Computational Fluid Dynamics

Principles and Applications

Jiri Blazek

• Publisher: Elsevier Science
• Year: 2005
• ISBN:
  9780080529677
• Edition: 2
• Topic: Scienze fisiche
• Subtopic: Meccanica dei fluidi

Computational Fluid Dynamics (CFD) is an important design tool in engineering and also a substantial research tool in various physical sciences as well as in biology.

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. It is also intended for engineers and scientists starting to work in the field of CFD or for those who apply CFD codes. Due to the detailed index, the text can serve as a reference handbook too.

Each chapter includes an extensive bibliography, which provides an excellent basis for further studies.

Chapter 1

Introduction

The history of the Computational Fluid Dynamics, or CFD for short, started in the early 1970’s. Around that time, it became an acronym for the combination of physics, numerical mathematics, and, to some extent, computer sciences all 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 the 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 simulations 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).

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 very 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 the fidelity of flow simulations, grid generation methods became 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 the increasing geometrical complexity of the configurations, the grids had to be divided 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 imposed 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 the 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 improves not only 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 significantly reduced.

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, biology, oil recovery, and in architecture. Many numerical techniques developed for CFD are also used in the solution of the Maxwell equations or in aeroacoustics. Hence, CFD is becoming an increasingly important design tool in engineering, and also a substantial research tool in various sciences. Due to the advances in numerical solution methods and in the computer technology, geometrically and physically complex cases can be run even on PC’s or on PC clusters. Large scale simulations of viscous flows on grids consisting of dozens of millions of elements can be accomplished within only a few hours on today’s supercomputers. However, it would be completely wrong to think that CFD represents a mature technology now, like for example the finite element methods in solid mechanics. 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. The coupling between CFD and other disciplines (like the solid mechanics) requires further research as well. 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 fluid motion as well as with the interaction of the fluid with solid bodies. The equations governing the motion of an inviscid fluid (Euler equations) and of viscous fluid (Navier-Stokes equations) are derived in Chapter 2. 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 the spatial discretisation of the Euler and the Navier-Stokes equations. A unique feature of the present book is that it deals with both the structured (Chapter 4) as well as with the unstructured finite volume schemes (Chapter 5), because of their broad application possibilities, especially for the treatment of complex flow problems routinely encountered in an industrial environment. The attention is particularly devoted to the definition of the 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.

The methodologies for the temporal discretisation of the governing equations can be divided into two classes. One class comprises explicit time-stepping schemes (Section 6.1), and the other one consists of implicit 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 the explicit 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 challenge 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 enable 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.

In order to account for 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. Basically, there are two types of boundary conditions: physical and numerical. Chapter 8 deals with both types for different situations like solid walls, inlet, outlet, injection and farfield. Symmetry planes, periodic and block boundaries are treated as well.

In order to reduce the computer 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 the 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, the third part of Chapter 10 deals with stability analysis for various model equations.

One of the 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 depth.

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 and Navier-Stokes structured and unstructured flow solvers. Furthermore, source codes of 2-D structured algebraic and elliptic grid generators are included together with a converter from structured to unstructured grids. Furthermore, two programs are provided to conduct the linear stability analysis of explicit and implicit time-stepping schemes. The source codes are completed by a set of worked out examples including the grids, the input files and the results. The CD-ROM also contains the source code of a visualisation tool with an easy-to-use GUI. 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 a 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.

Chapter 2

Governing Equations

2.1 The Flow and its Mathematical Description

Before we begin with the derivation of the basic equations describing the behaviour of the fluid, it may be convenient to clarify what the term ‘fluid dynamics’ stands for. It is, in fact, the investigation of the interactive motion of a large number of individual particles. These are in our case molecules or atoms. That means, we assume the density of the fluid is high enough, so that it can be approximated as a continuum. It implies that even an infinitesimally small (in the sense of differential calculus) element of the fluid still contains a sufficient number of particles, for which we can specify mean velocity and mean kinetic energy. In this way, we are able to define velocity, pressure, temperature, density and other important quantities at each point of the fl...