计算流体力学.pdf

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E.5.3.3

第五章作业

5.4.4

P.7.1

7.1.24

FM-H6594.tex 8/5/2007 12: 31 Page xv Preface to the second edition This second, long (over)due, edition presents a major extension and restructur- ing of the initial two volumes edition, based on objective as well as subjective elements. The ﬁrst group of arguments is related to numerous requests we have received over the years after the initial publication, for enhancing the didactic structure of the two volumes in order to respond to the development of CFD courses, starting often now at an advanced undergraduate level. We decided therefore to adapt the ﬁrst volume, which was oriented at the fundamen- tals of numerical discretizations, toward a more self-contained and student-oriented ﬁrst course material for an introduction to CFD. This has led to the following changes in this second edition: • We have focused on a presentation of the essential components of a simulation system, at an introductory level to CFD, having in mind students who come in contact with the world of CFD for the ﬁrst time. The objective being to make the student aware of the main steps required by setting up a numerical simulation, and the various implications as well as the variety of options available. This will cover Chapters 1–10, while Chapters 11 and 12 are dedicated to the ﬁrst applications of the general methodology to inviscid simple ﬂows in Chapter 11 and to 2D incompressible, viscous ﬂows in Chapter 12. • Several chapters are subdivided into two parts: an introductory level written for a ﬁrst introductory course to CFD and a second, more advanced part, which is more suitable for a graduate and more advanced CFD course. We hope that by putting together the introductory presentation and the more advanced topics, the student will be stimulated by the ﬁrst approach and his/her curiosity for the more advanced level, which is closer to the practical world of CFD, will be aroused. We also hope by this way to avoid frightening off the student who would be totally new to CFD, by a too ‘brutal’ contact with an approach that might appear as too abstract and mathematical. • Each chapter is introduced by a section describing the ‘Objectives and guidelines to this Chapter’, and terminates by a section on ‘Conclusions and main topics to remember’, allowing the instructor or the student to establish his or her guide through the selected source material. • The chapter on ﬁnite differences has been extended with additional considera- tions given to discretizations formulas on non-uniform grids. • The chapters on ﬁnite element and ﬁnite volume methods have been merged, shifting the ﬁnite element description to the ‘advanced’ level, into Chapter 5 of this volume. • A new Chapter 6 has been added devoted to an overview of various grids used in practice, including some recommendations related to grid quality. xv

FM-H6594.tex 8/5/2007 12: 31 Page xvi xvi Preface • Chapters 7 and 8 of the ﬁrst edition, devoted to the analysis of numerical schemes for consistency and stability have been merged and simpliﬁed, forming the new Chapter 7. • Chapter 9 of the ﬁrst edition has been largely reorganized, simpliﬁed and extended with new material related to general scheme properties, in particu- lar the extremely important concept of monotonicity and the methodologies required to suppress numerical oscillations with higher order schemes, with the introduction of limiters. This is found in Chapter 8 of this volume. • The former Chapters 10 and 11 have been merged in the new Chapter 9, devoted to the time integration schemes and to the general methodologies resulting from the combination of a selected space discretization with a separate time integration method. • Parts of the second volume have been transferred to the ﬁrst volume; in partic- ular sections on potential ﬂows (presented in Chapter 11) and two-dimensional viscous ﬂows in Chapter 12. This should allow the student already to come in contact, at this introductory CFD level, with initial applications of ﬂuid ﬂow simulations. • The number of problems has been increased and complete solution manuals will be made available to the instructors. Also a computer program for the numerical solutions of simple 1D convection and convection–diffusion equations, with a large variety of schemes and test cases can be made available to the instructors, for use in classes and exercises sessions. The objective of this option is to provide a tool allowing the students to develop their own ‘feeling’ and experience with various schemes, including assessment of the different types and level of errors generated by the combination of schemes and test cases. Many of the ﬁgures in the two volumes have been generated with these programs. The second group of elements is connected to the considerable evolution and exten- sion of Computational Fluid Dynamics (CFD) since the ﬁrst publication of these books. CFD is now an integral part of any ﬂuid-related research and industrial appli- cation, and is progressively reaching a mature stage. Its evolution, since the initial publication of this book, has been marked by signiﬁcant advancements, which we feel have to be covered, at least partly, in order to provide the reader with a reliable and up-to-date introduction and account of modern CFD. This relates in particular to: • Major developments of schemes and codes based on unstructured grids, which are today the ‘standard’, particularly with most of the commercial CFD packages, as unstructured codes take advantage of the availability of nearly automatic grid generation tools for complex geometries. • Advances in high-resolution algorithms, which have provided a deep insight in the general properties of numerical schemes, leading to a uniﬁed and elegant approach, where concepts of accuracy, stability, monotonicity can be deﬁned and applied to any type of equation. • Major developments in turbulence modeling, including Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES). • Applications of full 3D Navier–Stokes simulations to an extreme variety of com- plex industrial, environmental, bio-medical and other disciplines, where ﬂuids

FM-H6594.tex 8/5/2007 12: 31 Page xvii Preface xvii play a role in their properties and evolution. This has led to a considerable overall experience accumulated over the last decade, on schemes and models. • The awareness of the importance of veriﬁcation and validation of CFD codes and the development of related methodologies. This has given rise to the deﬁnition and evaluation of families of test cases including the related quality assessment issues. • The wide availability of commercial CFD codes, which are increasingly being used as teaching tools, to support the understanding of ﬂuid mechanics and/or to generate simple ﬂow simulations. This puts a strong emphasis on the need for educating students in the use of codes and providing them with an awareness of possible inaccuracies, sources of errors, grid and modeling effects and, more generally, with some global Best Practice Guidelines. Many of these topics will be found in the second edition of Volume II. I have beneﬁted from the spontaneous input from many colleagues and students, who have been kind enough to send me notices about misprints in text and in formulas, helping hereby in improving the quality of the books and correcting errors. I am very grateful to all of them. I also have to thank many of my students and researchers, who have contributed at various levels; in particular: Dr. Zhu Zong–Wen for the many problem solutions; Cristian Dinescu for various corrections. Benoit Tartinville and Dr. Sergey Smirnov have contributed largely to the calculations and derivations in Chapters 11 and 12. Brussels, December 2006

FM-H6594.tex 8/5/2007 12: 31 Page xviii Nomenclature a A c cp cv D e e ex,ey,ez E E f fe F ( f , g, h) g G h H I J k k M n n p P Pr q qH Q r R Re s S S t T u U convection velocity or wave speed Jacobian of ﬂux function speed of sound speciﬁc heat at constant pressure speciﬁc heat at constant volume ﬁrst derivative operator internal energy per unit mass vector (column matrix) of solution errors unit vectors along the x, y, z directions total energy per unit volume ﬁnite difference displacement (shift) operator ﬂux function external force vector ﬂux vector with components f , g, h gravity acceleration ampliﬁcation factor/matrix enthalpy per unit mass total enthalpy rothalpy Jacobian coefﬁcient of thermal conductivity wave number Mach number normal distance normal vector pressure convergence or conditioning operator Prandtl number non homogeneous term heat source source term; matrix of non homogeneous terms gas constant per unit mass residual of iterative scheme Reynolds number entropy per unit mass space discretization operator surface vector time temperature dependent variable vector (column matrix) of dependent variables xviii

FM-H6594.tex 8/5/2007 12: 31 Page xix Nomenclature xix vector of conservative variables; velocity velocity vector with components u, v, w eigenvectors of space discretization matrix relative velocity weight function cartesian coordinates ampliﬁcation factor of time integration scheme diffusivity coefﬁcient dimensionless diffusion coefﬁcient β= αt/x, also called Von Neumann number speciﬁc heat ratio circulation; boundary of domain central-difference operator forward and backward difference operators Laplace operator time step variation of solution U between levels n+ 1 and n spatial mesh size in x and y directions error of numerical solution turbulence dissipation rate dissipation or diffusion error dispersion error vorticity vector parameter controlling type of difference scheme wave-number vector; wave propagation direction eigenvalue of ampliﬁcation matrix coefﬁcient of dynamic viscosity averaging difference operator real part of ampliﬁcation matrix imaginary part of ampliﬁcation matrix density; spectral radius Courant number shear stress tensor stress tensor kinematic viscosity velocity potential; phase angle in Von Neumann analysis phase angle of ampliﬁcation factor time frequency of plane wave; overrelaxation parameters eigenvalue of space discretization matrix; volume U v (u, v, w) V w W x, y, z z α β − , δ γ δ + δ t U x, y ε εv εD εφ ζ κ θ λ µ µ ξ η ρ σ σ τ ν φ ω Subscripts e i, j I, J J external variable mesh point locations in x, y directions nodal point index eigenvalue number

FM-H6594.tex 8/5/2007 12: 31 Page xx xx Nomenclature min max n o v x, y, z ∞ minimum maximum normal or normal component stagnation values viscous term components in x, y, z directions; partial differentiation with respect to x, y, z freestream value Superscripts n iteration level; time level

Intro-H6594.tex 9/5/2007 11: 42 Page 1 Introduction: An Initial Guide to CFD and to this Volume Computational Fluid Dynamics, known today as CFD, is deﬁned as the set of methodologies that enable the computer to provide us with a numerical simulation of ﬂuid ﬂows. We use the word ‘simulation’ to indicate that we use the computer to solve numer- ically the laws that govern the movement of ﬂuids, in or around a material system, where its geometry is also modeled on the computer. Hence, the whole system is transformed into a ‘virtual’ environment or virtual product. This can be opposed to an experimental investigation, characterized by a material model or prototype of the system, such as an aircraft or car model in a wind tunnel, or when measuring the ﬂow properties in a prototype of an engine. This terminology is also referring to the fact that we can visualize the whole system and its behavior, through computer visualization tools, with amazing levels of realism, as you certainly have experienced through the powerful computer games and/or movie animations, that provide a fascinating level of high-ﬁdelity rendering. Hence the complete system, such as a car, an airplane, a block of buildings, etc. can be ‘seen’ on a computer, before any part is ever constructed. I.1 THE POSITION OF CFD IN THE WORLD OF VIRTUAL PROTOTYPING To situate the role and importance of CFD in our contemporary technological world, it might be of interest to take you down the road to the global world of Computer-Assisted Engineering or CAE. CAE refers to the ensemble of simulation tools that support the work of the engineer between the initial design phase and the ﬁnal deﬁnition of the manufacturing process. The industrial production process is indeed subjected to an accelerated evolution toward the computerization of the whole production cycle, using various software tools. The most important of them are: Computer-Assisted Design (CAD), Computer- Assisted Engineering (CAE) and Computer-Assisted Manufacturing (CAM) soft- ware. The CAD/CAE/CAM software systems form the basis for the different phases of the virtual prototyping environment as shown in Figure I.1.1. This chart presents the different components of a computer-oriented environment, as used in industry to create, or modify toward better properties, a product. This product can be a single component such as a cooling jacket in a car engine, formed by a certain number of circular curved pipes, down to a complete car. In all cases the succession of steps and the related software tools are used in very much similar ways, the difference being the degree of complexity to which these tools are applied. 1

Intro-H6594.tex 9/5/2007 11: 42 Page 2 2 Introduction: An Initial Guide to CFD and to this Volume Simulation and analysis phase Definition phase CFD Product Specification Shape Definition (CAD) Virtual Prototyping Simulation and Analysis (CAE) CAA CEM CSM Manufacturing phase Manufacturing Cycle (CAM) Figure I.1.1 The structure of the virtual prototyping environment. I.1.1 The Deﬁnition Phase The ﬁrst step in the creation of the product is the deﬁnition phase, which covers the speciﬁcation and geometrical deﬁnition. It is based on CAD software, which allows creating and deﬁning the geometry of the system, in all its details. Typically, large industries can employ up to thousands of designers, working full time on CAD software. Their day-to-day task is to build the geometrical model on the computer screen, in interaction with the engineers of the simulation and analysis departments. This CAD deﬁnition of the geometry is the required and unavoidable input to the CFD simulation task. Figure I.1.2 shows several examples of CAD deﬁnitions of different models, for which we will see later results of CFD simulations. These examples cover a very wide range of applications, industrial, environmental and bio-medical.

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