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 ofscientists, engineers, and applied mathematicians. Often the aim is to develop meth-ods to predict the flow fields of practical devices. To that end, analytical models aredevised that can be solved in computational fluid dynamics codes. At the heart of thisendeavor is a broad body of research, spanning a range from experimental measurementto mathematical analysis. The intent of this text is to introduce some of the basic conceptsand theories that have proved productive in research on turbulent flow.

Advances in computer speed are leading to an increase in the number of applicationsof turbulent flow prediction. Computerized fluid flow analysis is becoming an integral partof the design process in many industries. As the use of turbulence models in computa-tional fluid dynamics increases, more sophisticated models will be needed to simulate therange of phenomena that arise. The increasing complexity of the applications will requirecreative research in engineering turbulence modeling. We have endeavored in writingthis 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 iscertainly 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 reviewedin Part III and eddy simulation is the topic of Part IV. In this way an endeavor hasbeen made to provide a complete course on turbulent flow. We start with a perspectiveon the problem of turbulence that is pertinent to this text. Readers not very familiar

Statistical Theory and Modeling for Turbulent Flows, Second Edition P. A. Durbin and B. A. Pettersson Reif© 2011 John Wiley & Sons, Ltd

COPYRIG

HTED M

ATERIAL

4 INTRODUCTION

with the subject might find some of the terminology unfamiliar; it will be explicated indue course.

1.1 The turbulence problem

The turbulence problem is an age-old topic of discussion among fluid dynamicists. Itis not a problem of physical law; it is a problem of description. Turbulence is a stateof fluid motion, governed by known dynamical laws – the Navier–Stokes equations incases 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 offluid 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 inthe atmosphere, by surface deformations in the wakes of boats, and by many laboratorytechniques involving smoke, bubbles, dyes, etc. Computer simulation and digital imageprocessing show intricate details of the flow. But engineers need numbers as well aspictures, and scientists need equations as well as impressions. How can the complexitybe fathomed? That is the turbulence problem.

Two characteristic features of turbulent motion are its ability to stir a fluid and itsability to dissipate kinetic energy. The former mixes heat or material introduced into theflow. Without turbulence, these substances would be carried along streamlines of the flowand slowly diffuse by molecular transport; with turbulence they rapidly disperse acrossthe flow. Energy dissipation by turbulent eddies increases resistance to flow through pipesand it increases the drag on objects in the flow. Turbulent motion is highly dissipativebecause it contains small eddies that have large velocity gradients, upon which viscosityacts. In fact, another characteristic of turbulence is its continuous range of scales. Thelargest size eddies carry the greatest kinetic energy. They spawn smaller eddies vianonlinear processes. The smaller eddies spawn smaller eddies, and so on in a cascadeof 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. Itis 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 pertur-bations 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 understand-ing of such phenomena does not exist. Certainly these phenomena are highly complexand serve to emphasize that the true problem of turbulence is one of analyzing anintricate phenomenon.

The term “eddy” may have invoked an image of swirling motion round a vortex. Insome cases that may be a suitable mental picture. However, the term is usually meant tobe more ambiguous. Velocity contours in a plane mixing layer display both large- andsmall-scale irregularities. Figure 1.1 illustrates an organization into large-scale featureswith smaller-scale random motion superimposed. The picture consists of contours of apassive 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, eddiesare an impression of features seen in a contour plot. Large eddies are the large lumps

THE TURBULENCE PROBLEM 5

U∞

U−∞

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

seen in the figure, and small eddies are the grainy background. Further examples of largeeddies 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 flowin a wind tunnel. Figure 1.2 contains a smoke visualization of the turbulence down-stream of the bars of a grid. The upper portion of the figure contains velocity contoursfrom a numerical simulation of grid turbulence. In both cases the impression is madethat, on average, the scale of the irregular velocity fluctuations increases with distancedownstream. In this sense the average size of eddies grows larger with distance fromthe grid.

Analyses of turbulent flow inevitably invoke a statistical description. Individual eddiesoccur randomly in space and time and consist of irregular regions of velocity or vor-ticity. At the statistical level, turbulent phenomena become reproducible and subject tosystematic study. Statistics, like the averaged velocity, or its variance, are orderly anddevelop regularly in space and time. They provide a basis for theoretical descriptions andfor a diversity of prediction methods. However, exact equations for the statistics do notexist. The objective of research in this field has been to develop mathematical modelsand physical concepts to stand in place of exact laws of motion. Statistical theory is away to fathom the complexity. Mathematical modeling is a way to predict flows. Hencethe title of this book: “Statistical theory and modeling for turbulent flows.”

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

The goal of turbulence theories and models is to describe turbulent motion by analyt-ical 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

6 INTRODUCTION

U

(a)

U

(b)

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

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

The first step in statistical theory is to greatly compress the information contentfrom that of a random field of eddies to that of a field of statistics. In particular, theturbulent velocity consists of a three-component field (u1, u2, u3) as a function of fourindependent variables (x1, x2, x3, t). This is a rapidly varying, irregular flow field, suchas might be seen embedded in the billows of a smoke stack, the eddying motion of thejet in Figure 1.3, or the more explosive example of Figure 1.4. In virtually all cases ofengineering interest, this is more information than could be used, even if complete datawere 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 informationin an averaged representation. The small-scale structure is smoothed by averaging. A trueaverage in this case would require repeating the explosion many times and summing theimages; even the largest eddies would be lost to smoothing. A stationary flow can be

THE TURBULENCE PROBLEM 7

Figure 1.3 Instantaneous and time-averaged views of a jet in cross flow. The jet exitsfrom 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 isblurred 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 isprovided by grid turbulence. When air flows through a grid of bars, the fluid velocityproduced is a complex, essentially random, three-component, three-dimensional, time-dependent field that defies analytical description (Figure 1.2). This velocity field might

8 INTRODUCTION

be described statistically by its variance, q2, as a function of distance downwind of thegrid; q2 is the average value of u2

1 + u22 + u2

3 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 goodapproximation by a power law: q2 ∝ x−n where n is about 1. The average length scaleof the eddies grows like L ∝ x1−n/2. This provides a simple formula that agrees with theimpression 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 isonly a simplification if the statistics somehow can be obtained without having first to solvefor the whole, complex velocity field and then compute averages. The task is to predictthe 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 empir-ical modeling becomes necessary. One might imagine that an equation for the averagevelocity could be obtained by averaging the equation for the instantaneous velocity. Thatwould only be the case if the equations were linear, which the Navier–Stokes equationsare not.

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

u = ξ − 1. (1.1.1)

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

u2 + 2u = ξ − 1, (1.1.2)

then the averaged equation is

u2 + 2u = ξ − 1 = −1/2. (1.1.3)

This is not a closed∗ equation for u because it contains u2: squaring, then averaging,is not equal to averaging, then squaring, that is, u2 �= u2. In this example, averagingproduces a single equation with two dependent variables, u and u2. The example is con-trived so that it first can be solved, then averaged: its solution is u = √

ξ − 1; the averageis then u = 1

2 (√

1 − 1) + 12 (

√0 − 1) = −1/2. Similarly u2 = 1/2, but this could not be

known without first solving the random equation, then computing the average. In the caseof the Navier–Stokes equations, one cannot resort to solving, then averaging. As in thissimple illustration, the average of the Navier–Stokes equations are equations for u thatcontain u2. Unclosed equations are inescapable.

∗ The terms “closure problem” and “closure model” are ubiquitous in the literature. Mathematically, thismeans that there are more unknowns than equations. A closure model simply provides extra equations tocomplete the unclosed set.

CLOSURE MODELING 9

1.2 Closure modeling

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

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

The allusion to “engineering flows” implies that the flow arises in a configurationthat has technological application. Interest might be in the pressure drop in flow througha bundle of heat-exchanger tubes or across a channel lined with ribs. The turbulencedissipates energy and increases the pressure drop. Alternatively, the concern might bewith 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 averagedNavier–Stokes equations. The general features of the flow against the surface, or theseparated flow behind the tubes, will be produced by these equations if the dissipativeand transport effects of the turbulence are represented by a model. The model must alsoclose 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 introducedby averaging, such as u2 in the above example, must be related to the primary variables,such as u. For instance, if u2 in Eq. (1.1.3) were modeled by u2 = au2, the equationwould be au2 + 2u = −1/2, where a is an “empirical” constant. In this case a = 2 givesthe correct answer u = −1/2.

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

Models for such general purposes are usually phrased in terms of differentialequations. 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 rateof dissipation, ε. From their solution, an eddy viscosity is created for the purpose ofpredicting the mean flow. Other models represent turbulent influences by a stress tensor,the Reynolds stress. Transport models, or algebraic formulas, are developed for thesestresses. The perspective here is analogous to constitutive modeling of material stresses,although there is a difference. Macroscopic material stresses are caused by molecularmotion and by molecular interactions. Reynolds stresses are not a material property:they are a property of fluid motion; they are an averaged representation of randomconvection. When modeling Reynolds stresses, the concern is to represent properties of

10 INTRODUCTION

the flow field, not properties of a material. For that reason, the analogy to constitutivemodeling should be tempered by some understanding of the aspects of turbulent motionthat models are meant to represent. The various topics covered in this book are intendedto provide a tempered introduction to turbulence modeling.

In practical situations, closure relations are not exact or derivable. They invokeempiricism. Consequently, any closure model has a limited range of use, implicitly cir-cumscribed by its empirical content. In the course of time, a number of very usefulsemi-empirical models has been developed to calculate engineering flows. However, thiscontinues 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 moti-vations, 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 afew basic types have been explored in the literature. Often the variation is simply to addparametric dependences to empirical coefficients. Such variants affect the predictions ofthe models, but they do not alter their basic analytical form. The theme in this book isthe 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 onturbulent flow. The categorization presented in this section is suited to the aims of thisbook on theory and modeling. An experimenter, for instance, might survey the range ofpossibilities 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 turbulenceare unaffected by an arbitrary positioning of the origin of the coordinate system;ideal homogeneity implies an unbounded flow. In a laboratory, only approximatehomogeneity can be established. For instance, the smoke puffs in Figure 1.2 arestatistically 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 idealizationis homogeneous , isotropic turbulence. Its high degree of statistical symmetry facilitatesanalysis. Isotropy means there is no directional preference. If one were to imagine run-ning a grid every which way through a big tank of water, the resulting turbulence wouldhave no directional preference, much as illustrated by Figure 1.5. This figure shows theinstantaneous vorticity field in a box of homogeneous isotropic turbulence, simulated ona computer. At any point and at any time, a velocity fluctuation in the x1 direction wouldbe as likely as a fluctuation in the x2, or any other, direction. Great mathematical simpli-fications follow. The basic concepts of homogeneous, isotropic turbulence are covered inthis book. A vast amount of theoretical research has focused on this idealized state; wewill 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, theintensity of the velocity fluctuations is not the same in all directions. Strictly, it could beeither the velocity, the length scale, or both that have directional dependence – usually it

CATEGORIES OF TURBULENT FLOW 11

Figure 1.5 Vorticity magnitude in a box of isotropic turbulence. The light regions arehigh vorticity. Courtesy of J. Jimenez (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 turbulenteddies. Eddies are stretched in the direction of positive rate of strain and compressed inthe direction of negative strain. In this illustration, it is best to think of the eddies asvortices that are distorted by the mean flow. Their elongated shapes are symptomatic ofboth 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 velocityshould 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

† The convention of summation over repeated indices is used herein: Aijxj ≡ ∑3j=1 Aijxj , or, in vector

notation, U = A · x + B.

12 INTRODUCTION

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 capitalletter and the fluctuation from this mean by a lower-case letter; the process of averagingis signified by an overbar. The total random quantity is represented as the sum of itsaverage 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 showsthat U = −1/2. Then u = ξ − 1 − U = ξ − 1/2. By definition, the fluctuation has zeroaverage: in the present notation, u = 0.

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

The simplest non-homogeneous flows are parallel or self-similar shear flows. Theterm “parallel” means that the velocity is not a function of the coordinate parallel to itsdirection. 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, butthey are not strictly parallel. Self-similar flows include jets, wakes, and mixing layers. Forinstance, the width, δ(x), of a mixing layer spreads with downstream distance. But if thecross-stream coordinate, y, is normalized by δ, the velocity becomes parallel in the newvariable: U as a function of y/δ ≡ η is independent of x. Again, there is dependence ononly 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 1.7shows the transition of the flow in a jet from a laminar state, at the left, to the turbulentstate. Whether it is a laminar jet undergoing transition, or a turbulent flow evolving intoa jet, the upstream region contains a central core into which turbulence will penetrate asthe flow evolves downstream. A fully developed state is reached only after the turbulencehas permeated the jet.

Shear flows away from walls, or free-shear flows, often contain some suggestionof large-scale eddying motion with more erratic small-scale motions superimposed; anexample is the turbulent wake illustrated by Figure 1.8. All of these scales of irregularmotion 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. Theregular pattern of disturbances at the left evolves into the disorderly pattern at the right.Courtesy of B. J. Boersma.

CATEGORIES OF TURBULENT FLOW 13

lower Reynolds number higher Reynolds number

Turbulent wake behind a bullet, visualized by Schlireren photography (Corrsin and Kisler, 1954).Flow is from left to right.

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

of scales is called the spectrum of the turbulence. Fully turbulent flow has a continuousspectrum, ranging from the largest, most energetic scales, that cause the main indentationsin Figure 1.8, to the smallest eddies, nibbling at the edges. An extreme case is providedby the dust cloud of an explosion in Figure 1.4. A wide range of scales can be seen inthe plume rising from the surface. The more recognizable large eddies have acquired thename “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 largereddies 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 casesa background of irregular motion is present, as in Figure 1.8. Despite endeavors toidentify 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 shearlayers, boundary layers subject to curvature or transverse strain, three-dimensional thinshear layers, and the like. Alternatively, it might mean quite complex, and run the wholegamut. From a theoretician’s standpoint, complex flows are those in which statisticsdepend on more than one coordinate, and possibly on time. These include perturbations tobasic shear layers, and constitute the case of relatively complex turbulence. The categoryquite complex flows includes real engineering flows: impinging jets, separated boundarylayers, flow around obstacles, and so on. For present purposes, it will suffice to lump quiteand relatively complex flows into one category of complex turbulent flows . The models

14 INTRODUCTION

discussed in Chapters 6 and 7 are intended for computing such flows. However, theemphasis in this book is on describing the underlying principles and the processes ofmodel development, rather than on surveying applications. The basic forms of practicalmodels have been developed by reference to simple, canonical flows; fundamental dataare integrated into the model to create a robust prediction method for more complex appli-cations. A wealth of computational studies can be found in the literature: many archivaljournals 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 nonlin-ear 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

u3 + 3u2 + 3u = (u + 1)3 − 1 = 7ξ.

Show that, if u3 = u3 and u2 = u2 were correct, then the mean value of u would beu = (9/2)1/3 − 1. By contrast, show that the correct value is u = 1/2. Explain why thesediffer, and how this illustrates the “closure problem.”

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

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