffusion. Finally, we consider the influence on the optimum pressure of a constraint on the deposition uniformity. In this case, the optimum pressure maximizing centerline deposition rates is obtained in closed form using the method of Lagrange multipliers. 

Particle in Cell Methods. An alternative to the direct finite-diflFerence solution of (7) is the so-called particle in cell (PIC) technique. The distinguishing feature of the PIC technique is that the continuous concentration field is treated as a collection of mass points, each representing a given amount of pollutant and each located at the center of mass of the volume of material it represents. The mass points, or particles, are moved by advection and diffusion. It is convenient but not necessary, to have each of the particles of a given contaminant represent the same mass of material. The application of the PIC technique in hydrodynamic calculations is discussed by Harlow (32). Here we consider the use of the PIC technique in the numerical solution of (7). 

Smith GD (1985) Numerical Solution of Partial Differential Equations Finite Difference Methods. Third edition. Clarendon Press, Oxford Smolarkiewicz PK (1983) A simple positive definite advection scheme with small implicit diffusion. Mon Wea Rev 11 479-486... 

Analytical solutions for cases of temperature-dependent thermal conductivity are available [22, 23]. In cases where the solid s thermophysical properties vary significantly with temperature, or when phase changes (solid-liquid or solid-vapor) occur, approximate analytical, integral, or numerical solutions are oftentimes used to estimate the material thermal response. In the context of the present discussion, the most common and useful approximation is to utilize transient onedimensional semi-infinite solutions in which the beam impingement time is set equal to the dwell time of the moving solid beneath the beam. The consequences of this approximation have been addressed for the case of a top hat beam, p 1 = K = 0 material without phase change [29] and the ratios of maximum temperatures predicted by the steady-state 2D analysis. Transient ID analyses have also been determined. Specifically, at Pe > 1, the diffusion in the x direction is negligible compared to advection, and the ID analysis yields predictions of Umax to within 10 percent of those associated with the 2D analysis. 

The solution to this problem consists in the application of numerical solutions when diagenetic processes are modeled. Such numerical solutions always divide the continuum of reaction space and reaction time into discrete cells and discrete time intervals. If one divides up the continuum of space and time to a sufficient degree into discrete cells and time steps (which is not the decisive problem with the possibilities given by today s computers), one will be able to apply much simpler and better manageable conditions within the corresponding cells, and with regard to the expansion of a time interval, so that, in their entirety, they still will describe a complex system. Thus, it is possible, for example, to apply the two-step-procedure (Schulz and Reardon 1983), in which the individual observation of physical transport (advection, dispersion, diffusion) or any geochemical multiple component reaction is made feasible within one interval of time. 

Equations (25.13) and (25.14) describe mathematically the concentration of species above a given area assuming that the corresponding airshed is well mixed, accounting for emissions, chemical reactions, removal, advection of material in and out of the airshed, and entrainment of material during growth of the mixed layer. These equations cannot be solved analytically if one uses a realistic gas-phase chemical mechanism for the calculation of the ft, terms. Numerical solutions will be discussed subsequently. 

A third problem plaguing numerical solution of advection problems can be seen if the upwind scheme is modified to include the updated information in the cell i — 1. Replacing c" with in (25.128), we get... 

While this problem admits only advection in the fracture we use an upwind scheme to avoid oscillations in the numerical solution (Hughes and Brooks, 1982). Figure 6.11 gives plots versus distance along the fracture of the concentration distributions of the parent (solid lines) and daughter product (dashed lines) at t = 1000 days and t = 10000 days. The analytical solution of Cormenzana (2000) and the numerical solution of Rockflow fit very well. 

SMART is applicable if integral information on contaminant behaviour in groundwater is sufficient. If point information is needed a conventional FD or FE model has to be used. Although it is obvious that the streamtube approach is not as flexible as real 3D models , decoupling of conservative transport and physico-chemical processes allow to model three-dimensional contaminant transport in a convenient and computationally efficient way, especially if only one representative streamtube must be modelled. Computation times, as observed by Peter et al. (chapter 14) are much lower compared to MT3D simulations. It should also be mentioned, that the streamtube approach possesses some advantages compared to real 3D models even if each and every streamtube has to be modelled by means of a numerical model in order to evaluate F. Since only one dimensional advective-reactive transport must be modelled, numerical solutions based on discrete or mechanistical approaches, free of numerical dispersion, can be applied. In SMART this is done by a so-called parceltracking approach where contaminant transport is described by means of a continuous series of water volumes ( parcels ) as described in Finkel et al. (1998). 

The computational approach is based on a colocated, finite-volume, energy-con-serving numerical scheme on unstructured grids [10] and solves the low-Mach number, variable density gas-phase flow equations. Numerical solution of the governing equations of continuum phase and droplet phase are staggered in time to maintain time-centered, second-order advection of the fluid equations. Denoting the time level by a superscript index, the velocities are located at time level f and... 

For times approaching or exceeding these times under either advectively dominated or diffusion dominated conditions, a more complete model that includes the transport processes at the upper boundary is necessary to accurately predict fluxes and contaminant concentrations. Typically, a numerical solution is necessary but it is also possible to take a conservative approach and develop an analytical solution for the case of steady-state behavior. This model is discussed in more detail below. 

Fig. 7. Comparison of various transport schemes for advecting a cone-shaped puff in a rotating windfield after one complete rotation (a), the exact solution (b), obtained by an accurate numerical technique (c), the effect of numerical diffusion where the peak height of the cone has been severely tmncated and (d), where the predicted concentration field is very bumpy, showing the effects of artificial dispersion. In the case of (d), spurious waves are...

SCRAM (28) is a TDE dynamic, numerical finite difference soil model, with a TDE flow module and a TDE solute module. It can handle moisture behavior, surface runoff, organic pollutant advection, dispersion, adsorption, and is designed to handle (i.e., no computer code has been developed) volatilization and degradation. This model may not have received great attention by users because of the large number of input data required. 

MMT (32) is a 1- or 2-dimensional solute transport numerical groundwater model, to be driven off-line by a flow transport, such as VTT (Variable Thickness Transport). MMT employs the random-walk numerical method and was originally developed for radionuclide transport. The model accounts for advection, sorption and decay. 

In the first stage of the solution process, the advective control model seeks a pumping scheme in which the capture zone fully encompasses all control points representing the contaminant plume. The capture zone is simulated by tracking particles from extraction wells backwards through the velocity field. To represent the plume capture constraints numerically, a distance measure is used in which the minimum distance between each plume control point and all particles (see Figure 1) is constrained. When the distance between a control point and particle pathline equals zero then the plume control point lies within the capture zone. To ensure capture of the entire plume, the constraint function must equal zero for all control points. The reverse tracking formulation is stated as... 

Because of the success encountered by finite elements in the solution of elliptic problems, it was extended (in the 80s) to the advection or transport equation which is a hyperbolic equation with only one real characteristic. This equation can be solved naturally for an analytical velocity field by solving a time differential equation. It appeared important, when the velocity field was numerically obtained, to be able to solve simultaneously propagation and diffusion equations at low cost. By introducing upwinding in test functions or in the discretization scheme, the particular nature of the transport equation was considered. In this case, a particular direction is given at each point (the direction of the convecting flow) and boundary conditions are only considered on the part of the boundary where the flow is entrant. 

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