Basics of Computational Fluid Dynamics and Numerical Heat Transfer
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Contents |
Overview
The steady and transient heat conductions discussed in the preceding two sections are for linear problems with simple geometric configurations and boundary conditions. Even for the cases where analytical solution is available, the interpretation of the solution is often difficult because the solution is expressed as a very complicated series. Numerical solution therefore becomes a desirable approach for heat conduction under non-linear, complex geometric configurations, or complex boundary conditions. Instead of obtaining the analytical expression of the temperature distribution, the results of numerical solutions are given at discrete points. The numerical solution involves three steps: (1) discretization of the computational domain, (2) discretization of the governing equations, and (3) solution of the algebraic equations (Murthy et al., 2006).
Discretization of Computational Domain and Governing Equations
The discretization of the computational domain is a process that divides the computational domain into many control volumes as shown in Fig. 3.22 (Patankar, 1980). Each subdomain is bounded by faces represented by the dashed lines. The physical properties of the cell are defined at the grid points. In Practice A, the faces are always located midway between the grid points. On the contrary, the grid points are always located in the center of the control volume in Practice B. Obviously, if a uniform grid is used, the two practices result in identical grid size and therefore, any discussion on the differences between two practices are meaningful only if the grid size is not uniform. Since the faces in Practice A are located in the midway between grid points, the heat flux across the faces can be more accurately calculated. The disadvantage of Practice A is that the properties of the entire control volume are represented by a point not located in the center of the control volume, which will result in inaccuracy. On the other hand, the properties of the entire control volume in Practice B are represented by the grid point at the center which is a better representation. In addition, Practice
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(a) Practice A (b) Practice B
Figure 1: Discritization of the computational domain.(Patankar, 1980)
B can easily handle discontinuity of thermophysical properties or boundary conditions.
The discretization of governing equations can be done by local or point-wise representation of the partial differential equations (finite difference method; FDM), or weighted integral of the partial differential equations (finite element method; FEM). Patankar (1980) proposed a finite volume method (FVM) involving obtaining the discretized equation by performing integration on the governing equation over the small region. While the resultant algebraic equations for FVM and FDM are often similar, the FVM can guarantee conservation of the mass, momentum and energy on each cell, regardless of the size of the cell. The FVM can also be very easily extended to convective heat transfer because mature numerical methods have been developed in the last three decades.
Other Numerical Methods
Other notable methods for numerical solution of heat conduction problems include boundary element method (BEM) that volume integrals are converted to surface integrals, as well as spectral methods that approximate solutions by a series of functions. The drawback of these methods is that they are very difficult to extend to convective heat transfer. To maintain uniformity of the numerical method for all heat transfer modes and build a solid foundation for students to learn other methods, we will focus on FVM. This section will cover numerical solutions of one-dimensional steady and transient conduction problems, as well as multidimensional transient heat conduction problems. Its extension to convection and radiation problems will be discussed in detail in later chapters of this text.