Boundary fitted coordinates

Thompson, ). F., Warsi, Z. U., Mastin, C. W., Boundary-fitted coordinate systems for numerical solution of partial differential equations, ). Comput. Phys. 

R.E. Smith and B.L. Weigel. Analytic and approximate boundary fitted coordinate systems for fluid flow simulation. AIAA, 80 192, 1980. 

B. C-J. Chen, T. H. Chien, W. T. Sha, and J. H. Kim, Solution of Flow in an Infinite Square Array of Circular Tubes by Using Boundary Fitted Coordinate Systems, Numerical Grid Generation, ed. J. F. Thompson, Elsevier, New York, pp. 619-632,1982. 

Subdivision or discretization of the flow domain into cells or elements. There are methods, called boundary element methods, in which the surface of the flow domain, rather than the volume, is discretized, but the vast majority of CFD work uses volume discretization. Discretization produces a set of grid lines or cuives which define a mesh and a set of nodes at which the flow variables are to be calculated. The equations of motion are solved approximately on a domain defined by the grid. Curvilinear or body-fitted coordinate system grids may be used to ensure that the discretized domain accurately represents the true problem domain. 

These equations were derived for flow over a plane surface. They may be applied to flow over a curved surface provided that the boundary layer thickness remains small compared to the radius of curvature of the surface. When applied to flow over a curved surface, x is measured along the surface and y is measured normal to it at all points as shown in Fig. 2.15, i.e., body-fitted coordinates are used. 

Keywords Finite element Finite volume Finite difference Volume of fluid Level set Interface tracking Free surface flows Fixed mesh Boimdary-fitted coordinates Boundary integral Marker and cell Immersed boxmdary Volume tracking Surface tracking Surface capturing Interfacial flow modeling... 

The interface position is tracked by introducing a curvilinear interface-fitted non-orthogonal coordinate system. By means of a coordinate transformation, the physical domain is converted to a computational domain with known boundaries that are coordinate isoUnes. A boundary-fitted grid is generated around the deformed interface at each iteration by a pair of Poisson equations associated with spacing control functions. 

As is the case with all differential equations, the boundary conditions of the problem are an important consideration since they determine the fit of the solution. Many problems are set up to have a high level of symmetry and thereby simplify their boundary descriptions. This is the situation in the viscometers that we discussed above and that could be described by cylindrical symmetry. Note that the cone-and-plate viscometer —in which the angle from the axis of rotation had to be considered —is a case for which we skipped the analysis and went straight for the final result, a complicated result at that. Because it is often solved for problems with symmetrical geometry, the equation of motion is frequently encountered in cylindrical and spherical coordinates, which complicates its appearance but simplifies its solution. We base the following discussion on rectangular coordinates, which may not be particularly convenient for problems of interest but are easily visualized. 

Using the solution of the Laplace equation for diffusion in cylindrical coordinates given by Eq. 5.10, fitting it to the boundary conditions given by Eq. 16.80, and employing Eq. 13.3 for the flux, the total diffusion current of atoms (per unit pipe length) passing radially from R,n to f out is... 

The continuous region or body is subdivided into a finite number of subregions or elements (Fig. 15.5). The elements may be of variable size and shape, and they are so chosen because they closely fit the body. This is in sharp contrast to finite difference methods, which are characterized by a regular size mesh, describable by the coordinates that describe the boundaries of the body. 

Mathematically, geometric parameters can be described by using the Fourier Series in polar coordinates (p,9). Thus, given a set of boundary points (x, y) from an object of interest, they can be transformed into the polar coordinates with respect to its geometric center (x, y). A curve fitting technique in polar coordinates can be used to fit this set of points into a Fourier Series such that any point p(0) on this boundary can be expressed by... 

Figure 16. Concept of using a Fourier Series to represent the boundary of an object. The (x,y) coordinates of the boundary points of an object is transformed to polar coordinates. Each point on the boundary p(0) can be expressed by a Fourier Series obtained from curve fitting of the boundary points. (X,Y) are the coordinates of the center of gravity.

The use of the square wave production rate distribution is an obvious approximation and a Gaussian distribution for the production rate, such as used for Mn by Robbins and Callender (1975), would be more realistic. A smoothly increasing, then decreasing, Fe production-rate distribution would result in better fits to the actual data at the boundaries of zone 2 and lower estimates of ki. k would also be slightly smaller if the cylindrical coordinate model were used as previously illustrated for Mn. However, because of the very approximate nature of the production rates and the simplifications made concerning the kinetics of oxic and anoxic precipitation, more sophisticated modeling is not warranted. 

It should be clear that under conditions of very weak coupling, where 2/f < A, AG AGth and Fit Thus, estimates of AGth for a given metal ion and coordination sphere can be derived from the intervalence band energies. Equation (5) also gives the conditions for the boundary between class II and class III behavior, 2/fad =... 

Two-Stage Countercurrent Batch Sorption. The sorbent/solution ratio for a fixed set of process conditions can be determined by trial and error using the graphical technique. This is carried out by graphically fitting two steps between the equilibrium curve and the initial and final solute-liquid concentration boundary concentrations. The operating line can then be drawn, and this provides the coordinates, of the intermediate concentration, the... 

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