Turbulence results from which of the following situations

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Publisher Summary

This chapter discusses turbulence in fluid movements. The theoretical problem of the stability of steady flow past bodies with finite dimensions has not yet been solved. For any problem of viscous flow under given steady conditions, there is in principle an exact steady solution of the equations of fluid dynamics. These solutions formally exist for all Reynolds numbers. However, not all these solutions can be contested to occur in nature, and even the ones that do must not only obey the equations of fluid dynamics but must also be stable. Steady flow is stable for sufficiently small Reynolds numbers. The experimental data seem to indicate that when the Reynolds number increases, it eventually reaches the value of a critical Reynolds number beyond which the flow is unstable with respect to infinitesimal disturbances. For sufficiently large Reynolds numbers, steady flow past solid bodies is impossible. The critical Reynolds number is not a universal constant but takes a different value for each type of flow.

Abstract

Turbulence is a very complex motion of a fluid which occurs at sufficiently high flow rates in a wide range of physical systems and has far-reaching technological implications. Although the underlying equations of fluid dynamics are well known, a comprehensive theoretical description of turbulence is a major unresolved problem of classical physics. A better understanding of turbulence might be gained by investigating systems which offer simplifications. Quantum turbulence is represented by the seemingly complex dynamics of a tangle of quantised vortex lines in a superfluid. At very low temperatures, there is no normal fluid component and therefore no associated viscosity. These are very simple conditions, both from conceptual and mathematical view points. We have recently developed techniques for detecting quantum turbulence in superfluid 3He-B in the low temperature limit. We use quasiparticle excitations to directly probe the superfluid flow field. We describe various experiments investigating the production, evolution and decay of quantum turbulence. The results obtained give the first empirical steps towards addressing a number of interesting questions, such as how closely does quantum turbulence resemble classical turbulence and how does quantum turbulence decay in the absence of the viscous dissipation.

5.5.3.5 Turbulence

Turbulence cannot be detected directly from satellite imagery; however, certain cloud patterns are unmistakably related to turbulence. The most common geographic location for turbulence-induced clouds is in mountainous terrain. The impingement of strong wind flow into a high mountain barrier frequently results in mountain wave turbulence and, if moisture conditions are right, in the formation of mountain wave clouds (Fig. 5.13). Wave clouds are associated with moderate to severe turbulence. Figure 5.14 shows a typical example of mountain wave turbulence over the Appalachians. Wave clouds normally form in bandsthat are parallel to the mountain barrier that induced them. The spacing between bandsis linearly related to the normal component of the wind at the barrier, though different authors give different relationships (Brandli, 1976, p. 106; Weber and Wilderotter, 1981, p. 15).

Another form of wave cloud is the billow cloud (Ludlam, 1967; Brandli, 1976), which is similar in appearance to the wave cloud but is a high cloud and it is not associated with the topographic barrier. In fact, wave clouds can form when moist air is forced to flow over a cooler (more dense) air mass. The result is a sinusoidal wave pattern propagating downstream and forming clouds in the upper portion of the wave. Billow clouds, because they are high clouds, show up best on an infrared image.

Turbulent Regimes

Turbulence is studied from many perspectives. The subject of “transition to turbulence” attempts to describe the initial mechanisms responsible for the generation of turbulence starting from a laminar motion in particular geometries. This transition can be followed with respect to position in space (e.g., the flow becomes more complicated as we look further downstream on a flow past an obstacle or over a flat plate) or to parameters (e.g., as we increase the angle of attack of a wing or the pressure gradient in a pipe). This subject is divided into two cases: wall-bounded and free-shear flows. In the former, the viscosity, which causes the fluid to adhere to the surface of the wall, is the primary cause of the instability in the transition process. In the latter, inviscid mechanisms such as mixing layers and jets are the main factors. The tools for studying the transition to turbulence include linearization of the equations of motion around the laminar solution, nonlinear amplitude equations, and bifurcation theory.

“Fully developed turbulence,” on the other hand, concerns turbulence which evolves without imposed constraints, such as boundaries and external forces. This can be thought of turbulence in its “pure” form, and it is somewhat a theoretical framework for research due to its idealized nature. Hypotheses of homogeneity (when the mean quantities associated with the statistical order characterizing a turbulent flow are independent in space), stationarity (idem in time), and isotropy (idem with respect to rotations in space) concern fully developed turbulent flows. The Kolmogorov theory was developed in this context and it is the most fundamental theory of turbulence. Current research is dedicated in great part to unveil the mechanisms behind a phenomenon called intermittency and how it affects the laws obtained from the conventional theory. Research is also dedicated to derive such laws as much from first principles as possible, minimizing the use of phenomenological and dimensional analysis.

Real turbulent flows involve various regimes at once. A typical flow past a blunt object, for instance, displays laminar motion at its upstream edge, a turbulent boundary layer further downstream, and the formation of a turbulent wake (Figure 2). The subject of turbulent boundary layer is a world in itself with current research aiming to determine mean properties of flows over rough surfaces and varied topography. Convective turbulence involves coupling with active scalars such as large heat gradients, occurring in the atmosphere, and large salinity gradients, in the ocean. Geophysical turbulence involves also stratification and the anisotropy generated by Earth’s rotation. Anisotropic turbulence is also crucial in astrophysics and plasma theory. Multiphase and multicomponent turbulence appear in flows with suspended particles or bubbles and in mixtures such as gas, water, and oil. Transonic and supersonic flows are also of great importance and fall into the category of compressible turbulence, much less explored than the incompressible case.

In all those real situations one would like, from the engineering point of view, to compute mean properties of the flow, such as drag and lift for more efficient designs of aircraft, ships, and other vehicles. Knowledge of the drag coefficient is also of fundamental importance in the design of pipes and pumps, from pipelines to artificial human organs. Mean turbulent diffusion coefficients of heat and other passive scalars – quantities advected by the flow without interfering on it, such as chemical products, nutrients, moisture, and pollutants – are also of major importance in industry, ecology, meteorology, and climatology, for instance. And in most of those cases a large amount of research is dedicated to the “control of turbulence,” either to increase mixing or reduce drag, for instance. From a theoretical point of view, one would like to fully understand and characterize the mechanisms involved in turbulent flows, clarifying this fascinating phenomenon. This could also improve practical applications and lead to a better control of turbulence.

The concept of “two-dimensional turbulence” is controversial. A two-dimensional flow may be irregular and display mixing, statistical order, and a wide range of active scales but definitely it does not involve vortex stretching since the velocity field is always perpendicular to the vorticity field. For this reason many researchers discard two-dimensional turbulence altogether. It is also argued that real two-dimensional flows are unstable at complicated regimes and soon develop into a three-dimensional flow. Nevertheless, many believe that two-dimensional turbulence, even lacking vortex stretching, is of fundamental theoretical importance. It may shed some light into the three-dimensional theory and modeling, and it can serve as an approximation to some situations such as the motion of the atmosphere and oceans in the large and meso scales and some magnetohydrodynamic flows. The relative shallowness of the atmosphere and oceans or the imposition of a strong uniform magnetic field may force the flow into two-dimensionality, at least for a certain range of scales.

“Chaos” serves as a paradigm for turbulence, in the sense that it is now accepted that turbulence is a dynamic processes in a sensitive deterministic system. But not all chaotic motions in fluids are termed turbulent for they may not display mixing and vortex stretching or involve a wide range of scales. An important such example appears in the dispersive, nonlinear interactions of waves.

1 INTRODUCTION

Turbulence in the superfluid component of 4He was first mentioned as a theoretical possibility by Feynman (1955), who suggested that it takes the form of a random tangle of quantised vortex lines. At about the same time, it was shown experimentally that counterflow of the two fluids, associated with a heat current, exhibits the characteristics of turbulent flow, and it was suggested that this turbulence is homogeneous and maintained in the superfluid component by the relative motion of the two fluids (Vinen, 1961). There followed much study of counterflow quantum turbulence, both experimentally(Tough, 1982)and theoretically, and an understanding of at least some of its principal characteristics owed much to the pioneering simulations of Schwarz (1988). Steady-state counterflow turbulence has no classical analogue, and it was not until the 1990s that there was serious study of forms of quantum turbulence that do have classical counterparts.

A particularly simple but important form of classical turbulence is produced by steady flow through a grid. At a significant distance downstream from the grid, the turbulence is at least approximately homogeneous and isotropic, and the development of an understanding of this simple form of turbulence has been important in contributing to our general understanding of turbulent flow (Batchelor, 1953; Frisch, 1995). The quantum analogue was first studied by Stalp et al. (1999); Skrbek et al. (2000); Skrbek and Stalp (2000), although a very important experiment by Maurer and Tabeling (1998) on turbulence generated by counter-rotating blades pointed the way towards an understanding of the experimental results. Although these experiments were concerned with superfluid 4He at temperatures above 1 K, the helium carried no heat current, so that at least on average the two fluids could flow without relative motion, in contrast to the situation in counterflow turbulence. Indeed, as has now become widely accepted (Vinen, 2000), the two fluids have velocity fields that are closely similar, except on length scales comparable with or less than the spacing between vortex lines. The statistical properties of this single velocity field prove to be very similar to those in classical grid flow, and to involve a Richardson cascade in which there is a flow of energy from large-scale to small-scale motion; the energy flow is due to the nonlinear terms in the Navier-Stokes equation. Correspondingly, in the inertial range of wave numbers, the energy spectrum has, to a good approximation, the Kolmogorov (K41) form

(1)Ek=Cε2/3k−5/3;

E(k)dk is the turbulent energy per unit mass associated with wave numbers in the range k to k + dk in a spatial Fourier analysis of the velocity field; ε is the rate of flow of energy per unit mass down the cascade, equal to the rate of dissipation of the total energy; and C (the Kolmogorov constant) is of order unity. By the inertial range, we mean the range of wave numbers in which there is negligible dissipation; dissipation must of course exist at higher wave numbers to provide a sink into which the energy flux ε can ultimately flow. In classical turbulence, there are small deviations from Kolmogorov scaling associated with, for example, intermittency and even these have been observed in the quantum case (Maurer and Tabeling, 1998). The physical picture we have of classical grid turbulence is therefore as follows. Steady flow through the grid produces eddy motion on a (large) scale, comparable with that of the confining boundaries (the mesh of the grid or the size of the channel in which flow takes place); nonlinear coupling transfers energy in a cascade to smaller and smaller scales until the scale is so small that dissipation occurs. In a classical fluid, dissipation is associated with viscosity and it sets in at wave numbers greater than the reciprocal of the Kolmogorov dissipation length, which is equal to (v3/ε)1/4, where v is the kinematic viscosity of the fluid. In superfluid 4He above 1 K, the dissipation occurs on a scale comparable with the vortex spacing and is due to a combination of normal fluid viscosity and mutual friction (Vinen and Niemela, 2002), mutual friction being the force that acts between a vortex line and the normal fluid when there is appropriate relative motion (Donnelly, 1991).

We see then that quantum grid turbulence can be very similar in its structure to its classical counterpart. The reasons have been discussed extensively in the literature (Vinen and Niemela, 2002). The turbulent superfluid component contains a more or less random array of quantised vortex lines. The flow of such a component, isolated from any normal fluid, is believed to be governed by principles very similar to those operating in an inertial range for a classical fluid, provided that the scale of the motion is large compared with the spacing, ℓ, between the vortex lines (Figure 1). A Richardson cascade can develop, characterised by a Kolmogorov energy spectrum. Flow of the superfluid on a scale larger than ℓ is achieved by some local polarisation of the vortex tangle so that the tangle ceases to be strictly random. If both fluids are present, each can support a classical Richardson cascade independently of the other, and mutual friction will serve to ensure that the two velocity fields are essentially identical. This picture can apply only if, as is the case for 4He, the normal fluid has a sufficiently small viscosity so that there is a wide inertial range in the normal fluid. Indeed, in the case of 4He above 1 K, there is typically dissipation in the normal fluid only at wave numbers greater than ℓ− 1. The situation in superfluid 3He-B is very different, because in this case the normal fluid has such a high viscosity that normal fluid turbulence is hardly possible on any laboratory scale (Finne et al., 2006).

The type of quantum turbulence that we have just described, in which the superfluid system is behaving like a single classical fluid, will be called quasiclassical. We emphasise that such quasiclassical behaviour can occur only on length scales larger than ℓ. On length scales less than ℓ, the discrete structure associated with the quantised vortices rules out behaviour that is similar to that of a classical fluid.

The observations of quantum turbulence that we have been describing lead to a number of interesting questions. The observation of quasiclassical behaviour has been confined for the most part to superfluid 4He at temperatures above about 1.1 K. Although the normal fluid fraction at these lowest temperatures is less than 2%, there exists the possibility that the classical normal fluid is somehow forcing quasiclassical behaviour on the superfluid component. This possibility, albeit remote, requires that we attempt to observe quasiclassical grid turbulence in the superfluid component, in both 4He or 3He-B, at temperatures so low that the fraction of normal fluid is negligible

A second, and very interesting, question relates to the mechanism by which homogeneous quantum turbulence can be dissipated in the absence of the normal fluid: the mechanisms operating in 4He above 1 K, which depend on the viscosity of the normal fluid and on the force of mutual friction between the vortices and the normal fluid, disappear at temperatures well below 1 K. The continuing existence of a Richardson cascade depends of course on the continuing existence of dissipation at a large wave number.

A third group of questions relate to the extent to which other types of flow can exhibit quasiclassical characteristics. A type of classical turbulent flow that is commonly studied is that occurring at high Reynolds number past some obstacle: a sphere, a cylinder or a plate (Figure 2). For example, can superflow past a sphere involve the shedding of quasiclassical vortices, similar to those seen in classical fluids, and if so, at what critical velocity, analogous to the critical Reynolds number, does the flow cease to be laminar (or potential)? Flow in the immediate neighbourhood of a grid is also interesting, before the flow settles into the homogeneous isotropic type of turbulence that we find well downstream of the grid. These questions are perhaps especially interesting and perhaps easier to answer at the lowest temperatures, where the processes cannot be influenced by any normal fluid.

It is with studies relating to this last group of questions that this chapter is primarily concerned although, as we shall see, there is significant overlap between the three groups of questions. The plan of the chapter is as follows. It turns out, as we shall discuss in Section 2, that the study of quantum turbulence at the lowest temperatures is severely influenced by experimental difficulties. Consideration of these difficulties has led to the extensive experimental study of oscillating structures (spheres, cylinders or grids) in a superfluid at the lowest temperatures, in spite of associated difficulties in interpretation. Such experimental techniques as are available will be described. In Section 3, we summarise briefly what we know about the steady flow of a classical fluid past various forms of obstacle. As we shall see in Section 4, the behaviour in an oscillating classical flow, or the behaviour of a structure undergoing oscillatory movement in a classical fluid, can be quite complicated; more complicated, and probably less well understood, than is the case for steady flow. We shall attempt to give the reader some flavour of these complications, too much detail being at this stage unnecessary and undesirable. Nevertheless we devote significant space to the discussion of the various classical cases, because we believe that some understanding of these cases is important in developing an understanding of the corresponding quantum cases. In Section 5, we summarise the experimental results that have been obtained with oscillating structures in superfluids, focussing our attention significantly but not exclusively on the lowest temperatures. An important message in this Section is that the experimental results are often incomplete so that dependence on important parameters, such as the frequency of oscillation and the size of the structure, cannot be elucidated with any confidence. In Section 6, we shall attempt to understand the experimental results using classical analogies, simple physical arguments and the results of computer simulations. Overall, we shall be left with many unanswered questions, and we hope that this chapter will serve to stimulate more work.

2.1.4 Internal geophysics

Turbulence exists in the strongly heated liquid metal of Earth outer core. Here, the Rossby number may be calculated as follows in medium latitudes (Cardin [2003], private communication). One takes U = 10−3m/s, L = 1000 km, which gives Ro = 10−5. These flows are, because of their extreme slowness, the most rotation dominated of all those already considered up to now. They are electrically conductive and obey Magneto-Hydro-Dynamic (MHD) equations. Furthermore, they are submitted to a strong internal convection, whose result is certainly quasi 2D vortices of axis parallel to Earth axis of rotation.

What is Turbulence?

Turbulence is a highly nonlinear regime encountered in fluid flows. Such flows are described by continuous fields, for example, velocity or pressure, assuming that the characteristic scale of the fluid motions is much larger than the mean free path of the molecular motions. The prediction of the spacetime evolution of fluid flows from first principles is given by the solutions of the Navier–Stokes equations. The turbulent regime develops when the nonlinear term of Navier–Stokes equations strongly dominates the linear term; the ratio of the norms of both terms is the Reynolds number Re, which characterizes the level of turbulence. In this regime nonlinear instabilities dominate, which leads to the flow sensitivity to initial conditions and unpredictability.

The corresponding turbulent fields are highly fluctuating and their detailed motions cannot be predicted. However, if one assumes some statistical stability of the turbulence regime, averaged quantities, such as mean and variance, or other related quantities, for example, diffusion coefficients, lift or drag, may still be predicted.

When turbulent flows are statistically stationary (in time) or homogeneous (in space), as it is classically supposed, one studies their energy spectrum, given by the modulus of the Fourier transform of the velocity autocorrelation.

Unfortunately, since the Fourier representation spreads the information in physical space among the phases of all Fourier coefficients, the energy spectrum loses all structural information in time or space. This is a major limitation of the classical way of analyzing turbulent flows. This is why we have proposed to use the wavelet representation instead and define new analysis tools that are able to preserve time and space locality.

The same is true for computing turbulent flows. Indeed, the Fourier representation is well suited to study linear motions, for which the superposition principle holds and whose generic behavior is, either to persist at a given scale, or to spread to larger ones. In contrast, the superposition principle does not hold for nonlinear motions, their archetype being the turbulent regime, which therefore cannot be decomposed into a sum of independent motions that can be separately studied. Generically, their evolution involves a wide range of scales, exciting smaller and smaller ones, even leading to finite-time singularities, e.g., shocks. The “art” of predicting the evolution of such nonlinear phenomena consists of disentangling the active from the passive elements: the former should be deterministically computed, while the latter could either be discarded or their effect statistically modeled. The wavelet representation allows to analyze the dynamics in both space and scale, retaining only those degrees of freedom which are essential to predict the flow evolution. Our goal is to perform a kind of “distillation” and retain only the elements which are essential to compute the nonlinear dynamics.

CONCLUDING REMARKS

Turbulence characteristics of the wake around a V-shaped bluff body, modeling a V-gutter of the jet engine combustor, are investigated experimentally. Clarified also are the effects of the base configuration upon them as compared with the round triangular body. Main points obtained in the study are as follows.

The surface pressure exhibits some peculiar feature near the point which connects the round nose to the downstream straight side. It suggests an existence of a small separation bubble there.

There is no steady reverse flow inside the recirculation region, which extends 1.25d downstream from the base.

The turbulent stress, u′2¯ , v′2¯ and −u′v′2¯ exhibit the strong non-isotropy especially in the central part of the wake and it continues far downstream. In general, v′2¯ is very large compared to u′2¯ there.

The setting of the base plate as forming a round triangular bluff body brings about a decrease of the base pressure, an increase of the drag and also a shortening of the recirculation region compared to those of the V-shaped body.

1 INTRODUCTION

The effect of freestream turbulence on bluff-body mean flow is one of the most important problems in wind engineering. However, the problem has long been a puzzling one (ref. 1) because many experiments done to date showed that there is very little or no effect of changing turbulence scale, despite a significant effect of changing turbulence intensity. We have been concerned with this enigma for several years and found, in a series of wind-tunnel measurements, that bluff-body mean flow is indeed sensitive to changing turbulence scale. The present paper summarizes the results of our recent measurements on this subject which include the results on 3D square rods (refs 2,3), 2D rectangular cylinders (ref. 4) together with those on a 2D flat plate with rectangular leading edge geometry that have been obtained most recently.

Inviscid Flows and Turbulence

Loosely speaking, turbulence is the intricate motion of a slightly viscous flow. Going back to the first half of the last century, there are two main approaches to turbulence. The first is due to Leray. The dissipation of energy is a characteristic feature of three-dimensional turbulence, and Leray thought that, even if very small, the viscosity of the fluid plays an important role, so that to understand turbulence the first step is to study the Navier–Stokes equations. A radically different approach is due to Onsager. Onsager (1949) started with the fundamental remark that the 4/5 law of turbulence, which relates the dissipation of energy to the increments of the velocity field, does not involve viscosity. Furthermore, he observed that the proof of the conservation of energy for the solutions of Euler equations uses an integration by parts which supposes some regularity of the velocity field. He then imagined that an inviscid dissipation mechanism, due to a lack of regularity of the solutions, was at work in Euler equations. In modern terminology, he suggested to model turbulent flows by nonregular (weak) solutions satisfying the Euler equations in the sense of distributions. He also conjectured that if a solution satisfies a Hölder regularity condition of order >1/3, then the energy would be conserved.

Onsager’s views were revolutionary and forgotten for a long time. Recent works, such as the proof of Onsager’s conjecture, the construction of weak solutions with energy dissipation, and the discovery of the explicit local form of the energy dissipation for weak solutions, show a renewed interest in these views (see, e.g., Constantin and Titi (1994), Eyink (1994), Robert (2003), and Shnirelman (2003)).