Chapter 4 - CFD Techniques—The Basics

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This chapter presents some of the basic computational techniques that can be employed to solve the governing equations of fluid dynamics. The first stage of obtaining the computational solution involves the conversion of the governing equations into a system of algebraic equations. This is usually known as the discretization stage. It discusses some of the discretization tools, such as the finite- difference and finite-volume methods, which form the foundation of understanding the basic features of discretization. Both of these methods are abundant in many CFD applications. The second stage involves numerically solving the system of algebraic equations, which can be achieved by either the direct methods or iterative methods. Basic direct methods such as the Gaussian elimination and the Thomas algorithm are discussed. Simple iterative methods such as the point-by-point Jacobi and Gauss-Siedel methods are also described. Nevertheless, CFD problems are generally multidimensional and comprise a large system of equations to be solved. Efficient iterative methods such as the ADI or Stone's SIP are often applied to solve such a system of equations. To further enhance the convergence of the computational solution, precondition conjugate gradient methods or multigrid methods are employed to accelerate the iteration process. Finally, this chapter discusses the assessment of convergence. In practice, the algebraic equations that result from the discretization process yield the flow solution at each nodal point on a finite-grid layout.