Annual Review of Fluid Mechanics - Volume 34, 2002

This essay is based on the G.K. Batchelor Memorial Lecture that I delivered in May 2000 at the Institute for Theoretical Physics (ITP), Santa Barbara, where two parallel programs on Turbulence and Astrophysical Turbulence were in progress. It focuses on George Batchelor's major contributions to the theory of turbulence, particularly during the postwar years when the emphasis was on the statistical theory of homogeneous turbulence. In all, his contributions span the period 1946–1992 and are for the most part concerned with the Kolmogorov theory of the small scales of motion, the decay of homogeneous turbulence, turbulent diffusion of a passive scalar field, magnetohydrodynamic turbulence, rapid distortion theory, two-dimensional turbulence, and buoyancy-driven turbulence.

David Crighton, a greatly admired figure in fluid mechanics, Head of the Department of Applied Mathematics and Theoretical Physics at Cambridge, and Master of Jesus College, Cambridge, died at the peak of his career. He had made important contributions to the theory of waves generated by unsteady flow. Crighton's work was always characterized by the application of rigorous mathematical approximations to fluid mechanical idealizations of practically relevant problems. At the time of his death, he was certainly the most influential British applied mathematical figure, and his former collaborators and students form a strong school that continues his special style of mathematical application. Rigorous analysis of well-posed aeroacoustical problems was transformed by David Crighton.

Some applications of the study of outdoor acoustics and sets of data for sound-level spectra obtained close to the ground are described. Measurements and models of ground effects arising from the interaction between sound traveling directly from source to receiver and sound reflected from the ground are emphasized. Details are given concerning the influences of porosity, layering, small-scale surface roughness, and tall vegetation. Areas of related current and future research are outlined.

In this review we describe the discovery and development of understanding of the so-called elliptical instability. This is the name given to the linear instability mechanism that tends to break up regions of elliptical streamlines in a rotating flow. The instability is discussed in the three different contexts—an unbounded strained vortex, a localized strained vortex, and a triaxial ellipsoid—where it was originally discovered and then rediscovered. These make it clear that the instability is one of parametric resonance where a normal mode, or pair of normal modes, of the undistorted rotating flow resonates with the underlying strain field. The effects of additional physics on the instability process are examined before its nonlinear evolution is discussed. Various applications of the instability in nature are then reviewed.

A Lagrangian description of turbulence has unique physical advantages that are especially important in studies of mixing and dispersion. We focus on fundamental aspects, using mainly data from direct numerical simulations capable of great detail and precision when specific accuracy requirements are met. Differences between time evolution in Eulerian and Lagrangian frames illustrate the dominance of advective transport. We examine basic results in Kolmogorov similarity, giving an estimate of an inertial-range universal constant and the grid resolution and Reynolds number needed to attain the requisite scaling range of time lags. The Lagrangian statistics of passive scalars are discussed in view of current efforts in model development, with differential diffusion between multiple scalars being characterized by shorter timescales. We also note the need for new data in more complex flows and in other applications where a Lagrangian viewpoint is especially useful.

Cavitation in vortical structures is a common, albeit complex, problem in engineering applications. Cavitating vortical structures can be found on the blade surfaces, in the clearance passages, and at the hubs of various types of turbomachinery. Cavitating microvortices at the trailing edge of attached sheet cavitation can be highly erosive. Cavitating hub vortices in the draft tubes of hydroturbines can cause major surges and power swings. There is also mounting evidence that vortex cavitation is a dominant factor in the inception process in a broad range of turbulent flows. Most research has focused on the inception process, with limited attention paid to developed vortex cavitation. Wave-like disturbances on the surfaces of vapor cores are an important feature. Vortex core instabilities in microvortices are found to be important factors in the erosion mechanisms associated with sheet/cloud cavitation. Under certain circumstances, intense sound at discrete frequencies can result from a coupling between tip vortex disturbances and oscillating sheet cavitation. Vortex breakdown phenomena that have some commonalities are also noted, as are some differences with vortex breakdown in fully wetted flow. Simple vortex models can sometimes be used to describe the cavitation process in complex turbulent flows such as bluff body wakes and in plug valves. Although a vortex model for cavitation in jets does not exist, the mechanism of inception appears to be related to the process of vortex pairing. The pairing process can produce negative peaks in pressure that can exceed the rms value by a factor of ten, sometimes exceeding the dynamic pressure by a factor of two. A new and important issue is that cavitation is not only induced in vortical structures but is also a mechanism for vorticity generation.

Microstructure in an immiscible polymer blend consists of the size, shape, and orientation of the phases. Blends exhibit many interesting behaviors, including enhanced elasticity at small strains, drop-size hysteresis, enhanced shear thinning, and stress relaxation curves whose shapes are sensitive to deformation history. These behaviors are directly related to changes in the microstructure, which result from phase deformation, coalescence, retraction, and different types of breakup. These phenomena are reviewed, together with models that describe them. Rheological measurements can probe the microstructure because microstructure contributes directly to stress through interfacial tension. Rheo-optical experiments also provide important insights. Droplet theories explain most of the phenomena for Newtonian phases at low concentrations. Behaviors at high volume fractions or with strongly non-Newtonian phases are less well understood.

The coupling of fluid dynamics and biology at the level of the cell is an intensive area of investigation because of its critical role in normal physiology and disease. Microcirculatory flow has been a focus for years, owing to the complexity of cell-cell or cell-glycocalyx interactions. Noncirculating cells, particularly those that comprise the walls of the circulatory system, experience and respond biologically to fluid dynamic stresses. In this article, we review the more recent studies of circulating cells, with an emphasis on the role of the glycocalyx on red-cell motion in small capillaries and on the deformation of leukocytes passing through the microcirculation. We also discuss flows in the vicinity of noncirculating cells, the influence of fluid dynamic shear stress on cell biology, and diffusion in the lipid bi-layer, all in the context of the important fluid-dynamic phenomena.

Recent progress in modeling and simulation of the flow of nematic liquid crystals is presented. The Leslie-Ericksen (LE) theory has been successful in elucidating the flow of low molar-mass nematics. The theoretical framework for the flow of polymeric nematic liquid crystals is still evolving; extensions of the Doi theory capture qualitative features of the flow of polymeric nematics in simple geometries, but these theories have not been shown to predict texture development in flow. Mesoscopic theories for textured materials based on spatial averaging capture only some qualitative features of nonrectilinear liquid-crystalline polymer flow. Interfacial effects in liquid-crystalline systems have begun to receive attention in the context of interfacial viscoelasticity and the dynamics of dispersed liquid-crystalline polymers in immiscible blends.

We examine situations in which two droplets of the same liquid may come into apparent contact without coalescing or in which a droplet that normally wets a surface may deform against it without actually wetting it. The focus of this review is on cases driven by hydrodynamic lubrication, the lubricant provided either by surface motion or by evaporation.

The current understanding of boundary-layer receptivity to external acoustic and vortical disturbances is reviewed. Recent advances in theoretical modeling, numerical simulations, and experiments are discussed. It is shown that aspects of the theory have been validated and that the mechanisms by which freestream disturbances provide the initial conditions for unstable waves are better understood. Challenges remain, however, particularly with respect to freestream turbulence.

Turbulent flows driven by thermal buoyancy are featured by phenomena that pose a special challenge to conventional one-point closure models. Inherent unsteadiness, energy nonequilibrium, counter-gradient diffusion, strong pressure fluctuations, and lack of universal scaling, all believed to be associated with distinct large-scale coherent eddy structures, are hardly tractable by Reynolds-type averaging. Second-moment closures, though inadequate for providing information on eddy structure, offer better prospects than eddy-viscosity models for capturing at least some of the phenomena. For some configurations (e.g., with heating from below), unsteady computational solutions of ensemble-averaged equations, using a one-point closure as the subscale model, may be unavoidable for accurate prediction of flow details and wall heat transfer. This article reviews the rationale and some specific modeling issues related to buoyant flows within the realm of one-point closures. The inadequacy of isotropic eddy-diffusivity models is discussed first, followed by the rationale of the second-moment modeling and its term-by-term scrutiny based on direct numerical simulations (DNS). Algebraic models based on a rational truncation of the differential second-moment closure are proposed as the minimum closure level for complex flows. These closures are also recommended as subscale models for transient statistical modeling (T-RANS) and very large eddy simulations (VLES). Examples of applications illustrate some recent achievements.

Because the cost of large-eddy simulations (LES) of wall-bounded flows that resolve all the important eddies depends strongly on the Reynolds number, methods to bypass the wall layer are required to perform high-Reynolds-number LES at a reasonable cost. In this paper the available methodologies are reviewed, and their ranges of applicability are highlighted. Various unresolved issues in wall-layer modeling are presented, mostly in the context of engineering applications.

Filament-stretching rheometers are devices for measuring the extensional viscosity of moderately viscous non-Newtonian fluids such as polymer solutions. In these devices, a cylindrical liquid bridge is initially formed between two circular end-plates. The plates are then moved apart in a prescribed manner such that the fluid sample is subjected to a strong extensional deformation. Asymptotic analysis and numerical computation show that the resulting kinematics closely approximate those of an ideal homogeneous uniaxial elongation. The evolution in the tensile stress (measured mechanically) and the molecular conformation (measured optically) can be followed as functions of the rate of stretching and the total strain imposed. The resulting rheological measurements are a sensitive discriminant of molecularly based constitutive equations proposed for complex fluids. The dynamical response of the elongating filament is also coupled to the extensional rheology of the polymeric test fluid, and this can lead to complex viscoelastic-flow instabilities such as filament necking and rupture or elastic peeling from the rigid end-plates.

Recent advances in the computational modeling of molecular conformational and orientational effects in the flow of viscoelastic fluids are described. These advances involve the coupling of molecular models for the underlying microstructure of macromolecules with the macroscopic equations of change. The kinetic theory for polymeric liquids is described along with the most useful micromechanical models for computing the fluid flow of polymeric liquids. Three levels of description are covered for the computation of molecular orientation effects: methods for molecular models for which closed-form, continuum-like evolution equations for average quantities describing molecular conformations can be obtained, hybrid methods that involve coupling direct solution of the Fokker-Planck equation describing the distribution function for molecular orientations with the equations of change, and hybrid methods that couple stochastic simulations of individual molecule trajectories with the macroscopic equations of change. Illustrative results for rheometric flows (flows with homogeneous, fixed kinematics) and complex flows are given.

The Richtmyer-Meshkov instability arises when a shock wave interacts with an interface separating two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including nonlinear growth and subsequent transition to turbulence, across a wide range of Mach numbers. This review focuses on the basic physical processes underlying the onset and development of the Richtmyer-Meshkov instability in simple geometries. It examines the principal theoretical results along with their experimental and numerical validation. It also discusses the different experimental approaches and techniques and how they can be used to resolve outstanding issues in this field.

Remote observations of a surface ship wake using synthetic aperture radar (SAR) show distinct features such as a dark trailing centerline region, bright V-images aligned at some angle to the ship's path, and, sometimes, either the transverse or the diverging waves of the Kelvin-wave pattern. The dark region of relatively low radar backscatter is usually associated with a region that is relatively lacking in short wave components, whereas the bright line feature suggests a region of enhanced radar return within the apparent angular confines of the ship's usual Kelvin-wave pattern. This review provides a survey of remotely sensed wake images, the systems that have collected these images, and an overview of the theory of Kelvin wakes—a primary source of the phenomena that cause the dark centerline and bright V-images—with example predictions. The review concludes with a survey of the phenomena that cause the dark centerline returns and some example predictions of the radar reflectivity across these dark centerline returns.

The evolution of a synthetic (zero-net mass flux) jet and the flow mechanisms of its interaction with a cross flow are reviewed. An isolated synthetic jet is produced by the interactions of a train of vortices that are typically formed by alternating momentary ejection and suction of fluid across an orifice such that the net mass flux is zero. A unique feature of these jets is that they are formed entirely from the working fluid of the flow system in which they are deployed and, thus, can transfer linear momentum to the flow system without net mass injection across the flow boundary. Synthetic jets can be produced over a broad range of length and timescale, and their unique attributes make them attractive fluidic actuators for a number of flow control applications. The interaction of synthetic jets with an external cross flow over the surface in which they are mounted can displace the local streamlines and induce an apparent or virtual change in the shape of the surface, thereby effecting flow changes on length scales that are one to two orders of magnitude larger than the characteristic scale of the jets. This control approach emphasizes an actuation frequency that is high enough so that the interaction domain between the actuator and the cross flow is virtually invariant on the global timescale of the flow, and therefore, global effects such as changes in aerodynamic forces are effectively decoupled from the operating frequency of the actuators.