The World Ocean, considered as an active dynamical system, is in permanent motion. Most of its manifestations can be related to wave phenomena. Besides the well known surface waves, there are also waves of other nature; among these, internal gravity waves are particularly important. They exist due to the presence of vertical fluid stratification, they are permanently generated, and they evolve and are destroyed again in the deep ocean. The amplitudes of internal waves are usually much larger than those of surface waves, due to the weak returning force, and their amplitudes can sometimes reach values of 100 m and more (see refs. [5], [23], [120], [128], [180], [192], and [193]).

Numerous in situ measurements, carried out in all regions of the World Ocean (see, for instance, refs. [63], [119], [123], and [157]), have shown that internal gravity waves exist wherever a stable vertical stratification of a fluid is observed. They were discovered more than 100 years ago, and were understood by the scientists of the day to be a disappointing obstacle disturbing the “correct” structure and dynamics of the oceanic water masses. More than half a century passed before the importance of internal waves to the global dynamics of the ocean was realized.

This chapter is devoted to a study of the dynamics of infinitesimal waves in a continuously stratified ocean of variable depth. Small-amplitude baroclinic tides are usually generated when the intensity of the barotropic tidal forcing is small. As mentioned earlier, depending on the wave amplitude, the analysis can be carried out by means of either the linear or nonlinear theory. A quantitative estimation of the efficiency of the tidal generation of internal waves and the discussion of the necessity to use the full nonlinear system of equations to study baroclinic tides is given in refs. [73] and [75]. We return to this latter point in detail in Chapter 4. Here we simply assume that the amplitudes of the considered waves are so small that with sufficient accuracy the advective terms in the governing system (1.39) can be neglected.

Historically, the first linear models of baroclinic tides were developed for a two-layer ocean. In the models in refs. [199] and [265], the internal tide was generated as a result of the interaction of the barotropic tidal wave with a Heaviside-like bottom step, an obstacle, which approximates the transition zone between the deep and shallow parts of the ocean. In these works, reflection of the waves from the coastal line was used as a boundary condition.

In the previous chapters we have only considered the two-dimensional nature of baroclinic tides. This simplification of the three-dimensional behavior is a valid approximation in shelf zones, at least as a tendency, due to the oblongness of the continental margins. It is, however, often violated when one tries to study the wave-generation process near three-dimensional bottom features like oceanic banks or abrupt changes of the shelf break topography in the along shore direction. Such places constitute a remarkable sink of barotropic tidal energy into baroclinic wave components if only a substantial along slope tidal flux interacts with along shore bottom variations. Under such conditions, the correct prediction of the total scattering of the barotropic tidal energy into internal tides is impossible without using three-dimensional global tidal models, which can predict the dynamics in those shelf areas where such a remarkable along slope forcing of internal tides may exist. Figure 7.1 illustrates this point. The broad gray bands show regions where the tide, derived from the barotropic model of Schwiderski [214], propagates along the coast as a Kelvin-type wave, and where the amplitude of its elevation is larger than 0.4 m. If the depth of the ocean near shore is estimated typically as 2000 m, then the value of the maximum barotropic tidal flux along the slope is of the order of 60 m2 s-1.

In this chapter we consider the evolutionary stage of baroclinic tides, i.e. the behavior of the tidally generated internal waves beyond the source of generation. During propagation, the long baroclinic tidal waves which are radiated from this area of generation – usually bottom features – are subjected to nonlinear effects, as shown in Chapter 4. If the nonlinearity is sufficiently strong, these waves are usually transformed into a sequence of solitary internal waves or wave trains. So, we shall now concentrate on the dynamic structure of solitary internal waves. First, we will give a brief summary of a number of analytical theories and consider the stationary solutions of weakly nonlinear equations. Then, we will move on to consider strong solitary internal waves; even though any analytical theory fails to describe their structure and dynamics, nevertheless they are a common feature of the real ocean. Using several numerical procedures, the governing equations can be handled and a physical understanding can be extracted. Using this approach, we will study the spatial–temporal structure of strong waves and indicate their differences from strict analytical solutions of equations describing weaker nonlinear interactions. Finally, we will present the wave transformation appropriate over variable bottom topography that includes strong effects such as wave overturning and breaking.

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.