Acknowledgements.................................................................xi List of Symbols..................................................................xiii Abbreviations....................................................................xix 1 Introduction...................................................................1 2 Governing Equations............................................................5 3 Principles of Solution of the Governing Equations..............................29 4 Spatial Discretisation: Structured Finite Volume Schemes.......................75 5 Spatial Discretisation: Unstructured Finite Volume Schemes.....................129 6 Temporal Discretisation........................................................181 7 Turbulence Modelling...........................................................225 8 Boundary Conditions............................................................267 9 Acceleration Techniques........................................................299 10 Consistency, Accuracy and Stability...........................................331 11 Principles of Grid Generation.................................................353 12 Description of the Source Codes...............................................393 A Appendix.......................................................................401 Index............................................................................435
The history of Computational Fluid Dynamics, or CFD for short, started in the early 1970's. Around that time, it became an acronym for a combination of physics, numerical mathematics, and, to some extent, computer sciences employed to simulate fluid flows. The beginning of CFD was triggered by the availability of increasingly more powerful mainframes and the advances in CFD are still tightly coupled to the evolution of computer technology. Among the first applications of the CFD methods was the simulation of transonic flows based on the solution of the non-linear potential equation. With the beginning of the 1980's, the solution of first two-dimensional (2-D) and later also three-dimensional (3-D) Euler equations became feasible. Thanks to the rapidly increasing speed of supercomputers and due to the development of a variety of numerical acceleration techniques like multigrid, it was possible to compute inviscid flows past complete aircraft configurations or inside of turbomachines. With the mid 1980's, the focus started to shift to the significantly more demanding simulation of viscous flows governed by the Navier-Stokes equations. Together with this, a variety of turbulence models evolved with different degree of numerical complexity and accuracy. The leading edge in turbulence modelling is represented by the Direct Numerical Simulation (DNS) and the Large Eddy Simulation (LES). However, both approaches are still far away from being usable in engineering applications.
With the advances of the numerical methodologies, particularly of the implicit schemes, the solution of flow problems which require real gas modelling became also feasible by the end of 1980's. Among the first large scale application, 3-D hypersonic flow past re-entry vehicles, like the European HERMES shuttle, was computed using equilibrium and later non-equilibrium chemistry models. Many research activities were and still are devoted to the numerical simulation of combustion and particularly to flame modelling. These efforts are quite important for the development of low emission gas turbines and engines. Also the modelling of steam and in particular of condensing steam became a key for the design of efficient steam turbines.
Due to the steadily increasing demands on the complexity and fidelity of flow simulations, grid generation methods had to become more and more sophisticated. The development started first with relatively simple structured meshes constructed either by algebraic methods or by using partial differential equations. But with increasing geometrical complexity of the configurations, the grids had to be broken into a number of topologically simpler blocks (multiblock approach). The next logical step was to allow for non-matching interfaces between the grid blocks in order to relieve the constraints put on the grid generation in a single block. Finally, solution methodologies were introduced which can deal with grids overlapping each other (Chimera technique). This allowed for example to simulate the flow past the complete Space Shuttle vehicle with external tank and boosters attached. However, the generation of a structured, multiblock grid for a complicated geometry may still take weeks to accomplish. Therefore, the research also focused on the development of unstructured grid generators (and flow solvers), which promise significantly reduced setup times, with only a minor user intervention. Another very important feature of the unstructured methodology is the possibility of solution based grid adaptation. The first unstructured grids consisted exclusively of isotropic tetrahedra, which was fully sufficient for inviscid flows governed by the Euler equations. However, the solution of the Navier-Stokes equations requires for higher Reynolds numbers grids, which are highly stretched in the shear layers. Although such grids can also be constructed from tetrahedral elements, it is advisable to use prisms or hexahedra in the viscous flow regions and tetrahedra outside. This not only improves the solution accuracy, but it also saves the number of elements, faces and edges. Thus, the memory and run-time requirements of the simulation are reduced. In fact, today there is a very strong interest in unstructured, mixed-element grids and the corresponding flow solvers.
Nowadays, CFD methodologies are routinely employed in the fields of aircraft, turbomachinery, car, and ship design. Furthermore, CFD is also applied in meteorology, oceanography, astrophysics, in oil recovery, and also in architecture. Many numerical techniques developed for CFD are used in the solution of Maxwell equations as well. Hence, CFD is becoming an increasingly important design tool in engineering and also a substantial research tool in certain physical sciences. Due to the advances in numerical solution methods and computer technology, geometrically complex cases, like those which are often encountered in turbomachinery, can be treated. Also, large scale simulations of viscous flows can be accomplished within only a few hours on today's supercomputers, even for grids consisting of dozens of millions of grid cells. However, it would be completely wrong to think that CFD represents a mature technology now, like for example structural finite element methods. No, there are still many open questions like turbulence and combustion modelling, heat transfer, efficient solution techniques for viscous flows, robust but accurate discretisation methods, etc. Also the connection of CFD with other disciplines (like structural mechanics or heat conduction) requires further research. Quite new opportunities also arise in the design optimisation by using CFD.
The objective of this book is to provide university students with a solid foundation for understanding the numerical methods employed in today's CFD and to familiarise them with modern CFD codes by hands-on experience. The book is also intended for engineers and scientists starting to work in the field of CFD or who are applying CFD codes. The mathematics used is always connected to the underlying physics to facilitate the understanding of the matter. The text can serve as a reference handbook too. Each chapter contains an extensive bibliography, which may form the basis for further studies.
CFD methods are concerned with the solution of equations of motion of the fluid as well as with the interaction of the fluid with solid bodies. The equations of motion of an inviscid fluid (Euler equations) and of viscous fluid (Navier-Stokes equations), the so-called governing equations, are formulated in Chapter 2 in integral form. Additional thermodynamic relations for a perfect gas as well as for a real gas are also discussed. Chapter 3 deals with the principles of solution of the governing equations. The most important methodologies are briefly described and the corresponding references are included. Chapter 3 can be used together with Chapter 2 to get acquainted with the fundamental principles of CFD.
A series of different schemes was developed for an efficient solution of the Euler and the Navier-Stokes equations. A unique feature of the present book is that it deals with both structured (Chapter 4) as well as unstructured finite volume schemes (Chapter 5), because of their broad application possibilities, especially for the treatment of complex flow problems routinely encountered in industrial environment. Attention is particularly devoted to the definition of various types of control volumes together with spatial discretisation methodologies for convective and viscous fluxes. The 3-D finite volume formulations of the most popular central and upwind schemes are presented in detail.
Within the framework of the finite volume schemes, it is possible either to integrate the unsteady governing equations with respect to time (referred to as time-stepping schemes) or to solve the steady-state governing equations directly. The time-stepping can be split up into two classes. One class comprises explicit time-stepping schemes (Section 6.1), and the other consists of implicit time-stepping schemes (Section 6.2). In order to provide a more complete overview, recently developed solution methods based on the Newton-iteration as well as standard techniques like Runge-Kutta schemes are discussed.
Two qualitatively different types of viscous fluid flows are encountered in general: laminar and turbulent. The solution of the Navier-Stokes equations does not raise any fundamental difficulties in the case of laminar flows. However, the simulation of turbulent flows continues to present a significant problem as before. A relatively simple way of modelling the turbulence is offered by the so-called Reynolds-averaged Navier-Stokes equations. On the other hand, Reynolds stress models or LES allow considerably more accurate predictions of turbulent flows. In Chapter 7, various well-proven and widely applied turbulence models of varying level of complexity are presented in detail.
To take into account the specific features of a particular problem, and to obtain an unique solution of the governing equations, it is necessary to specify appropriate boundary conditions. There are basically two types of boundary conditions: physical and numerical. Chapter 8 deals with both types for different situations like solid walls, inlet, outlet and farfield. Symmetry planes, periodic and block boundaries are treated as well.
In order to shorten the time required to solve the governing equations for complex flow problems, it is quite essential to employ numerical acceleration technique. Chapter 9 deals extensively, among others, with approaches like implicit residual smoothing and multigrid. Another important technique, which is also described in Chapter 9 is preconditioning. It allows to use the same numerical scheme for flows, where the Mach number varies between nearly zero and transonic or higher values.
Each discretisation of the governing equations introduces a certain error — the discretisation error. Several consistency requirements have to be fulfilled by the discretisation scheme in order to ensure that the solution of the discretised equations closely approximates the solution of the original equations. This problem is addressed in the first two parts of Chapter 10. Before a particular numerical solution method is implemented, it is important to know, at least approximately, how the method will influence the stability and the convergence behaviour of the CFD code. It was frequently confirmed that the Von Neumann stability analysis can provide a good assessment of the properties of a numerical scheme. Therefore, in the third part of Chapter 10 it is dealt with stability analysis for various model equations.
One of the more challenging tasks in CFD is the generation of structured or unstructured body-fitted grids around complex geometries. The grid is used to discretise the governing equations in space. The accuracy of the flow solution is therefore tightly coupled to the quality of the grid. In Chapter 11, the most important methodologies for the generation of structured as well as unstructured grids are discussed.
In order to demonstrate the practical aspects of different numerical solution methodologies, various source codes are provided on the accompanying CD-ROM. Contained are the sources of quasi 1-D Euler as well as of 2-D Euler structured and unstructured flow solvers, respectively. Furthermore, source codes of 2-D structured algebraic and elliptic grid generators are included together with a convertor from structured to unstructured grids. Additionally, two programs are provided to conduct linear stability analysis of explicit and implicit time-stepping schemes. The source codes are completed by a set of worked out examples containing the grids, the input flies and the results. All source codes are written in standard FORTRAN-77. Chapter 12 describes the contents of the CD-ROM and the capabilities of the particular programs.
The present book is finalised by the Appendix and the Index. The Appendix contains the governing equations presented in differential form as well as their characteristic properties. Formulations of the governing equations in rotating frame of reference and for moving grids are discussed along with some simplified forms. Furthermore, Jacobian and transformation matrices from conservative to characteristic variables are presented for two and three dimensions. The GMRES conjugate gradient method for the solution of linear equations systems is described next. The Appendix closes with the explanation of the tensor notation.
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