This chapter provides an insight into the physical nature of turbulence and the mathematical framework that is used in numerical simulations of turbulent flows. The aim is to explain why turbulence must be modelled and how turbulence can be modelled, and also to explain what is modelled with different turbulence models. In addition, the limitations of the turbulence models are discussed. The intention is to give you such an understanding of turbulence modelling that you can actually suggest appropriate turbulence models for different kinds of turbulent flows depending upon their complexity and the required level of description.

The physics of fluid turbulence

Turbulence is encountered in most flows in nature and in industrial applications. Natural turbulent flows can be found in oceans, in rivers and in the atmosphere, whereas industrial turbulent flows can be found in heat exchangers, chemical reactors etc. Most flows encountered in industrial applications are turbulent, since turbulence significantly enhances heat- and mass-transfer rates. In industry a variety of turbulent multiphase flows can be encountered. Turbulence plays an important role in these types of flows since it affects processes such as break-up and coalescence of bubbles and drops, thereby controlling the interfacial area between the phases. Thus, turbulence modelling becomes one of the key elements in CFD.

The purpose of this chapter is to explain the input needed to solve CFD problems, e.g. CAD geometry, computational mesh, material properties, boundary conditions etc. The difficulty and accuracy of CFD simulations for various applications, such as laminar and turbulent flows, single-phase and multiphase flows and reactive systems are discussed briefly.

Modelling in engineering

Traditional modelling in engineering is heavily based on empirical or semi-empirical models. These models often work very well for well-known unit operations, but are not reliable for new process conditions. The development of new equipment and processes is dependent on the experience of experts, and scaling up from laboratory to full scale is very time-consuming and difficult. New design equations and new parameters in existing models must be determined when changing the equipment or the process conditions outside the validated experimental database. A new trend is that engineers are increasingly using computational fluid dynamics (CFD) to analyse flow and performance in the design of new equipment and processes. CFD allows a detailed analysis of the flow combined with mass and heat transfer. Modern CFD tools can also simulate transport of chemical species, chemical reactions, combustion, evaporation, condensation and crystallization.

Computational fluid dynamics (CFD) has become an indispensable tool for engineers. CFD simulations provide insight into the details of how products and processes work, and allow new products to be evaluated in the computer, even before prototypes have been built. It is also successfully used for problem shooting and optimization. The turnover time for a CFD analysis is continuously being reduced since computers are becoming ever more powerful and software uses ever more efficient algorithms. Low cost, satisfactory accuracy and short lead times allow CFD to compete with building physical prototypes, i.e. ‘virtual prototyping’.

There are many commercial programs available, which have become easy to use, and with many default settings, so that even an inexperienced user can obtain reliable results for simple problems. However, most applications require a deeper understanding of fluid dynamics, numerics and modelling. Since no models are universal, CFD engineers have to determine which models are most appropriate to the particular case. Furthermore, this deeper knowledge is required since it gives the skilled engineer the capability to judge the potential lack of accuracy in a CFD analysis. This is important since the analysis results are often used to make decisions about what prototypes and processes to build.

The purpose of this chapter is to give an introduction to problems faced by engineers wanting to use CFD for detailed modelling of turbulent reactive flows. After reading this chapter you should be able to describe the physical process of turbulent mixing and know why this can have an effect on the outcome of chemical reactions, e.g. combustion. The problem arises when the grid and time resolution is not sufficient to resolve the concentration and the average concentration in the cells is a poor estimation of the actual concentration as shown in Figure 5.1. The local concentration changes fast, and we need models that can predict the space- and time-average reaction rate in each computational cell.

The average concentration in a computational cell can be used to describe macromixing (large-scale mixing) in the reactor and is relatively straightforward to model. The concentration fluctuations, on the other hand, can be used to describe micromixing (small-scale mixing on the molecular level). To quantify micromixing, the variance of the concentration fluctuations is used. Chemical reactions can take place only at the smallest scales of the flow, after micromixing has occurred, because reactions occur only as molecules meet and interact. An expression for the instantaneous rate of chemical reactions is often known for homogeneous mixtures. However, the average rate of chemical reactions in a reactor subject to mixing will depend also on the rate of micromixing.

Computational fluid dynamics does not provide an exact solution to all problems, but is in many cases a reliable tool that can provide useful results when it is employed by an experienced user. An inexperienced user, on the other hand, may obtain very nice graphs that are very far from being a prediction of the stated problem. Some of the problems arise from the many default settings in commercial CFD codes, since the user may obtain results without knowing what the code is doing by accepting settings that are not appropriate for the specific problem. The user must make an active decision regarding each setting due to the fact that many problems can arise from a user failing to understand what the proper settings should be. This chapter provides some guidelines that can help a new user to avoid the most common mistakes. Many more recomendations selected by experienced CFD users can be found in the ‘Best Practice Guidelines’ for single-phase flows [20] and for dispersed multiphase flows [21] by the European Research Community on Flow Turbulence and Combustion (ERCOFTAC).

A CFD simulation contains both errors and uncertainties. An error is defined as a recognizable deficiency that is not due to a lack of knowledge, whereas an uncertainty is a potential deficiency that is due to a lack of knowledge.

Introduction

The flow considered in this chapter is assumed to be steady, incompressible, inviscid, and irrotational. The body immersed in the flow is assumed to be a body of revolution at zero angle of attack. An understanding of incompressible flow around bodies of revolution at zero or small angle of attack is important in several practical applications, including airships, aircraft and cruise-missile fuselages, submarine hulls, and torpedoes, as well as flows around aircraft engine nacelles and inlets. This type of flow problem is best handled in cylindrical coordinates (x, r), as shown in Fig. 7.1. Recall that r and θ lie in the y-z plane.

Because the flow fields discussed in this chapter are axisymmetric, the flow properties depend on only the axial distance x from the nose of the body (assumed to be at the origin in most cases) and the radial distance, r, away from this axis of symmetry. The flow properties are independent of the angle θ. As a result, we may examine the flow in any (x-r) plane because the flow in all such planes is identical due to the axial symmetry. It is convenient to develop the defining equations initially in cylindrical coordinates (i.e., dependence on x, r, and θ) and then to simplify them for axisymmetric flow (i.e., dependence on x, r only).

Preface

This textbook presents the fundamentals of aerodynamic analysis. Major emphasis is on inviscid flows whenever this simplification is appropriate, but viscous effects also are discussed in more detail than is usually found in a textbook at this level. There is continual attention to practical applications of the material. For example, the concluding chapter demonstrates how aerodynamic analysis can be used to predict and improve the performance of flight vehicles. The material is suitable for a semester course on aerodynamics or fluid mechanics at the junior/senior undergraduate level and for first-year graduate students. It is assumed that the student has a sound background in calculus, vector analysis, mechanics, and basic thermodynamics and physics. Access to a digital computer is required and an understanding of computer programming is desirable but not necessary. Computational methods are introduced as required to solve complex problems that cannot be handled analytically.

The objective of this textbook is to present in a clear and orderly manner the basic concepts underlying aerodynamic-prediction methodology. The ultimate goal is for the student to be able to use confidently various solution methods in the analysis of practical problems of current and future interest. Today, it is important for the student to master the basics because technology is advancing at such a rate that a more directed or specific approach often is rapidly outdated. In this book, the basic concepts are linked closely to physical principles so that they may be understood and retained and the limits of applicability of the concepts can be appreciated. Numerous example problems stress solution methods and the order of magnitude of key parameters. A comprehensive set of problems for home study is included at the end of each chapter.

Introduction

This chapter examines the role of viscosity in the flow of fluids and gases. Although the viscosity of air is small, it must be included in a flow model if we are to explain wing stall and frictional drag, for example. The four preceding chapters are concerned with the analysis of airfoils, wings, and bodies of revolution based on an assumption of inviscid flow (i.e., negligible viscous effects). The inviscid-flow model allowed analytical solutions to be developed for predicting, with satisfactory accuracy, the pressure distribution on bodies of small-thickness ratio at a modest (or zero) angle of attack. However, the inviscid-flow model leads to results that are at odds with experience, such as the prediction that the drag of two-dimensional airfoils and right-circular cylinders is zero. This contradiction is resolved by realizing that actual flows exhibit viscous effects.

Viscosity is discussed from a physical viewpoint in Chapter 2. In Chapters 5, 6, and 7, the existence of viscosity is acknowledged when it is necessary to advance an analytical derivation for an inviscid flow. Also, viscous effects are called on, with words like viscous drag and separation, when comparing the predicted and observed behavior of airfoils and wings. However, no analysis in this textbook has been developed thus far that provides the required detailed physical basis for these effects.

Introduction

Aerodynamics is the study of the flow of air around and within a moving object. Its main objective is understanding the creation of forces by the interaction of the gas motion with the surfaces of an object. Aerodynamics is closely related to hydrodynamics and gasdynamics, which represent the motion of liquid and compressible-gas flows, respectively.

Aerodynamics is the essence of flight and has been the focus of intensive research for about a century. Although this might seem to be a rather long period of development, it is really quite short considering the time span usually required for the formulation and full solution of basic scientific problems. In this relatively short time, mankind has advanced from the first gliding and primitive-powered airplane flights to interplanetary spaceflight.

Aerodynamic Forces

Because the objective of aerodynamics is the determination of forces acting on a flying object, it is necessary that we clearly identify their source. Lift and drag forces, for example, are the result of interactions between the airflow and vehicle surfaces. Part of the force must be a result of pressure variations from point to point along the surface; another part must be related to friction of gas particles as they scrub the surface. Clearly, the key to understanding these forces is found in details of the fluid motions. The application of simple molecular concepts provides considerable insight into these motions.

Modeling of Gas Motion

As a branch of fluid mechanics, aerodynamics is concerned with the motion of a continuously deformable medium. That is, when acted on by a constant shear force, a body of liquid or gas changes shape continuously until the force is removed. This is unlike a solid body, which only deforms until internal stresses come into equilibrium with the applied force; that is, a solid does not deform continuously.

Introduction

The preceding eight chapters take wholesale advantage of the assumption that the flow field for low-speed flight is incompressible. This allows considerable simplification in the formulation of the governing equations and in the solution of key aerodynamic problems. However, results of the calculations are limited in an important way that is emphasized in this summary chapter. What we attempt to do here is:

1. Summarize the most important elements of the first eight chapters.

2. Demonstrate how the results are incorporated in actual vehicle design.

3. Define the limits of application of the results.

Modeling of Airflows

What is accomplished to this point is the application of basic fluid mechanics in contructing detailed models for the airflow over aerodynamic surfaces (e.g., wings and bodies) at speeds low enough that compressibility effects do not seriously affect the results. These models are intended to provide accurate estimates of the aerodynamic forces and moments needed in solving the basic problem of aerodynamics as it was defined in Chapter 1. Although there is much discussion centered on the application of modern computational tools, for the most part, we rely on simplified mathematical representations. We try to emphasize the role of valid, simplifying assumptions in arriving at useful representations for the airflow. As Küchemann described the process in his famous book on the aerodynamic design of aircraft (Küchemann, 1978), “... the most drastic simplifying assumptions must be made before we can even think about the flow of gases and arrive at equations which are amenable to treatment. Our whole science lives on highly idealized concepts and ingenious abstractions and approximations.” First-class examples of this approach are demonstrated in this book, including Prandtl's elegant models describing the creation of lift by an airfoil, three-dimensional wing theory, and boundary-layer flows. These provide the backbone of the subject of aerodynamics.