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Modelling transport and mixing by turbulence in complex flows are huge challenges for computational fluid dynamics (CFD). This highly readable book introduces readers to modelling levels that respect the physical complexity of turbulent flows. It examines the hierarchy of Reynolds-averaged Navier-Stokes (RANS) closures in various situations ranging from fundamental flows to three-dimensional industrial and environmental applications. The general second-moment closure is simplified to linear eddy-viscosity models, demonstrating how to assess the applicability of simpler schemes and the conditions under which they give satisfactory predictions. The principal changes for the second edition reflect the impact of computing power: a new chapter devoted to unsteady RANS and another on how large-eddy simulation, LES, and RANS strategies can be effectively combined for particular applications. This book will remain the standard for those in industry and academia seeking expert guidance on the modelling options available, and for graduate students in physics, applied mathematics and engineering entering the world of turbulent flow CFD.
The present chapter deals with the FVM as applied to the diffusion equation in one and two dimensions. In the FVM, the domain Ω is divided into a collection of nonoverlapping subdomains, called control volumes and the collection is called a mesh or grid.
In this chapter, we will focus on solving the PDEs governing laminar flows of viscous incompressible fluids using the FVM (this chapter is a counterpart of Chapter 6 on FEM, where velocity–pressure and penalty finite element models of two dimensional flows of viscous incompressible fluids were presented). These equations are expressed in terms of the primitive variable, namely, the velocity field and the pressure. To begin with, we will consider isothermal flows (flows without the presence of the temperature effect), and demonstrate the use of the FVM for two-dimensional laminar flows of viscous incompressible fluids. Then cases of non-isothermal flows with both forced convection and natural convection will be considered in the sequel.
In Chapter 4 we considered finite element analysis of steady state heat transfer. When external stimuli (e.g., boundary conditions and internal heat generation) are independent of time, heat transfer in a medium may attain a steady state; otherwise, the temperature field changes with time (i.e., unsteady state). The governing equations of unsteady heat transfer are obtained using the principle of balance of energy. When unsteady equations are solved the temperature field reaches a steady state if the external stimuli are independent of time (i.e., the time dependence decays with time).