Background

In 1999, a major programme on turbulence was held at the Isaac Newton Institute (INI) at Cambridge, England, which was aimed at taking an overview of the current situation on turbulent flows with particular reference to the prediction of such flows in engineering systems. Though the programme spanned the range from the very fundamental to the applied, a very important feature was the involvement and support (through the UK Royal Academy of Engineering) of key players from industry. This volume, which has evolved from the INI programme, aims to address the needs of people in industry and academia who carry out calculations on turbulent systems.

It should be recognised that the prediction of turbulent flows is now of paramount importance in the development of complex engineering systems involving flow, heat and mass transfer and chemical reactions (including combustion). Whereas, in the past, the developer had to rely on experimental studies, based usually on small scale model systems, more and more emphasis is being placed nowadays on the use of computation, often through the use of commercial computational fluid dynamics (CFD) codes. Superficially, the use of such computational methods seems ideal; they allow painless extension to large scale and can often give information on fine details of the flow that are not economically accessible to experimental measurement. Furthermore, the results can be presented in an easily accessible and attractive form using the sophisticated computer graphics now generally available.

Summary

This chapter begins with a review of the principles underlying general purpose turbulence models and the assumptions and procedures involved in applying them to calculate the kind of complex flows that are analysed in practical engineering and environmental problems. Secondly we develop, from considerations of basic mechanisms of turbulence and the different types of statistical turbulence model, a new guideline ‘map’ based on characteristic statistical parameters, which can be derived from standard models. This indicates in principle which types of turbulent flow can and cannot be approximately calculated with the current generation of ‘CFD’, one-point turbulence models, including those using k–ε and second order closure equations. No attempt is made to identify any one optimum model scheme. Thirdly, the proposed guidelines for the likely accuracy of turbulent modelling are tested by comparing them with the results of previous test-case studies for a range of complex turbulent flows, where standard models fail or need special adaptation. These include thermal convection, free stream turbulence, aeronautical flows and flows round bluff bodies. The relative merits of advanced models (e.g. involving two-point statistics) and numerical simulations are also discussed, but the CFD practitioner should note that the emphasis here is on why current models will not work in all circumstances. The technical level of this chapter is most suitable for readers with some formal training in fluid dynamics. These general guidelines are complementary to user guidelines for computational fluid dynamics codes.

Abstract

Recent research is making progress in framing more precisely the basic dynamical and statistical questions about turbulence and in answering them. It is helping both to define the likely limits to current methods for modelling industrial and environmental turbulent flows, and to suggest new approaches to overcome these limitations. This chapter had its basis in the new results that emerged from more than 300 presentations during the programme held in 1999 at the Isaac Newton Institute, Cambridge, UK, and on research reported elsewhere. The objective of including this material (which is a revised form of an article which appeared in the Journal of Fluid Mechanics – Hunt et al., 2001) in the present volume is to give a background to the current state of the art. The emphasis is on the physics of turbulence and on how this relates to modelling. A general conclusion is that, although turbulence is not a universal state of nature, there are certain statistical measures and kinematic features of the small-scale flow field that occur in most turbulent flows, while the large-scale eddy motions have qualitative similarities within particular types of turbulence defined by the mean flow, initial or boundary conditions, and in some cases, the range of Reynolds numbers involved. The forced transition to turbulence of laminar flows caused by strong external disturbances was shown to be highly dependent on their amplitude, location, and the type of flow.

Introduction

Over 80% of the world's energy is generated by the combustion of hydrocarbon fuels, and this is likely to remain the case for the foreseeable future. In addition to the release of heat, this combustion is accompanied by the emission, in the exhaust stream, of combustion generated pollutants such as carbon monoxide, unburnt hydrocarbons and oxides of nitrogen, NOx. The former two quantities arise as a result of incomplete combustion, whereas NOx is formed from the reaction of nitrogen present in the air or fuel with oxygen, usually at high temperatures. An unavoidable outcome of the burning of hydrocarbon fuels is the formation of carbon dioxide, CO2, which is a ‘greenhouse’ gas that may contribute to global warming. While the amount of carbon dioxide generated depends on the fuel burnt, any improvements which can be achieved to combustion efficiency will clearly contribute to an overall reduction in the emissions of CO2. Because of the growing need to reduce the emissions of combustion generated pollutants and improve combustion efficiencies, there is increased interest in accurate methods for predicting the properties of combustion systems. The combustion in the vast majority of practical systems is turbulent and this poses a number of difficulties for prediction. The development of accurate methods for predicting turbulent combusting flows remains a largely unresolved problem which continues to attract a large number of researchers.

Introduction

Computer simulation of turbulent flows is becoming increasingly attractive due to the greater physical realism relative to conventional modelling, at a cost that is reducing with continuing advances in computer hardware and algorithms. The first turbulence simulations appeared over 30 years ago, since when we have seen increases in computer performance of over four orders of magnitude such that many of the canonical turbulent flows first studied by laboratory experiments can now be reliably simulated by computer. Examples include turbulent channels (Kim et al., 1987), turbulent boundary layers (Spalart, 1988), mixing layers (Rogers & Moser, 1994), subsonic and supersonic jets (Freund, 2001, Freund et al., 2000) and backward-facing steps (Le et al., 1997). Where simulations can be reliably made, they provide more data than are available from laboratory experiments, even with modern non-intrusive flow diagnostics. In these situations they provide insight into the basic fluid mechanics. This can be at a very simple flow visualisation level, where a conceptual picture of what is happening in a flow can be quickly obtained from computer animations of key features, or at more advanced levels where the simulations provide statistical data to assist Reynolds-averaged model development. Indeed, several important recent turbulence models have come out of groups who do both simulations and modelling, examples being the Spalart & Allmaras (1994) model and Durbin's K-ε-ν2 (Durbin, 1995), and it is rare to come across a turbulence modelling paper that has not used simulation data as a reference.

Introduction

The aim of this chapter is to provide, in plain English, a guide to the capabilities and shortcomings of turbulence models for reproducing satisfactorily engineering flows where buoyant or stratification effects are important. While these two descriptors are often used interchangeably in the literature, in the present chapter buoyant is used to denote a situation where the effect of gravity is to cause a force field whose primary effect is on the mean flow, while stratified implies that the principal effects on the flow arise from gravitational action on the turbulent fluctuating velocities. The distinction is neither pedantic nor unimportant; for, a stratified flow will ordinarily require a more rigorous modelling of gravitational effects than a buoyant flow. Put another way, gravitational effects on horizontal flows are more troublesome than on vertical flows. The account may hopefully also be useful where the flows of interest are affected by other types of force field, perhaps particularly flows affected by Coriolis forces or swirl.

The chapter gives especial attention to two-equation models of turbulence as this is currently the main level of commercial CFD. Linear two-equation eddyviscosity models are considered first, beginning with the situation where buoyant/stratification effects are absent. This is important to enable the reader to assess whether, for the flows of interest, a linear eddy viscosity model would be suitable even in a uniform density situation. Thereafter, the treatment of buoyant flows with linear two-equation models is considered.

Abstract

The problems concerning the understanding and prediction of strongly-distorted turbulent boundary layers are reviewed. Some of the views expressed emanate from the Isaac Newton Institute (INI) programme on turbulence in 1999; others are an experimentalist's view of current modelling techniques and the requirements for future work. The purpose of this chapter is, first, to pinpoint research that has already been carried out in this area and, second, to highlight gaps in our knowledge where there is a need for specific experiments to enable the subsequent development of existing turbulence models. Attention is not restricted to Reynolds-averaged Navier–Stokes (RANS) solvers but also considers the problems regarding the modelling for large-eddy simulation (LES). The subject of this chapter is vast, and therefore ample use is made of existing reviews by authors who are experts in specific subject areas. These include ‘extra rates of strain’, changes in boundary conditions, as well as even more complex phenomena such as shock/boundary-layer interaction and boundary layers with a variety of embedded vortices. While we have many of the numerical tools required for design and prediction, it is clear that physical understanding of the dominant mechanisms is lacking and therefore our ability to predict them is also. As far as our existing knowledge is concerned, emphasis is placed on an empirical approach for reasons of pragmatism. Some ‘application challenges’ presented during the course of the INI programme on turbulence are addressed.

Introduction

Computational modelling is assuming a greater and greater role in the study of multi-phase flows. Although it is not yet feasible to predict complex multi-phase flow fields over the full range of velocities and flow patterns, computational methods are helpful for a variety of reasons which include:

1. They enable insights to be obtained on the nature and relative importance of phenomena and are a natural aid to experimental measurement. Indeed, it is often possible to compute quantities which cannot be readily measured.

2. When coupled with experimental observations and empirical relationships, computational methods can give predictions which are reaching the stage of being useful in the design and operation of systems involving multi-phase flows, particularly for dispersed flow situations. This fact is reflected in the growing number of commercial computer codes which are available for application in this field.

In this chapter, we will deal first with the application of single-phase prediction methods in the interpretation of two-phase flows. Here, a brief description is given of the available turbulence models and examples cited of the application of this approach (flows in coiled tubes, horizontal annular flow and waves in annular flow).

An important class of two-phase flows is that where one of the phases is dispersed in the other, for example dispersions of bubbles in a liquid (bubble flow), dispersions of solid particles in a gas or liquid (gas–solids or liquid–solids dispersed flows) and dispersions of droplets of one liquid in another liquid (liquid–liquid dispersed flow).

Introduction

The theory of liquid sloshing dynamics in partially filled containers is based on developing the fluid field equations, estimating the fluid free-surface motion, and the resulting hydrodynamic forces and moments. Explicit solutions are possible only for a few special cases such as upright cylindrical and rectangular containers. The boundary value problem is usually solved for modal analysis and for the dynamic response characteristics to external excitations. The modal analysis of a liquid free-surface motion in a partially filled container estimates the natural frequencies and the corresponding mode shapes. The knowledge of the natural frequencies is essential in the design process of liquid tanks and in implementing active control systems in space vehicles. The natural frequencies of the free liquid surface appear in the combined boundary condition (kinematic and dynamic) rather than in the fluid continuity (Laplace's) equation.

For an open surface, which does not completely enclose the field, the boundary conditions usually specify the value of the field at every point on the boundary surface or the normal gradient to the container surface, or both. The boundary conditions may be classified into three classes (Morse and Fesbach, 1953):

1. the Dirichlet boundary conditions, which fix the value of the field on the surface;

2. the Neumann boundary conditions, which fix the value of the normal gradient on the surface; and

3. the Cauchy conditions, which fix both value of the field and normal gradient on the surface.

Each class is appropriate for different types of equations and different boundary surfaces.

Introduction

The previous chapter treated the free oscillations of liquid free surface in different container geometries. The fluid field equations were developed and the natural frequencies were determined from the free-surface boundary conditions. The knowledge of liquid-free-surface natural frequencies is important in the design of liquid containers subjected to different types of excitations. In the design process, it is important to keep the liquid natural frequencies away from all normal and nonlinear resonance conditions. The excitation can be impulsive, sinusoidal, periodic, or random. Its orientation with respect to the tank can be lateral, parametric, pitching/ yaw, roll, or a combination. Under forced excitation, it is important to determine the hydrodynamic loads acting on the container, and their phase with respect to the excitation. The hydrodynamic forces are estimated by integrating the pressure distribution over the wetted area. One should also determine the free-surface wave height, which affects the location of the center of mass. In the neighborhood of resonance, the free surface experiences different types of nonlinear behavior and this will be considered in Chapters 4, 6, and 7.

Under impulsive excitation, liquid sloshing dynamics was studied by Morris (1938), Werner and Sundquist (1949), Jacobsen (1949), Jacobsen and Ayre (1951), and Hoskins and Jacobsen (1957). Bauer (1965a) considered the response of liquid propellant to a single pulse excitation. Transient and steady state response to sinusoidal excitation of a liquid free surface was studied by Sogabe and Shibata (1974a, b).

Introduction

The linear theory of liquid sloshing is adequate for determining the natural frequencies and wave height of the free surface. Under translational excitation, the linear theory is also useful for predicting the liquid hydrodynamic pressure, forces, and moments as long as the free surface maintains a planar shape with a nodal diameter that remains perpendicular to the line of excitation. However, it does not take into account the important vertical displacement of the center of gravity of the liquid for large amplitudes of free-surface motion. It also fails to predict complex surface phenomena observed experimentally near resonance. These phenomena include nonplanar unstable motion of the free surface associated with rotation of the nodal diameter (rotary sloshing) and chaotic sloshing. These phenomena can be uncovered using the theory of weakly nonlinear oscillations for quantitative analysis and the modern theory of nonlinear dynamics for stability analysis. The main sources of nonlinearity in the fluid field equations appear in the free-surface boundary conditions.

The early work of nonlinear lateral sloshing is based on asymptotic expansion techniques. For example, Penney and Price (1952) carried out a successive approximation approach where the potential function is expressed as a Fourier series in space with coefficients that are functions of time. These coefficients are approximated by Fourier time series using the method of perturbation. The resulting solution is expressed in terms of a double Fourier series in space and time. This method was applied by Skalak and Yarymovich (1962) and Dodge, et al. (1965).

Our aim is now to introduce the concepts of real power and virtual power produced by forces and to present some applications. We first consider the very simple cases of a material point and of a system of material points (Section 4.1). We then study more complex situations (Section 4.2) before finally defining and studying the power of internal forces for a continuum medium in Section 4.3. This eventually leads to the virtual power theorem and to the kinetic energy theorem.

From the standpoint of mechanics, this chapter does not present much new material, but it gives very useful and different perspectives on the concepts and notions already introduced.

Study of a system of material points

Before considering the case of a system of material points, we start by considering that of a single material point. All that we say for a point or even for a system of points is simple and sometimes naive; it is, however, instructive.

The case of a material point

Definition 4.1.For a force F applied to a material point M, the (real) power produced by F at time t and for the given frame of reference is the scalar product F U (U being the velocity of M at time t in the considered frame of reference).