Overview

A body in a moving fluid has a boundary layer on its surface (Figure 9.1). Near the leading edge this boundary layer is laminar; it becomes turbulent at a downstream distance xtran where the local Reynolds number U∞xtran/v exceeds about 5 × 105. This turbulent boundary layer can be seen with flow-visualization techniques (Van Dyke, 1982).

In the early 1970s the acoustic sounding technique pioneered by McAllister et al. (1969) and Little (1969) made the atmospheric boundary layer (ABL) visible as well (Figure 9.2). Its instantaneous top (Figure 2.2) can be even thinner than we can easily measure. Neff et al. (2008) discuss the use of the acoustic sounder in the quasi-steady stable ABL at the South Pole.

The ABL has several important features:

• It is a region of continuous turbulence, instantaneously bounded by a thin, sharp top, the convoluted, ever-changing interface between the turbulent motion and the stably stratified, nonturbulent flow above. As a free-stream boundary (Figure 2.1) of a turbulent flow, its local, instantaneous thickness could be of the order of the Kolmogorov microscale η (Corrsin and Kistler, 1955; Corrsin, 1972).

• Averaging produces a much thicker interfacial layer at the ABL top; in it the mean potential temperature increases smoothly with height, the mean-wind components transition to their free-stream values, and the turbulence level decreases to near zero. Its thickness can be as much as 20–50% of the mean depth of the boundary layer. […]

Turbulence, its community, and our approach

Even if you have not studied turbulence, you already know a lot about it. You have seen the chaotic, ever-changing, three-dimensional nature of chimney plumes and flowing streams. You know that turbulence is a good mixer. You might have come across an article that described the intrigue it holds for mathematicians and physicists.

Unless a fluid flow has a low Reynolds number or very stable stratification (less dense fluid over more dense fluid), it is turbulent. Most flows in engineering, in the lower atmosphere, and in the upper ocean are turbulent. Because of its “mathematical intractability” – turbulence does not yield exact mathematical solutions – its study has always involved observations. But over the past three decades numerical approaches have proliferated; today they are a dominant means of studying turbulent flows.

Turbulence has long been studied in both engineering and geophysics. G. I. Taylor's contributions spanned both (Batchelor, 1996). The Lumley and Panofsky (1964) work was my introduction to that breadth, but as Lumley later commented, their parts of that text “just…touch.” Today the turbulence field seems more coherent than it was in 1964, although it still has subcommunities and dialects (Lumley and Yaglom, 2001).

In Part I of this book we focus on the physical understanding of turbulence, surveying its key properties. We'll use its governing equations to guide our discussions and inferences. We shall also discuss the main types of numerical approaches to turbulence.

Introduction and background

We saw in Chapter 1 that direct numerical solutions of the turbulence equations can be done only at relatively small values of the turbulence Reynolds number Rt. Calculations of the vastly larger Rt flows in engineering and geophysical applications use averaged forms of these equations. The averaging produces important turbulent fluxes that must be specified before the equations can be solved numerically. That specification allows today's calculations of averaged turbulence fields in applications ranging from flow in the atmospheric boundary layer to the general circulation of the earth's atmosphere and convection on the sun.

In this chapter we derive, interpret, and scale the conservation equations for several covariances, including turbulent fluxes, that arise from ensemble averaging. At very coarse resolution ergodicity (the property that any unbiased average converges to the ensemble average, Chapter 2) blurs the distinction between ensemble and space averaging, so these covariance equations are used also in traditional mesoscale-modeling and weather-forecasting applications.

Monin and Yaglom (1971) credit a 1924 paper by Keller and Friedmann as the first to present a method of deriving turbulence moment equations, of which the turbulent-flux budgets are an example. Because they contain unknown terms, the turbulence moment equations were little used until the late 1960s, when computers were large enough to solve approximate versions of them.

The literature on models of these flux budgets can be bewildering. The rationale for their closure approximations is not always discussed, and the models go under several names – e.g., “second-order closure,” “Reynolds-averaged Navier–Stokes (RANS),” “single-point closure,” “higher-order closure,” and “invariant modeling.”

Average and instantaneous properties contrasted

Figure 2.1 is a famous snapshot of a turbulent wake, the region downstream of a body in a moving fluid. The instantaneously thin, irregular boundary between the turbulent and nonturbulent flow is continuously deformed by the turbulent eddies, so that under averaging it becomes a broad, smooth transition region. Figure 2.2 illustrates this same feature at the top of the atmospheric boundary layer.

We have had access to instantaneous turbulence fields through remote sensing and numerical simulation only since the 1970s. Perhaps that is why our descriptive terms for turbulence tend to refer to its statistical properties, not its instantaneous ones. For example,

• Homogeneous turbulence has spatially uniform statistical properties (with the exception of mean pressure). A turbulent flow can be homogeneous in zero, one, two, or three directions. A sphere wake is an example of the first. The turbulent boundary layer near the leading edge of a flat plate, or downstream of a change in surface conditions, can be homogeneous in one direction (the lateral) but is inhomogeneous in the wall-normal and streamwise directions. The turbulent boundary layer over a uniform surface can be homogeneous in two directions, those in the plane parallel to the surface, but is necessarily inhomogeneous in the normal direction. The grid turbulence produced by a grating of bars spanning the cross section of a wind tunnel is homogeneous in the cross-stream plane but inhomogeneous in the streamwise direction because it decays as it goes downstream (Chapter 5). […]

Introduction

The stable boundary layer (SBL) is as different from the convective boundary layer (CBL) as night is from day. The SBL is typically much thinner and much less diffusive; chimney plumes in the stable air just after sunrise can travel intact for long distances, quite unlike those in midday. The decrease in surface wind speed around sunset on a clear day, which is prominent in wind climatologies (Arya, 2001), is also evident to casual observers. Figure 12.1 shows the root cause: in a given mean horizontal pressure gradient the surface mean wind speed is lower in stable stratification than in neutral or unstable stratification. Thus the lower nocturnal wind speeds tend to persist until the initiation of a CBL at sunrise.

We gave examples of SBLs in Chapter 9. The two sketched in Figure 9.7 are made stably stratified by flow over a cooler surface and by entrainment of warmer air aloft. Perhaps the most common example is the nocturnal SBL over land in fair weather. Another is the “long-lived” SBL at the South Pole; it is caused by a combination of a temperature inversion over a cooled, sloped surface and the typically downslope orientation of the mean pressure gradient (Neff et al., 2008).

As we'll see, turbulence in the SBL tends to be in a delicate dynamical balance, and so SBL structure tends to be more difficult to study and to parameterize than that in the CBL.

The spray equations have been studied and solved for many applications: single-component and multicomponent liquids, high-temperature and low-temperature gas environments, monodisperse and polydisperse droplet-size distributions, steady and unsteady flows, one-dimensional and multidimensional flows, laminar and turbulent regimes, subcritical and supercritical thermodynamic regimes, and recirculating (strongly elliptical) and nonrecirculating (hyperbolic, parabolic, or weakly elliptic) flows. The analyses discussed here will not be totally inclusive of all of the interesting analyses that have been performed; rather, only a selection is presented.

Spray flows can be classified in various ways. One important issue concerns whether the gas is turbulent or laminar. In this chapter, only laminar flows are considered; the turbulent situation is discussed in Chapter 10. Another issue concerns whether thermodynamic conditions are subcritical, on the one hand, or near critical to supercritical, on the other hand.

In the most general spray case, the gas and the droplets are not in thermal and kinematic equilibria, that is, the droplet temperature and the droplet velocity differ from those properties of the surrounding gas. Of course, heat transfer and drag forces result in the tendency to move toward equilibrium. The equilibrium case is sometimes described as a locally homogeneous flow. It is possible to have thermal equilibrium or kinematic equilibrium without the other. When thermal equilibrium exists, the analysis described in Chapter 7 is simplified because the droplet temperature Tl can be set equal to the gas temperature T and Eq. (7.82) or its alternative forms, Eq. (7.83) or Eq. (7.87), can be avoided.

The fluid dynamics and transport of sprays comprise an exciting field of broad importance. There are many interesting applications of spray theory related to energy and power, propulsion, heat exchange, and materials processing. Spray phenomena also have natural occurrences. Spray and droplet behaviors have a strong impact on vital economic and military issues. Examples include the diesel engine and gas-turbine engine for automotive, power-generation, and aerospace applications. Manufacturing technologies including droplet-based net form processing, coating, and painting are important applications. Applications involving medication, pesticides and insecticides, and other consumer uses add to the impressive list of important industries that use spray and droplet technologies. These industries involve annual production certainly measured in tens of billions of dollars and possibly higher. Many applications are still under development. The potentials for improved performance, improved market shares, reduced costs, and new products and applications are immense. Continuing effort is needed to optimize the designs of spray and droplet applications and to develop strategies and technologies for active control of sprays in order to achieve the huge potential.

In the first edition of this book and in this second edition, I have attempted to provide some scientific foundation for movement toward the goals of optimal design and effective application of active controls. The book, however, does not focus on design and controls. Rather, I discuss the fluid mechanics and transport phenomena that govern the behavior of sprays and droplets in the many important applications.