Finite Volume Method I: Introduction

This chapter introduces the finite volume method (FVM) for solving conservation laws in computational fluid dynamics. The method is derived from the integral form of governing equations and ensures local conservation over discretized control volumes. The formulation includes treatment of convective and diffusive fluxes, source terms, and time discretization using schemes such as Crank-Nicolson. Various interpolation methods for convective fluxes like upwind, central difference, hybrid, and QUICK are compared with respect to accuracy, considering the influence of the control volume Peclet number. The chapter emphasizes spatial discretization over hexahedral control volumes, illustrating how convective and diffusive terms are approximated at faces using midpoint rules. Source terms and pressure gradients are evaluated using divergence theorems and interpolation. The Crank-Nicolson scheme is introduced for time-stepping in unsteady problems. Implementation aspects in OpenFOAM are outlined via examples of gradient, divergence, and laplacian schemes.