Statistical Theory and Modeling for Turbulent Flows

Part I

FUNDAMENTALS OF TURBULENCE

1

Introduction

Where under this beautiful chaos can there lie a simple numerical structure?

– Jacob Bronowski

Turbulence is a ubiquitous phenomenon in the dynamics of fluid flow. For decades, comprehending and modeling turbulent fluid motion has stimulated the creativity of scientists, engineers, and applied mathematicians. Often the aim is to develop methods to predict the flow fields of practical devices. To that end, analytical models are devised that can be solved in computational fluid dynamics codes. At the heart of this endeavor is a broad body of research, spanning a range from experimental measurement to mathematical analysis. The intent of this text is to introduce some of the basic concepts and theories that have proved productive in research on turbulent flow.

Advances in computer speed are leading to an increase in the number of applications of turbulent flow prediction. Computerized fluid flow analysis is becoming an integral part of the design process in many industries. As the use of turbulence models in computational fluid dynamics increases, more sophisticated models will be needed to simulate the range of phenomena that arise. The increasing complexity of the applications will require creative research in engineering turbulence modeling. We have endeavored in writing this book both to provide an introduction to the subject of turbulence closure modeling, and to bring the reader up to the state of the art in this field. The scope of this book is certainly not restricted to closure modeling, but the bias is decidedly in that direction.

To flesh out the subject, the spectral theory of homogeneous turbulence is reviewed in Part III and eddy simulation is the topic of Part IV. In this way an endeavor has been made to provide a complete course on turbulent flow. We start with a perspective on the problem of turbulence that is pertinent to this text. Readers not very familiar with the subject might find some of the terminology unfamiliar; it will be explicated in due course.

1.1 The turbulence problem

The turbulence problem is an age-old topic of discussion among fluid dynamicists. It is not a problem of physical law; it is a problem of description. Turbulence is a state of fluid motion, governed by known dynamical laws – the Navier–Stokes equations in cases of interest here. In principle, turbulence is simply a solution to those equations. The turbulent state of motion is defined by the complexity of such hypothetical solutions. The challenge of description lies in the complexity: How can this intriguing behavior of fluid motion be represented in a manner suited to the needs of science and engineering?

Turbulent motion is fascinating to watch: it is made visible by smoke billows in the atmosphere, by surface deformations in the wakes of boats, and by many laboratory techniques involving smoke, bubbles, dyes, etc. Computer simulation and digital image processing show intricate details of the flow. But engineers need numbers as well as pictures, and scientists need equations as well as impressions. How can the complexity be fathomed? That is the turbulence problem.

Two characteristic features of turbulent motion are its ability to stir a fluid and its ability to dissipate kinetic energy. The former mixes heat or material introduced into the flow. Without turbulence, these substances would be carried along streamlines of the flow and slowly diffuse by molecular transport; with turbulence they rapidly disperse across the flow. Energy dissipation by turbulent eddies increases resistance to flow through pipes and it increases the drag on objects in the flow. Turbulent motion is highly dissipative because it contains small eddies that have large velocity gradients, upon which viscosity acts. In fact, another characteristic of turbulence is its continuous range of scales. The largest size eddies carry the greatest kinetic energy. They spawn smaller eddies via nonlinear processes. The smaller eddies spawn smaller eddies, and so on in a cascade of energy to smaller and smaller scales. The smallest eddies are dissipated by viscosity. The grinding down to smaller and smaller scales is referred to as the energy cascade. It is a central concept in our understanding of stirring and dissipation in turbulent flow.

The energy that cascades is first produced from orderly, mean motion. Small perturbations extract energy from the mean flow and produce irregular, turbulent fluctuations. These are able to maintain themselves, and to propagate by further extraction of energy. This is referred to as production and transport of turbulence. A detailed understanding of such phenomena does not exist. Certainly these phenomena are highly complex and serve to emphasize that the true problem of turbulence is one of analyzing an intricate phenomenon.

The term eddy may have invoked an image of swirling motion round a vortex. In some cases that may be a suitable mental picture. However, the term is usually meant to be more ambiguous. Velocity contours in a plane mixing layer display both large- and small-scale irregularities. Figure 1.1 illustrates an organization into large-scale features with smaller-scale random motion superimposed. The picture consists of contours of a passive scalar introduced into a mixing layer. Very often the image behind the term eddy is this sort of perspective on scales of motion. Instead of vortical whorls, eddies are an impression of features seen in a contour plot. Large eddies are the large lumps

Figure 1.1 Turbulent eddies in a plane mixing layer subjected to periodic forcing. From Rogers and Moser (1994), reproduced with permission.

seen in the figure, and small eddies are the grainy background. Further examples of large eddies are discussed in Chapter 5 of this book on coherent and vortical structures.

A simple method to produce turbulence is by placing a grid normal to the flow in a wind tunnel. Figure 1.2 contains a smoke visualization of the turbulence downstream of the bars of a grid. The upper portion of the figure contains velocity contours from a numerical simulation of grid turbulence. In both cases the impression is made that, on average, the scale of the irregular velocity fluctuations increases with distance downstream. In this sense the average size of eddies grows larger with distance from the grid.

Analyses of turbulent flow inevitably invoke a statistical description. Individual eddies occur randomly in space and time and consist of irregular regions of velocity or vorticity. At the statistical level, turbulent phenomena become reproducible and subject to systematic study. Statistics, like the averaged velocity, or its variance, are orderly and develop regularly in space and time. They provide a basis for theoretical descriptions and for a diversity of prediction methods. However, exact equations for the statistics do not exist. The objective of research in this field has been to develop mathematical models and physical concepts to stand in place of exact laws of motion. Statistical theory is a way to fathom the complexity. Mathematical modeling is a way to predict flows. Hence the title of this book: Statistical theory and modeling for turbulent flows.

The alternative to modeling would be to solve the three-dimensional, time-dependent Navier–Stokes equations to obtain the chaotic flow field, and then to average the solutions in order to obtain statistics. Such an approach is referred to as direct numerical simulation (DNS). Direct numerical simulation is not practicable in most flows of engineering interest. Engineering models are meant to bypass the chaotic details and to predict statistics of turbulent flows directly. A great demand is placed on these engineering closure models: they must predict the averaged properties of the flow without requiring access to the random field; they must do so in complex geometries for which detailed experimental data do not exist; they must be tractable numerically; and they must not require excessive computing time. These challenges make statistical turbulence modeling an exciting field.

The goal of turbulence theories and models is to describe turbulent motion by analytical methods. The particular methods that have been adopted depend on the objectives: whether it is to understand how chaotic motion follows from the governing equations, to

Figure 1.2 (a) Grid turbulence schematic, showing contours of streamwise velocity from a numerical simulation. (b) Turbulence produced by flow through a grid. The bars of the grid would be to the left of the picture, and flow is from left to right. Visualization by smoke wire of laboratory flow. Courtesy of T. Corke and H. Nagib.

construct phenomenological analogs of turbulent motion, to deduce statistical properties of the random motion, or to develop semi-empirical calculational tools. The latter two are the subject of this book.

The first step in statistical theory is to greatly compress the information content from that of a random field of eddies to that of a field of statistics. In particular, the turbulent velocity consists of a three-component field (u1, u2, u3) as a function of four independent variables (x1, x2, x3, t). This is a rapidly varying, irregular flow field, such as might be seen embedded in the billows of a smoke stack, the eddying motion of the jet in Figure 1.3, or the more explosive example of Figure 1.4. In virtually all cases of engineering interest, this is more information than could be used, even if complete data were available. It must be reduced to a few useful numbers, or functions, by averaging. The picture to the right of Figure 1.4 has been blurred to suggest the reduced information in an averaged representation. The small-scale structure is smoothed by averaging. A true average in this case would require repeating the explosion many times and summing the images; even the largest eddies would be lost to smoothing. A stationary flow can be

Figure 1.3 Instantaneous and time-averaged views of a jet in cross flow. The jet exits from the wall at left into a stream flowing from bottom to top (Su and Mungal 1999).

Figure 1.4 Large- and small-scale structure in a plume. The picture at the right is blurred to suggest the effect of ensemble averaging.

averaged in time, as illustrated by the time-lapse photograph on the right of Figure 1.3. Again, all semblance of eddying motion is lost in the averaged view.

An example of the greatly simplified representation invoked by statistical theory is provided by grid turbulence. When air flows through a grid of bars, the fluid velocity produced is a complex, essentially random, three-component, three-dimensional, timedependent field that defies analytical description (Figure 1.2). This velocity field might be described statistically by its variance, q², as a function of distance downwind of the grid; q² is the average value of c01_inline_equation001 over planes perpendicular to the flow. This statistic provides a smooth function that characterizes the complex field. In fact, the dependence of q2 on distance downstream of the grid is usually represented to good approximation by a power law: c01_inline_equation002 where n is about 1. The average length scale of the eddies grows like c01_inline_equation003 . This provides a simple formula that agrees with the impression created by Figure 1.2 of eddy size increasing with x.

The catch to the simplification that a statistical description seems to offer is that it is only a simplification if the statistics somehow can be obtained without having first to solve for the whole, complex velocity field and then compute averages. The task is to predict the smooth jet at the right of Figure 1.3 without access to the eddying motion at the left. Unfortunately, there are no exact governing equations for the averaged flow, and empirical modeling becomes necessary. One might imagine that an equation for the average velocity could be obtained by averaging the equation for the instantaneous velocity. That would only be the case if the equations were linear, which the Navier–Stokes equations are not.

The role of nonlinearity can be explained quite simply. Consider a random process generated by flipping a coin, assigning the value 1 to heads and 0 to tails. Denote this value by ξ. The average value of ξ is 1/2. Let a velocity, u, be related to ξ by the linear equation

(1.1.1)  c01_equation001

The average of u is the average of ξ − 1. Since ξ − 1 has probability 1/2 of being 0 and probability 1/2 of being −1, the average of u is −1/2. Denote this average by c01_inline_equation004 . The equation for c01_inline_equation005 can be obtained by averaging the exact equation: c01_inline_equation006 But if u satisfies a nonlinear equation

(1.1.2)  c01_equation002

then the averaged equation is

(1.1.3)  c01_equation003

This is not a closed* equation for c01_inline_equation007 because it contains c01_inline_equation008 : squaring, then averaging, is not equal to averaging, then squaring, that is, c01_inline_equation009 . In this example, averaging produces a single equation with two dependent variables, c01_inline_equation010 and c01_inline_equation011 . The example is contrived so that it first can be solved, then averaged: its solution is c01_inline_equation012 ; the average is then c01_inline_equation013 . Similarly c01_inline_equation014 , but this could not be known without first solving the random equation, then computing the average. In the case of the Navier–Stokes equations, one cannot resort to solving, then averaging. As in this simple illustration, the average of the Navier–Stokes equations are equations for c01_inline_equation015 that contain c01_inline_equation016 Unclosed equations are inescapable.

1.2 Closure modeling

Statistical theories of turbulence attempt to obtain statistical information either by systematic approximations to the averaged, unclosed governing equations, or by intuition and analogy. Usually, the latter has been the more successful: the Kolmogoroff theory of the inertial subrange and the log law for boundary layers are famous examples of intuition.

Engineering closure models are in this same vein of invoking systematic analysis in combination with intuition and analogy to close the equations. For example, Prandtl drew an analogy between the turbulent transport of averaged momentum by turbulent eddies and the kinetic theory of gases when he proposed his mixing length model. Thereby he obtained a useful model for predicting turbulent boundary layers.

The allusion to engineering flows implies that the flow arises in a configuration that has technological application. Interest might be in the pressure drop in flow through a bundle of heat-exchanger tubes or across a channel lined with ribs. The turbulence dissipates energy and increases the pressure drop. Alternatively, the concern might be with heat transfer to a cooling jet. The turbulence in the jet scours an impingement surface, enhancing the cooling. Much of the physics in these flows is retained in the averaged Navier–Stokes equations. The general features of the flow against the surface, or the separated flow behind the tubes, will be produced by these equations if the dissipative and transport effects of the turbulence are represented by a model. The model must also close the set of equations – the number of unknowns must equal the number of equations.

In order to obtain closed equations, the extra dependent variables that are introduced by averaging, such as c01_inline_equation017 in the above example, must be related to the primary variables, such as c01_inline_equation018 . For instance, if c01_inline_equation019 in Eq. (1.1.3) were modeled by c01_inline_equation020 , the equation would be c01_inline_equation021 , where a is an empirical constant. In this case a = 2 gives the correct answer c01_inline_equation022 .

Predicting an averaged flow field, such as that suggested by the time-averaged view in Figure 1.3, is not so easy. Conceptually, the averaged field is strongly affected by the irregular motion, which is no longer present in the blurred view. The influence of this irregular, turbulent motion must be represented if the mean flow is to be accurately predicted. The representation must be constructed in a manner that permits a wide range of applications. In unsteady flows, like Figure 1.4, it is unreasonable to repeat the experiment over and over to obtain statistics; nevertheless, there is no conceptual difficulty in developing a statistical prediction method. The subject of turbulence modeling is certainly ambitious in its goals.

Models for such general purposes are usually phrased in terms of differential equations. For instance, a widely used model for computing engineering flows, the k –ε model, consists of differential transport equations for the turbulent energy, k, and its rate of dissipation, ε. From their solution, an eddy viscosity is created for the purpose of predicting the mean flow. Other models represent turbulent influences by a stress tensor, the Reynolds stress. Transport models, or algebraic formulas, are developed for these stresses. The perspective here is analogous to constitutive modeling of material stresses, although there is a difference. Macroscopic material stresses are caused by molecular motion and by molecular interactions. Reynolds stresses are not a material property: they are a property of fluid motion; they are an averaged representation of random convection. When modeling Reynolds stresses, the concern is to represent properties of the flow field, not properties of a material. For that reason, the analogy to constitutive modeling should be tempered by some understanding of the aspects of turbulent motion that models are meant to represent. The various topics covered in this book are intended to provide a tempered introduction to turbulence modeling.

In practical situations, closure relations are not exact or derivable. They invoke empiricism. Consequently, any closure model has a limited range of use, implicitly circumscribed by its empirical content. In the course of time, a number of very useful semi-empirical models has been developed to calculate engineering flows. However, this continues to be an active and productive area of research. As computing power increases, more elaborate and more flexible models become feasible. A variety of models, their motivations, range of applicability, and some of their properties, are discussed in this book; but this is not meant to be a comprehensive survey of models. Many variations on a few basic types have been explored in the literature. Often the variation is simply to add parametric dependences to empirical coefficients. Such variants affect the predictions of the models, but they do not alter their basic analytical form. The theme in this book is the essence of the models and their mathematical properties.

1.3 Categories of turbulent flow

Broad categories can be delineated for the purpose of organizing an exposition on turbulent flow. The categorization presented in this section is suited to the aims of this book on theory and modeling. An experimenter, for instance, might survey the range of possibilities differently.

The broadest distinction is between homogeneous and non-homogeneous flows. The definition of spatial homogeneity is that statistics are not functions of position. Homogeneity in time is called stationarity. The statistics of homogeneous turbulence are unaffected by an arbitrary positioning of the origin of the coordinate system; ideal homogeneity implies an unbounded flow. In a laboratory, only approximate homogeneity can be established. For instance, the smoke puffs in Figure 1.2 are statistically homogeneous in the y direction: their average size is independent of y. Their size increases with x, so x is not a direction of homogeneity.

Idealized flows are used to formulate theories and models. The archetypal idealization is homogeneous, isotropic turbulence. Its high degree of statistical symmetry facilitates analysis. Isotropy means there is no directional preference. If one were to imagine running a grid every which way through a big tank of water, the resulting turbulence would have no directional preference, much as illustrated by Figure 1.5. This figure shows the instantaneous vorticity field in a box of homogeneous isotropic turbulence, simulated on a computer. At any point and at any time, a velocity fluctuation in the x1 direction would be as likely as a fluctuation in the x2, or any other, direction. Great mathematical simplifications follow. The basic concepts of homogeneous, isotropic turbulence are covered in this book. A vast amount of theoretical research has focused on this idealized state; we will only scratch the surface. A number of relevant monographs exist (McComb 1990) as well as the comprehensive survey by Monin and Yaglom (1975).

The next level of complexity is homogeneous, anisotropic turbulence. In this case, the intensity of the velocity fluctuations is not the same in all directions. Strictly, it could be either the velocity, the length scale, or both that have directional dependence – usually it

Figure 1.5 Vorticity magnitude in a box of isotropic turbulence. The light regions are high vorticity. Courtesy of J. Jiménez (Jimenez 1999).

Figure 1.6 Schematic suggesting eddies distorted by a uniform straining flow.

is both. Anisotropy can be produced by a mean rate of strain, as suggested by Figure 1.6. Figure 1.6 shows schematically how a homogeneous rate of strain will distort turbulent eddies. Eddies are stretched in the direction of positive rate of strain and compressed in the direction of negative strain. In this illustration, it is best to think of the eddies as vortices that are distorted by the mean flow. Their elongated shapes are symptomatic of both velocity and length scale anisotropy.

To preserve homogeneity, the rate of strain must be uniform in space. In general, homogeneity requires that the mean flow have a constant gradient; that is, the velocity should be of the form Ui = Aijxj + Bi, where Aij and Bi are independent of position (but are allowed to be functions of time).† The mean flow gradients impose rates of

rotation and strain on the turbulence, but these distortions are independent of position, so the turbulence remains homogeneous.

Throughout this book, the mean, or average, of a quantity will be denoted by a capital letter and the fluctuation from this mean by a lower-case letter; the process of averaging is signified by an overbar. The total random quantity is represented as the sum of its average plus a fluctuation. This prescription is to write U + u for the velocity, with U being the average and u the fluctuation. In a previous illustration, the velocity was ξ − 1, with ξ given by coin toss; thus, U + u = ξ − 1. Averaging the right-hand side shows that U = −1/2. Then u = ξ − 1 − U = ξ − 1/2. By definition, the fluctuation has zero average: in the present notation, u = 0.

A way to categorize non-homogeneous turbulent flows is by their mean velocity. A turbulent shear flow, such as a boundary layer or a jet, is so named because it has a mean shear. In a separated flow, the mean streamlines separate from the surface. The turbulence always has shear and the flow around eddies near to walls will commonly include separation; so these names would be ambiguous unless they referred only to the mean flow.

The simplest non-homogeneous flows are parallel or self-similar shear flows. The term parallel means that the velocity is not a function of the coordinate parallel to its direction. The flow in a pipe, well downstream of its entrance, is a parallel flow, U(r). The mean flow is in the x direction and it is a function of the perpendicular direction, r. All statistics are functions of r only. Self-similar flows are analogous to parallel flow, but they are not strictly parallel. Self-similar flows include jets, wakes, and mixing layers. For instance, the width, δ(x), of a mixing layer spreads with downstream distance. But if the cross-stream coordinate, y, is normalized by δ, the velocity becomes parallel in the new variable: U as a function of y/δ ≡ η is independent of x. Again, there is dependence on only one coordinate, η; the dependence on downstream distance is parameterized by δ(x). Parallel and self-similar shear flows are also categorized as fully developed. Figure 1.7 shows the transition of the flow in a jet from a laminar state, at the left, to the turbulent state. Whether it is a laminar jet undergoing transition, or a turbulent flow evolving into a jet, the upstream region contains a central core into which turbulence will penetrate as the flow evolves downstream. A fully developed state is reached only after the turbulence has permeated the jet.

Shear flows away from walls, or free-shear flows, often contain some suggestion of large-scale eddying motion with more erratic small-scale motions superimposed; an example is the turbulent wake illustrated by Figure 1.8. All of these scales of irregular motion constitute the turbulence. The distribution of fluctuating velocity over the range

Figure 1.7 Transition from a laminar to a turbulent jet via computer simulation. The regular pattern of disturbances at the left evolves into the disorderly pattern at the right. Courtesy of B. J. Boersma.

Figure 1.8 Schematic suggesting large- and small-scale structure of a free-shear layer versus Reynolds number. The large scales are insensitive to Reynolds number; the smallest scales become smaller as Re increases.

of scales is called the spectrum of the turbulence. Fully turbulent flow has a continuous spectrum, ranging from the largest, most energetic scales, that cause the main indentations in Figure 1.8, to the smallest eddies, nibbling at the edges. An extreme case is provided by the dust cloud of an explosion in Figure 1.4. A wide range of scales can be seen in the plume rising from the surface. The more recognizable large eddies have acquired the name coherent structures.

Boundary layers, like free-shear flows, also contain a spectrum of eddying motion. However, the large scales appear less coherent than in the free-shear layers. The larger eddies in boundary layers are described as horseshoe or hairpin vortices (Figure 5.9, page 99). In free-shear layers, large eddies might be rolls lying across the flow and rib vortices, sloping in the streamwise direction (Figure 5.3, page 94). In all cases a background of irregular motion is present, as in Figure 1.8. Despite endeavors to identify recognizable eddies, the dominant feature of turbulent flow is its highly irregular, chaotic nature.

A category of complex flows is invariably included in a discussion of turbulence. This might mean relatively complex, including pressure-gradient effects on thin shear layers, boundary layers subject to curvature or transverse strain, three-dimensional thin shear layers, and the like. Alternatively, it might mean quite complex, and run the whole gamut. From a theoretician’s standpoint, complex flows are those in which statistics depend on more than one coordinate, and possibly on time. These include perturbations to basic shear layers, and constitute the case of relatively complex turbulence. The category quite complex flows includes real engineering flows: impinging jets, separated boundary layers, flow around obstacles, and so on. For present purposes, it will suffice to lump quite and relatively complex flows into one category of complex turbulent flows. The models discussed in Chapters 6 and 7 are intended for computing such flows. However, the emphasis in this book is on describing the underlying principles and the processes of model development, rather than on surveying applications. The basic forms of practical models have been developed by reference to simple, canonical flows; fundamental data are integrated into the model to create a robust prediction method for more complex applications. A wealth of computational studies can be found in the literature: many archival journals contain examples of the use of turbulence models in engineering problems.

Exercises

Exercise 1.1. Origin of the closure problem. The closure problem arises in any nonlinear system for which one attempts to derive an equation for the average value. Let ξ correspond to the result of coin tossing, as in the text, and let

Show that, if c01_inline_equation024 were correct, then the mean value of u would be c01_inline_equation025 . By contrast, show that the correct value is c01_inline_equation026 . Explain why these differ, and how this illustrates the closure problem.

Exercise 1.2. Eddies. Identify what you would consider to be large- and small-scale eddies in the photographic portions of Figures 1.4 and 1.8.

Exercise 1.3. Turbulence in practice. Discuss practical situations where turbulent flows might be unwanted or even an advantage. Why do you think golf balls have dimples?

* The terms closure problem and closure model are ubiquitous in the literature. Mathematically, this means that there are more unknowns than equations. A closure model simply provides extra equations to complete the unclosed set.

† The convention of summation over repeated indices is used herein: c01_inline_equation023 or, in vector notation, U = A · x + B.

2

Mathematical and statistical background

To understand God’s thoughts we must study statistics, for these are the measure of his purpose

– Florence Nightingale

While the primary purpose of this chapter is to introduce the mathematical tools that are used in single-point statistical analysis and modeling of turbulence, it also serves to introduce some important concepts in turbulence theory. Examples from turbulence theory are used to illustrate the particular mathematical and statistical material.

2.1 Dimensional analysis

One of the most important mathematical tools in turbulence theory and modeling is dimensional analysis. The primary principles of dimensional analysis are simply that all terms in an equation must have the same dimensions and that the arguments of functions can only be non-dimensional parameters: the Reynolds number UL/ν is an example of a non-dimensional parameter. This might seem trivial, but dimensional analysis, combined with fluid dynamical and statistical insight, has produced one of the most useful results in turbulence theory: the Kolmogoroff −5/3 law. The reasoning behind the −5/3 law is an archetype for turbulence scale analysis.

The insight comes in choosing the relevant dimensional quantities. Kolmogoroff’s insight originates in the idea of a turbulent energy cascade. This is a central conception in the current understanding of turbulent flow. The notion of the turbulent energy cascade pre-dates Kolmogoroff’s work (Kolmogoroff, 1941); the origin of the cascade as an analytical theory is usually attributed to Richardson (1922).

Consider a fully developed turbulent shear layer, such as illustrated by Figure 1.8. The largest-scale eddies are on the order of the thickness, δ, of the layer; δ can be used as a unit of length. The size of the smallest eddies is determined by viscosity, ν. If the eddies are very small, they are quickly diffused by viscosity, so viscous action sets a lower bound on eddy size. Another view is that the Reynolds number of the small eddies, uη/ν, is small compared to that of the large eddies, uδ/ν, so small scales are the most affected by viscous dissipation. For the time being, it will simply be supposed that there is a length scale η associated with the small eddies and that η ≪ δ.

The largest eddies are produced by the mean shear – which is why their length scale is comparable to the thickness of the shear layer. Thus we have the situation that the large scales are being generated by shear and the small scales are being dissipated by viscosity. There must be a mechanism by which the energy produced at large scales is transferred to small scales and then dissipated. Kolmogoroff reasoned that this requires an intermediate range of scales across which the energy is transferred, without being produced or dissipated. In equilibrium, the energy flux through this range must equal the rate at which energy is dissipated at small scales. This intermediate range is called the inertial subrange. The transfer of energy across this range is called the energy cascade. Energy cascades from large scale to small scale, across the inertial range. The physical mechanism of the energy cascade is somewhat nebulous. It may be a sort of instability process, whereby larger-scale regions of shear develop smaller-scale irregularities; or it might be nonlinear distortion and stretching of large-scale vorticity.

As already alluded to, the rate of transfer across the inertial range, from the large scales to the small, must equal the rate of energy dissipation at small scale. Denote the rate of dissipation per unit volume by ρε, where ρ is the density, and ε is the rate of energy dissipation per unit mass. The latter has dimensions of l²/t³, which follows because the kinetic energy per unit mass, c02_inline_equation001 has dimensions of l²/t², and its rate of change has another factor of t in the denominator. The rate ε plays a dual role: it is the rate of energy dissipation, and it is the rate at which energy cascades across the inertial range. These two are strictly equivalent only in equilibrium. In practice, an assumption of local equilibrium in the inertial and dissipation ranges is usually invoked. Even though the large scales of turbulence might depart from equilibrium, the small scales are assumed to adapt almost instantaneously to them. The validity of this assumption is sometimes challenged, but it has provided powerful guidance to theories and models of turbulence.

Now the application of dimensional reasoning: we want to infer how energy is distributed within the inertial range as a function of eddy size, E(r). The inertial range is an overlap between the large-scale, energetic range and the small-scale dissipative range. It is shared by both. Large scales are not directly affected by molecular dissipation. Because the inertial range is common to the large scales, it cannot depend on molecular viscosity ν. The small scales are assumed to be of universal form, not depending on the particulars of the large-scale flow geometry. Because the inertial range is common to the small scales, it cannot depend on the flow width δ. All that remains is the rate of energy cascade, ε.

Consider an eddy of characteristic size r lying in this intermediate range. Based on the reasoning of the previous paragraph, on dimensional grounds its energy is of order (εr)²/³. This is the essence of Kolmogoroff’s law: in the inertial subrange the energy of the eddies increases with their size to the 2/3 power.

Figure 2.1 Experimental spectra measured by Saddoughi and Veeravalli (1994) in the boundary layer of the NASA Ames 80 × 100 foot wind tunnel. This enormous wind tunnel gives a very high Reynolds number, so that the −5/3 law can be verified over several decades. In this figure, c02_inline_equation002 is the energetic range and c02_inline_equation003 is the dissipation range.

This 2/3 law becomes a −5/3 law in Fourier space; that is how it is more commonly known. One motive for the transformation is to obtain the most obvious form in which to verify Kolmogoroff’s theory by experimental measurements. The distribution of energy across the scales of eddies in physical space is the inverse Fourier transform of the spectral energy density in Fourier space. This is a loose definition of the energy spectral density, E(κ). The energy spectral density is readily measured. It is illustrated by the log–log plot in Figure 2.1.

Equating the inertial-range energy to the inverse transform of the inertial-range energy spectrum (cf. Section 2.2.2.2):

c02_equation001_1 (2.1.1)

Assume that c02_inline_equation004 in (2.1.1). Then

c02_equation002 (2.1.2)

Here, κ has dimensions of 1/l, so the integrand of the second expression is nondimensional. The final integral is just some number, independent of r. Equating the exponents on both sides of (2.1.2) gives n + 1 = −2/3 or n = −5/3. The more famous

statement of Kolmogoroff’s result is the "−5/3 law"

c02_equation001_3 (2.1.3)

Note that E(κ) has dimensions c02_inline_equation005 . Here, E(κ) is the energy density per unit wavenumber (per unit mass), so E(κ) dκ has dimensions of velocity squared. Because c02_inline_equation006 , large scales correspond to small κ and vice versa. Hence, the spectrum in Figure 2.1 shows how the energy declines as the eddies grow smaller.

These ideas about scaling turbulent spectra expand to a general approach to constructing length and time-scales for turbulent motion. Such scaling is essential both to turbulence modeling for engineering computation, and to more basic theories of fluid dynamical turbulence.

2.1.1 Scales of turbulence

The notion of large and small scales, with an intervening inertial range, begs the question: How are large and small defined? If the turbulent energy k is being dissipated at a rate ε, then a time-scale for energy dissipation is T = k/ε. In order of magnitude this is the time it would take to dissipate the existing energy. This time-scale is sometimes referred to as the eddy lifetime, or integral time-scale. Since it is formed from the overall energy and its rate of dissipation, T is a scale of the larger, more energetic eddies.

Formula (2.1.3) and Figure 2.1 show that the large scales make the biggest contribution to k. The very small scales of motion