Showing papers in "Annual Review of Fluid Mechanics in 1986"

Characteristic-based schemes for the euler equations

Abstract: P. L. RoeCollege of Aeronautics, Cranfield Institute of Technology,Cranfield MK43 0AL, EnglandIntroductionComputer simulations of fluid flow provide today the sort of detailedinformation concerning special cases that could previously only beobtained from experime.nts. The computer is attractive as a replacement forexperiments that are difficult, dangerous, or expensive, and as an alternativeto experiments that are impossible. Nevertheless, a computer simulationdoes not have quite the same status as a physical experiment because atpresent there usually remains some doubt about its accuracy. Even thoughthe computer code may be free of error to the extent that it operates exactlyas its author intended, it is seldom possible to give a rigorous proof thatthese intentions were in all respects correct. Most of the practical codeswritten to solve complicated problems contain empirical features, some-times in the form of "adjustable constants" whose values must be "tuned"by appeal to the experiments that the simulations are intended to displace.A computer code is described as being "robust" if it has the virtue ofgiving reliable answers to a wide range of problems without needing to beretuned. The ideal code would be one that fully met some declaredspecification of accuracy and problem range, and whose every line was anecessary contribution to that aim. Few codes yet approach that ideal; amajor impediment is that we presently have little idea what properties canbe specified without contradiction.In recent years, however, our understanding of computer codes for aparticular class of problems has advanced some way toward completeness.The problems are sufficiently complex that naive numerical techniques canproduce disaster, yet sufficiently simple that well-understood physics can3370066-4189/86/0115-0337502.00www.annualreviews.org/aronline Annual Reviews

The continental-shelf bottom boundary layer

Abstract: Etude de la dynamique de la couche limite au fond, au-dessus du plateau continental, lorsque la couche est sous l'effet indirect du vent

Critical Layers in Shear Flows

Abstract: The normal mode approach to investigating the stability of a parallel shear flow involves the superposition of a small wavelike perturbation on the basic flow. Its evolution in space and/or time is then determined. In the linear inviscid theory, if ū(y) is the basic velocity profile, then a singularity occurs at critical points yc, where ū = c, the perturbation phase speed. This is plausible intuitively because energy can be exchanged most efficiently where the wave and mean flow are travelling at the same speed. The problem is of the singular perturbation type; when viscosity or nonlinearity, for example, are restored to the governing equations, the singularity is removed. In this lecture, the classical viscous theory is first outlined before presenting a newer perturbation approach using a nonlinear critical layer (i.e., nonlinear terms are restored within a thin layer). The application to the case of a density stratified shear flow is discussed and, finally, the results are compared qualitatively with radar observations and also with recent numerical simulations of the full equations. ∗Address for correspondence: Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 2K6, Canada. e-mail: maslowe@math.mcgill.ca

Regular and Mach Reflection of Shock Waves

Abstract: The properties that distinguish a shock wave from other waves are that its thickness is negligible compared with other characteristic lengths, and that the state of the medium is changed irreversibly by the passage of the wave. Shocks are therefore a highly nonlinear phenomenon, and parameter changes may be expected to lead to numerous bifurcations. The multitude of possibilities is compounded when more than one shock occurs, such as in the interaction of a shock with a solid surface or symmetry plane (i.e. in shock reflection). The subject of shock reflection is so complicated that it is necessary to introduce it at some length. In the interest of conciseness, this is done according to a logical rather than a historical sequence, with explicit reference to the important authors omitted in the text unless their work is relatively recent. Salient points in the early development of the subject should be mentioned here, however. These may be found in the experiments of Mach (1878), the theoretical work of von Neumann(1943), the experimental and theoretical work of the group around Bleakney at Princeton (e.g. Bleakney & Taub 1949, Smith 1945), the experiments of Kawamura & Saito (1956), Smith (1959), and Bryson & Gross (1961), and the theoretical work Lighthill (1949) and Jones et al. (195l). A review by Pack (1964) gives detailed references. Of the work since 1960, that of the group around Glass at Toronto (e.g. Law & Glass 1971, Ben-Dor & Glass 1979, 1980) and around Henderson at Sydney (e.g. Henderson & Lozzi 1975, 1979, Henderson & Gray 1981) stand out. A review by Griffith (1981) gives more detailed references. In this review the discussion is restricted to plane flow. This is because relatively little work has been done on three-dimensional situations, and because a compromise had to be made between depth and breadth of the field covered.

Interaction of Flows with the Crystal-Melt Interface

Abstract: The coupling between a crystal-melt interface and fluid flow is investigated. The solidification boundary conditions at the crystal-melt interface are described. The influences of morphological and double-diffusive instabilities during directional solidification on the interface are studied. The experiments by Glicksman and Mickalonis (1982) and Fang et al. (1985) which examine the relationship between the hydrodynamic state of the phase melt and phase-change interface are analyzed. The Rayleigh-Benard problem, and the effect of crystal-melt interaction on the Rayleigh number are examined. Various applications for melt-flow interactions with solid-liquid interface to engineering and welding are discussed.

Steady and Unsteady Boundary-Layer Separation

Abstract: The inspiring review oflaminar separation by Brown & Stewartson (1969) begins thus: "The phenomenon of separation is one of the most interesting features of the motion of an incompressible fluid past a bluff body at high Reynolds number. Here the main stream, which has hitherto been in close contact with the body, suddenly, and for no obvious reason, breaks away, and downstream a region of eddying flow, which is usually turbulent even if the flow elsewhere is laminar, is set up." These opening lines-which were followed by the above authors' brilliant and timely discourse emphasizing the difficulties associated with separation in classical boundary layers­ make an apt beginning for the present article on the current state of the art. The main feature, to which Brown & Stewarts on in fact alluded strongly, is that given the failure [due to singularities of the S. Goldstein (1948) type] in classical theory at separation, a fresh nonclassical start is necessary for high-Reynolds-number theory to encompass separating flow. This start was made by Stewartson & Williams (1969), Neiland (1969), and Messiter (1970). Our aim is to go from the significant advances made since 1969 to the recent developments and the possible areas of progress in the near future for the rational theory 'of steady and unsteady separation. The physical and theoretical understanding that can emerge from the study of flow structures at large characteristic Reynolds numbers Re is valuable for many reasons. Three major ones are the helpful comparisons with experiments and numerical results, the foundation such study gives to

Eddies, Waves, Circulation, and Mixing: Statistical Geofluid Mechanics

Abstract: The complexity of geophysical flows, from scales of planetary radius down to scales of molecular diffusion, has long posed a fascinating and frustrat­ ing challenge to fluid dynamicists. It is not only a tantalizing theo­ retical question but also one of practical importance. Despite determined study, understanding of the oceans and atmosphere and, especially, the pre­ diction of responses to our trespasses upon these environments remain dangerously suspect. Much of the dynamical difficulty arises from nonlinear coupling across many scales of motion. Occasionally one refers to the "turbulent atmo­ sphere" or the "turbulent ocean." Just as often it is remarked that these systems are not altogether "turbulent" if that adjective is taken to connote a condition that is highly chaotic, dissipative, diffusive, or possessed of whatever other attributes that one may assign to "turbulence." Examples of wavelike phenomena abound but often are partly obscured by nonlinear interactions. Persistent, coherent, finite-amplitude flow features also are observed. Moreover there is the disturbingly nontrivial problem of distinguishing mean and fluctuating fields. Geophysical flows often are characterized by spectra, both in frequency and in wave number, that are continuous and "red" in character. Then the definition of a mean field becomes more or less

Flows Far From Equilibrium Via Molecular Dynamics

Abstract: The basic concepts-mass, force, and acceleration-were first correctly interrelated by Newton. Three hundred years ago he explained the regular motion of the planets about the Sun on the basis of pairwise-additive gravitational forces. The possibility of solving Newton's equations of motion for N-body systems of atoms or molecules had to wait unti11953, when suitable computers had become available at Los Alamos. On a human time scale, the night sky has a relatively stable appearance. It is amusing that, from a mathematical viewpoint, gravitational N-body systems are much less "stable" than the molecular systems studied by molecular dynamics. Mathematical stability can be monitored by observ­ ing the separation between two neighboring trajectories. In a stable case, the separation grows linearly with time. In the typical unstable case, this distance increases exponentially with time. This "Lyapunov" instability is a ubiquitous feature of interesting dynamical systems. Whether or not it affects other properties of such systems in any important way is a

Strange Attractors in Fluids: Another View

Abstract: According to Lanford (1982), the fundamental idea of the strange-attractor theory of turbulence is that "turbulent time dependence is not an exceptional feature of particular equations of motion but a property shared by a broad class of typical differential equations." Lanford (1982) proceeds to describe features of "typical differential equations" that are of potential relevance to understanding turbulent flows. This article reviews the attempts that have been made to confirm or refute the theory and presents a critical assessment of it. Because results of dynamical-systems theory do not always have a clear interpretation for experiments, it is necessary to spend some time discussing matters that appear remote from the solution of the Navier-Stokes equations. To make predictions about fluid flow using the strange-attractor theory, a dictionary of dynamical behavior must be developed that is representative of the "typical differential equations" referred to by Lanford. Using this dictionary, one can then make comparisons with experimental observa­ tions. There are three possible uses of the theory. First, one may observe new aspects of typical dynamical behavior, which then suggest specific experimental observations. A good example of such a prediction is the period-doubling route for the transition to chaotic behavior (Feigenbaum 1983). Second, one may discover that certain types of dynamical behavior are atypical within a given class of systems and predict on this basis that such behavior will not be observed in fluid systems. Ruelle & Takens (1971) made predictions of this kind in their formulation of the strange-attractor

Three-Dimensional and Unsteady Boundary- Layer Computations

Abstract: From an engineering point of view, several methods are relatively successful in calculating two-dimensional, incompressible, steady turbulent boundary layers on a smooth surface, although considerable work (both computation­ ally and experimentally) needs to be done for flows with separation. The most advanced calculation methods reproduce a few characteristics of the flow. In practice, the features of interest are the skin friction, the displacement thickness, (occasionally) the mean velocity profiles, and (rarely) the turbulent kinetic energy or the Reynolds stresses. This situation could be considered to be satisfactory; however, improvements are needed, and it should be recognized that the practical calculation methods include few physical inputs. Thus, their extensions to new problems require careful studies and extensive comparisons with experimental data. This is certainly the case for three-dimensional and unsteady boundary layers, which are discussed in this review. Indeed, these flows introduce a third dimension and therefore new parameters. In order to focus the discussion on a more precise objective, we examine here both the three-dimensional boundary layers that develop, for example, either on a wing or on a fuselage of an aircraft and the unsteady boundary layers that develop under the influence of an imposed unsteadiness (oscillating flow, for example). In both cases, the flow is assumed incompressible. Three main problems are discussed. The first is related to the numerical aspects. This includes the numerical scheme for use in solving the equations, the choice and construction of a coordinate system (especially for the three-