PROLOGUE

In contrast to the early chapters, in the present chapter I will emphasize the frontiers of the field of flow control, pondering mostly the control of turbulent flows. I will review the important advances in the field that have taken place during the past few years and are anticipated to dominate progress in the future, essentially covering the fifth era outlined in Section 1.2. By comparison with laminar flow control or separation prevention, the control of turbulent flow remains a very challenging problem. Flow instabilities quickly magnify near critical flow regimes, and therefore delaying transition or separation is a relatively easier task. In contrast, classical control strategies are often ineffective for fully turbulent flows. Newer ideas for turbulent flow control to achieve skin-friction drag reduction, for example, focus on the direct onslaught on coherent structures. Spurred by the recent developments in chaos control, microfabrication, and soft computing tools, reactive control of turbulent flows through which sensors detect oncoming coherent structures and actuators attempt to modulate these quasi-periodic events favorably is now in the realm of the possible for future practical devices.

Fossil fuels remain the main source of energy for domestic heating, power generation, and transportation. Other energy sources such as solar and wind energy or nuclear energy still account for less than 20% of total energy consumption. Therefore combustion of fossil fuels, being humanity's oldest technology, remains a key technology today and for the foreseeable future. It is well known that combustion not only generates heat, which can be converted into power, but also produces pollutants such as oxides of nitrogen (NOx), soot, and unburnt hydrocarbons (HC). Ever more stringent regulations are forcing manufacturers of automotives and power plants to reduce pollutant emissions, for the sake of our environment. In addition, unavoidable emissions of CO2 are believed to contribute to global warming. These emissions will be reduced by improving the efficiency of the combustion process, thereby increasing fuel economy.

In technical processes, combustion nearly always takes place within a turbulent rather than a laminar flow field. The reason for this is twofold: First, turbulence increases the mixing processes and thereby enhances combustion. Second, combustion releases heat and thereby generates flow instability by buoyancy and gas expansion, which then enhances the transition to turbulence.

This book addresses gaseous turbulent flows only. Although two-phase turbulent flows such as fuel sprays are also of much practical interest, they are omitted here, because their fundamentals are even less well understood than those of turbulent combustion. We also restrict ourselves to low Mach number flows, because high speed turbulent combustion is an area of its own, with practical applications in supersonic and hypersonic aviation only.

Introduction

Premixed combustion requires that fuel and oxidizer be completely mixed before combustion is allowed to take place. Examples of practical applications are spark-ignition engines, lean-burn gas turbines, and household burners. In all three cases fuel and air are mixed before they enter into the combustion chamber. Such a premixing is only possible at sufficiently low temperatures where the chain-branching mechanism that drives the reaction chain in hydrogen and hydrocarbon oxidation is unable to compete with the effect of three-body chain-breaking reactions. Under such low temperature conditions combustion reactions are said to be “frozen.” At ambient pressures the crossover from chain-branching to chain-breaking happens when the temperature decreases to values lower than approximately 1,000K for hydrogen flames or lower than approximately 1,300K for hydrocarbon flames (cf. Peters, 1997). The frozen state is metastable, because a sufficiently strong heat source, a spark for example, can raise the temperature beyond the crossover temperature and initiate combustion.

Once fuel and oxidizer have homogeneously been mixed and a heat source is supplied it becomes possible for a flame front to propagate through the mixture. This will happen if the fuel-to-air ratio lies between the flammability limits: Flammable mixtures range typically from approximately ϕ = 0.5 to ϕ = 1.5, where ϕ is the fuel-air-equivalence ratio defined by (3.12) in Chapter 3. Owing to the temperature sensitivity of the reaction rates the gas behind the flame front rapidly approaches the burnt gas state close to chemical equilibrium, while the mixture in front of the flame typically remains in the unburnt state. Therefore, the combustion system on the whole contains two stable states, the unburnt (index u) and the burnt gas state (index b).

PROLOGUE

Among the goals of external flow control are separation postponement, lift enhancement, transition delay or advancement, and drag reduction. These objectives are not necessarily mutually exclusive. For low-Reynolds-number lifting surfaces, where the formation of a laminar separation bubble may have a dominant effect on the flow- field, the interrelation between the preceding goals is particularly salient, presenting an additional degree of complexity when flow control is attempted to achieve, say, maximum lift-to-drag ratio. This chapter discusses the aerodynamics of low- Reynolds-number lifting surfaces—particularly the formation and control of separation bubbles.

Introduction

Insects, birds and bats have perfected the art of flight through millions of years of evolution. Man's dream of flying dates back to the early Greek myth of Daedalus and his son Icarus, but the first successful heavier-than-air flight took place less than a century ago. Today, the Reynolds numbers for natural and man-made fliers span the amazing range from 102 to 109, insects being at the low end of this spectrum and huge airships occupying the high end (Carmichael 1981).

The nature of turbulent flows

There are many opportunities to observe turbulent flows in our everyday surroundings, whether it be smoke from a chimney, water in a river or waterfall, or the buffeting of a strong wind. In observing a waterfall, we immediately see that the flow is unsteady, irregular, seemingly random and chaotic, and surely the motion of every eddy or droplet is unpredictable. In the plume formed by a solid rocket motor (see Fig. 1.1), turbulent motions of many scales can be observed, from eddies and bulges comparable in size to the width of the plume, to the smallest scales the camera can resolve. The features mentioned in these two examples are common to all turbulent flows.

More detailed and careful observations can be made in laboratory experiments. Figure 1.2 shows planar images of a turbulent jet at two different Reynolds numbers. Again, the concentration fields are irregular, and a large range of length scales can be observed.

As implied by the above discussion, an essential feature of turbulent flows is that the fluid velocity field varies significantly and irregularly in both position and time. The velocity field (which is properly introduced in Section 2.1) is denoted by U(x, t), where x is the position and t is time.

Figure 1.3 shows the time history U1(t) of the axial component of velocity measured on the centerline of a turbulent jet (similar to that shown in Fig. 1.2).

From vector calculus we are familiar with scalars and vectors. A scalar has a single value, which is the same in any coordinate system. A vector has a magnitude and a direction, and (in any given coordinate system) it has three components. With Cartesian tensors, we can represent not only scalar and vectors, but also quantities with more directions associated with them. Specifically, an Nth-order tensor (N ≥ 0) has N directions associated with it, and (in a given Cartesian coordinate system) it has 3N components. A zeroth-order tensor is a scalar, and a first-order tensor is a vector. Before defining higher-order tensors, we briefly review the representation of vectors in Cartesian coordinates.

Cartesian coordinates and vectors

Fluid flows (and other phenomena in classical mechanics) take place in the three-dimensional, Euclidean, physical space. As sketched in Fig. A.1, let E denote a Cartesian coordinate system in physical space. This is defined by the position of the origin O, and by the directions of the three mutually perpendicular axes. The unit vectors in the three coordinate directions are denoted by e1, e2, and e3. We write ei to refer to any one of these, with the understanding that the suffix i (or any other suffix) takes the value 1, 2, or 3.

The basic properties of the unit vectors ei are succinctly expressed in terms of the Kronecker delta δij.

The most commonly studied turbulent free shear flows are jets, wakes, and mixing layers. As the name ‘free’ implies, these flows are remote from walls, and the turbulent flow arises because of mean-velocity differences.

We begin by examining the round jet. By combining experimental observations (Section 5.1) with the Reynolds equations (Section 5.2), a good deal can be learned, not only about the round jet, but also about the behavior of turbulent flows in general. In Section 5.3, we study the turbulent kinetic energy in the round jet, and the important processes of production and dissipation of energy. Other self-similar free shear flows are briefly described in Section 5.4; and further observations about the behavior of free shear flows are made in Section 5.5.

The round jet: experimental observations

A description of the flow

We have already encountered the round jet in Chapter 1, for example, Figs. 1.1–1.4. The ideal experimental configuration and the coordinate system employed are shown in Fig. 5.1. A Newtonian fluid steadily flows through a nozzle of diameter d, which produces (approximately) a flat-topped velocity profile, with velocity UJ. The jet from the nozzle flows into an ambient of the same fluid, which is at rest at infinity. The flow is statistically stationary and axisymmetric. Hence statistics depend on the axial and radial coordinates (x and r), but are independent of time and of the circumferential coordinate, θ.

The mean velocity 〈U(x, t)〉 and the Reynolds stresses 〈uiuj〉 are the first and second moments of the Eulerian PDF of velocity f(V; x, t) (Eq. (3.153)). In PDF methods, a model transport equation is solved for a PDF such as f(V; x, t).

The exact transport equation for f(V; x, t) is derived from the Navier–Stokes equations in Appendix H, and discussed in Section 12.1. In this equation, all convective transport is in closed form – in contrast to the term ∂〈uiuj〉/∂xi in the mean-momentum equation, and ∂〈uiuj〉/∂xi in the Reynolds-stress equation. A closed model equation for the PDF – based on the generalized Langevin model (GLM) – is given in Section 12.2, and it is shown how this is closely related to models for the pressure–rate-of-strain tensor, ℛij.

Central to PDF methods are stochastic Lagrangian models, which involve new concepts and require additional mathematical tools. The necessary background on diffusion processes and stochastic differential equations is given in Appendix J. The simplest stochastic Lagrangian model is the Langevin equation, which provides a model for the velocity following a fluid particle. This model is introduced and examined in Section 12.3.

A closure cannot be based on the PDF of velocity alone, because this PDF contains no information on the turbulence timescale. One way to obtain closure is to supplement the PDF equation with the model dissipation equation. A superior way, described in Section 12.5, is to consider the joint PDF of velocity and a turbulence frequency.