Mixed convection refers to conditions when forced and natural (buoyancy-driven) effects are both important and neither one can be neglected. Situations in which forced and buoyancy-driven convection terms are of similar orders of magnitude obviously fall in the mixed-convection flow category. However, in many applications we deal with either a predominantly forced convective flow in which buoyancy-driven effects are small but considerable or a predominantly buoyancy-driven flow in which a nonnegligible forced-flow contribution is also present.

Mixed convection is relatively common in nature. In more recent applications, it occurs in rotating flow loops and in the cooling minichannels in the blades of modern gas turbines. In these flow loops, Coriolis centripetal forces arise because of the rotation. When the fluid is compressible, secondary flow caused by the centripetal effect contributes to the wall–fluid heat transfer.

Mixed-convection effects are not always undesirable. In some applications we may intentionally seek buoyancy effect in order to augment heat transfer. Some recent applications of supercritical fluids are examples to this point. The very large compressibility of these fluids, which is achieved without a phase change (although a pseudo–phase change does occur for near-critical fluids) is very useful.

In situations that are predominantly forced flow, buoyancy-driven effects have four types of impact on the overall flow field:

1. They contribute (assist, resist, or do both at different parts of the flow field) to the forced-flow velocity field.

2. They cause secondary flows. The secondary flows can enhance or reduce the heat transfer rate.

3. […]

General Remarks

In the previous chapters we noted that the dimensionless boundary-layer conservation equations for momentum, thermal energy, and mass species are mathematically similar. This similarity among these dimensionless equations suggests that the mathematical solution for one equation should provide the solution of the other equations. One may argue that the empirical correlations for friction factor, heat transfer coefficient, and mass transfer coefficient represent empirical solutions to the momentum, energy, and mass-species conservation equations, respectively. Thus a correlation for friction factor of the form f = f (Re) is the empirical solution to the momentum conservation equation for a specific system and flow configuration, whereas an empirical correlation of the form Nu = Nu (Re, Pr) for the same system is an empirical solution to the energy equation and an empirical correlation of the form Sh = Sh(Re, Sc). Thus, using the analogy arguments, knowing an empirical correlation for either of the three parameters f, Nu, or Sh for a specific system will allow us to derive empirical correlations for the remaining two parameters.

The usefulness of the analogy approach becomes clear by noting that measurement of friction factor is usually much simpler than the measurement of heat or mass transfer coefficients.

The conservation equations for fluids were derived in the previous chapter. Because of viscosity, the velocity boundary condition on a solid–fluid interface in common applications is no-slip. Velocity slip occurs during gas flow when the gas molecular mean free path is not negligible in comparison with the characteristic dimension of the flow passage. It is discussed in Chapter 13.

The complete solution of viscous flow conservation equations for an entire flow field, it seems, is in principle needed in order to calculate what actually takes place on the surface of an object in contact with a fluid. The complete solution of the entire flow field is impractical, however, and is fortunately unnecessary. The breakthrough simplification that made the analysis of the flow field at the vicinity of surfaces practical was introduced by Ludwig Prandtl in 1904. He suggested that any object that moves while submerged in a low-viscosity fluid will be surrounded by a thin boundary layer. The impact of the no-slip boundary condition at the surface of the object will extend only through this thin layer of fluid, and beyond it the fluid acts essentially as an inviscid fluid. In other words, outside the boundary layer the flow field does not feel the viscous effect caused by the presence of the object. It feels only the blockage caused by the presence of the object, as a result of which the streamlines in the flow field become curved around the object.

Laminar flow in low-viscosity fluids is relatively rare in nature and industry. Turbulent flow is among the most complicated and intriguing natural phenomena and is not well understood, despite more than a century of study. Nevertheless, out of necessity, investigators developed simple models that can be used for engineering design and analysis.

Turbulent flows at relatively high Reynolds numbers (fully turbulent flows) are characterized by extremely irregular fluctuations in velocity, temperature, pressure, and other properties. At each point the velocity and other properties fluctuate around a mean value.

Turbulent flows are characterized by eddies and vortices. Chunks of fluid covering a wide size range move randomly around with respect to the mean flow. Fluid particles move on irregular paths, and the result is very effective mixing. Even the smallest eddies are typically orders of magnitude larger than the molecular mean free path (MMFP) (in gases) and the intermolecular distances. Within the small eddies, molecular (laminar) transport processes take place, but the interaction among eddies often dominates the overall transport processes and make molecular transport effects unimportant.

With respect to analysis, the Navier–Stokes equations discussed earlier in principle can be applied to turbulent flow as well. However, to obtain a meaningful solution, these equations must be solved in such a way that the largest and smallest eddies in the flow field are resolved. This approach [direct numerical simulation (DNS)] is extremely computational intensive, and it is possible at this time only for simple flow configurations and low Reynolds numbers.

General Remarks

Near-wall phenomena in internal turbulent flow has much in common with external turbulent flow, and the discussions of property fluctuations and near-wall phenomena in the previous chapter all apply to internal flow as well. The confined nature of the flow field, however, implies that, unlike external flow in which the free-stream conditions are not affected by what happens at the wall, the transport phenomena at the wall do affect the mean flow properties.

Consider fully developed internal flow in a smooth pipe, shown in Fig. 7.1. Similar to external flow, the entire flow field in the pipe can be divided into three zones: the viscous sublayer, the buffer zone, and the turbulent core. The mean thickness of the viscous sublayer is equal to y+ = 5, where y+ = yUτ/v is the distance from the wall in wall units and the buffer zone extends to y+ = 30. Close to the wall, where the effect of wall curvature is small and the fluid is not aware that the overall flow field is actually confined, the universal velocity profile presented in Eqs. (6.5.1)–(6.5.3) apply. Only when we approach the centerline does Eq. (6.5.3) deviate from measurements. Similar observations can be made about noncircular ducts.

We live in an era of unprecedented transition in science and technology education caused by the proliferation of computing power and information. Like most other science and technology fields, convective heat and mass transfer is already too vast to be covered in a semester-level course even at an outline level and is yet undergoing exponential expansion. The expansion is both quantitative and qualitative. On the quantitative side, novel and hitherto unexplored areas are now subject to investigation, not just by virtue of their intellectual challenge and our curiosity, but because of their current and potential technological applications. And on the qualitative side, massive sources of Internet-based information, powerful personal computers, and robust and flexible software and other computational tools are now easily accessible to even novice engineers and engineering students. This makes the designing of a syllabus for courses such as convection heat and mass transfer all the more challenging. Perhaps the two biggest challenges for an instructor of a graduate-level course in convection are defining a scope for the course and striking a reasonable balance between the now-classical analytic methods and the recently developing modern areas. Although the importance of modern topics and methods is evident, the coverage of these topics should not be at the expense of basics and classical methods.

Laminar Flamelets and the Bray, Moss, and Libby Model

This section is concerned with premixed turbulent combustion in circumstances in which the heat release reactions can be considered ‘fast’, a situation that is common in practical combustion systems. Several dimensionless parameters can help to identify the meaning of fast chemistry in this context. A Damköhler number, Da = τT/τc, compares a characteristic turbulent flow time scale with a laminar flame time. As explained in Chapter 1, when Da ≫ 1, combustion occurs in thin wrinkled interfaces separating unburned reactants from fully burned products. If also a Karlovitz number Ka = τL/τK is sufficiently small, where τK is the Kolmogorov time characterising the smallest eddies in a turbulent flow, these reacting interfaces will resemble unstretched laminar flames. It is then possible to divide a Reynolds-averaged Navier–Stokes (RANS) simulation or large-eddy simulation (LES) into two separate parts: a chemical kinetic analysis of a laminar flame and a subsequent fluid flow calculation. A third dimensionless quantity is, where τ = (Tb/Tu) − 1 is a heat release parameter and Tu and Tb are the temperatures in unburned reactants and fully burned combustion products, respectively. As explained in Chapter 1, gradient scalar transport expressions are found [1] to fail unless.

Aims and Coverage

Currently the energy required for domestic and industrial use and for transportation is predominantly met by burning fossil fuels. Although alternative sources are evolving, for example, by harvesting wind and solar energies, energy production by means of combustion is expected to remain dominant for many decades to come, especially for high-power density applications. Thus pollutant emission regulations for power-producing devices are set with the aim of reducing the impact of combustion on the environment, both by curtailing pollutant emissions and by increasing the efficiency of combustion equipment. Conventional combustion technologies are unable to achieve these two demands simultaneously but fuel-lean combustion, in which the fuel–air mixture contains a controlled excess of air, has the potential to fulfil both requirements.

The aim of this book is to bring together a review of the physics of lean combustion and its current modelling practices, together with a description of scientific challenges to be faced and ways to achieve stable lean combustion in practical devices. Additional material on non-reacting turbulent flows may be found in [1–5], and basic theories of combustion and turbulent reacting flows and their governing equations are presented in [6–9].

The present chapter sets the scene for the remainder of this volume.

Application of Lean Flames in Internal Combustion Engines

One of the two main ‘drivers’ for the adoption of lean flames in internal combustion (IC) engines has been the potential to reduce the emission of some pollutants. Since Bernard Lewis wrote these words in 1970, the same year as the enactment of the landmark extension to the Clean Air Act (CAA) – the so-called ‘Muskie’ Act – the design and technology behind IC engines for automobile application have seen many changes that indeed greatly reduced pollution. These include the widespread introduction of gasoline injection systems, three-way catalysts, the high-speed direct injection (DI) diesel engine, turbocharging, and high-pressure commonrail technology, to name a few at random. The related science has also progressed in ways amply described elsewhere in this volume: As to whether this has constituted ‘no special feat’, the reader will decide. Be that as it may, research into combustion in IC engines is now arguably going through one of its most exciting phases in the quest to ‘burn materials completely’, that is, with as little pollution as possible and with as much thermodynamic efficiency as possible. This part of the chapter describes the way in which engineering design can exploit combustion, predominantly lean combustion, in IC engines not only to reduce pollution, but also – by promoting fuel-conversion efficiency, the other main ‘driver’ – to husband the finite resource of petroleum-derived fuels [2].