Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 6
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Because the fluxentering a given volume is identical to that leaving the adjacent volume, the FVM isstrictly conservative. This inherent conservation property of the FVM makes it thepreferred method in CFD. Another important attribute of the FVM is that it can beformulated in the physical space on unstructured polygonal meshes. Finally in theFVM it is quite easy to implement a variety of boundary conditions in a noninvasive manner, since the unknown variables are evaluated at the centroids of thevolume elements, not at their boundary faces.These characteristics have made the Finite Volume Method quite suitable for thenumerical simulation of a variety of applications involving fluid flow and heat andmass transfer, and developments in the method have been closely entwined withadvances in CFD. From a limited potential at inception confined to solving simplephysics and geometry over structured grids, the FVM is now capable of dealingwith all kinds of complex physics and applications.1.3 This Book1.35This BookThis 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 overalmost 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 stepfashion and illustrations are used extensively in the book to clarify concepts. Inaddition, a number of solved examples and exercises are also provided. Eachchapter ends with a section denoted by “Computational Pointers” that providespertinent details on implementation issues for two codes.
The first, denoted byuFVM, is a Matlab®-based unstructured finite volume CFD educational codedeveloped by the authors; while the second is OpenFOAM®, an open source finitevolume code written in C++ capable of solving industrial type problems. Generallythe numerics in each chapter are first presented for a one dimensional grid andprogress 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 canbe used as a textbook for a senior undergraduate course covering the fundamentalsof the finite volume discretization. It can also be deployed as a textbook for agraduate course on the application of the finite volume method and its use incomputational fluid dynamics.
It is also a handy reference book for workers inCFD, numerical heat transfer, and transport phenomena in general.The content of the book falls into 20 chapters that may be grouped under thefollowing four categories: (i) Foundation (Chaps. 2 through 7), (ii) Numerics(Chaps. 8 through 14), (iii) Algorithms (Chaps.
15 and 16), and (iv) Applications(17 through 19). Chapter 20 presents some closing remarks.The uFVM Matlab® computer program, the OpenFOAM® developed routines,and the prepared lecture presentations can be downloaded from the book webpageat the following URL: “https://feaweb.aub.edu.lb/research/cfd”A summary of the material covered in the forthcoming chapters is presentednext.1.3.1FoundationThis part, covered in Chaps. 2 through 7, provides the necessary background forintroducing the FVM.Chapter 2 presents a short introduction of the elements of linear algebraincluding vectors, matrices, tensors, and their practices.
This is in addition to anexamination of the fundamental theorems of vector calculus.Chapter 3 overviews the conservation principles governing fluid flow and relatedtransport phenomena. It describes the derivations of the continuity, momentum, and61Introductionenergy equations (collectively known as the Navier-Stokes equations). This isfollowed by the development of a typical conservation equation for a general scalar,vector, or tensor quantity. This equation forms the cornerstone for the developmentspresented in the numerics section.
The conservation equation is shown to becomposed of a transient, convection, diffusion, and source term. The discretizationof each of these terms is presented in a separate chapter.Chapter 4 summarizes the various steps of the discretization process, whichinclude: (i) modeling the geometric domain and the physical phenomena, (ii) thediscretization of the modeled geometric domain into a grid system, (iii) the discretization of the partial differential equation into an equivalent system of algebraicequations defined over each of the elements of the computational domain, and(iv) the solution of the system of equations.Chapter 5 transforms the partial differential equation into a set of semi-discretized equations and presents a broad review of the numerical issues pertaining to thefinite volume method.
This provides a solid foundation for the chapters that follow.Chapter 6 is devoted to the finite volume mesh. It starts with mesh discretizationthat replaces the geometric domain by a set of non overlapping elements. Then itproceeds with the computation of geometric information relevant to the variousentities of the computational mesh in addition to the topological information thatdescribes the arrangement and inter-relations of these entities.Chapter 7 outlines the design decisions that shape the implementation of the twoCFD codes, uFVM and OpenFOAM®. First the data structure and memory management schemes of the two codes are presented, then a sample test case is presented. Finally the format of the system of equations generated by each of the twocodes are detailed.1.3.2NumericsThe material relevant to this part is covered in Chaps.
8 through 14. Each chapterspecializes in the discretization of one of the terms in the general conservationequation derived in Chap. 2, with the exception of Chap. 10, which deals withlinear solvers of algebraic systems of equations.Chapter 8 describes the discretization of the diffusion term. The developmentsstart on a structured Cartesian mesh and progress to unstructured non-orthogonalgrid, while explaining the adopted treatment of the non-orthogonal cross-diffusionterm.
The chapter continues with a discussion on the used interpolation profiles andthe rules that should be satisfied by the coefficients of the discretized system ofalgebraic equations. It also details the implementation of boundary conditions inaddition to the under-relaxation procedure needed for highly non-linear problems.Chapter 9 describes several techniques for evaluating gradients on a generalmesh topology following either the Green-Gauss or the least square approach. Italso presents methods to interpolate the gradient to element faces.1.3 This Book7Chapter 10 deals with solvers of systems of algebraic equations.
Both direct anditerative solvers are discussed with emphasis on iterative solvers because directsolvers are rarely used in CFD applications. The direct methods presented includethe Gauss elimination and LU factorization. The concept of preconditioning ispresented and the performance and limitations of some iterative methods arereviewed. This include the Jacobi, Gauss-Siedel, Incomplete LU factorization, andthe conjugate gradient methods. The chapter also introduces the algebraic multigridmethod, which is generally used in combination with iterative solvers to acceleratetheir convergence.Chapter 11 proceeds with the discretization of the convection term assuming aknown flow field.
The shortcomings of using a symmetrical linear profile for thediscretization of the convection term are delineated and a remedy is suggestedthrough the use of an upwind profile. The high diffusion error associated with theupwind scheme is pointed out and upwind-biased higher order schemes aresuggested.Chapter 12 continues the developments of convection schemes and discussesapproaches by which the dispersion error (unboundedness of the interpolationprofile) affecting High Order (HO) schemes is resolved.
This is achieved byenforcing a Convection Boundedness Criterion (CBC) on the HO profiles resultingin the group of High Resolution (HR) schemes. The Normalized VariableFormulation (NVF) and the Total Variation Diminishing (TVD) frameworks forconstructing these HR schemes are presented. The commonality between the twoapproaches is explained through the Normalized Variable Diagram (NVD) andSweby’s diagram used in the NVF and TVD formulation, respectively.
Manyschemes are presented in the context of both formulations. Techniques for theimplementation of HR schemes in structured and unstructured grids are reported.Chapter 13 focusses on the discretization of the unsteady term that arises in thesimulation of transient problems. Several transient schemes are developed following two different approaches.
In the first, a finite difference approximation (viaTaylor expansion) is used. In the second approach, the finite volume method is usedon a temporal element in a similar fashion to what was done to the convection term.Chapter 14 is devoted to a number of “small” numerical details that may have“big” effects on the convergence behavior. First the linearization of the source termwhen it is solution dependent is discussed. Then explicit and implicit techniques forunder relaxing the algebraic equations are presented. The chapter ends with anexamination of convergence indicators.1.3.3AlgorithmsThe previous chapters solved the general conservation equation assuming a givenflow field. In general, the flow field is not known and has to be computed.
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