7. Finite-Volume Methods for Navier-Stokes Equations

Flows of engineering interest can only be predicted by using numerical solution methods, because the governing equations cannot be solved analytically. Among many possible approaches, finite-volume methods have become popular in this field and are the basis for most commercial and public computer codes used to simulate fluid flow. One of the most widely used methods, which is applicable to complex geometries and arbitrary polyhedral computational grids, is described in this chapter. The solution method was originally developed for incompressible flows and later extended to compressible flows at any speed. It is also described how to deal with moving grids which follow the motion of solid bodies. The applicability of the method to various flow types is demonstrated by simulating flow around sphere at various Reynolds numbers, ranging from a creeping flow at Re = 5 to a supersonic flow at Re = 5,000,000. The final example involves a sphere oscillating in water at 6000 Hz with a very small amplitude, which requires that compressibility of water is taken into account. The fact that the same method can be applied to such a wide class of flows, including a wide range of turbulence-modeling approaches, is the main reason why it is used in general-purpose commercial codes.