Numerical methods with other boundary conditions

Analytical solution methods such as those presented in Chapter 2 are based on solving the governing differential equation together with the boundary conditions. Tliey result in solution functions for the temperature at every point in the medium. Numerical methods, on the other hand, are based on replacing the difi erential equation by a set of n algebraic equations for the unknown temperatures at n selected points in the medium, and the simultaneous solution of these equations results in the temperature values at those discrete points. 

Discussions in Chapter 2 may be referred to for explanations of the various symbols. It is straightforward to apply such conservation equations to single-phase flows. In the case of multiphase flows also, in principle, it is possible to use these equations with appropriate boundary conditions at the interface between different phases. In such cases, however, density, viscosity and all the other relevant properties will have to change abruptly at the location of the interface. These methods, which describe and track the time-dependent behavior of the interface itself, are called front tracking methods. Numerical solution of such a set of equations is extremely difficult and enormously computation intensive. The main difficulty arises from the interaction between the moving interface and the Eulerian grid employed to solve the flow field (more discussion about numerical solutions is given in Chapters 6 and 7). 

At the outlet, extrapolation of the velocity to the boundary (zero gradient at the outlet boundary) can usually be used. At impermeable walls, the normal velocity is set to zero. The wall shear stress is then included in the source terms. In the case of turbulent flows, wall functions are used near walls instead of resolving gradients near the wall (refer to the discussion in Chapter 3). Careful linearization of source terms arising due to these wall functions is necessary for efficient numerical implementation. Other boundary conditions such as symmetry, periodic or cyclic can be implemented by combining the formulations discussed in Chapter 2 with the ideas of finite volume method discussed here. More details on numerical implementation of boundary conditions may be found in Patankar (1980) and Versteeg and Malalasekara (1995). 

The boundary condition implementations play a very critical role in the accuracy of the numerical simulations. The hydrodynamic boundary conditions for the LBM have been smdied extensively. The conventional bounce-back rule is the most popular method used to treat the velocity boundary condition at the solid-fluid interface due to its easy implementation, where momentum from an incoming fluid particle is bounced back in the opposite direction as it hits the wall [20]. However, the conventional bounce-back rale has two main disadvantages. First, it requires the dimensionless relaxation time to be strictly within the range (0.5, 2) otherwise, the prediction will deviate from the correct result. Second, the nonslip boundary implemented by the conventional bounce-back rule is not located exactly on the boundary nodes, as mentioned before, which will lead to inconsistence when coupling with other partial differential equation (PDF) solvers on a same grid set [17]. 

Numerical solutions are also possible and in view of the complexity of the analytical solutions often desirable. One method is to replace the Lame equations (16) by a set of difference equations for points on an array over an r-z section of the cylinder. The boundary conditions are then used to remove undefined points at the boundaries. The solution is obtained by iteration (23) or by solving the equations directly by a matrix technique (24). The other common method is the use of finite elements, which have been applied widely to axisymmetric elastic and thermoelastic problems (23). This technique breaks the r-z section of the cylinder into regions or elements where the properties and conditions can be assumed to be approximately uniform. At the junctions of the elements, the nodes, displacements, and forces are defined. These displacements and forces are connected, using the elastic and thermal properties of the material, by minimizing the energy of the system. A set of linear equations in terms of the displacements is then obtained by matching the nodal forces and displacements from element to element, together with the boundary conditions. The set of linear equations is then solved in the same way as for the finite difference approximation. 

A number of authors have considered channel cross-sections other than rectangular [102-104]. Figure 2.17 shows some examples of cross-sections for which friction factors and Nusselt numbers were computed. In general, an analytical solution of the Navier-Stokes and the enthalpy equations in such channel geometries would be involved owing to the implementation of the wall boundary condition. For this reason, usually numerical methods are employed to study laminar flow and heat transfer in channels with arbitrary cross-sectional geometry. 

Calculation of the electric field dependence of the escape probability for boundary conditions other than Eq. (11b) with 7 = 0 poses a serious theoretical problem. For the partially reflecting boundary condition imposed at a nonzero R, some analytical treatments were presented by Hong and Noolandi [11]. However, their theory was not developed to the level, where concrete results of (p(ro,F) for the partially diffusion-controlled geminate recombination could be obtained. Also, in the most general case, where the reaction is represented by a sink term, the analytical treatment is very complicated, and the only practical way to calculate the field dependence of the escape probability is to use numerical methods. 

It would appear that the study of the diffusion equation subject to a phase change at one boundary is in a relatively satisfactory state, provided simple boundary conditions of the first, second, or third kind are specified. From a mathematical point of view, the interesting features of the problem arise from the nonlinearity, exhibited for all but a few particular boundary motions. A wide variety of approximate and numerical methods have been employed, and it has frequently been difficult for workers in one specialized field of activity to become conversant with similar approaches made by investigators in other areas. It is hoped that the present work will, to some extent, alleviate this problem. 

All numerical techniques require application of sampling theory. Briefly stated, one chooses a representative sample of points within the region of interest and at each point attempts to calculate iteratively the most accurate solution possible, guided by self-consistency of local solutions with each other and with the specified boundary conditions. We describe two seemingly contrasting techniques finite-difference and finite-element methods (1,2). 

The solution of the concentration profile ( >(z) should be specified for given temperature T, film thickness D, and the average blend composition in this film <(( . The parameters T and <( , important in experiments, might be translated [60] into interaction parameter % and the chemical potential difference Ap.more convenient in calculations. Thus, for say D, %, and Ap known and kept constant, the profile ( >(z) may be obtained (Eq. 50) by varying the reservoir concentration (]>b until the boundary conditions (Eq. 51) are met. If a few solutions exist, the relevant ones are those with minimal overall free energy F (Eq. 49). Such a shooting procedure was developed by Flebbe et al. [60]. A numerical method which starts from an arbitrary assumed profile and modifies its discretized form until conditions equivalent to Eqs. (50), (51) and (53) are met has also been proposed recently by Eggleton [222]. The solutions yielded by this technique may however correspond to metastable states. Concentration profiles in thin films were also evaluated by other theoretical treatments [93,118,177,219,221]. 

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