F. Moukalled L. Mangani - The Finite Volume Method in Computational Fluid Dynamics

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We are literally at a significant point in history. A third branch of the scientific method, computer simulation, is emerging as a day - to - day tool. It is taking its place next to experimental development and mathematical theory as a way to new discoveries in science and engineering . This was part of the speech of John Rollwagen, chairman and CEO of Cray Research, to the opening session of Supercomputing 89.

While it is common to refer to this or similar statements about the importance of simulation tools and techniques to the advancement of science and technology in general, it is now very clear that the use of simulation tools has become crucial to the development of a wide range of everyday technologies. In fact, numerical simulation tools nowadays play the role of technology enablers.

Computational Fluid Dynamics (CFD) is one such tool. Even though the impetus to its development was initially provided by some sections within the aeronautics and aerospace industry, it has grown to become an essential tool in a range of other design intensive industries such as the automotive, power generation, chemical, nuclear, and marine industries, to cite a few. Over the past decade newer industries have joined the ranks of heavy CFD users: for example in the electronics industry CFD is employed to optimize energy systems and heat transfer for the cooling of electronic devices, in the biomedical industry CFD is now a core development and validation tool for medical applications, and in the building industry CFD is used in HVAC (heating, ventilating, and air conditioning), in fire simulation, and in air-quality assessment.

This has happened in little over two decades since the statement of John Rollwagen was made and over four decades since the development of the seminal SIMPLE algorithm by the CFD group of Brian Spalding at Imperial College in the early 70s.

Computational Fluid Dynamics is just one of the later Computer Aided Engineering tools that has gone mainstream. It has joined a well-established set of tools, such as the Finite Element Analysis (FEA) for Solid Mechanics and Vibration that has been part of the engineering design cycle since the mid 80s. The reason for this delay is the complexity of the equations that need to be solved. At their center is the Navier-Stokes equation that amazingly enough models accurately a whole set of flow phenomena from turbulent or laminar single phase incompressible flows, to compressible all-speed flows, and all the way to multiphase flows.

Amongst the numerical methods used to implement CFD, the Finite Volume Method has come to play a unique role.

The Finite Volume Method (FVM) is a numerical technique that transforms the partial differential equations representing conservation laws over differential volumes into discrete algebraic equations over finite volumes (or elements or cells). In a similar fashion to the finite difference or finite element method, the first step in the solution process is the discretization of the geometric domain, which, in the FVM, is discretized into non-overlapping elements or finite volumes. The partial differential equations are then discretized/transformed into algebraic equations by integrating them over each discrete element. The system of algebraic equations is then solved to compute the values of the dependent variable for each of the elements.

In the finite volume method, some of the terms in the conservation equation are turned into face fluxes and evaluated at the finite volume faces. Because the flux entering a given volume is identical to that leaving the adjacent volume, the FVM is strictly conservative. This inherent conservation property of the FVM makes it the preferred method in CFD. Another important attribute of the FVM is that it can be formulated in the physical space on unstructured polygonal meshes. Finally in the FVM it is quite easy to implement a variety of boundary conditions in a non-invasive manner, since the unknown variables are evaluated at the centroids of the volume elements, not at their boundary faces.

These characteristics have made the Finite Volume Method quite suitable for the numerical simulation of a variety of applications involving fluid flow and heat and mass transfer, and developments in the method have been closely entwined with advances in CFD. From a limited potential at inception confined to solving simple physics and geometry over structured grids, the FVM is now capable of dealing with all kinds of complex physics and applications.

This book is about the Finite Volume Method and Computational Fluid Dynamics. It incorporates the basic know how of the method as acquired by the authors over almost three decades of work in the area. The terminology was carefully chosen, and vector notation was used whenever possible to ensure conciseness and consistency across all topics covered. Derivations are presented in a step by step fashion and illustrations are used extensively in the book to clarify concepts. In addition, a number of solved examples and exercises are also provided. Each chapter ends with a section denoted by Computational Pointers that provides pertinent details on implementation issues for two codes. The first, denoted by uFVM, is a Matlab-based unstructured finite volume CFD educational code developed by the authors; while the second is OpenFOAM, an open source finite volume code written in C++ capable of solving industrial type problems. Generally the numerics in each chapter are first presented for a one dimensional grid and progress towards two and three dimensional unstructured grids, to ease the introduction of difficult techniques.

The material presented allows the book to be utilized in a variety of ways. It can be used as a textbook for a senior undergraduate course covering the fundamentals of the finite volume discretization. It can also be deployed as a textbook for a graduate course on the application of the finite volume method and its use in computational fluid dynamics. It is also a handy reference book for workers in CFD, numerical heat transfer, and transport phenomena in general.

The content of the book falls into 20 chapters that may be grouped under the following four categories: (i) Foundation (Chaps. presents some closing remarks.

The uFVM Matlab computer program, the OpenFOAM developed routines, and the prepared lecture presentations can be downloaded from the book webpage at the following URL: https://feaweb.aub.edu.lb/research/cfd