Advection problem, solution

In the case of the fast binary reaction we could eliminate the reaction term from the reaction-diffusion-advection equation. But in general this is not possible. In this chapter we consider another class of chemical and biological activity for which some explicit analysis is still feasible. We consider the case in which the local-reaction dynamics has a unique stable steady state at every point in space. If this steady state concentration was the same everywhere, then it would be a trivial spatially uniform solution of the full reaction-diffusion-advection problem. However, when the local chemical equilibrium is not uniform in space, due to an imposed external inhomogeneity, the competition between the chemical and transport dynamics may lead to a complex spatial structure of the concentration field. As we will see in this chapter, for this class of chemical or biological systems the dominant processes that determine the main characteristics of the solutions are the advection and the reaction dynamics, while diffusion does not play a major role in the large Peclet number limit considered here. Thus diffusion can be neglected in a first approximation. 

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... 

All the problems encountered during application of the algorithms presented above illustrate the difficulty of solution of the advection problem in atmospheric transport models. A number of techniques have been developed to treat advection accurately, including flux-corrected transport (FCT) algorithms (Boris and Book, 1973), spectral and finite element methods [for reviews, see Oran and Boris (1987), Rood (1987), and Dabdub and Seinfeld (1994). Bott (1989, 1992), Prather (1986), Yamartino (1992), Park and Liggett (1991), and others have developed schemes specifically for atmospheric transport models. 

All of the problems encountered during application of the above algorithms illustrate the difficulty of solution of the advection problem in atmospheric transport models. A number of techniques have been developed to treat advection accurately including flux corrected transport (FCT) algorithms (Boris and Book, 1973), spectral and finite element... 

The closed-form solution for the advection-dispersion equation expressed in dimensionless group that is referred to as the Graetz-Nusselt problem solution is given as... 

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. 

Taylor s dispersion is one of the most well-known examples of the role of transport in dispersing a flow carrying a dissolved solute. The simplest setting for observing it is the injection of a solute into a slit channel. The solute is transported by Poiseuille s flow. In fact this problem could be studied in three distinct regimes (a) diffusion-dominated mixing, (b) Taylor dispersion-mediated mixing and (c) chaotic advection. 

A two-dimensional example problem is also developed to demonstrate the advective control model. The example problem is solved for both confined and unconfined conditions and the solutions are compared. In this example problem, the aquifer is homogeneous and isotropic, with no flow conditions imposed at the top and bottom boundaries and constant head conditions along the left and right boundaries. The head on the constant head boundaries slopes downward toward the bottom of the domain. The domain is 3100 m by 3100 m and is discretized into 49 rows and 58 columns. A river runs through the domain as shown in Figure 6. 

Begin by noting that C() satisfy the Neumann problem given by (5) and boundary conditions whose solution is Cj(x, y, t) = Cj(x, t). We now derive the overall macroscopic mass balance for the species. To this end we begin by deriving the closure problem for Cj. Given Cf(x,t), combine (6) with boundary conditions and neglect the advection induced by du/dt, to obtain the local Neumann problem... 

For offline ACTMs the choice of advection schemes is an independent and critically important problem. Additionally using NWP input data with nonconservative and not harmonized schemes (due to different schemes, grids, time steps in NWP and ACT models) offline ACTMs can get a dramatic problem with an explosive solution for the chemical part. For onhne coupling it is not a problem if one uses the same mass-conservative scheme for chemicals, cloud water and humidity. 

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. 

The fluid flow problem requires the solution of all these equations. The dependent variables are p, p, T and Vi. The governing equations are all coupled together. For instance, the momentum equation contains the density and pressure variables, and the viscosity parameter. These quantities depend on the local temperature. The temperature in turn is governed by the energy equation, which contains the velocity in the advective and dissipation terms. However, not all the terms in the equations have the same importance in determining the flow solution. 

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. 

Three-dimensional models offer more realism, at least apparently, but with the cost of greater complexity, a more limited number of simulations, and a higher probability of crucial regional errors in the base solutions, which may compromise direct, quantitative model-data comparisons. Ocean GCM solutions, however, should be exploited to address exactly those problems that are intractable for simpler conceptual and reduced dimensional models. For example, two key assumptions of the 1-D ad-vection-diffusion model presented in Figure 2 are that the upwelling occurs uniformly in the horizontal and vertical and that mid-depth horizontal advection is not significant. Ocean GCMs and tracer data, by contrast, show a rich three-dimensional circulation pattern in the deep Pacific. 

When solving how problems numerically, a Neumann boundary is described as an insulated boundary (or impermeable boundary), which means that there is no fiux at the boundary, while a Dirichlet boundary indicates that the value of head (potential, concentration, etc.) is constant at the boundary. Constant boundary conditions are not able to describe the nature of the electrokinetic transport realistically due to the existence of fiux boundaries caused by the electrode reactions and advection of fluid. In Cao s model, the boundary conditions apphed at the inlet and outlet of the soil column maintained the equahty between the flux of solute at the inside of the column and the flux of solute immediately outside of the column. The following boundary condition was used at the inlet (Lafolie and Hayot, 1993) ... 

The solution of equations (2.40) and (2.41) with the total accounting of all acting factors is quite complex. So the problem is simplified by excluding secondary factors, which may be disregarded. The exact solution of the migration problem of individual component in conditions of advective-dispersive mass transport without approximations is called the analytical solution. 

Then the hydrogeochemical part for the distribution of individual components dissolved in ground water is solved. The analytical solution of mass transport problems includes the introduction of edge and initial conditions of the hydrochemical object and selection of advection-dispersion mass transport equations matching the assigned conditions and mathematical solution of the equation themselves. 

It follows from this that all solutions of mass transport problems taking into account only advective-dispersive dispersion are also applicable for cases of linear sorption (Henry s sorption isotherm). It is sufficient for... 

---