The fact of turbulent flow

Man has evolved within a world where air and water are, by far, the most common fluids encountered. The scales of the environment around him and of the machines and artefacts his ingenuity has created mean that, given their relatively low kinematic viscosities, the relevant global Reynolds number, Re, associated with the motion of both fluids is, in most cases, sufficiently high that the resultant flow is of the continually time-varying, spatially irregular kind we call turbulent.

If, however, our Reynolds number is chosen not by the overall physical dimension of the body of interest – an aircraft wing, say – and the fluid velocity past it but by the smallest distance over which the velocity found within a turbulent eddy changes appreciably and the time over which such a velocity change will occur, its value then turns out to be of order unity. Indeed, one might observe that if this last Reynolds number, traditionally called the micro-scale Reynolds number, Reη, were significantly greater than unity, the rate at which the turbulent kinetic energy is destroyed by viscous dissipation could not balance the overall rate at which turbulence ‘captures’ kinetic energy from the mean flow.

The nature of viscous and wall effects: options for modelling

The turbulence models considered in earlier chapters were based on the assumption that the turbulent Reynolds numbers were high enough everywhere to permit the neglect of viscous effects. Thus, they are not applicable to flows with a low bulk Reynolds number (where the effects of viscosity may permeate the whole flow) or to the viscosity-affected regions adjacent to solid walls (commonly referred to as the viscous sublayer and buffer regions but which we shall normally collectively refer to as the viscous region) which always exist on a smooth wall irrespective of how high the bulk Reynolds number may be. In other words, while at high Reynolds number, viscous effects on the energy-containing turbulent motions are indeed negligible throughout most of the flow, the condition of no-slip at solid interfaces always ensures that, in the immediate vicinity of a wall, viscous contributions will be influential, perhaps dominant. Figure 6.1 shows the typical ‘layered’ composition for a near-wall turbulent flow (though with an expanded scale for the near-wall region) as found in a constant-pressure boundary layer, channel or pipe flow. Although the thickness of this viscosity-affected zone is usually two or more orders of magnitude less than the overall width of the flow (and decreases as the Reynolds number increases), its effects extend over the whole flow field since, typically, half of the velocity change from the wall to the free stream occurs in this region.

Because viscosity dampens velocity fluctuations equally in all directions, one may argue that viscosity has a ‘scalar’ effect. However, turbulence in the proximity of a solid wall or a phase interface is also subjected to non-viscous damping arising from the impermeability of the wall and the consequent reflection of pressure fluctuations. This ‘wall-blocking’ effect, which is also felt outside the viscous layer well into the fully turbulent wall region, directly dampens the velocity fluctuations in the wall-normal direction and thus it has a ‘vector’ character. A good illustration of this effect is the reduction of the surface-normal velocity fluctuations that has been observed in flow regions close to a phase interface, where there are no viscous effects, for example the DNS of Perot and Moin (1995).

Scientific papers on how to represent in mathematical form the types of fluid motion we call turbulent flow have been appearing for over a century while, for the last sixty years or so, a sufficient body of knowledge has been accumulated to tempt a succession of authors to collect, systematize and distil a proportion of that knowledge into textbooks. From the start a bewildering variety of approaches has been advocated: thus, even in the 1970s, the algebraic mixing-length models presented in the book by Cebeci and Smith jostled on the book-shelves with Leslie's manful attempt to make comprehensible to a less specialized readership the direct-interaction approach developed by Kraichnan and colleagues. As the progressive advance in computing power made it possible to apply the emerging strategy of computational fluid dynamics to an ever-widening array of industrially important flows, however, eddy-viscosity models (EVMs) based on the solution of two transport equations for scalar properties of turbulence (essentially, length and time scales of the energy-containing eddies) emerged as the modelling strategy of choice and, correspondingly, have been the principal focus in several textbooks on the modelling of turbulent flows (for example, Launder and Spalding, Wilcox and Piquet).

Today, two-equation EVMs remain the work-horse of industrial CFD and are applied through commercially marketed software to flows of a quite bewildering complexity, though often with uncertain accuracy. However, there has been a major shift among the modelling research community to abandon approaches based on the Reynolds-averaged Navier–Stokes (RANS) equations in favour of large-eddy simulation (LES) where the numerical solution for any flow adopts a three-dimensional, time-dependent discretization of the Navier–Stokes equations using a model to account simply for the effects of turbulent motions too fine in scale to be resolved with the mesh adopted – that is, a sub-grid-scale (or sgs) model. While acknowledging that LES offers the prospects of tackling turbulence problems beyond the scope of RANS, a further major driver for this changeover has been the manifold inadequacies of the stress-strain hypothesis adopted by linear eddy-viscosity models. While such a simple linkage between mean strain rate and turbulent stress seemed adequate for a large proportion of two-dimensional, nearly parallel flows, its weaknesses became abundantly clear as attention shifted to recirculating, impinging and three-dimensional shear flows. Although an LES approach will, most probably, also adopt an sgs model of eddy-viscosity type, the consequences are less serious for two reasons. First, the majority of the transport caused by the turbulent motion will be directly resolved by the large eddies and secondly, the finer scale eddies that must still be resolved by the sub-grid-scale model of turbulence will arguably be a good deal closer to isotropy. Thus, adopting an isotropic eddy viscosity as the sgs model may not significantly impair the accuracy of the solution.

Overview

Thermodynamics is unquestionably the most powerful and most elegant of the engineering sciences. Its power arises from the fact that it can be applied to any discipline, technology, application, or process. The origins of thermodynamics can be traced to the development of the steam engine in the 1700's, and thermodynamic principles do govern the performance of these types of machines. However, the power of thermodynamics lies in its generality. Thermodynamics is used to understand the energy exchanges accompanying a wide range of mechanical, chemical, and biological processes that bear little resemblance to the engines that gave birth to the discipline. Thermodynamics has even been used to study the energy exchanges that are involved in nuclear phenomena and it has been helpful in identifying sub-atomic particles. The elegance of thermodynamics is the simplicity of its basic postulates. There are two primary ‘laws’ of thermodynamics, the First Law and the Second Law, and they always apply with no exceptions. No other engineering science achieves such a broad range of applicability based on such a simple set of postulates.

So, what is thermodynamics? We can begin to answer this question by dissecting the word into its roots: ‘thermo’ and ‘dynamics’. The term ‘thermo’ originates from a Greek word meaning warm or hot, which is related to temperature. This suggests a concept that is related to temperature and referred to as heat. The concept of heat will receive much attention in this text. ‘Dynamics’ suggests motion or movement. Thus the term ‘thermodynamics’ may be loosely interpreted as ‘heat motion’. This interpretation of the word reflects the origins of the science. Thermodynamics was developed in order to explain how heat, usually generated from combusting a fuel, can be provided to a machine in order to generate mechanical power or ‘motion’. However, as noted above, thermodynamics has since matured into a more general science that can be applied to a wide range of situations, including those for which heat is not involved at all. The term thermodynamics is sometimes criticized because the science of thermodynamics is ordinarily limited to systems that are in equilibrium. Systems in equilibrium are not ‘dynamic’. This fact has prompted some to suggest that the science would be better named ‘thermostatics’ (Tribus, 1961).

Combustion is the reaction of a fuel with oxygen. It is a subset of the more general subject of chemical equilibrium, which is considered in Chapter 14. Combustion is treated as a separate topic because combustion reactions tend to progress until the fuel is completely consumed. Combustion of various hydrocarbon fuels is the major source of useful energy (i.e., exergy) for transportation, electrical generation, space conditioning, water heating, and industrial processes.

Introduction to Combustion

Table 13-1 summarizes the sources of the energy that are consumed in the United States. The nation consumes about 102 quads (1.02 × 1017 Btu or 1.08 × 1011 GJ) of useful energy each year, which represents about 25% of the world energy consumption. The majority of this energy is provided by combustion. There are two major concerns with this current situation. First, nearly all combustible fuels contain carbon, which results in the generation of carbon dioxide during combustion, as shown in this chapter. Combustion in the U.S. alone resulted in the production of 5,990 million metric tons (5.9 × 1012 kg) of carbon dioxide in 2007 according to the U.S. Department of Energy (2009). Carbon dioxide in the atmosphere absorbs a portion of the thermal energy that is re-radiated from the earth and thereby contributes to global warming. Second, reserves of petroleum and natural gas, which account for over 60% of our useful energy supply, are finite. Although the extent of these reserves is a subject of debate, it is likely that they will become depleted in less than 50 years at the present rate of use. There are hundreds of years of coal reserves, but coal generates more carbon dioxide per unit energy than natural gas and coal also produces other contaminants when combusted. The sustainability of our energy supply and the associated problem of global warming are perhaps the most serious problems that the human race has ever faced.

The specific internal energy, enthalpy and entropy at 1 atm pressure for common combustion gases are provided as a function of temperature in the following tables.

1. Table F-1: Ideal gas properties of CO2

2. Table F-2: Ideal gas properties of CO

3. Table F-3: Ideal gas properties of O2

4. Table F-4: Ideal gas properties of N2

5. Table F-5: Ideal gas properties of H2O

The data in these tables were obtained from EES. The reference state for specific enthalpy is based on the enthalpy of formation relative to the elements at 25°C. The reference state for specific entropy is based on the Third Law of Thermodynamics. The reference values are from:

1. Bonnie J. McBride, Michael J. Zehe, and Sanford Gordon

2. “NASA Glenn Coefficients for CalculatingThermodynamic Properties of Individual Species”

3. NASA/TP-2002-211556, Sept. 2002

4. http://www.lerc.nasa.gov/WWW/CEAWeb/

Note that these tables can be printed from the website associated with this text, www.cambridge.org/kleinandnellis, for use during closed book examinations.